Foundations of Grothendieck duality for diagrams of schemes

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1960 Joseph Lipman...

Author: Joseph Lipman | Mitsuyasu Hashimoto

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1960

Joseph Lipman · Mitsuyasu Hashimoto

Foundations of Grothendieck Duality for Diagrams of Schemes

ABC

Joseph Lipman

Mitsuyasu Hashimoto

Mathematics Department Purdue University West Lafayette, IN 47907 USA [email protected]

Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602 Japan [email protected]

ISBN: 978-3-540-85419-7 e-ISBN: 978-3-540-85420-3 DOI: 10.1007/978-3-540-85420-3 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008935627 Mathematics Subject Classification (2000): 14A20, 18E30, 14F99, 18A99, 18F99, 14L30 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

This volume contains two related, though independently written, monographs. In Notes on Derived Functors and Grothendieck Duality the ﬁrst three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded complexes, of the sheaf functors ⊗, Hom, f∗ and f ∗ where f is a ringed-space map. Included are some enhancements, for concentrated (i.e., quasi-compact and quasi-separated) schemes, of classical results such as the projection and K¨ unneth isomorphisms. The fourth chapter presents the abstract foundations of Grothendieck Duality—existence and tor-independent base change for the right adjoint of the derived functor Rf∗ when f is a quasi-proper map of concentrated schemes, the twisted inverse image pseudofunctor for separated ﬁnite-type maps of noetherian schemes, reﬁnements for maps of ﬁnite tor-dimension, and a brief discussion of dualizing complexes. In Equivariant Twisted Inverses the theory is extended to the context of diagrams of schemes, and in particular, to schemes with a group-scheme action. An equivariant version of the twisted inverse-image pseudofunctor is deﬁned, and equivariant versions of some of its important properties are proved, including Grothendieck duality for proper morphisms, and ﬂat base change. Also, equivariant dualizing complexes are dealt with. As an application, a generalized version of Watanabe’s theorem on the Gorenstein property of rings of invariants is proved. More detailed overviews are given in the respective Introductions.

v

Contents

Part I Joseph Lipman: Notes on Derived Functors and Grothendieck Duality Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1

Derived and Triangulated Categories . . . . . . . . . . . . . . . . . . . . . 1.1 The Homotopy Category K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Derived Category D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mapping Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Triangulated Categories (Δ-Categories) . . . . . . . . . . . . . . . . . . . 1.5 Triangle-Preserving Functors (Δ-Functors) . . . . . . . . . . . . . . . . . 1.6 Δ-Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Localizing Subcategories of K; Δ-Equivalent Categories . . . . . 1.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Complexes with Homology in a Plump Subcategory . . . . . . . . . 1.10 Truncation Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Bounded Functors; Way-Out Lemma . . . . . . . . . . . . . . . . . . . . . .

11 12 13 15 16 25 29 30 33 35 36 38

2

Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Deﬁnition of Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Existence of Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Right-Derived Functors via Injective Resolutions . . . . . . . . . . . 2.4 Derived Homomorphism Functors . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Derived Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Adjoint Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Acyclic Objects; Finite-Dimensional Derived Functors . . . . . . .

43 43 45 52 56 60 65 71

3

Derived Direct and Inverse Image . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2 Adjointness of Derived Direct and Inverse Image . . . . . . . . . . . 89 vii

viii

Contents

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4

Δ-Adjoint Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjoint Functors between Monoidal Categories . . . . . . . . . . . . . Adjoint Functors between Closed Categories . . . . . . . . . . . . . . . Adjoint Monoidal Δ-Pseudofunctors . . . . . . . . . . . . . . . . . . . . . . More Formal Consequences: Projection, Base Change . . . . . . . Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentrated Scheme-Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Independent Squares; K¨ unneth Isomorphism . . . . . . . . . . . . . . .

97 101 110 118 124 131 132 144

Abstract Grothendieck Duality for Schemes . . . . . . . . . . . . . . 4.1 Global Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sheaﬁﬁed Duality—Preliminary Form . . . . . . . . . . . . . . . . . . . . . 4.3 Pseudo-Coherence and Quasi-Properness . . . . . . . . . . . . . . . . . . 4.4 Sheaﬁﬁed Duality, Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Proof of Duality and Base Change: Outline . . . . . . . . . . . . . . . . 4.6 Steps in the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Quasi-Perfect Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Two Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Perfect Maps of Noetherian Schemes . . . . . . . . . . . . . . . . . . . . . . 4.10 Appendix: Dualizing Complexes . . . . . . . . . . . . . . . . . . . . . . . . . .

159 160 169 171 177 179 179 190 203 230 239

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Part II Mitsuyasu Hashimoto: Equivariant Twisted Inverses Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 1

Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors . . . . . . . . . . . . . . . . . . 271

2

Sheaves on Ringed Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

3

Derived Categories and Derived Functors of Sheaves on Ringed Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

4

Sheaves over a Diagram of S-Schemes . . . . . . . . . . . . . . . . . . . . 321

5

The Left and Right Inductions and the Direct and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6

Operations on Sheaves Via the Structure Data . . . . . . . . . . . 331

7

Quasi-Coherent Sheaves Over a Diagram of Schemes . . . . . 345

Contents

ix

8

Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

9

Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10 Descent Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 11 Local Noetherian Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 12 Groupoid of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 13 B¨ okstedt–Neeman Resolutions and HyperExt Sheaves . . . . 381 14 The Right Adjoint of the Derived Direct Image Functor . . 385 15 Comparison of Local Ext Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 393 16 The Composition of Two Almost-Pseudofunctors . . . . . . . . . 395 17 The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 18 Commutativity of Twisted Inverse with Restrictions . . . . . . 405 19 Open Immersion Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 20 The Existence of Compactiﬁcation and Composition Data for Diagrams of Schemes Over an Ordered Finite Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 21 Flat Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 22 Preservation of Quasi-Coherent Cohomology . . . . . . . . . . . . . . 423 23 Compatibility with Derived Direct Images . . . . . . . . . . . . . . . . 425 24 Compatibility with Derived Right Inductions . . . . . . . . . . . . . 427 25 Equivariant Grothendieck’s Duality . . . . . . . . . . . . . . . . . . . . . . . 429 26 Morphisms of Finite Flat Dimension . . . . . . . . . . . . . . . . . . . . . . 431 27 Cartesian Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 29 Group Schemes Flat of Finite Type . . . . . . . . . . . . . . . . . . . . . . . 445 30 Compatibility with Derived G-Invariance . . . . . . . . . . . . . . . . . 449

x

Contents

31 Equivariant Dualizing Complexes and Canonical Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 32 A Generalization of Watanabe’s Theorem . . . . . . . . . . . . . . . . . 457 33 Other Examples of Diagrams of Schemes . . . . . . . . . . . . . . . . . 463 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Introduction

(0.1) The ﬁrst three chapters of these notes1 treat the basics of derived categories and functors, and of the formalism of four of Grothendieck’s “six operations” ([Ay], [Mb]), over, say, the category of ringed spaces (topological spaces equipped with a sheaf of rings)—namely the derived functors, for complexes which need not be bounded, of the sheaf functors ⊗, Hom, and of the direct and inverse image functors f∗ and f ∗ relative to a map f . The axioms of this formalism are summarized in §3.6 under the rubric adjoint monoidal Δ-pseudofunctors, with values in closed categories (§3.5). Chapter 4 develops the abstract theory of the twisted inverse image functor f ! associated to a ﬁnite-type separated map of schemes f : X → Y . (Suppose for now that Y is noetherian and separated, though for much of what we do, weaker hypotheses will suﬃce.) This functor maps the derived category of cohomologically bounded-below OY -complexes with quasicoherent homology to the analogous category over X. Its characterizing properties are: – Duality. If f is proper then f ! is right-adjoint to the derived direct image functor Rf∗. – Localization. If f is an open immersion (or even ´etale), then f ! is the usual inverse image functor f ∗. – Pseudofunctoriality (or 2-functoriality). To each composition f g X − → Y − → Z we can assign a natural functorial isomorphism ∼ (gf )! −→ f ! g ! , in such a way that a kind of associativity holds with respect to any composition of three maps, see §(3.6.5).

1

That are a polished version of notes written largely in the late 1980s, available in part since then from < www.math.purdue.edu/~lipman >. I am grateful to Bradley Lucier for his patient instruction in some of the ﬁner points of TEX, and for setting up the appearance macros in those days when canned style ﬁles were not common—and when compilation was several thousand times slower than nowadays. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

5

6

Introduction

Additional basic properties of f ! are its compatibility with ﬂat base change (Theorems (4.4.3), (4.8.3)), and the existence of canonical functorial maps, for OY -complexes E and F having quasi-coherent homology: RHom(Lf ∗E, f ! F ) → f ! RHom(E, F ) Lf ∗E ⊗ f ! F → f ! (E ⊗ F) = = denotes the left-derived tensor product), of which the ﬁrst is an (where ⊗ = isomorphism when E is cohomologically bounded above, with coherent homology, and F is cohomologically bounded below, (Exercise (4.9.3)(b)), and the second is an isomorphism whenever f has ﬁnite tor-dimension (Theorem (4.9.4)) or E is a bounded ﬂat complex (Exercise (4.9.6)(a)). The existence and uniqueness, up to isomorphism, of the twisted inverse image pseudofunctor is given by Theorem (4.8.1), and compatibility with ﬂat base change by Theorem (4.8.3). These are culminating results in the notes. Various approximations to these theorems have been known for decades, see, e.g., [H, p. 383, 3.4]. At present, however, the proofs of the theorems, as stated here, seem to need, among other things, a compactiﬁcation theorem of Nagata, that any ﬁnite-type separable map of noetherian schemes factors as an open immersion followed by a proper map, a fact whose proof was barely accessible before the appearance of [Lt] and [C′] (see also [Vj]); and even with that compactiﬁcation theorem, I am not aware of any complete, detailed exposition of the proofs in print prior to the recent one by Nayak [Nk].2 There must be a more illuminating treatment of this awesome pseudofunctor in the Plato-Erd¨ os Book! (0.2) The theory of f ! was conceived by Grothendieck [Gr′, pp. 112–115], as a generalization of Serre’s duality theorems for smooth projective varieties over ﬁelds. Grothendieck also applied his ideas in the context of ´etale cohomology. The fundamental technique of derived categories was developed by Verdier, who used it in establishing a duality theorem for locally compact spaces that generalizes classical duality theorems for topological manifolds. Deligne further developed the methods of Grothendieck and Verdier (cf. [De′ ] and its references). Hartshorne gave an account of the theory in [H]. The method there is to treat separately several distinctive special situations, such as smooth maps, ﬁnite maps, and regular immersions (local complete intersections), where f ! has a nice explicit description; and then to do the general case by pasting together special ones (locally, a general f can be factored as smooth ◦ ﬁnite). The fact that this approach works is indicative of considerable depth in the underlying structure, in that the special cases, that don’t a priori have to 2 In fact Nayak’s methods, which are less dependent on compactiﬁcations, apply to other contexts as well, for example ﬂat ﬁnitely-presentable separated maps of notnecessarily-noetherian schemes, or separated maps of noetherian formal schemes, see [Nk, §7 ]. See also the summary of Nayak’s work in [S′, §§3.1–3.3].

Introduction

7

be related at all, can in fact be melded; and in that the reduction from general to special involves several choices (for example, in the just-mentioned factorization) of which the ﬁnal results turn out to be independent. Proving the existence of f ! and its basic properties in this manner involves many compatibilities among those properties in their various epiphanies, a notable example being the “Residue Isomorphism” [H, p. 185]. The proof in [H] also makes essential use of a pseudofunctorial theory of dualizing complexes,3 so that it does not apply, e.g., to arbitrary separated noetherian schemes. On ﬁrst acquaintance, [De′] appears to oﬀer a neat way to cut through the complexity—a direct abstract proof of the existence of f !, with indications about how to derive the concrete special situations (which, after all, motivate and enliven the abstract formalism). Such an impression is bolstered by Verdier’s paper [V′ ]. Verdier gives a reasonably short proof of the ﬂat base change theorem, sketches some corollaries (for example, the ﬁnite tordimension case is treated in half a page [ibid., p. 396], as is the smooth case [ibid., pp. 397–398]), and states in conclusion that “all the results of [H], except the theory of dualizing and residual complexes, are easy consequences of the existence theorem.” In short, Verdier’s concise summary of the main features, together with some background from [H] and a little patience, should suﬃce for most users of the duality machine. Personally speaking, it was in this spirit—not unlike that in which many scientists use mathematics—that I worked on algebraic and geometric applications in the late 1970s and early 1980s. But eventually I wanted to gain a better understanding of the foundations, and began digging beneath the surface. The present notes are part of the result. They show, I believe, that there is more to the abstract theory than ﬁrst meets the eye. (0.3) There are a number of treatments of Grothendieck duality for the Zariski topology (not to mention other contexts, see e.g., [Gl′], [De], [LO]), for example, Neeman’s approach via Brown representability [N], Hashimoto’s treatment of duality for diagrams of schemes (in particular, schemes with group actions) [Hsh], duality for formal schemes [AJL′ ], as well as various substantial enhancements of material in Hartshorne’s classic [H], such as [C], [S], [LNS] and [YZ]. Still, some basic results in these notes, such as Theorem (3.10.3) and Theorem (4.4.1) are diﬃcult, if not impossible, to ﬁnd elsewhere, at least in the present generality and detail. And, as indicated below, there are in these notes some signiﬁcant diﬀerences in emphasis. It should be clariﬁed that the word “Notes” in the title indicates that the present exposition is neither entirely self-contained nor completely polished. The goal is, basically, to guide the willing reader along one path to an understanding of all that needs to be done to prove the fundamental Theorems (4.8.1) and (4.8.3), and of how to go about doing it. The intent is to provide enough in the way of foundations, yoga, and references so that the reader can, 3 This enlightening theory—touched on in §4.10 below—is generalized to Cousin complexes over formal schemes in [LNS]. A novel approach, via “rigidity,” is given in [YZ], at least for schemes of ﬁnite type over a ﬁxed regular one.

8

Introduction

more or less mechanically, ﬁll in as much of what is missing as motivation and patience allow. So what is meant by “foundations and yoga”? There are innumerable interconnections among the various properties of the twisted inverse image, often expressible via commutativity of some diagram of natural maps. In this way one can encode, within a formal functorial language, relationships involving higher direct images of quasi-coherent sheaves, or, more generally, of complexes with quasi-coherent homology, relationships whose treatment might otherwise, on the whole, prove discouragingly complicated. As a strategy for coping with duality theory, disengaging the underlying category-theoretic skeleton from the algebra and geometry which it supports has the usual advantages of simpliﬁcation, clariﬁcation, and generality. Nevertheless, the resulting fertile formalism of adjoint monoidal pseudofunctors soon sprouts a thicket of rather complicated diagrams whose commutativity is an essential part of the development—as may be seen, for example, in the later parts of Chapters 3 and 4. Verifying such commutativities, fun to begin with, soon becomes a tedious, time-consuming, chore. Such chores must, eventually, be attended to.4 Thus, these notes emphasize purely formal considerations, and attention to detail. On the whole, statements are made, whenever possible, in precise category-theoretic terms, canonical isomorphisms are not usually treated as equalities, and commutativity of diagrams of natural maps—a matter of paramount importance—is not taken for granted unless explicitly proved or straightforward to verify. The desire is to lay down transparently secure foundations for the main results. A perusal of §2.6, which treats the basic and RHom, relation “adjoint associativity” between the derived functors ⊗ = and of §3.10, which treats various avatars of the tor-independence condition on squares of quasi-compact maps of quasi-separated schemes, will illustrate the point. (In both cases, total understanding requires a good deal of preceding material.) Computer-aided proofs are often more convincing than many standard proofs based on diagrams which are claimed to commute, arrows which are supposed to be the same, and arguments which are left to the reader. —J.-P. Serre [R, pp. 212–213]. In practice, the techniques used to decompose diagrams successively into simpler ones until one reaches those whose commutativity is axiomatic do not seem to be too varied or diﬃcult, suggesting that sooner or later a computer might be trained to become a skilled assistant in this exhausting task. (For the general idea, see e.g., [Sm].) Or, there might be found a theorem in 4

Cf. [H, pp. 117–119], which takes note of the problem, but entices readers to relax their guard so as to make feasible a hike over the seemingly solid crust of a glacier.

Introduction

9

the vein of “coherence in categories” which would help even more.5 Though I have been saying this publicly for a long time, I have not yet made a serious enough eﬀort to pursue the matter, but do hope that someone else will ﬁnd it worthwhile to try. (0.4) Finally, the present exposition is incomplete in that it does not include that part of the “Ideal Theorem” of [H, pp. 6–7] involving concrete realizations of the twisted inverse image, particularly through diﬀerential forms. Such interpretations are clearly important for applications. Moreover, connections between diﬀerent such realizations—isomorphisms forced by the uniqueness properties of the twisted inverse image—give rise to fascinating maps, such as residues, with subtle properties reﬂecting pseudofunctoriality and base change (see [H, pp. 197–199], [L′]). Indeed, the theory as a whole has two complementary aspects. Without the enlivening concrete interpretations, the abstract functorial approach can be rather austere—though when it comes to treating complex relationships, it can be quite advantageous. While the theory can be based on either aspect (see e.g., [H] and [C] for the concrete foundations), bridging the concrete and abstract aspects is not a trivial matter. For a simple example (recommended as an exercise), over the category of open-and-closed immersions f, it is easily seen that the functor f ! is naturally isomorphic to the inverse image functor f ∗ ; but making this isomorphism pseudofunctorial, and proving that the ﬂat base-change isomorphism is the “obvious one,” though not diﬃcult, requires some eﬀort. More generally, consider smooth maps, say with d-dimensional ﬁbers. For such f : X → Y , and a complex A• of OY -modules, there is a natural isomorphism ∼ f !A• f ∗A• ⊗OX ΩdX/Y [d] −→ where ΩdX/Y [d] is the complex vanishing in all degrees except −d, at which it is the sheaf of relative d-forms (K¨ahler diﬀerentials).6 For proper such f , f ! is right-adjoint to Rf∗ , there is, correspondingly, a natural map where • (A ) : Rf∗ f ! A• → A• . In particular, when Y = Spec(k), k a ﬁeld, these data give Serre Duality, i.e., the existence of natural isomorphisms d ∼ Homk (H i (X, F ), k) −→ Extd−i X (F , ΩX/Y )

for quasi-coherent OX -modules F . Pseudofunctoriality of ! corresponds here to the standard isomorphism ∼ ΩdX/Y ⊗OX f ∗ ΩeY /Z −→ Ωd+e X/Z

5

Warning: see Exercise (3.4.4.1) below. A striking deﬁnition of this isomorphism was given by Verdier [V′, p. 397, Thm. 3]. See also [S′, §5.1] for a generalization to formal schemes.

6

10

Introduction f

g

attached to a pair of smooth maps X → Y → Z of respective relative dimensions d, e. For a map h : Y ′ → Y , and pX : X ′ := X ×Y Y ′ → X the projection, the abstractly deﬁned base change isomorphism ((4.4.3) below) corresponds to the natural isomorphism ∼ ∗ d pX ΩX/Y . ΩdX ′/Y ′ −→

The proofs of these down-to-earth statements are not easy, and will not appear in these notes. Thus, there is a canonical dualizing pair (f ! , : Rf∗ f ! → 1) when f is smooth; and there are explicit descriptions of its basic properties in terms of diﬀerential forms. But it is not at all clear that there is a canonical such pair for all f, let alone one which restricts to the preceding one on smooth maps. At the (homology) level of dualizing sheaves the case of varieties over a ﬁxed perfect ﬁeld is dealt with in [Lp, §10], and this treatment is generalized in [HS, §4] to generically smooth equidimensional maps of noetherian schemes without embedded components. All these facts should ﬁt into a general theory of the fundamental class of an arbitrary separated ﬁnite-type ﬂat map f : X → Y with d-dimensional ﬁbers, a canonical derived-category map ΩdX/Y [d] → f ! OY which globalizes the local residue map, and expresses the basic relation between diﬀerentials and duality. It is hoped that a “Residue Theorem” dealing with these questions in full generality will appear not too many years after these notes do.

Chapter 1

Derived and Triangulated Categories

In this chapter we review foundational material from [H, Chap. 1]1 (see also [De, §1]) insofar as seems necessary for understanding what follows. The main points are summarized in (1.9.1). Why derived categories? We postulate an interest in various homology objects and their functorial behavior. Homology is deﬁned by means of complexes in appropriate abelian categories; and we can often best understand relations among homology objects as shadows of relations among their deﬁning complexes. Derived categories provide a supple framework for doing so. To construct the derived category D(A) of an abelian category A, we begin with the category C = C(A) of complexes in A. Being interested basically in homology, we do not want to distinguish between homotopic maps of complexes; and we want to consider a morphism of complexes which induces homology isomorphisms (i.e., a quasi-isomorphism) to be an “equivalence” of complexes. So force these two considerations on C: ﬁrst factor out the homotopy-equivalence relation to get the category K(A) whose objects are those of C but whose morphisms are homotopy classes of maps of complexes; and then localize by formally adjoining an inverse morphism for each quasi-isomorphism. The resulting category is D(A), see §1.2 below. The category D(A) is no longer abelian; but it carries a supplementary structure given by triangles, which take the place of, and are functorially better-behaved than, exact sequence of complexes, see 1.4, 1.5.2 Restricting attention to complexes which are bounded (above, below, or both), or whose homology is bounded, or whose homology groups lie in some plump subcategory of A, we obtain corresponding derived categories, all 1 An expansion of some of [V], for which [Do] oﬀers some motivation. See the historical notes in [N′, pp. 70–71]. See also [I′ ]. Some details omitted in [H] can be found in more recent expos´es such as [Gl], [Iv, Chapter XI], [KS, Chapter I], [W, Chapter 10], [N′ , Chapters 1 and 2], and [Sm]. 2 All these constructs are Verdier quotients with respect to the triangulated subcategory of K(A) whose objects are the exact complexes, see [N′ , p. 74, 2.1.8].

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

11

12

1 Derived and Triangulated Categories

of which are in fact isomorphic to full triangulated subcategories of D(A), see 1.6, 1.7, and 1.9. In 1.8 we describe some equivalences among derived categories. For example, any choice of injective resolutions, one for each homologically bounded-below complex, gives a triangle-preserving equivalence from the derived category of such complexes to its full subcategory whose objects are bounded-below injective complexes (and whose morphisms can be identiﬁed with homotopy-equivalence classes of maps of complexes). Similarly, any choice of ﬂat resolutions gives a triangle-preserving equivalence from the derived category of homologically bounded-above complexes to its full subcategory whose objects are bounded-above ﬂat complexes. (For ﬂat complexes, however, quasi-isomorphisms need not have homotopy inverses). Such equivalences are useful, for example, in treating derived functors, also for unbounded complexes, see Chapter 2. The truncation functors of 1.10 and the “way-out” lemmas of 1.11 supply repeatedly useful techniques for working with derived categories and functors. These two sections may well be skipped until needed.

1.1 The Homotopy Category K Let A be an abelian category [M, p. 194]. K = K(A) denotes the additive category [M, p. 192] whose objects are complexes of objects in A: C•

dn−1

dn

· · · → C n−1 −−−→ C n −→ C n+1 → · · ·

(n ∈ Z, dn ◦ dn−1 = 0)

and whose morphisms are homotopy-equivalence classes of maps of complexes [H, p. 25]. (The maps d n are called the diﬀerentials in C • .) We always assume that A comes equipped with a speciﬁc choice of the zero-object, of a kernel and cokernel for each map, and of a direct sum for any two objects. Nevertheless we will often abuse notation by allowing the symbol 0 to stand for any initial object in A; thus for A ∈ A, A = 0 means only that A is isomorphic to the zero-object. For a complex C • as above, since d n ◦dn−1 = 0 therefore dn−1 induces a natural map C n−1 → (kernel of dn ) , the cokernel of which is deﬁned to be the homology H n (C • ). A map of complexes u : A• → B • obviously induces maps H n (u) : H n (A• ) → H n (B • )

(n ∈ Z),

and these maps depend only on the homotopy class of u. Thus we have a family of functors Hn : K → A (n ∈ Z).

1.2 The Derived Category D

13

We say that u (or its homotopy class u ¯, which is a morphism in K) is u) is an a quasi-isomorphism if for every n ∈ Z, the map H n (u) = H n (¯ isomorphism.

1.2 The Derived Category D The derived category D = D(A) is the category whose objects are the same as those of K, but in which each morphism A• → B • is the equivalence class f /s of a pair (s, f ) f

s

A• ←− C • −→ B • of morphisms in K, with s a quasi-isomorphism, where two such pairs (s, f ), (s′, f ′ ) are equivalent if there is a third such pair (s′′, f ′′ ) and a commutative diagram in K: C• s

A•

s′′

f

C ′′•

s′

f ′′

B•

f′

C ′• see [H, p. 30]. The composition of two morphisms f /s : A• → B • , f ′ /s′ : B • → B ′• , is f ′ g/st, where (t, g) is a pair (which always exists) such that f t = s′ g, see [H, pp. 30–31, 35–36]: C1• t

A•

s

g

C•

C ′•

f′

B ′•

s′

f

B• In particular, with (s, f ) as above and 1C • the homotopy class of the identity map of C •, we have f /s = (f /1C • )◦(1C • /s) = (f /1C • )◦(s/1C • )−1 .

14

1 Derived and Triangulated Categories

There is a natural functor Q : K → D with Q(A• ) = A• for each complex A• in K and Q(f ) = f /1A• for each map f : A• → B • in K. If f is a quasiisomorphism then Q(f ) = f /1A• is an isomorphism (with inverse 1A• /f ); and in this respect, Q is universal: any functor Q′ : K → E taking quasiisomorphisms to isomorphisms factors uniquely via Q, i.e., there is a unique ′ ◦Q (so that Q ′ (A• ) = Q′ (A• ) and ′ : D → E such that Q′ = Q functor Q ′ ′ ′ −1 Q (f /s) = Q (f )◦ Q (s) ). This characterizes the pair (D, Q) up to canonical isomorphism.3 Moreover [H, p. 33, Prop. 3.4]: any morphism Q′1 → Q′2 of such functors ′ → Q ′ . In other words, composition extends uniquely to a morphism Q 1 2 with Q gives, for any category E, an isomorphism of the functor category Hom(D, E) onto the full subcategory of Hom(K, E) whose objects are the functors K → E which transform quasi-isomorphisms in K into isomorphisms in E. One checks that the category D supports a unique additive structure such that the canonical functor Q : K → D is additive; and accordingly we will always regard D as an additive category. If the category E and the above ′ . functor Q′ : K → E are both additive, then so is Q

Remark 1.2.1. The homology functors H n : K → A deﬁned in (1.1) transform quasi-isomorphisms into isomorphisms, and hence may be regarded as functors on D. (1.2.2). A morphism f /s : A• → B • in D is an isomorphism if and only if H n (f /s) = H n (f ) ◦ H n (s)−1 : H n (A• ) → H n (B • ) is an isomorphism for all n ∈ Z. Indeed, if H n (f /s) is an isomorphism for all n, then so is H n (f ), i.e., f is a quasi-isomorphism; and then s/f is the inverse of f /s. (1.2.3). There is an isomorphism of A onto a full subcategory of D, taking any object X ∈ A to the complex X • which is X in degree zero and 0 elsewhere, and taking a map f : X → Y in A to f • /1X • , where f • : X • → Y • is the homotopy class whose sole member is the map of complexes which is f in degree zero. Bijectivity of the indicated map HomA (X, Y ) → HomD(A) (X •, Y • ) is a straightforward consequence of the existence of a natural functorial isomor∼ phism Z −→ H 0 (Z • ) (Z ∈ A).

3 The set Σ of quasi-isomorphisms in K admits a calculus of left and of right fractions, and D is, up to canonical isomorphism, the category of fractions K[Σ−1 ], see e.g., [Sc, Chapter 19.] The set-theoretic questions arising from the possibility that Σ is “too large,” i.e., a class rather than a set, are dealt with in loc. cit. Moreover, there is often a construction of a universal pair (D, Q) which gets around such questions (but may need the axiom of choice), cf. (2.3.2.2) and (2.3.5) below.

1.3 Mapping Cones

15

1.3 Mapping Cones An important construction is that of the mapping cone Cu• of a map of complexes u : A• → B • in A. (For this construction we need only assume that the category A is additive.) Cu• is the complex whose degree n component is Cun = B n ⊕ An+1 and whose diﬀerentials d n : Cun → Cun+1 satisfy d n |B n = dnB ,

d n |An+1 = u|An+1 − dn+1 A

(n ∈ Z)

where the vertical bars denote “restricted to,” and dB , dA are the diﬀerentials in B • , A• respectively. Cun+1 = B n+1 ⊕ An+2 ⏐ ⏐ ⏐ dB ⏐ u ⏐−dA d⏐ Cun = B n

⊕ An+1

For any complex A• , and m ∈ Z, A• [m] will denote the complex having degree-n component (A• [m])n = An+m

(n ∈ Z)

and in which the diﬀerentials An [m] → An+1 [m] are (−1)m times the corresponding diﬀerentials An+m → An+m+1 in A• . There is a natural “translation” functor T from the category of A-complexes into itself satisfying T A• = A• [1] for all complexes A• . To any map u as above, we can then associate the sequence of maps of complexes u

v

w

A• −→ B • −→ Cu• −→ A• [1]

(1.3.1)

where v (resp. w) is the natural inclusion (resp. projection) map. The sequence (1.3.1) could also be represented in the form Cu• (1.3.2)

[1]

A•

u

B•

and so we call such a sequence a standard triangle.

16

1 Derived and Triangulated Categories

A commutative diagram of maps of complexes u

A• −−−−→ ⏐ ⏐

B• ⏐ ⏐

A′• −−−− → B ′• ′ u

gives rise naturally to a commutative diagram of associated g triangles (each arrow representing a map of complexes): u

A• −−−−→ ⏐ ⏐

B • −−−−→ ⏐ ⏐

Cu• −−−−→ A• [1] ⏐ ⏐ ⏐ ⏐

A′• −−−− → B ′• −−−−→ Cu•′ −−−−→ A′• [1] ′ u

Most of the basic properties of standard triangles involve homotopy, and so are best stated in K(A). For example, the mapping cone C1• of the identity map A• → A• is homotopically equivalent to zero, a homotopy between the identity map of C1• and the zero map being as indicated: C1n+1 ⏐ ⏐ hn+1 C1n

= An+1 ⊕ An+2 1

An

=

⊕ An+1

(i.e., for each n ∈ Z, hn+1 restricts to the identity on An+1 and to 0 on An+2 ; and dn−1 hn + hn+1 d n is the identity of C1n ). Other properties can be found e.g., in [Bo, pp. 102–105], [Iv, pp. 22–33]. For subsequent developments we need to axiomatize them, as follows.

1.4 Triangulated Categories (Δ-Categories) A triangulation on an arbitrary additive category K consists of an additive automorphism T (the translation functor) of K, and a collection T of diagrams of the form u

v

w

A −→ B −→ C −→ T A .

(1.4.1)

A triangle (with base u and summit C) is a diagram (1.4.1) in T . (See (1.3.2) for a more picturesque—but typographically less convenient—representation of a triangle.) The following conditions are required to hold:

1.4 Triangulated Categories (Δ-Categories)

17

(Δ1)′

Every diagram of the following form is a triangle:

(Δ1)′′

A −−−−−→ A −−−−→ 0 −−−−→ T A . Given a commutative diagram

identity

A −−−−→ B −−−−→ C −−−−→ T A ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ γ α β T α

(Δ2)

A′ −−−−→ B ′ −−−−→ C ′ −−−−→ T A′ if α, β, γ are all isomorphisms and the top row is a triangle then the bottom row is a triangle. For any triangle (1.4.1) consider the corresponding inﬁnite diagram (1.4.1)∞ : −T −1 w

u

v

w

−T u

· · · −→ T −1 C −−−−−→ A −→ B −→ C −→ T A −−−→ T B −→ · · ·

(Δ3)′ (Δ3)′′

in which every arrow is obtained from the third preceding one by applying −T . Then any three successive maps in (1.4.1)∞ form a triangle. u Any morphism A −→ B in K is the base of a triangle (1.4.1). For any diagram u

A −−−−→ B −−−−→ C −−−−→ T A ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ α β (∃γ) T α A′ −−−− → B ′ −−−−→ C ′ −−−−→ T A′ ′ u

whose rows are triangles, and with maps α, β given such that βu = u′ α, there exists a morphism γ : C → C ′ making the entire diagram commute, i.e., making it a morphism of triangles.4 4

(Δ3)′′ is implied by a stronger “octahedral” axiom, which states that for a u

β

composition A −→ B −→ B ′ and triangles Δu , Δβu , Δβ with respective bases u, βu, β, there exist morphisms of triangles Δu → Δβu → Δβ extending the diagram u

A −−−−→

A −−−−→ βu ⏐ ⏐ u

B ⏐ ⏐β

B′

B −−−−→ B ′ β

and such that the induced maps on summits Cu → Cβu → Cβ are themselves the sides of a triangle, whose third side is the composed map Cβ → T B → T Cu . This axiom is incompletely stated in [H, p. 21], see [V, p. 3] or [Iv, pp. 453–455]. We omit it here because it plays no role in these notes (nor, as far as I can tell, in [H]). Thus the adjective “pre-triangulated” may be substituted for “triangulated” throughout, see [N′, p. 51, Deﬁnition 1.3.13 and p. 60, Remark 1.4.7].

18

1 Derived and Triangulated Categories

As a consequence of these conditions we have [H, p. 23, Prop. 1.1 c]: (Δ3)∗ If in (Δ3)′′ both α and β are isomorphisms, then so is γ. Thus, and by (Δ3)′ : u Every morphism A −→ B is the base of a triangle, uniquely determined up to isomorphism by u. Deﬁnition 1.4.2. A triangulated category (Δ-category for short) is an additive category together with a triangulation. Exercise 1.4.2.1. (Cf. [N′ , pp. 42–45].) For any triangle u

v

w

A −→ B −→ C −→ T A in a Δ-category K, and any object M , the induced sequence of abelian groups Hom(M , A) → Hom(M , B) → Hom(M , C) is exact [H, p. 23, 1.1 b)]. Using this and (Δ2) (or otherwise), show that u is an isomorphism iff C ∼ = 0. More generally, the following conditions are equivalent: (a) u is a monomorphism. (b) v is an epimorphism. (c) w = 0. (d) There exist maps A ←− s C such that t B ←− 1A = tu,

1B = sv + ut,

1C = vs

(so that B ∼ = A ⊕ C). Consequently, in view of (Δ3)′ , any monomorphism in K has a left inverse and any epimorphism has a right inverse. And incidentally, the existence of ﬁnite direct sums in K follows from the other axioms of Δ-categories.

Examples 1.4.3. For any abelian (or just additive) category A, the homotopy category K := K(A) of (1.1) has a triangulation, with translation T such that (A• ∈ K) T A• = A• [1] (i.e., T is induced by the translation functor on complexes, see (1.3), a functor which respects homotopy), and with triangles all those diagrams (1.4.1) which are isomorphic (in the obvious sense, see (Δ3)∗ ) to the image in K of some standard triangle, see (1.3) again. The properties (Δ1)′, (Δ1)′′, and (Δ3)′ follow at once from the discussion in (1.3). To prove (Δ3)′′ we may assume that C = Cu• , C ′ = Cu•′ , and the rows of the diagram are standard triangles. By assumption, βu is homotopic to u′ α, i.e., there is a family of maps hn : An → B ′n−1 (n ∈ Z) such that n−1 n β n un − u′n αn = dB + hn+1 dnA . ′ h

1.4 Triangulated Categories (Δ-Categories)

19

Deﬁne γ by the family of maps γ n : C n = B n ⊕ An+1 −→ B ′n ⊕ A′n+1 = C ′n

(n ∈ Z)

such that for b ∈ B n and a ∈ An+1 , γ n (b, a) = β n (b) + hn+1 (a), αn+1 (a) ,

and then check that γ is as desired. For establishing the remaining property (Δ2), we recall some facts about cylinders of maps of complexes (see e.g., [B, §2.6]—modulo sign changes leading to isomorphic complexes). Let u : A• → B • be a map of complexes, and let w : Cu• → A• [1] be the u• , to be the complex natural map, see §1.3. We deﬁne the cylinder of u, C • := C • [−1] . C u w

• is also the cone of the map (−1, u) : A → A ⊕ B.) One checks that there (C u u• → B • given in degree n by the map is a map of complexes ϕ : C such that

un = An ⊕ B n ⊕ An+1 → B n ϕn : C ϕn (a, b, a′ ) = u(a) + b .

The map ϕ is a homotopy equivalence, with homotopy inverse ψ given in degree n by ψ n (b) = (0, b, 0) . n is given by n+1 → C n+1 is the diﬀerential and hn+1 : C n → C [If d n : C u u u u hn+1 (a, b, a′ ) = (0, 0, −a), then 1C n − ψ n ϕn = dn−1 hn + hn+1 d n . . .] u

There results a diagram of maps of complexes u ˜ u• −−−v˜−→ A• −−−−→ C ⏐ ⏐ ϕ

w

Cu• −−−−→ A• [1]

(1.4.3.1)

A• −−−−→ B • −−−−→ Cu• −−−−→ A• [1] u

v=v ˜ψ

w

in which u ˜ and v˜ are the natural maps, the bottom row is a standard triangle, the two outer squares commute, and the middle square is homotopycommutative, i.e., v˜ − vϕ = v˜(1 − ψϕ) is homotopic to 0.

20

1 Derived and Triangulated Categories

Now, (1.4.3.1) implies that the diagram −w[−1]

u

v

u ˜

v ˜

Cu• [−1] −−−−−→ A• −→ B • −→ Cu• is isomorphic in K to the diagram −w[−1]

u• −→ Cu• Cu• [−1] −−−−−→ A• −→ C

• = C • [−1] = C • which is a standard triangle, since C w u −w[−1] . Hence if u′

v′

w′

A• −→ B • −→ C • −→ A• [1] is any triangle in K, then −w′ [−1]

u′

v′

C • [−1] −−−−−→ A• −→ B • −→ C • is a triangle, and—by the same reasoning—so is −v ′ [−1]

−w′ [−1]

u′

B • [−1] −−−−−→ C • [−1] −−−−−→ A• −→ B • , and consequently so is v′

u′ [1]

−w′

B • −→ C • −−→ A• [1] −−−→ B • [1] • • • ∼ • (because if A• ∼ = C−v ′ [−1] = Cv ′ [−1], then A [1] = Cv ′ ), as is the isomorphic diagram v′

−u′ [1]

w′

B • −→ C • −→ A• [1] −−−−→ B • [1] . Property (Δ2) for K results.5 We will always consider K to be a Δ-category, with this triangulation. There is a close relation between triangles in K and certain exact sequences. For any exact sequence of complexes in an abelian category A u

v

0 −→ A• − → B• − → C • −→ 0 ,

(1.4.3.2)

if u0 is the isomorphism from A• onto the kernel of v induced by u, then we have a natural exact sequence of complexes inclusion

χ

→ C • −→ 0 0 −→ Cu•0 −−−−−→ Cu• −

5

(1.4.3.3)

For other treatments of (Δ2) and (Δ3)′′ see [Bo, pp. 102–104] or [Iv, p. 27, 4.16; and p. 30, 4.19]. And for the octahedral axiom, use triangle (4.22) in [Iv, p. 32], whose vertices are the cones of two composable maps and of their composition.

1.4 Triangulated Categories (Δ-Categories)

21

where χn : Cun → C n (n ∈ Z) is the composition natural

v

χn : Cun = B n ⊕ An+1 −−−−→ B n − → Cn (see (1.3)). It is easily checked—either directly, or because Cu•0 is isomorphic to the cone of the identity map of A• —that H n (Cu•0 ) = 0 for all n; and then from the long exact cohomology sequence associated to (1.4.3.3) we conclude that χ is a quasi-isomorphism. If the exact sequence (1.4.3.2) is semi-split, i.e., for every n ∈ Z, the restriction v n : B n → C n of v to B n has a left inverse, say ϕn , then with Φn = ϕn ⊕ (ϕn+1 dnC − dnB ϕn ) : C n → B n ⊕ An+1 (where An+1 is identiﬁed with ker(v n+1 ) via u), the map of complexes Φ := (Φn )n∈Z is a homotopy inverse for χ: χ ◦Φ is the identity map of C •, and also the map (1Cu• −Φ ◦ χ) : Cu• → ker(χ) = Cu•0 ∼ = 0 vanishes in K. [More explicitly, if hn+1 : Cun+1 → Cun is given by hn+1 (b, a) := b − φn+1 v n+1 b ∈ An+1 ⊂ Cun

(b ∈ B n+1 , a ∈ An+2 )

and d is the diﬀerential in Cu• , then 1Cun − Φn ◦χn = (dn−1 hn + hn+1 d n ).] Thus χ induces a natural isomorphism in K ∼ C• , Cu• −→

and hence by (Δ1)′′ we have a triangle ¯ u

¯ v

¯ w

A• −→ B • −→ C • −→ A• [1]

(1.4.3.4)

where u ¯, v¯ are the homotopy classes of u, v respectively, and w ¯ is the homotopy class of the composed map Φ

natural

(ϕn+1 dnC − dnB ϕn )n∈Z : C • − → Cu• −−−−→ A• [1] ,

(1.4.3.5)

a class independent of the choice of splitting maps ϕn, because χ does not depend on that choice, so that neither does its inverse Φ, up to homotopy. This w ¯ is called the homotopy invariant of (1.4.3.2) (assumed semi-split).6 6

The category A need only be additive for us to deﬁne the homotopy invariant of a u v semi-split sequence of complexes A• ⇄ B • ⇄ C • (i.e., B n ∼ = An ⊕ C n for all n, and ψ ϕ un , ψ n , v n , ϕn are the usual maps associated with a direct sum): it’s the homotopy class of the map ψ(ϕdC − dB ϕ) : C • → A• [1], a class depending, as above, only on u and v. [More directly, note that if ϕ′ is another family of splitting maps then ψ(ϕdC − dB ϕ) − ψ(ϕ′ dC − dB ϕ′ ) = dA[1] ψ(ϕ − ϕ′ ) + ψ(ϕ − ϕ′ )dC . ]

22

1 Derived and Triangulated Categories

Moreover, any triangle in K is isomorphic to one so obtained. This is shown by the image in K of (1.4.3.1) (in which the bottom row is any standard triangle, and the homotopy equivalence ϕ becomes an isomorphism) as soon as one checks that the top row is in fact of the form speciﬁed by (1.4.3.4) and (1.4.3.5). By way of illustration here is an often used fact, whose proof involves triangles. (See also [H, pp. 35–36].) s

f

Lemma 1.4.3.6. Any diagram A• ← C • → B • in K(A), with s a quasiisomorphism, can be embedded in a commutative diagram f

C • −−−−→ ⏐ ⏐ s

B• ⏐ ⏐′ s

(1.4.3.7)

A• −−−− → D• ′ f

with s′ a quasi-isomorphism. Proof. By (Δ3)′ there exists a triangle (s,−f )

C • −−−−→ A• ⊕ B • −→ D• −→ C • [1] .

(1.4.3.8)

If f ′ is the natural composition A• → A• ⊕ B • → D• , and s′ is the composition B • → A• ⊕ B • → D• , then commutativity of (1.4.3.7) results from the easily-veriﬁable fact that the composition of the ﬁrst two maps in a standard triangle is homotopic to 0.7 And if s is a quasi-isomorphism, then from (1.4.3.8) we get exact homology sequences 0 → H n (C • ) → H n (A• ) ⊕ H n (B • ) → H n (D• ) → 0

(n ∈ Z)

(see (1.4.5) below) which quickly yield that s′ is a quasi-isomorphism too. Examples 1.4.4. The above triangulation on K leads naturally to one on is determined by the derived category D of 1.2. The translation functor T the relation QT = T Q, where Q : K → D is the canonical functor, and T is the translation functor in K (see (1.4.3)): note that QT transforms quasiisomorphisms into isomorphisms, and use the universal property of Q given (A• ) = A• [1] for every complex A• ∈ D. (T is additive, in 1.2. In particular T by the remarks just before (1.2.1).) The triangles are those diagrams which are isomorphic—in the obvious sense, see (Δ3)∗ —to those coming from K via Q, i.e., diagrams isomorphic to natural images of standard triangles. Conditions (Δ1)′ , (Δ1)′′ , and (Δ2) are easily checked. 7

In fact in any Δ-category, any two successive maps in a triangle compose to 0 [H, p. 23, Prop. 1.1 a)].

1.4 Triangulated Categories (Δ-Categories)

23 s

f

Next, given f /s : A• → B • in D, represented by A• ← X • → B • in K g h f (see 1.2), we have, by (Δ3)′ for K, a triangle X • → B • → C • → X • [1] in K, whose image is the top row of a commutative diagram in D, as follows: Q(f )

Q(g)

Q(h)

X • −−−−→ B • −−−−→ C • −−−−→ X • [1] ⏐ ⏐ ⏐ ⏐ ≃T Q(s) Q(s)≃ A• −−−−→ B • −−−−→ C • −−−−→ X • [1]

(1.4.4.1)

f /s

Condition (Δ3)′ for D results. As for (Δ3)′′ , we can assume, via isomorphisms, that the rows of the diagram in question come from K, via Q. Then we check via deﬁnitions in 1.2 that the commutative diagram u

A• −−−−→ ⏐ ⏐ α

B• ⏐ ⏐ β

A′• −−−− → B ′• ′ u

in D can be expanded to a commutative diagram of the form u

A• −−−−→ B • ⏐ ⏐ α1 ⏐≃ β1 ⏐≃

X • −−−−→ Y • ⏐ ⏐ ⏐ ⏐ α2 β2

A′• −−−− → B ′• ′ u

(i.e., α = α2 α1−1, β = β2 β1−1 ), where all the arrows represent maps coming from K, i.e., maps of the form Q(f ). By (Δ3)′ and (Δ3)′′ for K, this diagram embeds into a larger commutative one whose middle row also comes from K: u

A• −−−−→ B • −−−−→ C • −−−−→ A• [1] ⏐ ⏐ ⏐ ⏐ γ1 ⏐ α1 ⏐≃ ≃⏐T α1 β1 ⏐≃

X • −−−−→ Y • −−−−→ Z • −−−−→ X • [1] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ γ2 α2 β2 T α2 A′• −−−− → B ′• −−−−→ C ′• −−−−→ A′• [1] ′ u

24

1 Derived and Triangulated Categories

Using (1.2.2) and the exact homology sequences associated to the top two rows (see (1.4.5) below), we ﬁnd that γ1 is an isomorphism. Then γ := γ2 γ1−1 fulﬁlls (Δ3)′′ . So we have indeed deﬁned a triangulation; and from (Δ1)′′ , (Δ3)∗ , and (1.4.4.1) we conclude that this is the unique triangulation on D with translation T and such that Q transforms triangles into triangles. We will always consider D to be a Δ-category, with this triangulation. Now for any exact sequence of complexes in A u

v

0 −→ A• −→ B • −→ C • −→ 0

(1.4.4.2)

the quasi-isomorphism χ of (1.4.3.3) becomes an isomorphism χ ˜ in D, so that in D there is a natural composed map χ ˜−1

w ˜ : C • −→ Cu• −→ A• [1] ; and then with u ˜ and v˜ corresponding to u and v respectively, the diagram u ˜

v ˜

w ˜

A• −→ B • −→ C • −→ A• [1]

(1.4.4.2)∼

is a triangle in D. If the sequence (1.4.4.2) is semi-split, then (1.4.4.2)∼ is the image in D of the triangle (1.4.3.4) in K. Since every triangle in K is isomorphic to one coming from a semi-split exact sequence (see end of example (1.4.3)), therefore every triangle in D is isomorphic to one of the form (1.4.4.2)∼ arising from an exact sequence of complexes in A (in fact, from a semi-split such sequence). u

v

w

(1.4.5). To any triangle A• −→ B • −→ C • −→ A• [1] in K or in D, we can apply the homology functors H n (see (1.2.1)) to obtain an associated exact homology sequence H i−1 (w)

H i (u)

· · · −→ H i−1 (C • ) −−−−−−→ H i (A• ) −−−−→ H i (B • ) H i (v)

H i (w)

−−−−→ H i (C • ) −−−−→ H i+1 (A• ) −→ · · ·

(1.4.5)H Exactness is veriﬁed by reduction to the case of standard triangles. For an exact sequence (1.4.4.2), the usual connecting homomorphism H i (C • ) → H i+1 (A• )

(i ∈ Z)

is easily seen to be −H i (w) ˜ (see (1.4.4.2)∼ ). Thus (1.4.5)H (for (1.4.4.2)∼ ) is, except for signs, the usual homology sequence associated to (1.4.4.2). It should now be clear why it is that we can replace exact sequences of complexes in A by triangles in D. And the following notion of “Δ-functor ” will eventually make it quite advantageous to do so.

1.5 Triangle-Preserving Functors (Δ-Functors)

25

1.5 Triangle-Preserving Functors (Δ-Functors) Let K1 , K2 be Δ-categories (1.4.2) with translation functors T1 , T2 respectively. A (covariant) Δ-functor is deﬁned to be a pair (F , θ) consisting of an additive functor F : K1 → K2 together with an isomorphism of functors ∼ θ : F T1 −→ T2 F

such that for every triangle u

v

w

A −→ B −→ C −→ T1 A in K1 , the corresponding diagram Fu

Fu

θ ◦F w

F A −−→ F B −−→ F C −−−→ T2 F A is a triangle in K2 . These are the exact functors of [V, p. 4], and also the ∂-functors of [H, p. 22]; it should be kept in mind however that θ is not always the identity transformation (see Examples (1.5.3), (1.5.4) below—but see also Exercise (1.5.5)). In practice, for given F if there is some θ such that (F , θ) is a Δ-functor then there will usually be a natural one, and after specifying such a θ we will simply say (abusing language) that F is a Δ-functor. Let K3 be a third Δ-category, with translation T3 . If each one of (F , θ) : K1 → K2 and (H, χ) : K2 → K3 is a Δ-functor, then so is (H ◦F , χ ◦θ) : K1 → K3 where χ ◦θ is deﬁned to be the composition via χ

via θ

HF T1 −−−→ HT2 F −−−→ T3 HF . A morphism η : (F , θ) → (G, ψ) of Δ-functors (from K1 to K2 ) is a morphism of functors η : F → G such that for all objects X in K1 , the following diagram commutes: θ(X)

F T1 (X) −−−−→ T2 F (X) ⏐ ⏐ ⏐T (η(X)) ⏐ η(T1 (X)) 2 GT1 (X) −−−−→ T2 G(X) ψ(X)

The set of all such η can be made, in an obvious way, into an abelian group. If μ : (G, ψ) → (G′, ψ ′ ) is also a morphism of Δ-functors, then so is the composition μη : (F , θ) → (G′, ψ ′ ). And if (H, χ) : K2 → K3 [respectively (H ′, χ′ ) : K3 → K1 ] is, as above, another Δ-functor then η naturally induces

26

1 Derived and Triangulated Categories

a morphism of composed Δ-functors (H ◦F , χ ◦θ) → (H ◦ G, χ ◦ψ) [ respectively (F ◦H ′, θ ◦χ′ ) → (G ◦H ′, ψ ◦ χ′ ) ] . We ﬁnd then that: Proposition. The Δ-functors from K1 to K2 , and their morphisms, form an additive category Hom∆ (K1 , K2 ); and the composition operation Hom∆ (K1 , K2 ) × Hom∆ (K2 , K3 ) −→ Hom∆ (K1 , K3 ) is a biadditive functor. A morphism η as above has an inverse in Hom∆ (K1 , K2 ) if and only if η(X) is an isomorphism in K2 for every X ∈ K1 . We call such an η a Δ-functorial isomorphism. Similarly, a contravariant Δ-functor is a pair (F , θ) with F : K1 → K2 a contravariant additive functor and ∼ θ : T2−1 F −→ F T1

an isomorphism of functors such that for every triangle in K1 as above, the corresponding diagram Fu

Fv

−F w ◦ θ

F A ←−− F B ←−− F C ←−−−−− T2−1 F A is a triangle in K2 . Composition and morphisms etc. of contravariant Δ-functors are introduced in the obvious way. Exercise. A contravariant Δ-functor is the same thing as a covariant Δ-functor on the opposite (dual) category Kop 1 [M, p. 33], suitably triangulated. (For example, D(A)op is Δ-isomorphic to D(Aop ), see (1.4.4).)

Examples. (1.5.1). (see [H, p. 33, Prop. 3.4]). By (1.4.4), the natural functor Q : K → D of §1.2, together with θ = identity, is a Δ-functor. Moreover, as in 1.2: composition with Q gives, for any Δ-category E, an isomorphism of the category of Δ-functors Hom∆ (D, E) onto the full subcategory of Hom∆ (K, E) whose objects are the Δ-functors (F , θ) such that F transforms quasi-isomorphisms in K to isomorphisms in E.8 Equivalently (∗): F (C • ) ∼ = 0 for every exact complex C • ∈ K. (“C • exact” means H i (C • ) = 0 for all i, i.e., the zero map C • → 0 is a quasi-isomorphism). Exactness of the homology sequence (1.4.5)H of a standard triangle shows that a map u in K • is exact. Also, the base of a triangle is an is a quasi-isomorphism iﬀ the cone Cu • isomorphism iﬀ the summit is 0, see (1.4.2.1). So since F (Cu ) is the summit of a triangle with base F (u), (∗) implies that if u is a quasi-isomorphism then F (u) is an isomorphism. 8

1.5 Triangle-Preserving Functors (Δ-Functors)

27

(1.5.2). Let F : A1 → A2 be an additive functor of abelian categories, and set K1 = K(A1 ), K2 = K(A2 ). Then F extends in an obvious way to an additive functor F¯ : K1 → K2 which commutes with translation, and which (together with θ = identity) is easily seen to be a Δ-functor, essentially because F¯ takes cones to cones, i.e., for any map u of complexes in A1 , F¯ (Cu• ) = CF•¯ (u) .

(1.5.2.1)

(1.5.3) (expanding [H, p. 64, line 7] and illustrating [De, p. 265, Prop. 1.1.7]). For complexes A• , B • in the abelian category A, the complex of abelian groups Hom• (A• , B • ) is given in degree n by

Hom(Aj , B j+n ) Homn (A• , B • ) = Homgr (A• [−n], B • ) = j∈Z

(“Homgr ” denotes “homomorphisms of graded groups”) and the diﬀerential d n : Homn → Homn+1 takes f ∈ Homgr (A• [−n], B • ) to d n (f ) := (dB ◦f )[−1] + f ◦ dA[−n−1] ∈ Homgr (A• [−n − 1], B • ). In other words, if f = (f j )j∈Z with f j ∈ Hom(Aj , B j+n ) then n+j j j d n (f ) = dB ◦f + (−1)n+1 f j+1 ◦ dA .9 j∈Z

For ﬁxed C • , the additive functor of complexes

F1 (A• ) = Hom• (C • , A• ) preserves homotopy, and so gives an additive functor (still denoted by F1 ) from K = K(A) into K(Ab) (where Ab is the category of abelian groups). One checks that F1 T = T∗ F1 , (T = translation in K, T∗ = translation in K(Ab)) and that F1 takes cones to cones (cf. (1.5.2.1)); and hence F1 (together with θ1 = identity) is a Δ-functor. Similarly, for ﬁxed D• , F2 (A• ) = Hom• (A• , D• ) gives a contravariant additive functor from K into K(Ab). But now we run into sign complications: the complexes T∗−1 F2 (A• ) and F2 T (A• ), while coinciding as graded objects, are not equal, the diﬀerential in one being the negative of the diﬀerential in the other. We deﬁne a functorial isomorphism ∼ F2 T (A• ) θ2 (A• ) : T∗−1 F2 (A• ) −→

to be multiplication in each degree n by (−1)n , and claim that the pair (F2 , θ2 ) is a contravariant Δ-functor. 9

This standard d n diﬀers from the one in [H, p. 64] by a factor of (−1)n+1.

28

1 Derived and Triangulated Categories

Indeed, if u : A• → B • is a morphism of complexes in A, then we check (by writing everything out explicitly) that, with F = F2 , θ = θ2 , the map of graded objects T∗ (θ(A• ))⊕(−1)

CF• u = F A• ⊕ T∗ F B • −−−−−−−−−−→ T∗ F T A• ⊕ T∗ F B • = T∗ F Cu• is an isomorphism of complexes, whence, v : B • → Cu• and w : Cu• → T A• being the canonical maps, the diagram (T∗ F w)◦T∗ (θ(A• ))

Fu

−T F v

F B • −−→ F A• −−−−−−−−−−−−→ T∗ F Cu• −−−∗−→ T∗ F B • is a triangle in K(Ab), i.e., (−F w)◦θ(A• )

Fv

Fu

T∗−1 F A• −−−−−−−−−→ F (Cu• ) −−→ F B • −−→ F A• is a triangle (see (Δ2) in §1.4); and the claim follows. (1.5.4) (see again [De, p. 265, Prop. 1.1.7]). Let U be a topological space, O a sheaf of rings—say, for simplicity, commutative—and A the abelian category of sheaves of O-modules. For complexes A• , B • in A, the complex A• ⊗ B • is given in degree n by (Ap ⊗ B n−p ) (⊗ = ⊗O ) (A• ⊗ B • )n = p∈Z

and the diﬀerential d n : (A• ⊗ B • )n → (A• ⊗ B • )n+1 is the unique map whose restriction to Ap ⊗ B n−p is p n−p d n |(Ap ⊗ B n−p ) = dA ⊗ 1 + (−1)p ⊗ dB

(p ∈ Z).

With the usual translation functor T , we have for each i, j ∈ Z a unique isomorphism of complexes ∼ θij : T i A• ⊗ T j B • −→ T i+j (A• ⊗ B • )

satisfying, for every p, q ∈ Z, θij |(Ap+i ⊗ B q+j ) = multiplication by (−1)pj . [Note that Ap+i ⊗ B q+j is contained both in (T i A• ⊗ T j B • )p+q and in (T i+j (A ⊗ B))p+q .] For ﬁxed A• , we ﬁnd then that the functor of complexes taking B • to • B ⊗ A• preserves homotopy and takes cones to cones, giving an additive functor from K(A) into itself, which, together with θ10 = identity, is a Δ-functor.

1.6 Δ-Subcategories

29

Similarly, for ﬁxed A• the functor taking B • to A• ⊗ B • induces a functor of K(A) into itself which, together with θ01 = identity, is a Δ-functor. And for ﬁxed A• , the family of isomorphisms ∼ θ(B • ) : A• ⊗ B • −→ B • ⊗ A•

(1.5.4.1)

deﬁned locally by θ(B • )(a ⊗ b) = (−1)pq (b ⊗ a)

(a ∈ Ap , b ∈ B q )

constitutes an isomorphism of Δ-functors. Exercise 1.5.5. Let K1 and K2 be Δ-categories with respective translation functors T1 and T2 ; and let (F , θ) : K1 → K2 be a Δ-functor. An object A in K1 is periodic if there is an integer m > 0 such that T1m (A) = A. Suppose that 0 is the only periodic object in K1 . (For example, K1 could be any one of the Δ-categories K* of §1.6 below.) Then we can choose a function ν : (objects of K1 ) → Z such that ν(0) = 0 and ν(T1 A) = ν(A) − 1 for all A = 0; and using θ, we can deﬁne isomorphisms −ν(A)

∼ ηA : F (A) −→ T2

ν(A)

F (T1

A) =: f (A)

(A ∈ K1 ).

Note that f (T1 A) = T2 f (A). Verify that there is a unique way of extending f to ∼ a functor such that the ηA form an isomorphism of Δ-functors (F , θ) −→ (f , identity).

1.6 Δ-Subcategories A full additive subcategory K′ of a Δ-category K carries at most one triangulation for which the translation is the restriction of that on K, and such that the inclusion functor ι : K′ ֒→ K (together with the identity transformation from ιT to T ι) is a Δ-functor. For the existence of such a triangulation it is necessary and suﬃcient that K′ be stable under the translation automorphism and its inverse, and that the summit of any triangle in K with base in K′ be isomorphic to an object in K′ ; the triangles in K′ are then precisely the triangles of K whose vertices are all in K′ . (Details left to the reader.) Such a K′ is called a Δ-subcategory of K. For example, if K = K(A) is as in (1.4.3), then a full additive subcategory K′ is a Δ-subcategory if and only if: (i) for every complex A• ∈ K we have A• ∈ K′ ⇔ A• [1] ∈ K′ , and (ii) the mapping cone of any A-morphism of complexes u : A• → B • with A• and B • in K′ is homotopically equivalent to a complex in K′ . Examples 1.6.1. We consider various full additive subcategories K+, K−, Kb, K+, K−, Kb, of K = K(A). The objects of K+ are complexes A• which are bounded below, i.e., there is an integer n0 (depending on A• ) such that An = 0 for n < n0 . The objects

30

1 Derived and Triangulated Categories

of K+ are complexes B • whose homology is bounded below, i.e., H m (B • ) = 0 for all m < m0 (B • ). The objects of K− and K− (respectively Kb and Kb) are speciﬁed similarly, with “bounded above” (resp. “bounded above and below ”) in place of “bounded below.” We have, obviously, Kb = K+ ∩ K− ,

Kb = K+ ∩ K− ;

and if * stands for any one of + , − , or b , then K* ⊂ K* . Using the natural exact sequence (see (1.3)) 0 → B • → Cu• → A• [1] → 0

(1.6.2)

associated with a morphism u : A• → B • of complexes in A, we ﬁnd that if both A• and B • satisfy one of the above boundedness conditions then so does the cone Cu• , whence K* and K* are Δ-subcategories of K. Remark 1.6.3. In (1.4.3.6) and its proof, we can replace K(A) by any Δ-subcategory.

1.7 Localizing Subcategories of K; Δ-Equivalent Categories In the description of the derived category D given in §1.2, we can replace K by any Δ-subcategory L, and obtain a derived category DL together with a functor QL : L → DL which is universal among all functors transforming quasi-isomorphisms into isomorphisms. (Here, as in 1.2, for checking details one needs [H, p. 35, Prop. 4.2].) Then, just as in (1.4.4), DL has a unique triangulation for which the translation functor is the obvious one and for which QL is a Δ-functor; and (1.5.1) remains valid with QL in place of Q. If L′ ⊂ L′′ are Δ-subcategories of K and j : L′ → L′′ is the inclusion, then there exists a natural commutative diagram of Δ-functors j

L′ −−−−→ ⏐ ⏐ Q′:= QL′

L′′ ⏐ ⏐Q =: Q′′ L′′

D′ := DL′ −−−−→ DL′′ =: D′′ ˜j

Note that on objects of D′ (= objects of L′ ), j˜ is just the inclusion map to objects of D′′ . Recalling that passage to derived categories is a kind of localization in categories (§1.2, footnote), we say that L′ localizes to a Δ-subcategory of D′′,

1.7 Localizing Subcategories of K; Δ-Equivalent Categories

31

or more brieﬂy, that L′ is a localizing subcategory of L′′ , if the functor j˜ is fully faithful, i.e., the natural map is an isomorphism ∼ HomD′ (A•, B • ) −→ HomD′′ (˜ jA•, j˜B • )

for all A• and B • in D′ . When this condition holds, j˜ is an additive isomorphism of D′ onto the full subcategory j˜(D′ ) of D′′, so j˜ carries the triangulation on D′ over to a triangulation on j˜(D′ ); and then since j˜ is a Δ-functor, the inclusion functor j˜(D′ ) ֒→ D′′ , together with θ = identity, is a Δ-functor, i.e., j˜(D′ ) is a Δ-subcategory of D′′ . Thus if L′ is localizing in L′′ , then we can identify D′ with the Δ-subcategory of D′′ whose objects are the complexes in L′ , and Q′ with the restriction of Q′′ to L′ . (1.7.1). From deﬁnitions in §1.2, we deduce easily the following simple suﬃcient condition for L′ to be localizing in L′′ : For every quasi-isomorphism X • → B • in L′′ with B • in L′, there exists a quasi-isomorphism A• → X • with A• in L′ . (1.7.1)op . A “dual” argument (see [H, p. 32, proof of 3.2]) yields: The same condition with arrows reversed is also suﬃcient. For example, if the objects in L′ are precisely those complexes in K which satisfy some condition on their homology (for instance, if L′ is any one of the categories K* of (1.6.1)), then L′ is localizing in L′′ . This follows at once from (1.7.1) (take A• = X • ). The following results will provide a useful interpretation of various kinds of resolutions (injective, ﬂat, ﬂasque, etc.) as deﬁning an equivalence of Δ-categories. (1.7.2). If for every X • ∈ L′′ there exists a quasi-isomorphism A• → X • with A• ∈ L′ then j˜ is an equivalence of categories, i.e., there exists a functor ρ : D′′ → D′ together with functorial isomorphisms ∼ 1D′′ −→ j˜ρ ,

∼ 1D′ −→ ρ˜ j

(1.7.2.1)

(see [M, p. 91]). Moreover, for the usual translation T there is then a unique functorial isomorphism ∼ θ : ρ T −→ Tρ such that the pair (ρ, θ) is a Δ-functor and the isomorphisms (1.7.2.1) are isomorphisms of Δ-functors (§1.5). We say then that j˜ and ρ—or more precisely (˜ j, identity) and (ρ, θ)—are Δ-equivalences of categories, quasi-inverse to each other. (1.7.2)op . Same as (1.7.2), with A• → X • replaced by X • → A• . To prove (1.7.2)op , for example, suppose that we have a family of quasiisomorphisms (“right L′ -resolutions”) ϕX • : X • → A•X • ∈ L′

(X • ∈ L′′ ) .

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1 Derived and Triangulated Categories

Then by (1.7.1)op , L′ is localizing in L′′ . So ﬁnding an additive functor ρ with isomorphisms (1.7.2.1) is equivalent to ﬁnding for each object X • of D′′ an isomorphism to an object in D′ ⊂ D′′ (see [M, p. 92, (iii)⇒(ii)]). But Q′′ (ϕX • ) is such an isomorphism. Thus we have ρ : D′′ → D′ with ρ(X • ) = A•X •

(X • ∈ D′′ ) .

Next, deﬁne θ(X • ) to be the unique map making the following diagram (with all arrows representing isomorphisms in D′′ ) commute: T X• Q′′ (ϕTX• )

ρ T X • = A•TX •

T Q′′ (ϕX• )

θ(X•)

(1.7.2.2)

TA•X • = TρX •

Then, one checks, the family θ(X • ) constitutes an isomorphism of functors ∼ θ : ρT −→ Tρ. Furthermore, if u v w X • −→ Y • −→ Z • −→ T X • is a triangle in D′′ , then (Δ1)′′ (see §1.4) applied to the commutative diagram in D′′ u v w X • −−−−→ Y • −−−−→ Z • −−−−−−−→ T X • ⏐ ⏐ ⏐ ⏐ ⏐Q′′ ϕ ⏐T Q′′ ϕ ⏐ ⏐ Q′′ ϕY • Q′′ ϕX• X• Z• A•X • −−−−→ A•Y • −−−−→ A•Z • −−−− −−−→ T A•X • • ρ(u)

ρ(v)

θ(X )◦ρ(w)

guarantees that the bottom row is a triangle; and so (ρ, θ) is a Δ-functor. The fact that the isomorphisms in (1.7.2.1) (induced by the family ϕX• ) are isomorphisms of Δ-functors is nothing but the commutativity of (1.7.2.2). Thus the family θ := {θ(X • )} is the unique functorial isomorphism having the properties stated in (1.7.2)op . Remark 1.7.2.3. It is sometimes possible to choose the functor ρ so that ρ T = T ρ and θ = identity, i.e., to ﬁnd a family of quasi-isomorphisms ϕX• : X • → A•X • commuting with translation (see (1.8.1.1), (1.8.2), and (1.8.3) below).

1.8 Examples

33

1.8 Examples (1.8.1). If L′ ⊂ K is any one of the Δ-subcategories K* of (1.6.1) and if L′′ is any Δ-subcategory of K containing L′ , then L′ is localizing in L′′ . The same holds for L′ = K+ or L′ = K−; and also for L′ = Kb if L′′ is localizing in K. For L′ = K* the assertion follows at once from (1.7.1). For the rest (and for other purposes) we need the truncation operators τ + , τ − , deﬁned as follows: For any B • ∈ K, set i = i(B • ) := inf{ m | H m (B • ) = 0 } and let τ + (B • ) be the complex · · · → 0 → 0 → coker(B i−1 → B i ) → B i+1 → B i+2 → · · · . (When i = ∞, i.e., when B • is exact, this means τ + (B • ) = 0• ; and when i = −∞, τ + (B • ) = B • .) There is an obvious quasi-isomorphism B • → τ + (B • ) .

(1.8.1)+

Dually, for any C • ∈ K set s = s(C • ) := sup{ n | H n (C • ) = 0 } and let τ − (C • ) be the complex · · · → C s−2 → C s−1 → ker(C s → C s+1 ) → 0 → 0 → · · · . There is an obvious quasi-isomorphism τ − (C • ) → C • .

(1.8.1)−

Now if C • → B • is a quasi-isomorphism in L′′ with B • ∈ K− then C ∈ K−, and we have the quasi-isomorphism (1.8.1)− with τ − (C • ) ∈ K− . So (1.7.1) with L′ = K− ⊂ L′′ shows that K− is localizing in L′′ . Dually, via (1.8.1)+ , (1.7.1)op implies that K+ is localizing in any Δ-subcategory L′′ of K containing K+ . And again via (1.8.1)− , (1.7.1) shows that Kb is localizing in K+ ; and since as above K+ is localizing in K, the natural functors Db → D+ → D between the corresponding derived categories are both fully faithful, whence so is their composition, i.e., Kb is localizing in K. It follows at once that Kb is localizing in any L′′ ⊃ Kb such that L′′ is localizing in K. Consequently, as in (1.7): the derived category D* (resp. D*) of K* (resp. K*) can be identiﬁed in a natural way with a Δ-subcategory of D. •

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1 Derived and Triangulated Categories

Then the inclusion D+ ֒→ D+ is a Δ-equivalence of categories. Indeed, as in the proof of (1.7.2)op , with L′ = K+ , L′′ = K+, and ϕB • = (1.8.1)+ , we can see that τ + —which commutes with translation—extends to a Δ-functor (τ +, 1) : D+ → D+

(1.8.1.1)

which is quasi-inverse to the inclusion. Similarly the inclusions D− ֒→ D−, Db ֒→ Db are Δ-equivalences, with respective quasi-inverses τ − and τ b = τ − ◦ τ + = τ + ◦ τ − . More precisely, τ b is the composition τ τ− Db −→ Db ∩ D+ −→ D− ∩ D+ = Db . +

(1.8.2) Let I be a full additive subcategory of A such that every object of A admits a monomorphism into an object in I. Then there exists a family of quasi-isomorphisms • • ϕB • : B • → IB B ∈ K+ = K+(A) •

• n where each I • = IB ∈ I for all n, • is a bounded-below I-complex (i.e., I n and I = (0) for n ≪ 0); and such that moreover with the usual translation functor T we have • • ITB • = T IB • ,

ϕTB • = T (ϕB • ) .

(1.8.2.1)

To see this, ﬁrst construct quasi-isomorphisms ϕB • as in [H, p. 42, 4.6, 1)] for those B • such that H 0 (B • ) = 0 and B m = 0 for m < 0. Then (1.8.2.1) forces the deﬁnition of ϕB • for any B • such that there exists i ∈ Z with H i (B • ) = 0 and B m = 0 for all m < i (i.e., 0• = B • = τ + B • , see (1.8.1)). Set I0• = 0• , and ﬁnally for any B • ∈ K+ set ϕB • = (ϕτ

+

+

B • )◦ (1.8.1)

.

+ Now let K+ I be the full subcategory of K whose objects are the boundedbelow I-complexes. Since the additive subcategory I ⊂ A is closed under + ﬁnite direct sums, one sees that K+ I is a Δ-subcategory of K . Accord+ + op ing to (1.7.2) , the derived category DI of KI can be identiﬁed with a Δ-subcategory of D+, and the above family ϕB • gives rise to an I-resolution functor ρ : D+ → D+ I

(1.8.2.2)

which is, together with θ = identity, a Δ-equivalence of categories, quasiinverse to the inclusion DI+ ֒→ D+. For example, if I is the full subcategory of A whose objects are all the injectives in A, then by [H, p. 41, Lemma 4.5] every quasi-isomorphism in KI+ is an isomorphism, so that K+ I can be identiﬁed with its derived

1.9 Complexes with Homology in a Plump Subcategory

35

category D+ I . Thus, if A has enough injectives (i.e., every object of A admits a monomorphism into an injective object), then the natural composition + + + D+ I = KI ֒→ K → D is a Δ-equivalence, having as quasi-inverse an injective resolution functor (1.8.2.2) (cf. [H, p. 46, Prop. 4.7]). (1.8.3). Let P be a full additive subcategory of A such that for every object B ∈ A there exists an epimorphism PB → B with PB ∈ P. An argument dual to that in (1.8.2) yields that there exists a family of quasiisomorphisms • B ∈ K−(A) ψB • : PB• • → B •

commuting with translation, and such that each PB• • is a bounded-above P-complex. According to (1.7.2), we have then a P-resolution functor which is a Δ-equivalence into D−(A) from its Δ-subcategory whose objects are boundedabove P-complexes. For example, if U is a topological space, O is a sheaf of rings on U , and A is the abelian category of (sheaves of) left O-modules, then we can take P to be the full subcategory of A whose objects are all the ﬂat O-modules [H, p. 86, Prop. 1.2].

1.9 Complexes with Homology in a Plump Subcategory (1.9.1). Here, in brief, are some essential basic facts. Let A# be a plump subcategory of the abelian category A, i.e., a full subcategory containing 0 and such that for every exact sequence in A X1 → X2 → X → X3 → X4 , if X1 , X2 , X3 , and X4 all lie in A# then so does X. Then the kernel and cokernel (in A) of any map in A# must lie in A# (whence A# is abelian), and any object of A isomorphic to an object in A# must itself be in A# . Considering only complexes in A whose homology objects all lie in A#, we * * * obtain full subcategories K# of K, K* # of K , and K# of K (see (1.6.1)). H Via the exact homology sequence (1.4.5) of a standard triangle (1.3.1), we ﬁnd that these subcategories are all Δ-subcategories (see (i) and (ii) in §1.6), and indeed, by (1.7.1), localizing subcategories. From (1.8.1) it follows then * that K# , K* # , and K# are localizing subcategories of K, from which we derive * Δ-subcategories D# , D* # , and D# of D, with universal properties analogous * to (1.5.1). As in (1.8.1) the inclusion D* # ֒→ D# is a Δ-equivalence of ∗ categories, with quasi-inverse τ .

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1 Derived and Triangulated Categories

(1.9.2). The following isomorphism test will be useful. Lemma. If A# is a plump subcategory of A, and u : A•1 → A•2 is a map in • b D+ # such that for all B ∈ D# the induced map HomD (B •, A•1 ) → HomD (B •, A•2 ) is an isomorphism, then u is an isomorphism. Proof. Let C • ∈ D+ # be the summit of a triangle with base u, so that by (1.4.2.1), u is an isomorphism iﬀ C • ∼ = 0, i.e., iﬀ τ +(C • ) = 0• , see (1.8.1) and (1.2.2). For each m ∈ Z and each object M ∈ A# we have, by (1.4.2.1) and (Δ2) in §1.4, an exact sequence (with Hom = HomD): −−→ Hom(M [−m], A•2 ) −→ Hom(M [−m], C • ) Hom(M [−m], A•1 ) −− via u

−→ Hom(M [−m], A•1 [1]) −−− −−→ Hom(M [−m], A•2 [1]). via −u[1]

The two labeled maps are, by hypothesis, isomorphisms, and hence Hom(M [−m], C • ) = 0 . Were τ +(C • ) = 0• , then with m := i(C • ) (see (1.8.1) and M := H m (C • ) = ker τ +(C • )m → τ +(C • )m+1 = 0 ,

the inclusion M ֒→ τ +(C • )m would lead to a map j : M [−m] → τ +(C • ) with H m (j) the (non-zero) identity map of M , so we’d have Hom M [−m], C • −− −−→ Hom M [−m], τ +(C • ) = 0 , (1.8.1)+

a contradiction. Thus τ +(C • ) = 0• .

Q.E.D.

1.10 Truncation Functors Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ≤n A• be the truncated complex · · · −→ An−2 −→ An−1 −→ ker(An → An+1 ) −→ 0 −→ 0 −→ · · · , and dually we let τ≥n A be the complex · · · −→ 0 −→ 0 −→ coker(An−1 → An ) −→ An+1 −→ An+2 −→ · · · .

1.10 Truncation Functors

37

Note that H m (τ≤n A• ) = H m (A• )

if m ≤ n ,

=0

if m > n ,

and that H m (τ≥n A• ) = H m (A• )

if m ≥ n ,

=0

if m < n .

One checks that τ≥n (respectively τ≤n ) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D, see §1.2. In fact if D≤n (resp. D≥n ) is the full subcategory of D whose objects are the complexes A• such that H m (A• ) = 0 for m > n (resp. m < n) then we have additive functors τ≤n : D −→ D≤n ⊂ D τ≥n : D −→ D≥n ⊂ D together with obvious functorial maps inA : τ≤n A• −→ A• n jA : A• −→ τ≥n A• . n Proposition 1.10.1. The preceding maps inA , jA induce functorial isomorphisms ∼ HomD≤n (B • , τ≤n A• ) −→ HomD (B • , A• )

(B • ∈ D≤n ),

(1.10.1.1)

∼ HomD≥n (τ≥n A• , C • ) −→ HomD (A• , C • )

(C • ∈ D≥n ).

(1.10.1.2)

Proof. Bijectivity of (1.10.1.1) means that any map ϕ : B • → A• (in D) n with B • ∈ D≤n factors uniquely via iA := iA . Given ϕ, we have a commutative diagram τ≤n ϕ

τ≤n B • −−−−→ τ≤n A• ⏐ ⏐ ⏐i ⏐ iB A B•

−−− −→ ϕ

A•

and since B • ∈ D≤n , therefore iB is an isomorphism in D, see (1.2.2), so we −1 can write ϕ = iA ◦(τ≤n ϕ ◦ iB ), and thus (1.10.1.1) is surjective.

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1 Derived and Triangulated Categories

To prove that (1.10.1.1) is also injective, we assume that iA ◦ τ≤n ϕ = 0 and deduce that τ≤n ϕ = 0. As in §1.2, the assumption means that there is a commutative diagram in K(A) f

s

C • −−−−→ τ≤n A• ⏐ ⏐i ⏐ A ⏐

τ≤n B • ←−− −− C ′′• −−−−→ ′′ s

0

A•

where s and s′′ are quasi-isomorphisms, and f /s = τ≤n ϕ. Applying the (idempotent) functor τ≤n , we get a commutative diagram τ≤n f

τ≤n s

τ≤n C • −−−−→ τ≤n A• ⏐ ⏐ 0

τ≤n B • ←−−−′′− τ≤n C ′′• τ≤n s

Since τ≤n s and τ≤n s′′ are quasi-isomorphisms, we have τ≤n ϕ = τ≤n f /τ≤n s = 0/τ≤n s′′ = 0 , as desired. A similar argument proves the bijectivity of (1.10.1.2). Remarks 1.10.2. Let n ∈ Z, A• ∈ D(A). (i) There exist natural isomorphisms τ≤n τ≥n A• ∼ = H n (A• )[−n] ∼ = τ≥n τ≤n A• . n−1 (ii) The cokernel of iA : τ≤n−1 A• → A• maps quasi-isomorphically to τ≥n A• ; and hence there are natural triangles in D(A) (see (1.4.4.2)∼ ): in−1

jn

τ≤n−1 A• −−A−−→ A• −−−A−→ τ≥n A• −→ (τ≤n−1 A• )[1] ,

(1.10.2.1)

τ≤n−1 A• −→ τ≤n A• −→ H n (A• )[−n] −→ (τ≤n−1 A• )[1] .

(1.10.2.2)

Details are left to the reader.

1.11 Bounded Functors; Way-Out Lemma Many of the main results in subsequent chapters will be to the eﬀect that some natural map or other is a functorial isomorphism. So we’ll need isomorphism criteria. In (1.11.3) we review some commonly used ones (“Lemma on way-out functors,” [H, p. 68, Prop. 7.1]).

1.11 Bounded Functors; Way-Out Lemma

39

Throughout this section, A and B are abelian categories, A# is a plump # subcategory of A, and D* # (A) ⊂ D(A) is as in (1.9.1). We identify A with (A), see (1.2.3). a full subcategory of D* # For a subcategory E of D(A), E≤n (resp. E≥n ) will denote the full subcategory of E whose objects are those complexes A• such that H m (A• ) = 0 for m > n (resp. m < n). Deﬁnition 1.11.1. Let E be a subcategory of D(A), and let F (resp. F ′ ) be a covariant (resp. contravariant) additive functor from E to D(B). The upper dimension dim+ and lower dimension dim− of these functors are:

dim+ F := inf d F (E≤n ) ⊂ D≤n+d (B) for all n ∈ Z ,

dim+ F ′ := inf d F ′ (E≥−n ) ⊂ D≤n+d (B) for all n ∈ Z ,

dim− F := inf d F (E≥n ) ⊂ D≥n−d (B) for all n ∈ Z ,

dim− F ′ := inf d F ′ (E≤−n ) ⊂ D≥n−d (B) for all n ∈ Z .

The functor F is bounded above 10 (resp. bounded below)11 if dim+ F < ∞

(resp. dim− F < ∞); and similarly for F ′ . F (resp. F ′ ) is bounded if it is both bounded-above and bounded-below. Remarks 1.11.2. (i) Let T1 and T2 be the translation functors in D(A) and D(B) respectively. Suppose that T1 E = E and that there is a functorial ∼ ∼ T2 F (resp. T2−1 F ′ −→ F ′ T1 ). (For example, E could isomorphism F T1 −→ ′ be a Δ-subcategory of D(A) and F a Δ-functor.) Then, for instance, F ′ (E≥−n ) ⊂ D≤n+d (B) holds for all n ∈ Z as soon as it holds for one single n. (ii) If E is a Δ-subcategory of D(A) such that for all n ∈ Z, τ≤n E ⊂ E ′ and τ≥n E ⊂ E (e.g., E = D* # (A)), and if F (resp. F ) is a Δ-functor, then: dim+ F ≤ d ⇐⇒ H i F (A• ) − −→ H i F (τ≥n A• ) n jA

for all A• ∈ E, n ∈ Z, and i ≥ n + d.

n ) (see §1.10) is an isomorphism; (The display signiﬁes that the map H i (jA and as in (i), we can restrict attention to a single n.) The implication ⇒ follows from the exact homology sequence (1.4.5)H of the triangle gotten by applying F to (1.10.2.1); while ⇐ is obtained by taking A• to be an arbitrary complex in E≤n−1 . An equivalent condition is that if α : A•1 → A•2 is a map in E such that H i (α) is an isomorphism for all i ≥ n, (that is, if α induces ∼ τ≥n A•2 ), then H i (F α) is an isomorphism for an isomorphism τ≥n A•1 −→ all i ≥ n + d.

10 11

way-out left in the terminology of [H, p. 68]. way-out right.

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1 Derived and Triangulated Categories

Similarly: dim+ F ′ ≤ d

⇐⇒ H i F ′ (A• )

dim− F ≤ d

⇐⇒ H i F (τ≤n A• )

dim− F ′ ≤ d

⇐⇒ H i F ′ (τ≥−n A• ) −− −−→

−− −−→ H i F ′ (τ≤−n A• ) i−n A

−− −n−→ iA

−n jA

(i ≥ n + d),

H i F (A• )

(i ≤ n − d),

H i F ′ (A• )

(i ≤ n − d).

(iii) If E = A# (so that E≥0 = E = E≤0 ), then dim+ F ≤ d ⇔ H j F (A) = 0 for all j > d and all A ∈ A# . Similarly, dim− F ≤ d ⇔ H j F (A) = 0 for j < −d and A ∈ A# . These assertions remain true when F is replaced by F ′ . + + (iv) If E = D+ # (A) and F is a Δ-functor, then dim F = dim F0 where F0 ′ − is the restriction F |A# . A similar statement holds for dim F ; and analogous statements hold for dim− F or dim+ F ′ when E = D− # (A). Here is a typical proof: we deal with dim− F ′ when E = D+ # (A). Obviously dim− F ′ ≥ dim− F0′ . To prove the opposite inequality, suppose that dim− F0′ ≤ d < ∞, ﬁx an n ∈ Z, and let us show for any A• ∈ E≤−n that H j F ′ (A• ) = 0 whenever j < n − d. We proceed by induction on the number ν = ν(A• ) of non-vanishing homology objects of A• , the case ν = 0 being trivial. If ν = 1, say H −m (A• ) =: H = 0 (m ≥ n), then A• ∼ = τ − τ + A• ∼ = H[m] (see (1.8.1)), and ′ since F is a contravariant Δ-functor, F ′ (A• ) ∼ = F ′ (H)[−m]; so by deﬁnition of dim− F0′ , H j F ′ (A• ) ∼ = H j−m F ′ (H) = 0

if j − m < −d,

whence the conclusion. When ν > 1, choose any integer s such that there exist integers p < s ≤ q with H p (A• ) = 0, H q (A• ) = 0 (so that ν(τ≤s−1 A• ) < ν(A• ) and ν(τ≥s A• ) < ν(A• )). Then apply F ′ to (1.10.2.1) to get a triangle F ′ (τ≤s−1 A• ) ←− F ′ (A• ) ←− F ′ (τ≥s A• ) ←− F ′ (τ≤s−1 A• )[−1] whose associated homology sequence (1.4.5)H yields the inductive step. Lemma 1.11.3. Let (F , θ) and (G, ψ) be covariant Δ-functors from D* # (A) to D(B), and assume one of the following sets of conditions: (i) * = b. (ii) * = + and both F and G are bounded below. (iii) * = − and both F and G are bounded above. (iv) * = blank and F and G are bounded above and below. Then for a morphism η : F → G of Δ-functors to be an isomorphism it suﬃces that η(X) be an isomorphism for all objects X ∈ A# . A similar assertion holds for contravariant functors if we interchange “bounded above” and “bounded below.”

1.11 Bounded Functors; Way-Out Lemma

41

Complement 1.11.3.1. Let I (resp. P) be a set of objects in A# such that every object in A# admits a monomorphism into one in I (resp. is the target of an epimorphism out of one in P). If * = + and F and G are bounded below (resp. * = − and F and G are bounded above) and if η(X) is an isomorphism for all objects X ∈ I (resp. X ∈ P), then η is an isomorphism. A similar assertion holds for contravariant functors if we interchange “bounded above” and “bounded below.” Proof. We deal ﬁrst with the covariant case. (i) Using the deﬁnition of “morphism of Δ-functors” (§1.5) we see by induction on |n| that η(X[−n]) is an isomorphism for all X ∈ A# and n ∈ Z. In showing that η(A• ) is an isomorphism for all A• ∈ Db# (A), we may replace A• by the isomorphic complex τ − (A• ) = τ≤n A• with n := s(A• ), see (1.8.1). From (1.10.2.2), and (Δ2) of §1.4, we get a map of triangles, induced by η: F (H n (A• )[−n − 1]) −→ F (τ≤n−1 A• ) −→ F (τ≤n A• ) −→ F (H n (A• )[−n]) ⏐ ⏐ ⏐ ⏐

G(H n (A• )[−n − 1]) −→ G(τ≤n−1 A• ) −→ G(τ≤n A• ) −→ G(H n (A• )[−n]) and then we can conclude by (Δ3)∗ of §1.4 and induction on the number of non-vanishing homology objects of A• (a number which is less for τ≤n−1 A• than for A• whenever n is ﬁnite). (ii) By (1.2.2), it suﬃces to show that η(A• ) induces an isomorphism from H i F (A• ) to H i G(A• ) for all A• ∈ D+ (A) and all i ∈ Z. For this, #

remark (1.11.2)(ii) allows us to replace A• by τ≤i+d A• ∈ Db# (A) for any d ≥ max(dim− F , dim− G), and then (i) applies. (iii) Similar to (ii). (iv) As in the proof of (i), (1.10.2.1) with n = 0 gives rise to a map of triangles, induced by η: F (τ≥0 A• )[−1]) −−→ F (τ≤−1 A• ) −−→ F (A• ) −−→ F (τ≥0 A• ) ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐≃ ⏐ ⏐ ≃ ?

G((τ≥0 A• )[−1]) −−→ G(τ≤−1 A• ) −−→ G(A• ) −−→ G(τ≥0 A• ) in which the maps other than ? are isomorphisms by (ii) and (iii), whence, by (Δ3)∗ of §1.4, so is ?. For (1.11.3.1), it now suﬃces to show that η(X) is an isomorphism for all objects X ∈ A# . By a standard resolution argument (see [H, p. 43]), X is isomorphic in D# (A) to a bounded-below complex I • of objects of I (resp. bounded-above complex P • of objects of P), and so it suﬃces to show that η(I • ) (resp. η(P • )) is an isomorphism for any such I • (resp. P • ). This is done as above, except that in the inductive step in (i), say for bounded I •, one uses instead of (1.10.2.2) the triangle associated as in (1.4.3) to the natural

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1 Derived and Triangulated Categories

semi-split exact sequence 0 −→ I n [−n] −→ τ≤′ n I • −→ τ≤′ n−1 I • −→ 0 where for any A• and m ∈ Z, τ≤′ m A• is the complex · · · −→ Am−2 −→ Am−1 −→ Am −→ 0 −→ 0 −→ · · · ; and in (ii), for example, one replaces I • by the bounded complex τ≤′ i+d+1 I •. Similar arguments settle the contravariant case. (Or, use the exercise just before (1.5.1).) Q.E.D.

Chapter 2

Derived Functors

Derived functors are Δ-functors out of derived categories, giving rise, upon application of homology, to functors such as Ext, Tor, and their sheaftheoretic variants—in particular sheaf cohomology. Derived functors are characterized in §2.1 below by a universal property, and conditions for their existence are given in 2.2, leading up to the construction of right-derived functors via injective resolutions in 2.3 and, dually, of some left-derived functors via ﬂat resolutions in 2.5. We use ideas of Spaltenstein [Sp] to deal throughout with unbounded complexes. The basic examples RHom• and ⊗ = are described in 2.4 and 2.5 respectively. Illustrating all that has gone before, their relation “adjoint associativity” is given in 2.6, which also includes an abbreviated discussion of what is, in all conscience, involved in constructing natural transformations of multivariate derived functors: a host of underlying category-theoretic trivialities, usually ignored, but of whose existence one should at least be aware. The last section 2.7 develops further reﬁnements.

2.1 Deﬁnition of Derived Functors Fix an abelian category A, let J be a Δ-subcategory of K(A), let DJ be the corresponding derived category, and let Q = QJ : J → DJ be the canonical Δ-functor (see (1.7)). For any Δ-functors F and G from J to another Δ-category E, or from DJ to E, Hom(F , G) will denote the abelian group of Δ-functor morphisms from F to G. Deﬁnition 2.1.1. A Δ-functor F : J → E is right-derivable if there exists a Δ-functor RF : DJ → E J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

43

44

2 Derived Functors

and a morphism of Δ-functors ζ : F → RF ◦Q such that for every Δ-functor G : DJ → E the composed map via ζ

natural

Hom(RF , G) −−−−→ Hom(RF ◦Q, G ◦ Q) −−−→ Hom(F , G ◦ Q) is an isomorphism (i.e., by (1.5.1), the map “via ζ” is an isomorphism). The Δ-functor F is left-derivable if there exists a Δ-functor LF : DJ → E and a morphism of Δ-functors ξ : LF ◦Q → F such that for every Δ-functor G : DJ → E the composed map natural

via ξ

Hom(G, LF ) −−−−→ Hom(G ◦ Q, LF ◦Q) −−−→ Hom(G ◦Q, F ) is an isomorphism (i.e., by (1.5.1), the map “via ξ” is an isomorphism). Such a pair (RF , ζ) respectively: (LF , ξ) is called a right-derived (respectively: left-derived ) functor of F . As in (1.5.1), composition with Q gives an embedding of Δ-functor categories Hom∆ (DJ , E) ֒→ Hom∆ (J, E),

(2.1.1.1)

with image the full subcategory whose objects are the Δ-functors which transform quasi-isomorphisms into isomorphisms. Consequently we can regard a right-(left-)derived functor of F as an initial (terminal ) object [M, p. 20] in the category of Δ-functor morphisms F → G′ (G′ → F ) where G′ ranges over all Δ-functors from J to E which transform quasi-isomorphisms into isomorphisms. As such, the pair (RF , ζ) (or (Lf , ξ))—if it exists—is unique up to canonical isomorphism. Complement 2.1.2. Let A′ be another abelian category. Any additive functor F : A → A′ extends to a Δ-functor F¯ : K(A) → K(A′ ) (see (1.5.2)). Q′ : K(A′ ) → D(A′ ) being the canonical map, we will refer to derived functors of Q′F¯, or of the restriction of Q′F¯ to some speciﬁed Δ-subcategory J of K(A), as being “derived functors of F ” and denote them by RF or LF . Example 2.1.3. If F : J → E transforms quasi-isomorphisms into isomorphisms then F = F ◦Q for a unique F : DJ → E; and (F, identity) is both a right-derived and a left-derived functor of F .

2.2 Existence of Derived Functors

45

Remarks 2.1.4. Let A′ be an abelian category, and in (2.1.1) suppose that E is a Δ-subcategory of K(A′ ) or of D(A′ ). If RF exists we can set RiF (A) := H i (RF (A))

(A ∈ J, i ∈ Z).

Since RF is a Δ-functor, any triangle A → B → C → A[1] in J is transformed by RF into a triangle in E, and hence we have an exact homology sequence (see (1.4.5)H ): · · · → Ri−1F (C) → RiF (A) → RiF (B) → RiF (C) → Ri+1F (A) → · · · (2.1.4)H This applies in particular to the triangle (1.4.4.2)∼ associated to an exact sequence of A-complexes 0→A→B→C→0

(A, B, C ∈ J).

A similar remark can be made for LF .

2.2 Existence of Derived Functors Derivability of a given functor is often proved by reduction, via suitable Δ-equivalences of categories, to the trivial example (2.1.3), as we now explain—and summarize in (2.2.6). We consider, as in (1.7), a diagram j

J′ −−−−→ ⏐ ⏐ Q′

J′′ ⏐ ⏐ ′′ Q

D′ −−−−→ D′′ ˜j

where J′ ⊂ J′′ are Δ-subcategories of K(A), D′ and D′′ are the corresponding derived categories, Q′ and Q′′ are the canonical Δ-functors, j is the inclusion, and j˜ is the unique Δ-functor making the diagram commute; and we assume that the conditions of (1.7.2) or of (1.7.2)op obtain. In other words we have a family of quasi-isomorphisms ψX : AX → X,

X ∈ J′′ , AX ∈ J′ ,

(see (1.7.2)),

(2.2.1)

(see (1.7.2)op ).

(2.2.1)op

or a family of quasi-isomorphisms ϕX : X → AX ,

X ∈ J′′ , AX ∈ J′ ,

46

2 Derived Functors

In either situation, j˜ identiﬁes D′ with a Δ-subcategory of D′′ ; there is a Δ-functor (ρ, θ) : D′′ → D′ with (X ∈ J′′ );

ρ(X) = AX

and there are isomorphisms of Δ-functors ∼ 1D′′ −→ j˜ρ,

∼ 1D′ −→ ρ˜ j

(2.2.2)

induced by ψ or by ϕ. Proposition 2.2.3. With preceding notation, let E be a Δ-category, let F : J′′ → E be a Δ-functor, and suppose that the restricted functor F ′ := F ◦ j : J′ → E has a right-derived functor RF ′ : D′ → E,

ζ ′ : F ′ → RF ′ ◦ Q′ .

If there exists a family ϕX : X → AX as in (2.2.1)op , whence a functor ρ as above, then F has the right-derived functor (RF , ζ) where RF = RF ′ ◦ ρ : D′′ → E so that RF (X) = RF ′ (AX )

(X ∈ J′′ ),

and where for each X ∈ J′′ , ζ(X) is the composition ζ ′ (AX )

F (ϕ )

X F (X) −−−− → F (AX ) = F ′ (AX ) −−−−→ RF ′ (AX ) = RF (X) .

A similar statement holds for left-derived functors when there exists a family ψX as in (2.2). Proof. We check ﬁrst that ζ is actually a morphism of Δ-functors. Consider a map u : X → Y in J′′ . Since Q′′ (ϕX ) is an isomorphism, there is a unique map u ˜ : AX → AY in D′′ (and hence in the full subcategory D′ ) making the following D′′ -diagram commute: Q′′(ϕ )

X X −−−−− → ⏐ ⏐ Q′′ (u)

AX ⏐ ⏐ u˜

Y −−′′−−−→ AY Q (ϕY )

2.2 Existence of Derived Functors

47

By the deﬁnition of the functor ρ (see proof of (1.7.2)), that ζ is a morphism of functors means that the following diagram D(u) commutes for all u: ζ ′ (AX )

F (ϕ )

X F (X) −−−− → F (AX ) −−−−→ RF ′ (AX ) ⏐ ⏐ ⏐ ′ ⏐ F (u) ? RF (˜u)

F (Y ) −−−−→ F (AY ) −− −−→ RF ′ (AY ) ′ F (ϕY )

ζ (AY )

If there were a J′ -map u′ : AX → AY such that u′ ϕX = ϕY u, whence Q (u′ )Q′′ (ϕX ) = Q′′ (ϕY )Q′′ (u) and u ˜ = Q′′ (u′ ) = Q′ (u′ ), then the broken arrow in D(u) could be replaced by the map F (u′ ), making both resulting subdiagrams of D(u), and hence D(u) itself, commute. We don’t know that such a u′ exists; but, I claim, there exists a quasi-isomorphism v : Y → Z such that (with self-explanatory notation) both v ′ and (vu)′ exist. This being so, both diagrams D(v) and D(vu) commute; and since v˜ is an isomorphism (because v is a quasi-isomorphism), therefore RF ′ (˜ v ) is an isomorphism, and it follows easily that D(u) also commutes, as desired. To verify the claim, use (1.6.3) to construct in J′′ a commutative diagram ′′

ϕ

X −−−X−→ AX ⏐ ⏐ u

w

Y −−−−→ AY −−−−→ Z −−−−→ AZ ϕ

ϕY

ϕZ

with ϕ a quasi-isomorphism, and set v := ϕ ◦ϕY v ′ := ϕZ ◦ ϕ (vu)′ := ϕZ ◦ w. Then v ′ ϕY = ϕZ v and (vu)′ ϕX = ϕZ (vu), as desired. Thus ζ is a morphism of functors; and it is straightforward to check, via commutativity of (1.7.2.2), that ζ is in fact a morphism of Δ-functors. Now we need to show (see (2.1.1)) that for every Δ-functor G : D′′ → E the composed map (1.5.1)

via ζ

Hom(RF , G) −−−−→ Hom(RF ◦Q′′ , G ◦Q′′ ) −−−−→ Hom(F , G ◦ Q′′ ) is bijective. For this it suﬃces to check that the following natural composition is an inverse map:

48

2 Derived Functors

Hom(F , G ◦Q′′ ) −−−−→ Hom(F ◦j, G ◦Q′′ ◦j) Hom(F ′, G ◦ j˜◦Q′ ) (2.1.1)

−−−−→ Hom(RF ′, G ◦ j˜) −−−−→ Hom(RF ′ ◦ρ, G ◦ j˜◦ρ) (2.2.2)

−−−−→ Hom(RF ′ ◦ρ, G) Hom(RF , G) . This checking is left to the reader, as is the proof for left-derived functors. Q.E.D. Example 2.2.4 [H, p. 53, Thm. 5.1]. Let j : J′ ֒→ J′′, F : J′′ → E, and ϕX : X → AX be as above, and suppose that the restricted functor F ′ := F ◦j transforms quasi-isomorphisms into isomorphisms (or, equivalently, F (C) ∼ =0 for every exact complex C ∈ J′ , see (1.5.1)). Then by (2.1.3), F ′ has a rightderived functor (RF ′ , 1) where F ′ = RF ′ ◦Q′ and 1 is the identity morphism of F ′. So by (2.2.3), F has a right-derived functor (RF , ζ) with RF (X) = F (AX ) and ζ(X) = F (ϕX ) : F (X) → F (AX ) = RF (X) ′′

for all X ∈ J . Note that if X ∈ J′ then ϕX is a quasi-isomorphism in J′ , whence ζ(X) is an isomorphism. The action of RF on maps can be described thus: if u : X → Y is a map in J′′ then with v ′ and (vu)′ as in the preceding proof, RF (u/1) = F (v ′ )−1 ◦F ((vu)′ ) ; and for any map f /s in D′′ (see §1.2), we have RF (f /s) = RF (f /1)◦ RF (s/1)−1 . As for the Δ-structure on RF , one has for each X the isomorphism θ(X) : RF (X[1])=F (AX[1] ) −− −−→ F (AX [1]) − −→ F (AX )[1] = RF (X)[1] F (ηX)

θF

where

ηX := Q′′ (ϕX [1])◦ Q′′ (ϕX[1] )−1 : AX[1] − −→ AX [1] ,

and where the isomorphism θF comes from the Δ-functoriality of F .

2.2 Existence of Derived Functors

49

(2.2.5). Let A be an abelian category, let J be a Δ-subcategory of K(A), and let F be a Δ-functor from J to a Δ-category E. We say that a complex X in J is right-F -acyclic if for each quasi-isomorphism u : X → Y in J there is a quasi-isomorphism v : Y → Z in J such that the map F (vu) : F (X) → F (Z) is an isomorphism. Left-F -acyclicity is deﬁned similarly, with arrows reversed. For example, if J := J′′ in (2.2.4), then every complex X ∈ J′ is rightF -acyclic—just take Z := AY and v := ϕY . Conversely: Lemma 2.2.5.1. The right-F -acyclic complexes in J are the objects of a localizing subcategory (§1.7). Moreover, the restriction of F to this subcategory transforms quasi-isomorphisms into isomorphisms; in other words, if the complex X is both exact and right-F -acyclic, then F (X) ∼ = 0 (see (1.5.1)). Proof. Since F commutes with translation—up to isomorphism—it is clear that X is right-F -acyclic iﬀ so is X[1]. Next, suppose we have a triangle X → X1 → X2 → X[1] in which X1 and X2 are right-F -acyclic. We will show that then X is right-F -acyclic. Any quasi-isomorphism u : X → Y can be embedded into a map of triangles X −−−−→ X1 −−−−→ X2 −−−−→ X[1] ⏐ ⏐ ⏐ ⏐ ⏐u[1] ⏐ ⏐ ⏐ u2 u1 u Y −−−−→ Y1 −−−−→ Y2 −−−−→ Y [1]

where u1 is a quasi-isomorphism whose existence is given by (1.6.3), and where u2 is then given by (Δ3)′ and (Δ3)′′ in §1.4. Such a u2 is also a quasi-isomorphism, as one sees by applying the ﬁve-lemma to the natural map between the homology sequences of the two triangles (see (1.4.5)H ). Similarly, from the deﬁnition of right-F -acyclic we deduce a triangle-map Y1 −−−−→ Y2 −−−−→ Y [1] −−−−→ Y1 [1] ⏐ ⏐ ⏐ ⏐ ⏐v [1] ⏐ ⏐ ⏐ v2 v1 v[1] 1 Z1 −−−−→ Z2 −−−−→ Z[1] −−−−→ Z1 [1]

where v1 , v2 , and v are quasi-isomorphisms such that F (v1 u1 ) and F (v2 u2 ) are isomorphisms. (Here (Δ2) in §1.4 should be kept in mind.) We can then apply the Δ-functor F to the map of triangles X1 −−−−→ X2 −−−−→ X[1] −−−−→ X1 [1] ⏐ ⏐ ⏐ ⏐ ⏐(v u )[1] ⏐ ⏐ ⏐ v2 u2 v1 u1 (vu)[1] 1 1 Z1 −−−−→ Z2 −−−−→ Z[1] −−−−→ Z1 [1]

and deduce from (Δ3)∗ that F ((vu)[1]), and hence F (vu), is also an isomorphism. Thus X is indeed right-F -acyclic.

50

2 Derived Functors

In particular, the direct sum of two right-F -acyclic complexes is rightF -acyclic, because the direct sum is the summit of a triangle whose base is the zero-map from one to the other, see (1.4.2.1). Also, 0 ∈ J is clearly rightF -acyclic. We see then that the right-F -acyclic complexes are the objects of a Δ-subcategory of J. For this subcategory to be localizing it suﬃces, by (1.7.1)op , that if X → Y → Z is as in the deﬁnition of right-F -acyclic, then Z is rightF -acyclic; and this follows from: Lemma 2.2.5.2. If X is right-F -acyclic and if there exists a quasiisomorphism α : X → Z such that F (α) : F (X) → F (Z) is an epimorphism, then Z is right-F -acyclic. Proof. Given a quasi-isomorphism Z → Y ′ , there exists a quasiisomorphism Y ′ → Z ′ such that F (X) → F (Z) → F (Z ′ ) is an isomorphism (since X is right-F -acyclic); and since F (X) → F (Z) is an epimorphism, Q.E.D. therefore F (Z) → F (Z ′ ) is an isomorphism. To justify the last assertion in (2.2.5.1), take Y := 0 in the deﬁnition of right-F -acyclicity. Q.E.D. We leave it to the reader to establish a corresponding statement for left-F -acyclic complexes. In summary: Proposition 2.2.6. Let A be an abelian category, let J be a Δ-subcategory of K(A), and let F be a Δ-functor from J to a Δ-category E. Suppose J contains a family of quasi-isomorphisms ϕX : X → AX (X ∈ J) such that AX is right-F -acyclic for all X, see (2.2.5). Then F has a right-derived functor (RF , ζ) such that for all X ∈ J, RF (X) = F (AX )

and

ζ(X) = F (ϕX ) : F (X) → F (AX ) = RF (X) .

Moreover, X is right-F -acyclic ⇔ ζ(X) is an isomorphism. Proof. Everything is contained in (2.2.4) and (2.2.5), except for the fact that if ζ(X) is an isomorphism then X is right-F -acyclic, which is proved by taking, in (2.2.5), Z := AY , v := ϕY , and noting that then F (vu) is the composite isomorphism F (X) − −→ RF (X) − −→ RF (Y ) = F (Z). ζ(X)

Q.E.D.

Corollary 2.2.6.1. With assumptions as in (2.2.6), if G : E → E′ is any Δ-functor then (G ◦ RF , G(ζ)) is a right-derived functor of GF . Proof. Clearly, right-F -acyclic complexes are right-(GF )-acyclic. It follows then from (2.2.4) and (2.2.5) that the assertion need only be proved for the restriction of F to the subcategory of right-F -acyclic complexes, in which case it follows from (2.1.3). Q.E.D.

2.2 Existence of Derived Functors

51

Corollary 2.2.7. Let A, A′ be abelian categories, let J ⊂ K(A), J′ ⊂ K(A′ ) be Δ-subcategories with canonical functors Q : J → DJ , Q′ : J′ → DJ′ to their respective derived categories, and let F : J → J′ and G : J′ → E be Δ-functors. Assume that G has a right-derived functor RG and that every complex X ∈ J admits a quasi-isomorphism into a right(Q′ F )-acyclic complex AX such that F (AX ) is right-G-acyclic. Then Q′ F and GF have right-derived functors, denoted RF and R(GF ), and there is a unique Δ-functorial isomorphism ∼ α : R(GF ) −→ RG RF

such that the following natural diagram commutes for all X ∈ J: GF (X) ⏐ ⏐

−−−−→ R(GF )(QX) ⏐ ⏐ ≃α(QX)

(2.2.7.1)

RGQ′ F (X) −−−−→ RGRF (QX)

Proof. Derivability of Q′ F results from (2.2.6). Derivability of GF results similarly once we show, as follows, that AX is right-(GF )-acyclic: note for any quasi-isomorphism AX → Y in J that, by (2.2.5.1), the resulting composed map F (AX ) → F (Y ) → F (AY ) is a quasi-isomorphism ∼ and so GF (AX ) −→ GF (AY ). The existence of a unique Δ-functorial α making (2.2.7.1) commute follows from the deﬁnition of right-derived functor. Since AX is right-(GF )-acyclic and right-(Q′ F )-acyclic, and F (AX ) is rightG-acyclic, (2.2.6) implies that α(QX) is isomorphic to the identity map of GF (AX ). Thus α is an isomorphism. Q.E.D. We leave the corresponding statements for left-F -acyclic complexes and left-derived functors to the reader. Incidentally, (2.2.6) generalizes in a simple way to triangulation-compatible multiplicative systems in any Δ-category (see [H, p. 31]). It is of course of little interest unless we can construct a family (ϕX ). That matter is addressed in the following sections. Exercises 2.2.8. (a) Verify that F transforms quasi-isomorphisms into isomorphisms iﬀ every complex X ∈ J is right-F -acyclic. (b) Verify that if X ∈ J is exact then X is right-F -acyclic iﬀ F (X) ∼ = 0. (c) Let F be a Δ-functor from J to a Δ-category E. Let J′ be the full subcategory of J whose objects are all the complexes in J admitting a quasi-isomorphism to a right-F -acyclic complex. Then J′ is a Δ-subcategory of J. (d) X is right-F -acyclic iﬀ every map C → X in J with C exact factors as C → C ′ → X with C ′ exact and F (C ′ ) ∼ = 0.

52

2 Derived Functors (e) X is said to be “unfolded for F ” if for every Z ∈ E the natural map HomE (Z, F (X)) → limHomE (Z, F (Y )) −→ X→Y

is an isomorphism, where the lim is taken over the category of all quasi-isomorphisms −→ X → Y in J [De, p. 274, (iv)]. Check that any right-F -acyclic X is unfolded for F ; and that the converse holds under the hypotheses of (2.2.6). (f) Show: X is unfolded for F iﬀ every map C → X in J with C exact factors as C → C ′ → X with C ′ exact and F (C) → F (C ′ ) the zero map. (For this, the octahedral axiom in E may be needed, see §1.4.)

2.3 Right-Derived Functors via Injective Resolutions The basic example of a family (ϕX ) as in (2.2.6) arises when A has enough injectives, i.e., every object of A admits a monomorphism into an injective object. Then every complex X ∈ K+(A) admits a quasi-isomorphism ϕX : X → IX into a bounded-below complex of injectives (see (1.8.2)); and by (2.3.4) and (2.3.2.1) below, this IX is right-F -acyclic for every Δ-functor F : K+(A) → E, whence F is right-derivable. Later on, however, it will become important for us to be able to deal with unbounded complexes; and for this purpose the following more general injectivity notion is, via (2.3.5), essential. Deﬁnition 2.3.1. Let A be an abelian category, and let J be a Δ-subcategory of K(A). A complex I ∈ J is said to be q-injective in J (or J-q-injective) if f s for every diagram Y ← −X− → I in J with s a quasi-isomorphism, there exists g : Y → I such that gs = f .1 Lemma 2.3.2. I ∈ J is J-q-injective iﬀ every quasi-isomorphism I → Y in J has a left inverse. Proof. In (2.3.1) take X := I and f := identity to see that if I is q-injective then the quasi-isomorphism s has a left inverse. Conversely, by (1.6.3) any s

f

−X− → I is part of a commutative diagram diagram Y ← f

X −−−−→ ⏐ ⏐ s

I ⏐ ⏐′ s

Y −−−− → Y′ ′ f

in which s′ is a quasi-isomorphism; and then if t is a left inverse for s′ and Q.E.D. g := tf ′ , we have gs = f . 1 Here “q” stands for the class of quasi-isomorphisms. The equivalent term “Kinjective” in [Sp, p. 127] seems to me less suggestive.

2.3 Right-Derived Functors via Injective Resolutions

53

Corollary 2.3.2.1. I ∈ J is J-q-injective iﬀ I is right-F -acyclic for every Δ-functor F : J → E. Proof. If any quasi-isomorphism I → Y has a left inverse, then setting X := I in (2.2.5) we see at once that I is right-F -acyclic. Conversely, if I is right-F -acyclic for the identity functor J → J, then every quasi-isomorphism I → Y has a left inverse. Q.E.D. Taking F := identity in (2.2.5.1), we deduce: Corollary 2.3.2.2. The J-q-injective complexes are the objects of a localizing subcategory I. Every quasi-isomorphism in I is an isomorphism, so the pair (I, identity) has the universal property of the derived category DI (§1.2), and therefore I ∼ = DI can be identiﬁed with a Δ-subcategory of DJ . Corollary 2.3.2.3. Suppose that there exists a family of q-injective resolutions ϕX : X → IX (X ∈ J), i.e., for each X, ϕX is a quasi-isomorphism and IX is J-q-injective. Then any Δ-functor F : J → E has a right-derived functor (RF , ζ)2 with RF (X) = F (IX )

and

ζ(X) = F (ϕX ) : F (X) → F (IX ) = RF (X) , s

f

and such that for any morphism f /s : X1 ← X → X2 in DJ , RF (f /s) = F (f ′ ) ◦ F (s′ )−1 where f ′ is the unique map in I making the following square in J commute ϕ

X −−−X−→ ⏐ ⏐ f

IX ⏐ ⏐ ′ f

X2 −−−−→ IX2 ϕX

2

and similarly for s′ . Proof. Since ϕX becomes an isomorphism in DJ , the map f ′ exists uniquely in DJ , hence in I (2.3.2.2). For the rest see (2.2.4), with J′ := I, J′′ := J, and v := identity. Q.E.D. Example 2.3.3. An object I in A is injective iﬀ when considered as a complex vanishing in all nonzero degrees it is q-injective in K(A) (or in Kb(A)). f s0 Suﬃciency: for any A-diagram Y 0 ←− X −→ I with s0 a monomorphism, take Y to be the complex which looks like the natural map Y 0 → coker(s0 ) in degrees 0 and 1, and vanishes elsewhere, and take s : X → Y to be the obvious quasi-isomorphism; then deduce from (2.3.1) that if I is q-injective there exists g 0 : Y 0 → I such that g 0 s0 = f —so that I is A-injective. 2

So the embedding functor (2.1.1.1) has a left adjoint, taking F to RF .

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2 Derived Functors

For necessity, use (2.3.2): to ﬁnd a left inverse in K(A) for a quasiisomorphism β : I → Y we may replace Y by the complex τ≥0 Y , to which Y maps quasi-isomorphically (§1.10), i.e., we may assume that Y vanishes in all negative degrees; then β induces a monomorphism (in A) β 0 : I → Y 0 , which has a left inverse if I is A-injective, and that gives rise, obviously, to a left inverse for β. (One could also use (iv) in (2.3.8) below.) Example 2.3.4. Any bounded-below complex I of A-injectives is q-injective in K(A). Indeed, by [H, p. 41, Lemma 4.5], I satisﬁes the condition in (2.3.2). (One could also use (2.3.8)(iv).) Thus (2.3.2.3) applies to J := K+(A) whenever A has enough injectives (see beginning of §2.3). In that case, further, every K+(A)-q-injective complex admits a quasi-isomorphism, hence, by (2.3.2.2), an isomorphism, to a bounded-below complex of A-injectives. Example 2.3.5. Let U be a topological space, O a sheaf of rings on U , and A the abelian category of left O-modules. Then a theorem of Spaltenstein [Sp, p. 138, Theorem 4.5] asserts that every complex in K(A) admits a q-injective resolution. Hence by (2.3.2.3), every Δ-functor out of K(A) is right-derivable. More generally, a q-injective resolution exists for every complex in any Grothendieck category, i.e., an abelian category with exact direct limits and having a generator [AJS, p. 243, Theorem 5.4]. For example, injective Cartan-Eilenberg resolutions [EGA, III, Chap. 0, (11.4.2)] always exist in Grothendieck categories; and their totalizations—which generally require countable direct products—give q-injective resolutions when such products of epimorphisms are epimorphisms (a condition which holds in the category of modules over a ﬁxed ring, but fails, for instance, in most categories of sheaves on topological spaces). Example 2.3.6. Let A1 , A2 be abelian categories, A1 having enough injectives. As in (1.5.2) any additive functor F : A1 → A2 extends to a Δ-functor F¯ : K+(A1 ) → K+(A2 ) which has, by (2.3.4), a right-derived functor R+F¯ : D+(A1 ) → K+(A2 ) satisfying, for a given family ϕX : X → IX of injective resolutions, R+F¯ (X) = F¯ (IX ) . We can extend the domain of R+F¯ to D+(A1 ) by composing with the equivalence τ + deﬁned in (1.8.1). Moreover, if every A1 -complex has a q-injective resolution, then there is a further extension to a derived functor RF¯ : D(A1 ) → K(A2 )—whose composition with the canonical map K(A2 ) → D(A2 ) is RF , see (2.1.2).

2.3 Right-Derived Functors via Injective Resolutions

55

With H i the usual homology functor, let Ri F : A1 → A2 (i ∈ Z) be the composition (1.2.2) R+ F H A1 −−−−→ D+(A1 ) −−−−→ K+(A2 ) −−−−→ A2 i

(cf. (2.1.4)). Then Ri F = 0 for i < 0, and there is a natural map of functors F → R0 F which is an isomorphism if and only if F is left-exact. Example 2.3.7. Let f : U1 → U2 be a continuous map of topological spaces. Let A i be the category of sheaves of abelian groups on Ui (i = 1, 2). Then A i is abelian, and has enough injectives. The direct image functor f∗ : A1 → A2 is left-exact, and has, as in (2.3.6), a derived functor R+f∗ : D+(A1 ) → K+ (A2 ) . f∗

Q

By (2.3.5), the composition K(A1 ) − → K(A2 ) − → D(A2 ) has a derived + functor Rf∗ , whose restriction to D (A1 ) is isomorphic to Q◦R+f∗ . In particular, when U2 is a single point then A2 = Ab, the category of abelian groups, and f∗ is the global section functor Γ = Γ(U1 , −). In this case one usually sets, for i ∈ Z, see (2.1.4), Rf∗ = RΓ,

Rif∗ = Ri Γ = Hi ,

Rif∗ (−) = H i (U1 , −) .

Here are some other characterizations of q-injectivity, see [Sp, p. 129, Prop. 1.5], [BN, Def. 2.6 etc.]. Proposition 2.3.8. Let A be an abelian category, and let J be a Δ-subcategory. of K(A). The following conditions on a complex I ∈ J are equivalent: (i) I is q-injective in J. s

f

− X − → I in J with s a quasi-isomorphism (i)′ For every diagram Y ← there is a unique g : Y → I such that gs = f . (ii) Every quasi-isomorphism I → Y in J has a left inverse. (ii)′ Every quasi-isomorphism I → Y in J is a monomorphism. (iii) I is right-F -acyclic for every Δ-functor F : J → E. (iii)′ I is right-F -acyclic for F the identity functor J → J. (iv) For every exact complex X ∈ J, we have Hom J(X, I) = 0. (iv)′ The Δ-functor Hom• (−, I ) : J → K(Ab) of (1.5.3) takes quasiisomorphisms into quasi-isomorphisms. (v) For every complex X ∈ J, the natural map Hom J(X, I ) → Hom DJ(X, I ) is bijective. Proof. The equivalence of (i), (ii), (iii) and (iii)′ has already been shown (see (2.3.2) and the proof of (2.3.2.1)). For (ii) ⇔ (ii)′ see (1.4.2.1). Taking Y := 0 in (2.3.1), we see that (i) ⇒ (iv). The equivalence of (iv) and (iv)′ results from the footnote in (1.5.1) and the easily-checked relation

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2 Derived Functors

H

n

Hom• (X, I ) ∼ = HomJ (X[−n], I )

(n ∈ Z, X ∈ J).

(2.3.8.1)

The implications (v) ⇒ (i)′ ⇒ (i) are simple to verify. We show next that (iv) ⇒ (ii). Let X be the summit of a triangle T in J whose base is a quasi-isomorphism I → Y . By [H, p. 23, 1.1 b)], the resulting sequence Hom(X, I ) → Hom(Y , I ) → Hom(I, I ) → Hom(X[−1], I ) is exact. Moreover, the exact homology sequence (1.4.5)H of T shows that X is exact. So if (iv) holds, then Hom(Y , I) → Hom(I, I) is bijective, and (ii) follows. Finally, we show (ii) ⇒ (v). For any map f /s : X → I in DJ , (1.6.3) yields a commutative diagram in J, with s′ a quasi-isomorphism: f

A −−−−→ ⏐ ⏐ s

I ⏐ ⏐′ s

X −−−− → B ′ f

If ts′ = identity, then f /s = (s′ /1)−1 (f ′ /1) = (tf ′ )/1, and so the map HomJ (X, I) → HomDJ (X, I) is surjective. For injectivity, given f : X → I in J, note that f /1 = 0 =⇒ there exists a quasi-isomorphism t : X ′ → X such that f t = 0 (see §1.2) =⇒ there exists a quasi-isomorphism s : I → Y such that sf = 0 [H, p. 37]; and if s has a left inverse, then sf = 0 =⇒ f = 0. Q.E.D. Exercise 2.3.9. Show that if A is a Grothendieck category then D(A) is equivalent to the homotopy category of q-injective complexes. Hence if A has inverse limits then so does D(A).

2.4 Derived Homomorphism Functors Let A be an abelian category, and let L be a Δ-subcategory of K(A) in which there exists a family of quasi-isomorphisms ϕX : X → IX (X ∈ L) such that IX ∈ L is q-injective in K(A) for every X. Then for any quasi-isomorphism s : X → Y with Y in K(A) there exists, by (2.3.1), a map g : Y → IX , necessarily a quasi-isomorphism, such that gs = ϕX ; and hence by (1.7.1)op , L is a localizing subcategory of K(A), i.e., the derived category DL identiﬁes naturally with a Δ-subcategory of D(A). For example, if A has enough injectives we could take L := K+(A), see (2.3.4). Or, if U is a topological space with a sheaf of rings O and A is the category of left O-modules, we could take L := K(A), see (2.3.5).

2.4 Derived Homomorphism Functors

57

By (2.3.2.3), every Δ-functor F : L → E is right-derivable. So for any ﬁxed object A ∈ K(A), the Δ-functor FA : L → K(Ab) given by FA (B) = Hom• (A, B)

(B ∈ L)

(see (1.5.3)) has a right-derived functor RFA : DL → K(Ab) with

RFA (B) = Hom• (A, IB ).

For ﬁxed B and variable A, Hom• (A, IB ) is a contravariant Δ-functor from K(A) to K(Ab) (see 1.5.3), which takes quasi-isomorphisms in K(A) to quasi-isomorphisms in K(Ab) ((2.3.8)(iv)′ ) and hence—after composition with the natural functor Q′ : K(Ab) → D(Ab)—to isomorphisms in D(Ab). So by (1.5.1)—and the exercise preceding it—there results a Δ-functor D(A)op → D(Ab). Thus we obtain a functor of two variables RHom• (A, B) : D(A)op × DL → D(Ab) which, together with appropriate θ (see (1.5.3)), is a Δ-functor in each variable separately: RHom• (A, B) = Q′ Hom• (A, IB )

(2.4.1)

for all objects A ∈ D(A)op , B ∈ DL ; and we leave it to the reader to make explicit the eﬀect of RHom• on morphisms in D(A)op and DL respectively. From (2.3.8)(v) and (2.3) (with J := K(A)), we deduce canonical isomorphisms (Yoneda theorem): ∼ H n (RHom• (X, B)) −→ HomD(A) (X, B[n])

(n ∈ Z).

(2.4.2)

This leads, in particular, to an elementary interpretation of the exact sequence (2.1.4)H when F := FX , see [H, p. 23, Prop. 1.1, b)]. (2.4.3). The variables A, B are treated quite diﬀerently in the above deﬁnition of RHom• . But there is a more symmetric characterization of this derived functor, analogous to the one in (2.1.1). This is given in (2.4.4), after the necessary preparation. Let K1 , K2 , E be Δ-categories, with respective translation functors T1 , T2 , T . A Δ-functor from K1 × K2 to E is deﬁned to be a triple (F , θ1 , θ2 ) with F : K1 × K2 → E a functor and ∼ θ1 : F ◦(T1 × 1) −→ T ◦F ,

∼ θ2 : F ◦(1 × T2 ) −→ T ◦F

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isomorphisms of functors, such that for each B ∈ K2 the functor FB (A) := F (A, B) together with θ1 is a Δ-functor from K1 to E, and for each A ∈ K1 the functor FA (B) := F (A, B) together with θ2 is a Δ-functor from K2 to E; and such that furthermore the composed functorial isomorphisms via θ

via θ

via θ

via θ

1 2 F (T1 × T2 ) = F (T1 × 1)(1 × T2 ) −−−−→ T F (1 × T2 ) −−−−→ TTF 2 1 T F (T1 × 1) −−−−→ TTF F (T1 × T2 ) = F (1 × T2 )(T1 × 1) −−−−→

are negatives of each other. Similarly, we can deﬁne Δ-functors of three or more variables—with a condition indicated by the equation (via θi )◦(via θj ) = −(via θj )◦(via θi )

(i = j).

Morphisms of Δ-functors are deﬁned in the obvious way, see (1.5). For example, let L ⊂ K := K(A) be as above, with respective derived categories DL ⊂ D, and consider the functor Hom• : Kop × L → K(Ab). As in the exercise preceding (1.5.1), we can consider the opposite category Kop to be triangulated, with translation inverse to that in K, in such a way that the canonical contravariant functor K → Kop and its inverse, together with θ = identity, are both Δ-functors. This being so, one checks then that Hom• is a Δ-functor (see (1.5.3)). Similarly RHom• : Dop × DL → D(Ab) is a Δ-functor. Furthermore, the q-injective resolution maps ϕB : B → IB induce a natural morphism of Δ-functors η : Q′ Hom• (A, B) → Q′ Hom• (A, IB )

(2.4.1)

=

RHom• (QA, QB)

where Q : K → D is the canonical functor. This η is, in the following sense, universal (hence unique up to isomorphism): Lemma 2.4.4. Let

G : Dop × DL → D(Ab)

be a Δ-functor, and let μ : Q′ Hom• (A, B) → G(QA, QB)

(A ∈ Kop , B ∈ L)

2.4 Derived Homomorphism Functors

59

be a morphism of Δ-functors. Then there exists a unique morphism of Δ-functors μ : RHom• → G such that μ = μη. Proof. μ is the composition μ

∼ RHom• (QA, QB) = Q′ Hom• (A, IB ) −→ G(QA, QIB ) −→ G(QA, QB) .

The rest is left to the reader. (See also (2.6.5) below.) (2.4.5). Next we discuss the sheaﬁﬁed version of the above. Let U be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. The “sheaf-hom” functor Hom : Aop × A → A extends naturally to a Δ-functor Hom• : K(A)op × K(A) → K(A) (essentially because everything in (1.5.3) is compatible with restriction to open subsets—details left to the reader). Taking note of the following Lemma, we can proceed as above to derive a Δ-functor RHom• : D(A)op × D(A) → D(A) .

Lemma 2.4.5.1. If I is a q-injective complex in K(A) then the functor Hom• (−, I ) takes quasi-isomorphisms to quasi-isomorphisms. Proof. For A ∈ K(A) and i ∈ Z, the homology H i (Hom• (A, I)) is the sheaf associated to the presheaf V → H i Γ(V , Hom• (A, I) = H i Hom• (A|V , I|V ) (V open in U ).

We can then apply (2.3.8)(iv)′ to the category AV of (O|V )-modules, as soon as we know:

Lemma 2.4.5.2. Let V be an open subset of U , with inclusion map i : V ֒→ U . Then for any q-injective complex I ∈ K(A), the restriction i∗I = I|V is q-injective in K(AV ). Proof. The extension by zero of an OV -module M is the sheaf i! M associated to the presheaf on U which assigns M (W ) to any open W ⊂ V and 0 to any open W V . The restriction i∗ i! M can be identiﬁed with M ; and the stalk of i! M at any point w ∈ / V is 0. So i! is an exact functor.

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2 Derived Functors s

f

Now from any diagram Y ← X → i∗I of maps of AV -complexes with s a quasi-isomorphism, we get the diagram is

i! f

α

i! Y ←!− i! X −→ i! i∗I ֒→ I where i! s is a quasi-isomorphism (since i! is exact) and α is the natural map. By (2.3.1), there exists a map g : i! X → I such that g ◦i! s = α ◦i! f in K(A); and then we have, in K(AV ), i∗g ◦s = i∗g ◦ i∗ i! s = i∗ α ◦i∗ i! f = 1◦ f = f . Thus i∗I is indeed q-injective.

Q.E.D.

(2.4.5.3). Similarly, any functor having an exact left adjoint preserves q-injectivity.

2.5 Derived Tensor Product Let U be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. Recall from (1.5.4) the deﬁnition of the tensor product (over O) of two complexes in K(A), and its Δ-functorial properties. The standard theory of the derived tensor product, via resolutions by complexes of ﬂat modules, applies to complexes in D−(A), see e.g., [H, p. 93]. Following Spaltenstein [Sp] we can use direct limits to extend the theory to arbitrary complexes in D(A). Before deﬁning, in (2.5.7), the derived tensor product, we need to develop an appropriate acyclicity notion, “q-ﬂatness.” Deﬁnition 2.5.1. A complex P ∈ K(A) is q-ﬂat if for every quasiisomorphism Q1 → Q2 in K(A), the resulting map P ⊗ Q1 → P ⊗ Q2 is also a quasi-isomorphism; or equivalently (see footnote under (1.5.1)), if for every exact complex Q ∈ K(A), the complex P ⊗ Q is also exact. Example 2.5.2. P ∈ K(A) is q-ﬂat iﬀ for each point x ∈ U , the stalk Px is q-ﬂat in K(Ax ), where Ax is the category of modules over the ring Ox . (In verifying this statement, note that an exact Ox -complex Qx is the stalk at x of the exact O-complex Q which associates Qx to those open subsets of U which contain x, and 0 to those which don’t.) For instance, a complex P which vanishes in all degrees but one (say n) is q-ﬂat if and only if tensoring with the degree n component P n is an exact functor in the category of O-modules, i.e., P n is a ﬂat O-module, i.e., for each x ∈ U, Pxn is a ﬂat Ox -module.

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61

Example 2.5.3. Tensoring with a ﬁxed complex Q is a Δ-functor, and so the exact homology sequence (1.4.5)H of a triangle yields that the q-ﬂat complexes are the objects of a Δ-subcategory of K(A). A bounded complex P :

··· → 0 → 0 → Pm → ··· → Pn → 0 → 0 → ···

ﬁts into a triangle P ′ → P → P ′′ → P ′ [1] where P ′ is P n in degree n and 0 elsewhere, and where P ′′ is the cokernel of the obvious map P ′ → P . So starting with (2.5.2) we see by induction on n − m that any bounded complex of ﬂat O-modules is q-ﬂat. Example 2.5.4. Since (ﬁltered) direct limits commute with both tensor product and homology, therefore any such limit of q-ﬂat complexes is again q-ﬂat. A bounded-above complex P :

··· → Pm → ··· → Pn → 0 → 0 → ···

is the limit of the direct system P0 → P1 → · · · → Pi → · · · where Pi is obtained from P by replacing all the components P j with j < n − i by 0, and the maps are the obvious ones. Hence, any bounded-above complex of ﬂat O-modules is q-ﬂat. A q-ﬂat resolution of an A-complex C is a quasi-isomorphism P → C where P is q-ﬂat. The totality of such resolutions (with variable P and C) is the class of objects of a category, whose morphisms are the obvious ones. Proposition 2.5.5. Every A-complex C is the target of a quasi-isomorphism ψC from a q-ﬂat complex PC , which can be constructed to depend functorially on C, and so that PC[1] = PC [1] and ψC[1] = ψC [1]. Proof. Every O-module is a quotient of a ﬂat one; in fact there exists a functor P0 from A to its full subcategory of ﬂat O-modules, together with a functorial epimorphism P0 (F) ։ F (F ∈ A). Indeed, for any open V ⊂ U let OV be the extension of O|V by zero, (i.e., the sheaf associated to the presheaf taking an open W to O(W ) if W ⊂ V and to 0 otherwise), so that OV is ﬂat, its stalk at x ∈ U being Ox if x ∈ V and 0 otherwise. There is a canonical isomorphism ∼ ψ : F(V ) −→ Hom(OV , F)

(F ∈ A)

such that ψ(λ) takes 1 ∈ OV (V ) to λ. With Oλ := OV for each λ ∈ F(V ), the maps ψ(λ) deﬁne an epimorphism, with ﬂat source, P0 (F) := Oλ ։ F, V open λ∈F(V )

and this epimorphism depends functorially on F.

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We deduce then, for each F, a functorial ﬂat resolution · · · → P2 (F) → P1 (F) → P0 (F) ։ F with P1 (F) := P0 ker(P0 (F) ։ F) , etc. Set Pn (F) = 0 if n < 0. Then to a complex C we associate the ﬂat complex P = PC such that r r+1 P r := ⊕m−n=r Pn (C m ) and the restriction of the diﬀerential P → P to Pn (C m ) is Pn (C m → C m+1 ) ⊕ (−1)m Pn (C m ) → Pn−1 (C m ) , together with the natural map of complexes P → C induced by the epimorphisms P0 (C m ) ։ C m (m ∈ Z). Elementary arguments, with or without spectral sequences, show that for the truncations τ≤m C of §1.10, the maps Pτ C → τ≤m C are quasi-isomorphisms. Since homology commutes with ≤m direct limits, the resulting map ψC : PC = lim Pτ≤m C → lim τ≤m C = C, −→ −→ m m (which depends functorially on C) is a quasi-isomorphism; and by (2.5.4), PC is q-ﬂat. That PC[1] = PC [1] and ψC[1] = ψC [1] is immediate. Q.E.D. Exercises 2.5.6. (a) Let P and Q be complexes of O-modules, and suppose that for all integers s, t, u, v the complex τ≤s τ≥t P ⊗O τ≤u τ≥v Q is exact. Then P ⊗ Q = lim τ≤s P ⊗ τ≤u Q − → s,u is exact. (b) If for all n ∈ Z the homology H n (P ) is a ﬂat O-module and furthermore, for all n the kernel of P n → P n+1 is a direct summand of P n (or, for all n the image of P n → P n+1 is a direct summand of P n+1 ), then P is q-ﬂat. (Use (a) to reduce to where P is bounded; then apply induction to the number of n such that P n = 0.)

(2.5.7). Let A be, as above, the category of O-modules, and let J′ ⊂ K := K(A) be the Δ-subcategory of K whose objects are all the q-ﬂat complexes, see (2.5.3). Fix B ∈ K and consider the Δ-functor FB : K → D := D(A) such that FB (A) = A ⊗ B

(see (1.5.4)).

If A is both q-ﬂat and exact, then A ⊗ B is exact: to see this, we may replace B by any quasi-isomorphic complex B ′ (since A is q-ﬂat), and by (2.5.5) we may assume that B ′ is q-ﬂat, whence, by (2.5.1), A ⊗ B ′ is exact. Hence the restriction of FB to J′ transforms quasi-isomorphisms into isomorphisms.

2.5 Derived Tensor Product

63

There exists, by (2.5.5), a functorial family of quasi-isomorphisms ψA : PA → A

(A ∈ K, PA ∈ J′ ).

with PA[1] = PA [1]. An argument dual to that in (2.2.4) (with J′′ := K) shows then that FB has a left-derived Δ-functor (LFB , identity) : D → D with

(2.5.7.1)

LFB (A) = PA ⊗ B ∼ = PA ⊗ PB ∼ = A ⊗ PB ,

the isomorphisms being the ones induced by ψA and ψB . Alternatively, PA is left-FB -acyclic for all A, B (see 2.5.10(d)), so one can apply (2.2.6). For ﬁxed A and variable B, PA ⊗B is a Δ-functor from K to D which takes quasi-isomorphisms to isomorphisms, so by (1.5.1) there results a Δ-functor from D to D. Hence there is a functor of two variables, called a derived tensor product, : D × D −→ D ⊗ = which together with appropriate θ (see (1.5.4)) is a Δ-functor in each variable separately (i.e., it is a Δ-functor as deﬁned in (2.4.3)). Though the variables A and B have been treated diﬀerently in the foregoing, their roles are essentially equivalent. Indeed, there is a universal property analogous to (the dual of) that in (2.4.4), characterizing the natural composite map of Δ-functors from K × K to D: ∼ QA ⊗ QB −→ Q(PA ⊗ PB ) −→ Q(A ⊗ B) . =

Hence, in view of (1.5.4.1), there is a canonical Δ-bifunctorial isomorphism ∼ B⊗ A −→ A⊗ B. = = ∼ This arises, in fact, from the natural isomorphism PB ⊗ PA −→ PA ⊗ PB .

(2.5.8). The local hypertor sheaves are deﬁned by B) Torn (A, B) = H −n (A ⊗ =

(n ∈ Z; A, B ∈ D).

As in (2.1.4), short exact sequences in either the A or B variable give rise to long exact hypertor sequences. We remark that when U is a scheme and O = OU , if the homology sheaves of the complexes A and B are all quasi-coherent then so are the sheaves Torn (A, B). This is clear, by reduction to the aﬃne case, if A and B are quasi-coherent OX -modules (i.e., complexes vanishing except in degree 0). In the general case, since A ⊗ B = lim τ≤s A ⊗ τ≤u B , − → s,u

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2 Derived Functors

we may assume A and B lie in D−, and then argue as in [H, p. 98, Prop. 4.3], or alternatively, use the K¨ unneth spectral sequence 2 Epq =

⊕

i+j=q

Torp (H −i (A), H −j (B)) ⇒ Tor• (A, B)

(as described e.g., in [B, p. 186, Exercise 9(b)], with ﬂat resolutions replacing projective ones). Thus, with notation as in (1.9), denoting by Dqc the Δ-subcategory D# ⊂ D with A# ⊂ A the subcategory of quasi-coherent OU -modules (which is plump, see [GD, p. 217, (2.2.2) (iii)]), we have a Δ-functor ⊗ : Dqc × Dqc −→ Dqc . =

(2.5.8.1)

(2.5.9). The deﬁnitions in (1.5.4) can be extended to three (or more) variables, to give a Δ-functor A ⊗ B ⊗ C from K × K × K to K. There exists a Δ-functor T3 : D×D×D → D together with a Δ-functorial map η : T3 (A, B, C) −→ A ⊗ B ⊗ C (A, B, C ∈ K) such that for any Δ-functor H : D × D × D → D and any Δ-functorial map μ : H(A, B, C) −→ A ⊗ B ⊗ C there is a unique Δ-functor map μ ¯ : H → T3 such that μ = η ◦ μ ¯. (The reader can ﬁll in the missing Q’s.) In fact there is such a T3 with T3 (A, B, C) = PA ⊗ PB ⊗ PC . We usually write T3 (A, B, C) = A ⊗ B⊗ C. = = There are canonical Δ-functorial isomorphisms ∼ ∼ (A ⊗ B) ⊗ C −→ A⊗ B⊗ C ←− A⊗ (B ⊗ C) . = = = = = =

Similar considerations hold for n > 3 variables. Details are left to the reader. (See, for example, (2.6.5) below.) Exercises 2.5.10. (a) Show that if A ∈ K(A) is q-ﬂat and B ∈ K(A) is q-injective then Hom• (A, B) is q-injective. (b) Let Γ : A → Ab be the global section functor. Show that there is a natural isomorphism of Δ-functors (of two variables, see (2.4.3)) ∼ RHom• (A, B) −→ RΓ RHom• (A, B).

(Use (a) and (2.2.7), or [Sp, 5.14, 5.12, 5.17].) (c) Let (Aα ) be a (small, directed) inductive system of A-complexes. Show that for any complex B ∈ D(A) there are natural isomorphisms ∼ lim Torn Aα , B −→ Torn (lim Aα ), B (n ∈ Z). −→ − → α α

2.6 Adjoint Associativity

65

(d) Show that for P to be q-ﬂat it is necessary that P be left-FB -acyclic for all B (FB as in (2.5.7)), and suﬃcient that P be left-FB -acyclic for all exact B. (For the last part, (2.2.6) could prove helpful.) Formulate and prove an analogous statement involving q-injectivity and Hom•. (See (2.3.8).)

2.6 Adjoint Associativity Again let U be a topological space, O a sheaf of commutative rings, and A the abelian category of O-modules. Set K := K(A), D := D(A). This section is devoted to (2.6.1)—or better, (2.6.1)∗ at the end—which expresses the basic adjointness relation between the Δ-functors RHom• : Dop × D → D : D × D → D deﬁned in (2.4.5) and (2.5.7) respectively. and ⊗ = Proposition 2.6.1. There is a natural isomorphism of Δ-functors ∼ RHom• (A ⊗ B, C) −→ RHom• (A, RHom• (B, C)) . =

Remarks. (i) In fact, the Δ-functors RHom• and ⊗ are deﬁned only up to = canonical isomorphism, by universal properties, as in (2.5.9). We leave it to the reader to verify that the map in (2.6.1) (to be constructed below) is compatible, in the obvious sense, with such canonical isomorphisms. (ii) A proof similar to the following one3 yields a natural isomorphism ∼ RHom• (A⊗ B, C) −→ RHom• (A, RHom• (B, C)) . =

Applying homology H 0 we have, by (2.4), the adjunction isomorphism ∼ HomD (A ⊗ B, C) −→ HomD (A, RHom• (B, C)) . =

(2.6.1′ )

(iii) Prop. (2.6.1) gives a derived-category upgrade of the standard sheaf isomorphism ∼ Hom (F ⊗ G, H) −→ Hom (F , Hom (G, H))

(F , G, H ∈ A).

(2.6.2)

Proof of (2.6.1). We discuss the proof at several levels of pedantry, beginning with the argument, in full, given in [I, p. 151, Lemme 7.4] (see also [Sp, p. 147, Prop. 6.6]): “Resolve C injectively and B ﬂatly.” This argument can be expanded as follows. Choose quasi-isomorphisms C → IC ,

3

PB → B

Or application of the functor RΓ to (2.6.1), see (2.5.10),

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where IC is q-injective and PB is q-ﬂat. It follows from (2.3.8)(iv) that the complex of sheaves Hom• (PB , IC ) is q-injective, since for any exact complex X ∈ K, the isomorphism of complexes ∼ Hom• (X ⊗ PB , IC ) − → Hom• (X, Hom• (PB , IC ))

coming out of (2.6) yields, upon application of homology H 0 , ∼ 0 = HomK (X ⊗ PB , IC ) − → HomK (X, Hom• (PB , IC )).

Now consider the natural sequence of D-maps RHom• (A ⊗ B, C) = ⏐ ⏐

RHom• (A, RHom• (B, C)) ⏐ ⏐

RHom• (A ⊗ B, IC ) = ⏐ ⏐

RHom• (A, RHom• (B, IC )) ⏐ ⏐

RHom• (A ⊗ PB , IC ) = ⏐ ⏐

RHom• (A ⊗ PB , IC ) ⏐ ⏐

RHom• (A, RHom• (PB , IC )) ⏐ ⏐

Hom• (A ⊗ PB , IC ) −−−−−−−→ from (2.6.2)

RHom• (A, Hom• (PB , IC )) ⏐ ⏐ Hom• (A, Hom• (PB , IC ))

Since PB is q-ﬂat, and IC and Hom• (PB , IC ) are q-injective, all these maps are isomorphisms (as follows, e.g., from the last assertion of (2.2.6)); so we can compose to get the isomorphism (2.6.1). But we really should check that this isomorphism does not depend on the chosen quasi-isomorphisms, and that it is in fact Δ-functorial. This can be quite tedious. The following remarks outline a method for managing such veriﬁcations. The basic point is (2.6.4) below. Let M be a set. An M-category is an additive category C plus a map t : M → Hom(C, C) from M into the set of additive functors from C to C, such that with Tm := t(m) it holds that Ti ◦Tj = Tj ◦Ti for all i, j ∈ M . Such an M-category will be denoted CM , the map f —or equivalently, the commuting family (Tm )m∈M —understood to have been speciﬁed; and when the context renders it superﬂuous, the subscript “M ” may be omitted. ′ An M-functor F : CM → CM is an additive functor F : C → C′ together with isomorphisms of functors ∼ θi : F ◦ Ti −→ Ti′ ◦ F

(i ∈ M )

2.6 Adjoint Associativity

67

′ (with (Tm )m∈M the commuting family of functors deﬁning the M-structure on C′ ) such that for all i = j, the following diagram commutes: T ′ (θj )

via θ

i F ◦Ti ◦Tj −−−−→ Ti′ ◦F ◦Tj −−i−−→ Ti′ ◦Tj′ ◦ F

F ◦Tj ◦Ti −−−−→ Tj′ ◦F ◦Ti −−−− −→ Tj′ ◦ Ti′ ◦ F ′ via θj

−Tj (θi )

where, for instance, Tj′ (θi ) is the isomorphism of functors such that for each object X ∈ C, [Tj′ (θi )](X) is the C′ -isomorphism ∼ Tj′ Ti′ F (X) . Tj′ θi (X) : Tj′ F Ti (X) −→

A morphism η : (F , {θi }) → (G, {ψi }) of M -functors is a morphism of functors η : F → G such that for every i ∈ M and every object X in C, the following diagram commutes: θi (X)

F Ti (X) −−−−→ Ti′ F (X) ⏐ ⏐ ⏐T ′ (η(X)) ⏐ η(Ti (X)) i GTi (X) −−−−→ Ti′ G(X) ψi (X)

Composition of such η being deﬁned in the obvious way, the M -functors from C to C′, and their morphisms, form a category H := HomM (C, C′ ). If ′ M ′ ⊃ M and CM ′ is viewed as an M-category via “restriction of scalars” then H is itself an M ′-category, with j ∈ M ′ being sent to the functor Tj# : H → H such that on objects of H, Tj# F , {θi } = Tj′ ◦ F , {−Tj′ (θi )} , where the isomorphism of functors

∼ Tj′ (θi ) : (Tj′ ◦F )◦Ti −→ Tj′ ◦ Ti′ ◦F = Ti′ ◦ (Tj′ ◦F )

is as above.4 The deﬁnition of Tj# η (η as above), and the veriﬁcation that H is thus an M ′-category, are straightforward. Suppose given such categories AM , BN , and CM ∪N , where the sets M and N are disjoint. A × B is considered to be an (M ∪ N )-category, with i ∈ M going to the functor Ti × 1 and j ∈ N to the functor 1 × Tj . Also, HomN (B, C) is considered, as above, to be an (M ∪ N )-category.

The reason for the minus sign in the deﬁnition of Tj# is hidden in the details of the proof of Lemma (2.6.3) below.

4

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Lemma 2.6.3. With preceding notation, there is a natural isomorphism of M ∪ N-categories ∼ HomM ∪N A × B, C −→ HomM A, HomN (B, C)

The proof, left to the reader, requires very little imagination, but a good deal of patience.

For any positive integer n, let △n be the set {1, 2, . . . , n}. From now on, we deal with Δ-categories, always considered to be △1 -categories via their translation functors. If C1 , . . . , Cn are Δ-categories, then the product category C = C1 × C2 × · · · × Cn becomes a △n -category by the product construction used in (2.6.3). A Δ-category E can also be made into an △n -category by sending each i ∈ △n to the translation functor of E. With these understandings, we see that the △n -functors from C1 × C2 × · · · × Cn to E are just the Δ-functors of (2.4.3) (categories of which we denote by Hom∆ ). For example, one checks that the source and target of the isomorphism in (2.6.1) are both △3 -functors. Now for 1 ≤ i ≤ n ﬁx abelian categories A i , and let Li be a Δ-subcategory of K(A i ), with corresponding derived category Di and canonical functor Qi : Li → Di . Let E be any Δ-category. We can generalize (1.5.1) as follows: Proposition 2.6.4. The canonical functor L1 × · · · × Ln −−−−−−−→ D1 × · · · × Dn Q1 ×...×Qn

induces an isomorphism from the category Hom∆ (D1 × D2 × · · · × Dn , E) to the full subcategory of Hom∆ (L1 × L2 × · · · × Ln , E) whose objects are the Δ-functors F such that for any quasi-isomorphisms α1 , . . . , αn in L1 , . . . , Ln respectively, F (α1 , . . . , αn ) is an isomorphism in E. Proof. The case n = 1 is contained in (1.5.1). We can then proceed by induction on n, using the natural isomorphism Hom△n C1 × C2 × · · · × Cn , E ∼ − → Hom△1 C1 , Hom△n−1 (C2 × . . . × Cn , E)

provided by (2.6.3) (with Ci := Di or Li ).

Q.E.D.

L′i

L′′i

Suppose next that we have pairs of Δ-subcategories ⊂ in K(A i ), with respective derived categories D′i , D′′i , and canonical functors Qi′ : L′i →D′i , Q′′i : L′′i → D′′i (1 ≤ i ≤ n). Suppose further that every complex A ∈ L′′i admits a quasi-isomorphism into a complex IA ∈ L′i . Then as in (1.7.2) the natural Δ-functors j˜i : D′i → D′′i are Δ-equivalences, having quasi-inverses ρi satisfying ρi (A) = IA (A ∈ L′′i ). There result functors

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69

j˜∗ : Hom∆ (D′′1 × · · · × D′′n , E) −→ Hom∆ (D′1 × · · · × D′n , E) ρ∗ : Hom∆ (D′1 × · · · × D′n , E) −→ Hom∆ (D′′1 × · · · × D′′n , E) together with functorial isomorphisms ∼ identity, j˜∗ρ∗ −→

∼ ρ∗j˜∗ −→ identity,

i.e., j˜∗ and ρ∗ are quasi-inverse equivalences of categories. We deduce the following variation on the theme of (2.2.3), thereby arriving at a general method for specifying maps between Δ-functors on products of derived categories:5 Corollary 2.6.5. With above notation let H : L′1 × · · · × L′n → E, F : D′′1 × · · · × D′′n → E, and G : D′′1 × · · · × D′′n → E be Δ-functors. Let ∼ ζ : H −→ F ◦ (˜ j1 Q′1 × . . . × j˜n Q′n ),

β : H −→ G ◦ (˜ j1 Q′1 × . . . × j˜n Q′n ) be Δ-functorial maps, with ζ an isomorphism. Then: (i) There exists a unique Δ-functorial map β¯ : F → G such that for all A1 ∈ L′1 , . . . , An ∈ L′n , β(A1 , . . . , An ) factors as ζ

β¯

H(A1 , . . . , An ) −−−−→ F (A1 , . . . , An ) −−−−→ G(A1 , . . . , An ). (2.6.5.1) ¯ Moreover, if β is an isomorphism then so is β. (ii) If H in (i) extends to a Δ-functor H : L′′1 × . . . × L′′n → E, and ζ (respectively β) to a Δ-functorial map ζ : H → F ◦ (˜ j1 Q′′1 × . . . × j˜n Q′′n ) ′′ ′′ (respectively β : H → G ◦ (˜ j1 Q1 ×. . .× j˜n Qn )), then the factorization (2.6.5.1) of β(A1 , . . . , An ) holds for all A1 ∈ L′′1 , . . . , An ∈ L′′n . Proof. (i) The assertion just means that β¯ is the unique map (resp. isomorphism) F → G in the category Hom∆ (D′′1 × . . . × D′′n , E) corresponding via the above equivalence j˜∗ and (2.6.4) to the map (resp. isomorphism) βζ −1 in the category Hom∆ (L′1 × . . . × L′n , E). (ii) Use quasi-isomorphisms Ai → IAi to map (2.6.5.1) into the corresponding diagram with IAi ∈ L′i in place of Ai . To this latter diagram (i) applies; and as the resulting map G(A1 , . . . , An ) → G(IA1 , . . . , IAn ) is an isomorphism, the rest is clear. Q.E.D.

5

This is no more (or less) than a careful formulation of the method used, e.g., throughout [H, Chapter II].

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We can now derive (2.6.1) as follows. Take n = 3, and set L′1 := K Δ-subcategory of K whose objects are L′2 := the q-ﬂat complexes (2.5.3). Δ-subcategory of K whose objects are L′3 := the q-injective complexes (2.3.2.2). Let D′1 , D′2 , D′3 be the corresponding derived categories, and set L′′i := K,

D′′i := D

(i = 1, 2, 3),

so that the natural maps ji : Di ′ → D′′i are Δ-equivalences, with quasiinverses obtained for i = 2 and i = 3 from q-ﬂat (resp. q-injective) resolutions, i.e., from families of quasi-isomorphisms PB → B

(B ∈ K, PB ∈ L′2 ),

C → IC

(C ∈ K, IC ∈ L′3 ).

In Corollary (2.6.5)(ii), let H : L′′1 × L′′2 × L′′3 → D be the Δ-functor H(A, B, C) := Hom• (A ⊗ B, C), let ζ be the natural composed Δ-functorial map Hom• (A ⊗ B, C) → RHom• (A ⊗ B, C) → RHom• (A ⊗ B, C), = and let β be the natural composed Δ-functorial map ∼ Hom• (A ⊗ B, C) −→ Hom• (A, Hom• (B, C)) (2.6.2)

−→ RHom• (A, Hom• (B, C)) −→ RHom• (A, RHom• (B, C)). (Meticulous readers may wish to insert the missing Q’s). We saw near the beginning of the proof of (2.6.1), that for (B, C) ∈ L′2 ×L′3 , the complex Hom• (B, C) is q-injective, and hence for such (B, C), ζ and β are isomorphisms. Modifying (2.6.5) in the obvious way to take contravariance into account, we deduce the following elaboration of (2.6.1): Proposition (2.6.1)* . There is a unique Δ-functorial isomorphism ∼ α : RHom• (A ⊗ B, C) −→ RHom• (A, RHom• (B, C)) =

2.7 Acyclic Objects; Finite-Dimensional Derived Functors

71

such that for all A, B, C ∈ D, the following natural diagram (in which H• stands for Hom• ) commutes: H• (A ⊗ B, C) ⏐ ⏐ via(2.6.2)

−−−−→

RH• (A ⊗ B, C)

−−−−→

RH• (A ⊗ B, C) ⏐= ⏐ ≃α

H• (A, H• (B, C)) −−−−→ RH• (A, H• (B, C)) −−−−→ RH• (A, RH• (B, C)) This Δ-functorial isomorphism is the same as the one described—noncanonically, via PB and IC —near the beginning of this section. See also exercise (3.5.3)(e) below. From (2.5.7.1) and (3.3.8) below (dualized), we deduce: Corollary 2.6.6. For ﬁxed A the Δ-functor FA (−) := Hom• (A, −) of §2.4 has a right-derived Δ-functor of the form (RFA , identity). Exercise 2.6.7 (see [De, §1.2]). Deﬁne derived functors of several variables, and generalize the relevant results from §§2.2–2.3.

2.7 Acyclic Objects; Finite-Dimensional Derived Functors This section contains additional results about acyclicity, used to get some more ways to construct derived functors, further illustrating (2.2.6). It can be skipped on ﬁrst reading. Let A, A′ be abelian categories, and let φ : A → A′ be an additive functor. We also denote by φ the composed Δ-functor K(φ)

Q

K(A) −−−−→ K(A′ ) −−−−→ D(A′ ) where K(φ) is the natural extension of the original φ to a Δ-functor. We say then that an object in A is right-(or left-)φ-acyclic if it is so when viewed as a complex vanishing outside degree zero (see (2.2.5) with J := K(A)). In this section we deal mainly with the “left” context, and so we abbreviate “leftφ-acyclic” to “φ-acyclic.” (The corresponding—dual—results in the “right” context are left to the reader. They are perhaps marginally less important because of the abundance of injectives in situations that we will deal with.) If X ∈ A and Z → X is a quasi-isomorphism in K(A), then the natural map τ≤0 Z → Z of §1.10 is a quasi-isomorphism. If furthermore the induced map φ(Z) → φ(X) is a quasi-isomorphism and the functor φ is either right exact or left exact, then, one checks, the natural composition φ(τ≤0 Z) → φ(Z) → φ(X) is also a quasi-isomorphism.

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One deduces the following characterization of φ-acyclicity: Lemma 2.7.1. If X ∈ A is such that every exact sequence · · · −−−−→ Y2 −−−−→ Y1 −−−−→ Y0 −−−−→ X −−−−→ 0 embeds into a commutative diagram in A · · · −−−−→ Z2 −−−−→ ⏐ ⏐

Z1 −−−−→ ⏐ ⏐

Z0 −−−−→ ⏐ ⏐

X −−−−→ 0

· · · −−−−→ Y2 −−−−→ Y1 −−−−→ Y0 −−−−→ X −−−−→ 0

with the top row and its image under φ both exact, then X is φ-acyclic; and the converse holds whenever φ is either right exact or left exact. Proposition 2.7.2. With preceding notation, let P be a class of objects in A such that (i) every object in A is a quotient of (i.e., target of an epimorphism from) one in P; (ii) if A and B are in P then so is A ⊕ B ; and (iii) for every exact sequence 0 → A → B → C → 0 in A, if B and C are in P, then A ∈ P and moreover the corresponding sequence 0 → φA → φB → φC → 0 in A′ is also exact. Then every bounded-above P-complex (i.e., complex with all components in P)—in particular every object in P—is φ-acyclic; the restriction φ− of φ to K−(A) has a left-derived functor Lφ− : D−(A) → D(A′ ); and if φ ∼

0 = then dim+ Lφ− = 0 (see (1.11.1)). Proof. Since P is nonempty—by (i)—therefore (iii) with B = C ∈ P shows that 0 ∈ P. Then (ii) implies that the P-complexes in K−(A) are the objects of a Δ-subcategory, see (1.6). Starting from (i), an inductive argument ([H, p. 42, 4.6, 1)], dualized—and with assistance, if desired, from [Iv, p. 34, Prop. 5.2]) shows that every complex in K−(A)—and so, via (1.8.1)−, in K−(A)—is the target of a quasi-isomorphism from a bounded-above Pcomplex. Hence, for the ﬁrst assertion it suﬃces to show that φ transforms quasi-isomorphisms between bounded-above P-complexes into isomorphisms, i.e., that for any bounded-above exact P-complex X • , φ(X • ) ∼ = 0 (see (1.5.1)). Using (iii), we ﬁnd by descending induction (starting with i0 such that X j = 0 for all j > i0 ) that for every i, the kernel K i of X i → X i+1 lies in P and the obvious sequence 0 → φ(K i ) → φ(X i ) → φ(K i+1 ) → 0 is exact. Consequently, the complex obtained by applying φ to X • is exact, i.e., φ(X • ) ∼ = 0 in D(A′ ).

2.7 Acyclic Objects; Finite-Dimensional Derived Functors

73

Now by (2.2.4) (dualized) we see that Lφ− exists and dim+ Lφ− ≤ 0, with equality if φ(A) ∼ = 0 for some A ∈ A, because there is a natural epimorphism Q.E.D. H 0 Lφ− A ։ φ(A). Exercise 2.7.2.1. Let φ : A → A′ be as above. Let (Λi )0≤i<∞ be a “homological functor” [Gr, p. 140], with Λ0 = φ. Let P consist of all objects B in A such that Λi (B) = 0 for all i > 0, and suppose that every object A ∈ A is a quotient of one in P. Then Lφ− exists, and the homological functors (Λi ) and (Λ′i ) := (H −i Lφ− ) are coeﬀaceable, hence universal [Gr, p. 141, Prop. 2.2], hence isomorphic to each other.

Example 2.7.3. A ringed space is a pair (X, OX ) with X a topological space and OX a sheaf of commutative rings on X; and a morphism of ringed spaces (f , θ) : (X, OX ) → (Y , OY ) is a continuous map f : X → Y together with a map θ : OY → f∗ OX of sheaves of rings. Any such (f , θ) gives rise to a (left-exact) direct image functor f∗ : {OX -modules} → {OY -modules} such that [f∗ M ](U ) = M (f −1 U ) for any OX -module M and any open set U ⊂ Y , the OY -module structure on f∗ M arising via θ; and also to a (rightexact) inverse image functor f ∗ : {OY -modules} → {OX -modules} deﬁned up to isomorphism as being left-adjoint to f∗ [GD, Chap. 0, §4]. For every OY -module N , the stalk (f ∗N )x at x ∈ X is OX,x ⊗OY ,f (x) Nf (x) . An OY -module F is ﬂat if the stalk Fy is a ﬂat OY,y -module for all y ∈ Y . The class P of ﬂat OY -modules satisﬁes the hypotheses of (2.7.2) when φ = f ∗ : (i) is given by [H, p. 86, Prop. 1.2], (ii) is easy, and for (iii) see [B′ , Chap. 1, §2, no. 5]. Thus the restriction f−∗ of f ∗ to K−(Y ) has a leftderived functor Lf−∗ : D−(Y ) → D(X) (where D(X) is the derived category of the category of OX -modules, etc.), deﬁned via resolutions (on the left) by complexes of ﬂat OY -modules. Using the family of quasi-isomorphisms ψA : PA → A (A ∈ D(Y )) with PA q-ﬂat (see (2.5.5)), we can, in view of (2.5.2) and (2.5.3), show as in (2.5.7) that Lf−∗ extends to a derived Δ-functor (Lf ∗, identity) : D(Y ) → D(X)

(2.7.3.1)

satisfying Lf ∗ (A) = f ∗ (PA ). For any OY -module N , the stalk of the homology Li f ∗ (N ) := H −i Lf ∗ (N ) O

(i ≥ 0)

at any x ∈ X is Tori Y ,f (x) (OX,x , Nf (x) ). So by the last assertion in (2.2.6) O (dualized), or in (2.7.4), N is f ∗-acyclic iﬀ Tori Y ,f (x) (OX,x , Nf (x) ) = 0

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for all x ∈ X and i > 0. (Note here that since f ∗ is right exact, the natural ∼ f ∗ (N ).) Thus—or by (2.7.2)—any ﬂat map is an isomorphism L0 f ∗ (N ) −→ OY -module is f ∗-acyclic. Recall that an OX -module M is ﬂasque (or ﬂabby) if the restriction map M (X) → M (U ) is surjective for every open subset U of X. For example, injective OX -modules are ﬂasque [G, p. 264, 7.3.2] (with L = OX ). The class of ﬂasque OX -modules satisﬁes the hypotheses of (2.7.2) (dual version) when φ = f∗ : for (i) see [G, p. 147], (ii) is easy, and (iii) follows from the fact that if 0→F →G→H→0 is an exact sequence of OX -modules, with F ﬂasque, then for all open sets V ⊂ X the sequence 0 → F (V ) → G(V ) → H(V ) → 0 is still exact [G, p. 148, Thm. 3.1.2]. So the restriction f∗+ of f ∗ to K+(X) has a right-derived functor Rf + : D+(X) → D(Y ) ∗

deﬁned via resolutions (on the right) by complexes of ﬂasque OX -modules. Of course we already know from (2.3.4), via (somewhat less elementary) injective resolutions, that Rf∗+ exists, and by (2.3.5) it extends to a derived functor Rf∗ : D(X) → D(Y ). (See also (2.3.7).) In fact, in view of (2.7.3.1), it follows from (3.2.1) and (3.3.8) (dualized) that: (2.7.3.2). The Δ-functor (f∗ , identity) has a derived Δ-functor of the form (Rf∗ , identity). An OX -module M is f∗ -acyclic iﬀ the “higher direct image” sheaves Rif∗ (M ) := H i Rf∗ (M )

(i ≥ 0)

vanish for all i > 0, see last assertion in (2.2.6) or in (2.7.4) (dualized). ∼ (Since f∗ is left-exact, the natural map is an isomorphism f∗ −→ R0f∗ .) Flasque sheaves are f∗ -acyclic. For more examples involving ﬂasque sheaves see [H, p. 225, Variations 6 and 7] (“cohomology with supports”). Proposition 2.7.4. Let A and A′ be abelian categories, and let φ : A → A′ be a right-exact additive functor. If C is φ-acyclic, then for every exact sequence 0 → A → B → C → 0 in A the corresponding sequence 0 → φA → φB → φC → 0 is also exact, and A is φ-acyclic iﬀ B is. So if every object in A is a quotient of a φ-acyclic one, then the conclusions of (2.7.2) hold with P the class of φ-acyclic objects; and then D ∈ A is φ-acyclic iﬀ the natural map Lφ− (D) → φ(D) is an isomorphism in D(A′ ), i.e., iﬀ H −i Lφ− (D) = 0 for all i > 0.

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75

Proof. For the ﬁrst assertion, note that by (2.7.1) there exists a commutative diagram δ

C2 −−−−→ ⏐ ⏐

γ

C1 −−−−→ ⏐ ⏐

C0 −−−−→ ⏐ ⏐β

C −−−−→ 0

0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0 α

such that the top row is exact and remains so after application of φ. There results a commutative diagram C2 ⏐ ⏐ δ

0 −−−−→ 0 −−−−→ ⏐ ⏐

C1 ⏐ ⏐ γ′

A −−−−→ ⏐ ⏐

B ⏐ ⏐

C2 ⏐ ⏐ δ

0 −−−−→ A −−−−→ C0 ×C B −−−−→ π ⏐ ⏐ 0

0

C1 −−−−→ 0 ⏐ ⏐γ

C0 −−−−→ 0 ⏐ ⏐

−−−−→ C −−−−→ 0 ⏐ ⏐ 0

with exact columns, in which the middle row is split exact, a right inverse for the projection π being given by the graph of the map β.6 (The coordinates of γ ′ are γ and 0.) Applying φ preserves split-exactness; and then, since φ is right-exact, so that e.g., φC = coker(φγ), the “snake lemma” yields an exact sequence 0 → ker(φγ ′ ) → ker(φγ) → φA → φB → φC → 0 . Since ker(φγ) = im(φδ) ⊂ ker(φγ ′ ) we conclude that 0 → φA → φB → φC → 0 is exact, as asserted in (2.7.4). In other words, if Z is the complex which looks like A → B in degrees −1 and 0 and which vanishes elsewhere, then the quasi-isomorphism Z → C given by the exact sequence 0 → A → B → C → 0 becomes, upon application of φ, an isomorphism in D(A′ ); and hence, by (2.2.5.2) (dualized), Z is a φ-acyclic complex. 6

Recall that C0 ×C B is the kernel of the map C0 ⊕ B → C whose restriction to C0 is αβ and to B is −α.

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The natural semi-split sequence 0 → B → Z → A[1] → 0 leads, as in (1.4.3), to a triangle B −→ Z −→ A[1] −→ B[1] ; and since the φ-acyclic complexes are the objects of a Δ-subcategory, see (2.2.5.1), it follows that A is φ-acyclic iﬀ B is. Since Δ-subcategories are closed under direct sum, it is clear now that (ii) and (iii) in (2.7.2) hold when P is the class of φ-acyclic objects, whence the second-last assertion in (2.7.4). In view of (2.7.2) and its proof, the last assertion of (2.7.4) is contained in (2.2.6). Q.E.D. The derived functor Lφ− of (2.7.4) satisﬁes dim+ Lφ− = 0 (unless φ ∼ = 0, see (2.7.2)). When its lower dimension satisﬁes dim− Lφ− < ∞, more can be said. Proposition 2.7.5. Let φ : A → A′ be a right-exact functor such that every object in A is a quotient of a φ-acyclic one, and let Lφ− be a left-derived functor of φ| K−(A), see (2.7.4). Then the following conditions on an integer d ≥ 0 are equivalent: (i) dim− Lφ− ≤ d. (ii) For any F ∈ A we have Lj φ(F ) := H −j Lφ− (F ) = 0

for all j > d.

(iii) In any exact sequence in A 0 → 0 → Bd → Bd−1 → . . . → B0 , if B0 , B1 , . . . , Bd−1 are all φ-acyclic then so is Bd .7 (iv) For any F ∈ A there is an exact sequence 0 → Bd → Bd−1 → . . . → B0 → F → 0 in which every Bi is φ-acyclic. (v) For any complex F • ∈ K(A) and integers m ≤ n, if F j = 0 for all j ∈ / [m, n] then there exists a quasi-isomorphism B • → F • where j B is φ-acyclic for all j and B j = 0 for j ∈ / [m − d, n]. (vi) For any complex F • ∈ K(A) and integer m, if F j = 0 for all j < m then there exists a quasi-isomorphism B • → F • where B j is φ-acyclic for all j and B j = 0 for all j < m − d.

7

For d = 0 this means that every B ∈ A is φ-acyclic, i.e., φ is an exact functor, see (2.7.4) (and then every F • ∈ K(A) is φ-acyclic, see (2.2.8(a)).

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77

When there exists an integer d ≥ 0 for which these conditions hold, then: (a) Every complex of φ-acyclic objects is a φ-acyclic complex. (b) Every complex in A is the target of a quasi-isomorphism from a φ-acyclic complex. (c) A left-derived functor Lφ : D(A) → D(A′ ) exists, dim+ Lφ = 0 (unless φ ∼ = 0) and dim− Lφ ≤ d. (d) The restriction Lφ| D*(A) is a left-derived functor of φ| K*(A) , and Lφ(D*(A)) ⊂ D*(A′ )

(∗ = +, −, or b).

Proof. (i)⇔(ii). This is given by (iii) and (iv) in (1.11.2). (iii)⇒(v)⇒(iv). Let F • and m ≤ n be as in (v). As in the proof of (2.7.2), there is a quasi-isomorphism P • → F • with P j φ-acyclic for all j and P j = 0 for j > n. Let B m−d be the cokernel of P m−d−1 → P m−d . If (iii) holds, then B m−d is φ-acyclic: this is trivial if d = 0, and otherwise follows from the exact sequence 0 → B m−d → P m−d+1 → · · · → P m−1 → P m . So all components of the complex B • = τ≥m−d P • (see (1.10)) are φ-acyclic, and clearly P • → F • factors naturally as P • → B • → F • = τ≥m−d F • where both arrows represent quasi-isomorphisms. Thus (iii)⇒(v); and (v)⇒(iv) is obvious. Recalling from (2.7.4) that B ∈ A is φ-acyclic iﬀ Li φ(B) = 0 for all i > 0, we easily deduce the implications (iv)⇒(ii)⇒(iii) from: Lemma 2.7.5.1. Let 0 = Bd+1 → Bd → Bd−1 → · · · → B0 → F → 0 be an exact sequence in A with B0 , B1 , . . . , Bd−1 all φ-acyclic, and let Kj be the cokernel of Bj+1 → Bj (0 ≤ j ≤ d). Then for any i > 0, there results a natural sequence of isomorphisms ∼ ∼ Li+d−1 φ(K1 ) −→ ··· Li+d φ(F ) = Li+d φ(K0 ) −→ ∼ ∼ ∼ · · · −→ Li+2 φ(Kd−2 ) −→ Li+1 φ(Kd−1 ) −→ Li φ(Kd ) = Li φ(Bd ) .

Proof. When d = 0, it’s obvious. If d > 0, apply (2.1.4)H (dualized) to the natural exact sequences 0 → Kj → Bj−1 → Kj−1 → 0

(0 < j ≤ d)

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to obtain exact sequences 0 = Li+d−j+1 φ(Bj−1 ) → Li+d−j+1 φ(Kj−1 ) → Li+d−j φ(Kj ) → Li+d−j φ(Bj−1 ) = 0.

Q.E.D.

(iii)⇒(vi). Condition (iii) coincides with condition (iii) of [H, p. 42, Lemma 4.6, 2)] (dualized, and with P the set of φ-acyclics in A). Condition (i) of loc. cit. holds by assumption, and condition (ii) of loc. cit. is contained in (2.7.4). So if (iii) holds, loc. cit. gives the existence of a quasi-isomorphism B • → F • with B j φ-acyclic for all j; and the recipe at the bottom of [H, p. 43] for constructing B • allows us, when F j = 0 for all j < m, to do so in such a way that B j = 0 for all j < m − d. (vi)⇒(ii). Assuming (vi), we can ﬁnd for each object F ∈ A a quasiisomorphism B • → F with all B j φ-acyclic and B j = 0 for j < −d. If K is the cokernel of B −1 → B 0 then the natural composition H 0 (B • ) −→ K −→ F is an isomorphism, whence so are the functorially induced compositions Lj φ(H 0 (B • )) −→ Lj φ(K) −→ Lj φ(F )

(j ∈ Z).

(2.7.5.2)

But for every j > d, (2.7.5.1) with K in place of F yields Lj φ(K) = 0, so that the isomorphism (2.7.5.2) is the zero-map. Thus (ii) holds. Now suppose that (i)–(vi) hold for some d ≥ 0. We have just seen, in proving that (iii)⇒(vi), that then every complex in A receives a quasiisomorphism from a complex B • of φ-acyclics; and so, as in the proof of (2.7.2), assertion (2.7.5)(a)—and hence (b)—will result if we can show that whenever such a B • is exact, then so is φ(B • ). But condition (iii) guarantees that when B • is exact, the kernel K i of B i → B i+1 is φ-acyclic for all i, whence by (2.7.4) we have exact sequences 0 → φ(K i−1 ) → φ(B i−1 ) → φ(K i ) → 0

(i ∈ Z)

which together show that φ(B • ) is indeed exact. The existence of Lφ, via resolutions by complexes of φ-acyclic objects, follows now from (2.2.6); and the dimension statements follow, after application of (1.8.1)+ or (1.8.1)− , from (v) with m = −∞ (obvious interpretation, see beginning of above proof that (iii)⇒(v)) and from (vi). Similar considerations yield (d). Q.E.D. Example 2.7.6. The dimension dim f of a map f : X → Y of ringed spaces is the upper dimension (1.11) of the functor Rf∗+ : D+(X) → D(Y ) of (2.7.3): dim f := dim+ Rf∗+ ,

2.7 Acyclic Objects; Finite-Dimensional Derived Functors

79

a nonnegative integer unless f∗ OX ∼ = 0, in which case dim f = −∞. When f has ﬁnite dimension, (2.7.5)(c) (dualized) gives the existence of a derived functor Rf∗ : D(X) → D(Y ) via resolutions (on the right) by complexes of f∗ -acyclic objects, and we have ∞ > dim f = dim+ Rf∗ . The tor-dimension (or ﬂat dimension) tor-dim f of a map f : X → Y of ringed spaces is deﬁned to be the lower dimension (see (1.11)) of the functor Lf−∗ : D−(Y ) → D(X) of (2.7.3): tor-dim f := dim− Lf−∗ , a nonnegative integer unless OX ∼ = 0, in which case tor-dim f = −∞. When f has ﬁnite tor-dimension, (2.7.5)(c) gives the existence of a derived functor Lf ∗ : D(X) → D(Y ) via resolutions (on the left) by complexes of f ∗-acyclic objects, and we have ∞ > tor-dim f = dim− Lf ∗ . Following [I, p. 241, D´eﬁnition 3.1] one says that an OX -complex E has ﬂat f-amplitude in [m, n] if for any OY -module F , H i (E ⊗ Lf ∗ F ) = 0 for all i ∈ / [m, n], = or equivalently, for the functor LE (F ) := E ⊗ Lf ∗ F of OY -module F , = dim+ L ≤ m and dim− L ≤ −n. This means that the stalk Ex at each x ∈ X is D(OY ,f (x))-isomorphic to a ﬂat complex vanishing in degrees outside [m, n], see [I, p. 242, 3.3], or argue as in (2.7.6.4) below. E has ﬁnite ﬂat f-amplitude if such m and n exist. It follows from (2.7.6.4) below and [I, p. 131, 5.1] that f has ﬁnite tordimension ⇐⇒ OX has ﬁnite ﬂat f-amplitude. (2.7.6.1). If X is a compact Hausdorﬀ space of dimension ≤ d (in the sense that each point has a neighborhood homeomorphic to a locally closed subspace of the Euclidean space Rd ), and OX is the constant sheaf Z, then dim f ≤ d. Indeed, if I • is a ﬂasque resolution of the abelian sheaf F , then for any open U ⊂ Y the restriction I • |f −1 (U ) is a ﬂasque resolution of F |f −1 (U ), and Rj f∗ (F ) is, up to isomorphism, the sheaf associated to the presheaf taking any such U to the group H j (Γ(f −1 (U ), I • |f −1 (U )) , a group isomorphic to H j (f −1 (U ), F |f −1 (U )) [G, p. 181, Thm. 4.7.1(a)], and hence vanishing for j > d, see [Iv, Chap. III, §9]. More generally, if X is locally compact and we assume only that the ﬁbers f −1 y (y ∈ Y ) are compact and have dimension ≤ d, then dim f ≤ d (because the stalk (Rj f∗ F )y is the cohomology H j (f −1 y, F |f −1 y), see [Iv, p. 315, Thm. 1.4], whose proof does not require any assumption on Y ). (2.7.6.2). (Grothendieck, see [H, p. 87]). If (X, OX ) is a noetherian scheme of ﬁnite Krull dimension d, then dim f ≤ d.

80

2 Derived Functors

(2.7.6.3). For a ringed-space map f : X → Y with OX ≇ 0, the following conditions are equivalent: (i) tor-dim f = 0. (i)′ Every OY -module is f ∗-acyclic. (i)′′ The functor f ∗ of OY -modules is exact. (ii) f is ﬂat (i.e., OX,x is a ﬂat OY ,f (x) -module for all x ∈ X). Proof. Since every OX -module is a quotient of a ﬂat one, which is f ∗ -acyclic (see (2.7.3)), the equivalence of (i), (i)′ , and (i)′′ is given, e.g., by that of (i) and (iii) in (2.7.5) (for d = 0). The equivalence of (i) and (ii) is the case d = 0 of: (2.7.6.4) Let f : X → Y be a ringed-space map and d ≥ 0 an integer. Then tor-dim f ≤ d ⇐⇒ for each x ∈ X there exists an exact sequence of OY ,f (x) modules 0 → Pd → Pd−1 → . . . → P 1 → P 0 → OX,x → 0

(*)

with Pi ﬂat over OY ,f (x) (0 ≤ i ≤ d). Proof. (“if ”) Let F be an OY -module and let Q• → F be a quasi-isomorphism with Q• a ﬂat complex (1.8.3). Then for j ≥ 0, the homology Lj f ∗ (F ) ∼ = H −j (f ∗ Q• )

(see (2.7.3))

vanishes iﬀ for each x ∈ X, with y = f (x), R = OY ,y , and S = OX,x we have R 0 = H −j (f ∗ Q• )x = H −j S ⊗R Q• y = Torj (S, Fy )

(where the last equality holds since Q• y → Fy is an R-ﬂat resolution of Fy ), whence the assertion. (“only if ”) Suppose only that Ld+1 f ∗ (F ) = 0 for all F , so that (see above) TorR d+1 (S, Fy ) = 0; and let · · · → P2′ → P1′ → P0′ → S → 0 be an R-ﬂat resolution of S. Then, I claim, the module ′ Pd := coker(Pd+1 → Pd′ )

is R-flat, whence we have (∗) with Pi = Pi′ for 0 ≤ i < d. Indeed, the ﬂatness of Pd is equivalent to the vanishing of TorR 1 (Pd , R/I) for all R-ideals I [B′ , §4, Prop. 1]. But any such I is Iy where I ⊂ OY is the OY -ideal such that for any open U ⊂ Y , I(U ) = { r ∈ OY (U ) | ry ∈ I }

=0

if y ∈ U if y ∈ / U;

so that if F = OY /I, then R/I = Fy ; and from the ﬂat resolution ′ ′ · · · → Pd+2 → Pd+1 → Pd′ → Pd → 0

of Pd , we get the desired vanishing: R R TorR 1 (Pd , R/I) = Tor1 (Pd , Fy ) = Tord+1 (S, Fy ) = 0.

2.7 Acyclic Objects; Finite-Dimensional Derived Functors

81

Exercise 2.7.6.5. (For amusement only.) If Y is a quasi-separated scheme, then f : X → Y satisﬁes tor-dim f ≤ d if (and only if) for every quasi-coherent OY -ideal I, we have Ld+1 f ∗ (OY /I) = 0. If in addition Y is quasi-compact or locally noetherian, then we need only consider finite-type quasi-coherent OY -ideals. [The following facts in [GD] can be of use here: p. 111, (5.2.8); p. 313, (6.7.1); p. 294, (6.1.9) (i); p. 295, (6.1.10)(iii); p. 318, (6.9.7).]

Chapter 3

Derived Direct and Inverse Image

A ringed space is a pair (X, OX ) with X a topological space and OX a sheaf of commutative rings on X; and a morphism (or map) of ringed spaces (f , θ) : (X, OX ) → (Y , OY ) is a continuous map f : X → Y together with a map θ : OY → f∗ OX of sheaves of rings. (Usually we will just denote such a morphism by f : X → Y , the accompanying θ understood to be standing by.) Associated with (f , θ) are the adjoint functors f∗

← {OY -modules} =: AY AX := {OX -modules} → f∗

and their respective derived functors Rf∗ , Lf ∗ , which are also adjoint—as Δ-functors, (3.2), (3.3). In this chapter we ﬁrst review the deﬁnitions and basic formal (i.e., category-theoretic) properties of these adjoint derived and RHom• , and their “pseudofunctorial” functors, their interactions with ⊗ = behavior with respect to composition of ringed-space maps (3.6), many of the main results being packaged in (3.6.10). A basic objective, in the spirit of Grothendieck’s philosophy of the “six operations,” is the categorical formalization of relations among functorial and RHom• .1 maps involving the four operations Rf∗ , Lf ∗, ⊗ = More explicitly (details in §§3.4, 3.5), if f : X → Y is a map of ringed spaces, then the derived categories D(AX ), D(AY ) have natural structures of symmetric monoidal closed categories, given by ⊗ and RHom• ; and the = ∗ adjoint Δ-functors Rf∗ and Lf respect these structures, as do the conjugate ∼ Rg∗ Rf∗ , isomorphisms, arising from a second map g : Y → Z, R(gf )∗ −→ ∗ ∗ ∼ ∗ Lf Lg −→ L(gf ) . We express all this by saying that R−∗ and L−∗ are adjoint monoidal Δ-pseudofunctors. Thus, relations among the four operations can be worked with as instances of category-theoretic relations involving adjoint monoidal functors between 1 A ﬁfth operation, “twisted inverse image,” is brought into play in Chapter 4, at least for schemes. The sixth, “direct image with proper supports” [De′, no3] will not appear here, except for proper scheme-maps, where it coincides with derived direct image.

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

83

84

3 Derived Direct and Inverse Image

closed categories. This eliminates excess baggage of resolutions of complexes, which would otherwise cause intolerable tedium later on, where proofs of major results depend heavily on involved manipulations of such relations.2 Even so, the situation is far from ideal—see the introductory remarks in §3.4, and, for example, the proof of Proposition (3.7.3), which addresses the interaction between the projection morphisms of (3.4.6) and “base change.” By way of illustration, consider the following basic functorial maps, with A, B ∈ D(AY ) and E, F ∈ D(AX ):3 Rf∗ RHom•X (Lf ∗B, E) → RHom•Y (B, Rf∗ E) , ∗

∗

∗

(3.2.3.2)

Lf A ⊗ Lf B ← Lf (A ⊗ B) , = =

(3.2.4)

Rf∗ (F ) → Rf∗ (E ⊗ F), Rf∗ (E) ⊗ = =

(3.2.4.2)

B → Rf∗ (E ⊗ Lf ∗B) . Rf∗ E ⊗ = =

(3.4.6)

The ﬁrst two can be deﬁned at the level of complexes, after replacing the arguments by appropriate resolutions. (The reduction is straightforward for the second, but not quite so for the ﬁrst.) At that level, one sees that they are both isomorphisms. For ﬁxed B, the source and target of the ﬁrst are leftadjoint, respectively, to the target and source of the second; and it turns out that the two maps are conjugate (3.3.5). This is shown by reduction to the analogous statement for the ordinary direct and inverse image functors for sheaves, which can be treated concretely (3.1.10) or formally (3.5.5). So each one of these isomorphisms determines the other from a purely categorical point of view. The second and third maps determine each other via Lf ∗ –Rf∗ adjunction (3.4.5), as do the third and fourth (3.4.6). When the ﬁrst map is given, the adjunction. second and third maps also determine each other via RHom• –⊗ = (This is not obvious, see Proposition (3.2.4).) Thus, any three of the four maps can be deduced category-theoretically from the remaining one. In (3.9) we consider the case when our ringed spaces are schemes. Under mild assumptions, we note that then Rf∗ and Lf ∗ “respect quasi-coherence” (3.9.1), (3.9.2). We also show that some previously introduced functorial morphisms become isomorphisms: (3.9.4) treats variants of the projection morphisms, while (3.9.5) signiﬁes that Rf∗ behaves well—even for unbounded complexes—with respect to ﬂat base change.4 More generally, in (3.10) we see that such good behavior of Rf∗ characterizes tor-independent base changes, 2 Cf. in this vein Hartshorne’s remarks on “compatibilities” [H, pp. 117–119]. Note however that the formalization became fully feasible only after Spaltenstein’s extension of the theory of derived functors in [H] to unbounded complexes [Sp]. 3 The ﬁrst is a sheaﬁﬁed version of Lf ∗ –Rf∗ adjunction (3.2.5)(f), the second and third underly monoidality of Lf ∗ and Rf∗ , and the fourth is “projection.” 4 Cf. [I, III, 3.7 and IV, 3.1].

3.1 Preliminaries

85

as does a certain K¨ unneth map’s being an isomorphism; the precise statement is given in (3.10.3), a culminating result for the chapter.

3.1 Preliminaries For any ringed space (X, OX ), let AX be the category of (sheaves of) OX modules—which is abelian, see e.g., [G, Chap. II, §2.2, §2.4, and §2.6], C(X) the category of AX -complexes, K(X) the category of AX -complexes with homotopy equivalence classes of maps of complexes as morphisms, and D(X) the derived category gotten by “localizing” K(X) with respect to quasiisomorphisms (see §§(1.1), (1.2)). To any ringed-space map (f , θ) : (X, OX ) → (Y , OY ) one can associate the additive direct image functor f∗ : AX → AY such that [f∗ M ](U ) = M (f −1 U ) for any OX -module M and any open set U ⊂ Y , the OY -module structure on f∗ M arising via θ; and also an inverse image functor f ∗ : AY → AX deﬁned up to isomorphism as a left-adjoint of f∗ , see [GD, p. 100, (4.4.3.1)] (where Ψ∗ (F) should be Ψ∗ (F)). Such an adjoint exists with, e.g., f ∗A := f −1 A ⊗f −1 OY OX

(A ∈ AY )

where f −1 A is the sheaf associated to the presheaf taking an open V ⊂ X to lim A(U ) with U running through all the open neighborhoods of f (V ) in Y . −→ In particular, if X is an open subset of Y , OX is the restriction of OY , f is the inclusion, and θ is the obvious map, then the functor “restriction to X” is left-adjoint to f∗ , so it is the natural choice for f ∗ . Being adjoint to an additive functor, f ∗ is also additive.5 From adjointness, or directly, one sees that f∗ is left-exact and f ∗ is right-exact. (The stalk (f ∗ N )x at x ∈ X is functorially isomorphic to OX,x ⊗OY ,f (x) Nf (x) .) Derived functors (see (2.1.1) and its complement) Rf∗ : D(X) → D(Y ),

Lf ∗ : D(Y ) → D(X)

can be constructed by means of q-injective and q-ﬂat resolutions, respectively, as follows. α

Additivity of f ∗ means that for any two maps A→ B in AY and any E ∈ AX , the → β ∗ sum of the induced maps Hom(f ∗B, E)→ →Hom(f A, E) is the map induced by α + β, a condition which follows from the additivity of f∗ via the adjunction isomorphisms (of abelian groups) Hom(f ∗ −, E) → Hom(f∗ f ∗ −, f∗ E) → Hom(−, f∗ E) .

5

86

3 Derived Direct and Inverse Image

Assume chosen once and for all, for each ringed space X, two families of quasi-isomorphisms A → IA ,

PA → A

(A ∈ K(X))

(3.1.1)

with each IA a q-injective complex and each PA q-ﬂat, see (2.3.5), (2.5.5), with A → IA the identity map when A is itself q-injective, and PA → A the identity when A is q-ﬂat. Then set Rf∗ (B) := f∗ (IB ) B ∈ D(X) , (3.1.2)

and for a map α in D(X) deﬁne Rf∗ (α) as indicated in (2.3.2.3) (with J := K(X)). The Δ-structure on Rf∗ is speciﬁed at the end of (2.2.4). Similar considerations apply to Lf ∗ , once one veriﬁes that f ∗ takes exact q-ﬂat complexes to exact complexes (for which argue as in (2.5.7), keeping in mind (2.5.2)). Proceeding as in (2.2.4) (dualized, with J′ ⊂ K(Y ) the Δ-subcategory whose objects are the q-ﬂat complexes, and J′′ := K(Y )), set Lf ∗ (A) := f ∗ (PA ) A ∈ D(Y ) , (3.1.3) etc. [See also (2.7.3).] Proposition (3.2.1) below says in particular that these derived functors are also adjoint. Before getting into that we review some elementary functorial sheaf maps, and their interconnections. For OX -modules E and F , there is a natural map of OY -modules φE,F : f∗ HomX (E, F ) → HomY (f∗ E, f∗ F )

(3.1.4)

taking a section of f∗ HomX (E, F ) over an open subset U of Y —i.e., a map α : E|f −1 U → F |f −1 U —to the section αφ of HomY (f∗ E, f∗ F ) given by the family of maps αφ (V ) : (f∗ E)(V ) → (f∗ F )(V ) (V open ⊂ U ) with αφ (V ) := α(f −1 V ) : E(f −1 V ) → F (f −1 V ) . Here is another description of φE,F (U ): given the commutative diagram j

f −1 U −−−−→ ⏐ ⏐ g U

X ⏐ ⏐f

−−−−→ Y i

where i and j are inclusions and g is the restriction f |f −1 U , and recalling that i∗ and j ∗ are restriction functors, one veriﬁes the functorial equalities f∗ j∗ j ∗ = i∗ g∗ j ∗ = i∗ i∗f∗

3.1 Preliminaries

87

and checks then that φE,F (U ) is the natural composition f∗ HomX (E, F )(U )

def

Hom(j ∗ E, j ∗ F )

∼ Hom(E, j∗ j ∗ F ) −→

−→ Hom(f∗ E, f∗ j∗ j ∗ F ) Hom(f∗ E, i∗ i∗ f∗ F ) ∼ −→ Hom(i∗ f∗ E, i∗ f∗ F )

def

HomY (f∗ E, f∗ F )(U ) .

Lemma 3.1.5. Let f : X → Y be a ringed-space map, A ∈ AY , B ∈ AX , φ := φf ∗A,B (see (3.1.4)). Let ηA : A → f∗ f ∗A be the map corresponding by adjunction to the identity map of f ∗A. Then the composition φ

via η

→ HomY (A, f∗ B) f∗ HomX (f ∗A, B) −−−−→ HomY (f∗ f ∗A, f∗ B) −−−−A is an isomorphism of additive bifunctors. Proof. The preceding description of φ identiﬁes (up to isomorphism) the sections over an open U ⊂ Y of the composite map in (3.1.5) with the natural composition via η

Hom(f ∗A, j∗ j ∗ B) −→ Hom(f∗ f ∗A, f∗ j∗ j ∗ B) −−−−A → Hom(A, f∗ j∗ j ∗ B) which is, by adjointness of f ∗ and f∗ , an isomorphism. Additive bifunctoriality of this isomorphism is easily veriﬁed. Q.E.D. (3.1.6). We leave it to the reader to elaborate the foregoing to get isomorphisms of complexes, functorial in A• ∈ C(Y ), B • ∈ C(X), ∼ Hom•Y (A• , f∗ B • ) , Hom•X (f ∗A• , B • ) −→ ∼ Hom•Y (A• , f∗ B • ) . f∗ Hom•X (f ∗A• , B • ) −→

(See (1.5.3) and (2.4.5) for the deﬁnitions of Hom• and Hom• .) Ditto for the maps in (3.1.7)–(3.1.9) below. For any two OX -modules E, F , the tensor product E ⊗X F is by deﬁnition the sheaf associated to the presheaf U → E(U ) ⊗OX (U ) F (U ) (U open ⊂ X), so there exist canonical maps E(U ) ⊗OX (U ) F (U ) → (E ⊗X F )(U ) from which, taking U = f −1 V (V open ⊂ Y ), one gets a canonical map f∗ E ⊗Y f∗ F → f∗ (E ⊗X F ) .

(3.1.7)

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3 Derived Direct and Inverse Image

(3.1.8). We will abbreviate by omitting the subscripts attached to ⊗, and by writing HZ (−, −) for HomOZ (−, −). The maps (3.1.4) and (3.1.7) are related via Hom-⊗ adjunction (2.6.2) as follows. After taking global sections of (2.6.2) (with F , G replaced by E, F respectively) one ﬁnds, corresponding to the identity map of E ⊗ F , a canonical map E → HX (F , E ⊗ F ) .

(3.1.8.1)

Similarly, corresponding to the identity map of HX (E, F ) one has a map HX (E, F ) ⊗ E → F .

(3.1.8.2)

Veriﬁcation of the following two assertions is left to the reader. —The map (3.1.7) is Hom-⊗ adjoint to the composition (3.1.8.1)

(3.1.4)

f∗ E −−−−−→ f∗ HX (F , E ⊗ F ) −−−−→ HY (f∗ F , f∗ (E ⊗ F )) . —The map (3.1.4) is Hom-⊗ adjoint to the composition (3.1.7)

(3.1.8.2)

f∗ HX (E, F ) ⊗ f∗ E −−−−→ f∗ (HX (E, F ) ⊗ E) −−−−−→ f∗ F . (3.1.9) Deﬁne the functorial map α

f ∗ (A ⊗ B) − → f ∗A ⊗ f ∗B

(A, B ∈ AY )

to be the adjoint of the composition natural

(3.1.7)

A ⊗ B −−−−→ f∗ f ∗A ⊗ f∗ f ∗B −−−−→ f∗ (f ∗A ⊗ f ∗B). Let x ∈ X, y = f (x), so that f induces a map of local rings Oy → OX , where OX is the stalk OX,x , and similarly for Oy . One checks that the stalk map αx is just the natural map → (Ay ⊗Oy OX ) ⊗OX (By ⊗Oy OX ) , (Ay ⊗Oy By ) ⊗Oy OX − whence α coincides with the standard isomorphism deﬁned, e.g., in [GD, p. 97, (4.3.3.1)]. Exercise 3.1.10. Show that the source and target of the map α in (3.1.9) are, as functors in the variable A, left-adjoint to the target and source (respectively) of the composed isomorphism—call it β—in (3.1.5), considered as functors in B; and that α and β are conjugate, see (3.3.5). (See also (3.5.5).) Work out the analog for complexes.

3.2 Adjointness of Derived Direct and Inverse Image

89

3.2 Adjointness of Derived Direct and Inverse Image We begin with a direct proof of adjointness of the derived direct and inverse image functors Rf∗ and Lf ∗ associated to a ringed-space map f : X → Y .6 A more elaborate localized formulation is given in (3.2.3). Proposition (3.2.4) introduces the basic maps connecting Rf∗ and Lf ∗ to ⊗ . It includes derived= category versions of part of (3.1.8) and of (3.1.10), as an illustration of the basic strategy for understanding relations among maps of derived functors through purely formal considerations (see 3.5.4). Proposition 3.2.1. For any ringed-space map f : X → Y , there is a natural bifunctorial isomorphism, ∼ HomD(Y ) (A, Rf∗ B) HomD(X) (Lf ∗A, B) −→ A ∈ D(Y ), B ∈ D(X) .

Proof. There is a simple equivalence between giving the adjunction isomorphism (3.2.1) and giving functorial morphisms η : 1 → Rf∗ Lf ∗ ,

ǫ : Lf ∗ Rf∗ → 1

(3.2.1.0)

(1 := identity) such that the corresponding compositions via η

via ǫ

Rf∗ −−−−→ Rf∗ Lf ∗ Rf∗ −−−−→ Rf∗ Lf ∗ −−−−→ Lf ∗ Rf∗ Lf ∗ −−−−→ Lf ∗ via η

(3.2.1.1)

via ǫ

are identity morphisms [M, p. 83, Thm. 2]. Indeed, η(A) (resp. ǫ(B)) corresponds under (3.2.1) to the identity map of Lf ∗A (resp. Rf∗ B); and conversely, (3.2.1) can be recovered from η and ǫ thus: to a map α : Lf ∗A → B associate the composed map η(A)

Rf α

∗ A −−−→ Rf∗ Lf ∗A −−− → Rf∗ B ,

and inversely, to a map β : A → Rf∗ B associate the composed map Lf ∗β

ǫ(B)

Lf ∗A −−−→ Lf ∗ Rf∗ B −−−→ B. Deﬁne ǫ to be the unique Δ-functorial map such that the following natural diagram in D(X) commutes for all B ∈ K(X):7 6

An ultra-generalization of this “trivial duality formula” is given in [De, p. 298, Thm. 2.3.7]. 7 Here, and elsewhere, we lighten notation by omitting Qs, so that, e.g., B sometimes denotes the (physically identical) image QB of B in D(X). This should not cause confusion.

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3 Derived Direct and Inverse Image

Lf ∗f∗ B −−−−→ Lf ∗ Rf∗ B ⏐ ⏐ ⏐ ⏐ǫ(B) f ∗f∗ B −−−−→

(3.2.1.2)

B

Such an ǫ exists because Lf ∗ Rf∗ is a right-derived functor of Lf ∗ QY f∗ (where QY : K(Y ) → D(Y ) is the canonical functor), and the natural composition Lf ∗ QY f∗ → QX f ∗f∗ → QX is Δ-functorial, see (2.1.1) and (2.2.6.1). (Alternatively, use (2.6.5), with n = 1, L′′ = K(X), L′ ⊂ L′′ the Δ-subcategory whose objects are the q-injective complexes, and β the preceding Δ-functorial composition.) Dually, deﬁne η to be the unique Δ-functorial map such that the following natural diagram commutes for all A ∈ K(Y ): Rf∗ f ∗A ←−−−− Rf∗ Lf ∗A ⏐η(A) ⏐ ⏐ ⏐ f∗ f ∗A ←−−−−

(3.2.1.3)

A

To see then that the ﬁrst row in (3.2.1.1) is the identity, i.e., that its composition with the canonical map ζ : f∗ → Rf∗ is just ζ itself, consider the diagram (with obvious maps) f∗

−−−−−−−−−−−−−−−−−−−→

Rf∗ ⏐ ⏐

f∗ ⏐ ⏐

−−−−→ Rf∗ Lf ∗ f∗ −−−−→ Rf∗ Lf ∗ Rf∗ ⏐ ⏐ ⏐ ⏐ 2 1

f∗

−−−−−−−−−−−−−−−−−−−→

f∗ f ∗f∗ −−−−→ Rf∗ f ∗f∗ −−−−→ ⏐ ⏐

Rf∗ Rf∗

Subdiagrams 1 and 2 commute by the deﬁnitions of η and ǫ. The top and bottom rectangles clearly commute. Thus the whole diagram commutes, giving the desired conclusion. A similar argument applies to the second row in (3.2.1.1). Q.E.D. Corollary 3.2.2. The adjunction isomorphism (3.2.1) is the unique functorial map ρ making the following natural diagram commute for all A ∈ K(Y ) and B ∈ K(X) :

3.2 Adjointness of Derived Direct and Inverse Image HomK(X) (f ∗A, B) −−−→ HomD(X) (f ∗A, B) −−−→ HomD(X) (Lf ∗A, B) ⏐ ⏐ ⏐ ⏐ρ H 0 (3.1.6)≃

91

(3.2.2.1)

ν

HomK(Y ) (A, f∗ B) −−−→ HomD(Y ) (A, f∗ B) −−−→ HomD(Y ) (A, Rf∗ B)

Moreover, ν is an isomorphism whenever A is left-f ∗-acyclic (e.g., q-ﬂat) and B is q-injective. Proof. Suppose ρ is the adjunction isomorphism. To show (3.2.2) commutes, chase a K(X)-map φ : f ∗A → B around it in both directions to reduce to showing that the following natural diagram commutes: via φ

Rf∗ Lf ∗A −−−−→ Rf∗ f ∗A −−−−→ Rf∗ B ⏐ ⏐ ⏐ η⏐ ⏐ ⏐ A

via φ

−−−−→ f∗ f ∗A −−−−→ f∗ B

Here the left square commutes by the deﬁnition of η, and the right square commutes by functoriality of the natural map f∗ → Rf∗ . If, furthermore, A is left-f ∗-acyclic (i.e., Lf ∗A → f ∗A is an isomorphism (2.2.6)) and B is q-injective, then all the maps in (3.2.2.1) other than ν are isomorphisms (see (2.3.8)(v)), so ν is an isomorphism too. Finally, to prove the uniqueness of a functorial map ρ(A, B) making diagram (3.2.2.1) commute, use the canonical maps PA → A and B → IB to map (3.2.2.1) to the corresponding diagram with PA in place of A and IB in place of B. As we have just seen, all the maps in this last diagram other than ρ(PA , IB ) are isomorphisms, so that ρ(PA , IB ) is uniquely determined by the commutativity condition; and since the sources and targets of ρ(PA , IB ) and ρ(A, B) are isomorphic, it follows that ρ(A, B) is uniquely determined. Q.E.D. Exercise. With ψA : PA → A (resp. ϕB : B → IB ) the canonical isomorphism in D(Y ) (resp. D(X)), see (3.1.1), η(A) and ǫ(B) are the respective compositions ψ −1

natural

f∗ (ϕf∗P )

A A −−A−→ PA −−−−−→ f∗ (f ∗PA ) −−−−−− → f∗ (If ∗PA ) = Rf∗ Lf ∗A ,

B −−−→ IB ←−−−−− f ∗ (f∗ IB ) ←− −−−−− f ∗ (Pf∗IB ) = Lf ∗ Rf∗ B . ∗ −1 ϕB

natural

f (ψf

∗IB

)

Recall from §2.4 the derived functors RHom• and RHom• . We write RHom•X and RHom•X to specify that we are working on the ringed space X. For E, F ∈ K(X), and IF as in (3.1.1), we have then, in D(X), RHom•X (E, F ) = Hom• (E, IF ), RHom•X (E, F ) = Hom• (E, IF ).

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3 Derived Direct and Inverse Image

Proposition 3.2.3. (see [Sp, p. 147]). Let f : X → Y be a ringed-space map. (i) There is a unique Δ-functorial isomorphism ∼ α : RHom•X (Lf ∗A, B) −→ RHom•Y (A, Rf∗ B) (3.2.3.1) A ∈ K(Y ), B ∈ K(X)

such that the following natural diagram in D(X)8 commutes:

Hom•X (f ∗A, B) −−−−→ RHom•X (f ∗A, B) −−−−→ RHom•X (Lf ∗A, B) ⏐ ⏐ ⏐ ⏐ ≃α (3.1.6)≃

Hom•Y (A, f∗ B) −−−−→ RHom•Y (A, f∗ B) −−−−→ RHom•Y (A, Rf∗ B).

Moreover, the induced homology map ∼ H 0 (α) : HomD(X) (Lf ∗A, B) −→ HomD(Y ) (A, Rf∗ B)

(see (2.4.2)) is just the adjunction isomorphism in (3.2.1). (ii) There is a unique Δ-functorial isomorphism ∼ β : Rf∗ RHom•X (Lf ∗A, B) −→ RHom•Y (A, Rf∗ B) (3.2.3.2) A ∈ K(Y ), B ∈ K(X) such that the following natural diagram commutes ∗ ∗ ∗ f∗ Hom• −−→ Rf∗ RHom• −−→ Rf∗ RHom• X (f A, B) − X (f A, B) − X (Lf A, B) ⏐ ⏐ ⏐ ⏐ (3.1.6)≃ ≃β

Hom• −−→ Y (A, f∗ B) −

RHom• Y (A, f∗ B)

−−−→

RHom• Y (A, Rf∗ B)

Proof. (i) For the ﬁrst assertion it suﬃces, as in (2.6.5), that in the derived category of abelian groups the natural compositions a

b

c

d

Hom•X (f ∗A, B) − → RHom•X (f ∗A, B) − → RHom•X (Lf ∗A, B) → RHom•Y (A, Rf∗ B) → RHom•Y (A, f∗ B) − Hom•Y (A, f∗ B) − be isomorphisms whenever A is q-ﬂat and B is q-injective. But in this case we have A = PA and B = IB , so that a, b, and d are identity maps. As for c, we need only note that by the last assertion of (3.2.2), the induced homology maps H i (c) : HomK(Y ) (A[−i], f∗ B) → HomD(Y ) (A[−i], f∗ B) are isomorphisms, see (1.2.2) and (2.4.2). Now apply the functor H 0 to the diagram and conclude by the uniqueness of ρ in (3.2.2) that H 0 (α) is as asserted. 8

With missing Q’s left to the reader.

3.2 Adjointness of Derived Direct and Inverse Image

93

(ii) As above, it comes down to showing that the natural maps a′

f∗ Hom•X (f ∗A, B) −→ Rf∗ Hom•X (f ∗A, B) c′

→ RHom•Y (A, f∗ B) = Hom•Y (A, If∗ B ) Hom•Y (A, f∗ B) − are isomorphisms (in D(X), D(Y ) respectively) whenever A is q-ﬂat and B is q-injective. The stalk (f ∗A)x (x ∈ X) being isomorphic to OX,x ⊗OY ,f (x) Af (x) , (2.5.2) shows that f ∗A is q-ﬂat, and then (2.3.8)(iv) shows (via (2.6.2)) that H := Hom•X (f ∗ A, B) is q-injective; so H = IH and a′ : f∗ H → f∗ IH is in fact an identity map. For c′ , it is enough to check that we get an isomorphism after applying the functor ΓU (sections over U ) for arbitrary open U ⊂ Y , since then c′ induces isomorphisms of the homology presheaves—and hence of the homology sheaves—of its source and target (see (1.2.2)). Let i : U → Y , j : f −1 U → X be the inclusion maps, and let g : f −1 U → U be the map induced by f . We have then by (2.3.1) a commutative diagram of quasi-isomorphisms i∗f∗ B −−−−→ i∗ If∗ B ⏐ ⏐γ

i∗f∗ B −−−−→ Ii∗f∗ B

Since i∗If∗ B is q-injective (2.4.5.2), γ is an isomorphism in K(U ) (2.3.2.2). Keeping in mind that i∗f∗ = g∗ j ∗ , consider the commutative diagram Γ (c′ )

ΓU Hom•Y (A, f∗ B) −−U−−→ ΓU Hom•Y (A, If∗ B )

Hom•U (i∗A, i∗f∗ B) −−−−→ Hom•U (i∗A, i∗ If∗ B ) ⏐ ⏐ ≃via γ Hom•U (i∗A, i∗f∗ B) −−−−→ Hom•U (i∗A, Ii∗f∗ B )

Hom•U (i∗A, g∗ j ∗ B) −−− −→ RHom•U (i∗A, g∗ j ∗ B) c U

As in the proof of (i), since j ∗B is q-injective and i∗A is q-ﬂat (see above), therefore cU is an isomorphism; and hence so is ΓU (c′ ). Q.E.D. Corollary 3.2.3.3. Let U ⊂ Y be open and let ΓU : AY → Ab be the abelian functor “sections over U .” Then for any q-injective B ∈ K(X), f∗ B is rightΓU -acyclic.

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3 Derived Direct and Inverse Image

Consequently, by (2.2.7) or (2.6.5), there is a unique Δ-functorial isomor∼ RΓU Rf∗ making the following natural diagram commute phism RΓf −1 U −→ for all B ∈ K(X): Γf −1 U B −−−−−−−−−−−−−−−−−−→ RΓf −1 U B ⏐ ⏐≃

ΓU f∗ B −−−−→ RΓU f∗ B −−−−→ RΓU Rf∗ B

′ Proof. Let OU ∈ AY be the “extension by zero” of OU ∈ AU , i.e., the sheaf associated to the presheaf taking an open V ⊂ Y to OU (V ) if V ⊂ U , and to 0 otherwise. Then there is a natural functorial identiﬁcation ′ ′ , −). Since OU is ﬂat, we have as in the proof of (3.2.3)(i) ΓU (−) = HomY (OU • ′ ′ that the map c : Hom (OU , f∗ B) → RHom• (OU , f∗ B) is an isomorphism, i.e., ΓU (f∗ B) → RΓU (f∗ B) is an isomorphism, whence the conclusion (see last assertion in (2.2.6)). Q.E.D.

Proposition 3.2.4. (i) For any ringed-space map f : X → Y , there is a unique Δ-bifunctorial isomorphism

∗ ∼ ∗ Lf ∗ (A ⊗ X Lf B Y B) −→ Lf A ⊗ = =

A, B ∈ D(Y )

making the following natural diagram commute for all A, B: ∗ −−→ Lf ∗A ⊗ Lf ∗ (A ⊗ X Lf B Y B) −− = = ⏐ ⏐ ⏐ ⏐ ∗

f (A ⊗Y B) −− −−→ (3.1.9)

∗

(3.2.4.1)

∗

f A ⊗X f B

This isomorphism is conjugate (3.3.5) to the isomorphism β in (3.2.3.2). (ii) With η ′ : E → RHom•X (F , E ⊗ F ) corresponding via (2.6.1)∗ to the = ∗ identity map of E ⊗ F , and ǫ : Lf Rf∗ → 1 as in (3.2.1.0), the (Δ-functorial) = map γ : Rf∗ (E) ⊗ Rf∗ (F ) −→ Rf∗ (E ⊗ F) = = adjoint to the composed map

E, F ∈ D(X)

∼ Lf ∗ Rf∗ E ⊗ Rf∗ F −→ Lf ∗ Rf∗ E ⊗ Lf ∗ Rf∗ F −−→ E ⊗ F = = = ǫ⊗ǫ =

(3.2.4.2)

(3.2.4.3)

3.2 Adjointness of Derived Direct and Inverse Image

95

corresponds via (2.6.1)∗ to the composed map Rf η ′

F) Rf∗ E −−−∗−→ Rf∗ RHom•X (F , E ⊗ = via ǫ

−−−−→ Rf∗ RHom•X (Lf ∗ Rf∗ F , E ⊗ F) = β F) . −−−−→ RHom•X Rf∗ F , Rf∗ (E ⊗ =

(3.2.4.4)

(3.2.3.2)

Proof. (i) For x ∈ X, the stalk (f ∗A)x is OX,x ⊗OY ,f (x) Af (x) , and so (2.5.2) shows that f ∗A is q-ﬂat whenever A is. Hence if A and B are both q-ﬂat (whence so, clearly, is A ⊗Y B), then the vertical arrows in (3.2.4.1) are isomorphisms, and the ﬁrst assertion follows from (2.6.5) (dualized). The second assertion amounts to commutativity, for any complexes E, F , G ∈ D(X), of the following diagram of natural isomorphisms: (2.6.1)∗ HomD(X) Lf ∗ E, RHom•X (Lf ∗ F , G) −−−→ HomD(X) Lf ∗ E ⊗ Lf ∗ F , G = ⏐ ⏐ ⏐≃ ⏐ (3.2.1) • HomD(Y ) E, Rf∗ RHomX (Lf ∗ F , G) HomD(X) Lf ∗ (E ⊗ F ), G = ⏐ ⏐ ⏐ ⏐ via β (3.2.1) HomD(Y ) E, RHom•Y (F , Rf∗ G) −−−→∗ HomD(Y ) E ⊗ F , Rf∗ G) = (2.6.1)

(3.2.4.5)

in proving which, we may replace E by PE , F by pf , and G by IG , i.e., we may assume E and F to be q-ﬂat and G to be q-injective. Using the commutativity in (2.6.1)∗ (after applying homology H 0 ), (3.2.2.1), (3.2.3.2), and (3.2.4.1), we ﬁnd that (3.2.4.5) is the target of a natural map, in the category of diagrams of abelian groups, coming from the diagram of isomorphisms (see (3.1.6), and recall that H 0 Hom•X = HomK(X) ): HomK(X) f ∗ E, Hom•X (f ∗ F , G) −−→ HomK(X) f ∗ E ⊗ f ∗ F , G ⏐ ⏐ ⏐ ⏐ HomK(Y ) E, f∗ Hom•X (f ∗ F , G) HomK(X) f ∗ (E ⊗ F ), G ⏐ ⏐ ⏐ ⏐ HomK(Y ) E, Hom•Y (F , f∗ G) −−→ HomK(X) E ⊗ F , f∗ G (3.2.4.6) ∗ ∗ ∼ ∗ ∗ Also, E and F are q-ﬂat (so that Lf E ⊗ Lf F −→ f E ⊗ f F ) and G is = ∗ Lf F → G is represented by a map of q-injective, so any D(X)-map Lf ∗ E ⊗ = complexes f ∗ E ⊗ f ∗ F → G, see (2.3.8)(v). Hence one need only show that (3.2.4.6) commutes. This is exercise (3.1.10), left to the reader.

96

3 Derived Direct and Inverse Image

(ii) With η : 1 → Rf∗ Lf ∗ as in (3.2.1.0), the map (3.2.4.2) is the composition Rf∗ (3.2.4.3)

η

Rf∗ (E) ⊗ Rf∗ (F ) − → Rf∗ Lf ∗ (Rf∗ (E) ⊗ Rf∗ (F )) −−−−−−−−→ Rf∗ (E ⊗ F) = = =

which is clearly Δ-functorial. The rest of the statement is best understood in the formal context of closed categories, see (3.5.4). In the present instance of that context—see (3.5.2)(d) and (3.4.4)(b)—the map (3.4.2.1) is just γ, and hence the adjoint (3.5.4.1) of (3.4.2.1) is the map in (i) above. Commutativity of (3.5.5.1) says that (3.4.5.1) is conjugate to the map (3.5.4.2), which must then, by (i), be β. Hence (ii) follows from the sentence preceding (3.5.4.2) and the description of (3.5.4.1) immediately following (3.5.4.2). Q.E.D. Remark. Commutativity of (3.2.4.5) yields another proof that β is an isomorphism, since the maps labeled (3.2.1) and (2.6.1)∗ are isomorphisms. Exercises 3.2.5. f : X → Y is a ringed-space map, A ∈ D(A), B ∈ D(X). (a) Show that the following two natural composed maps correspond under the adjunction isomorphism (3.2.1): Lf ∗ OY → f ∗ OY → OX ,

OY → f∗ OX → Rf∗ OX .

(b) Write τn for the truncation functor τ≥n of §1.10. Write f∗ (resp. f ∗ ) for Rf∗ (resp. Lf ∗ ). Deﬁne the functorial map ψ : f ∗ τn −→ τn f ∗ to be the adjoint of the natural composed map ∼ τn −→ τn f∗ f ∗ −→ τn f∗ τn f ∗ −→ f∗ τn f ∗ .

(The isomorphism obtains because f∗ D≥n (X) ⊂ D≥n (Y ), see (2.3.4).) Show that the following natural diagram commutes: f∗ ⏐ ⏐

τn f ∗

−−−→ f ∗ τn ⏐ ⏐ ψ

f ∗ τn ⏐ ⏐

τn f ∗ −−−→ τn f ∗ τn

(One way is to check commutativity of the diagram whose columns are adjoint to those of the one in question. For this, (1.10.1.2) may be found useful.) • (c) The natural map Hom• Y (A, f∗ B) → RHomY (A, Rf∗B) is an isomorphism for all q-injective B ∈ K(X) iﬀ Lf ∗A → f ∗A is an isomorphism. (d) Formulate and prove a statement to the eﬀect that the map β in (3.2.3.2) is compatible with open immersions U ֒→ Y . (e) With ΓY as in (3.2.3.3), show that the natural map ∗ • ∗ ΓY f∗ Hom• X (f A, B) → RΓY Rf∗ RHomX (Lf A, B)

is an isomorphism if A is q-ﬂat and B is q-injective.

3.3 Δ-Adjoint Functors

97

(f) Show that there is a natural diagram of isomorphisms ∗ • RΓY Rf∗ RHom• −−− −−→ RΓY RHomY (A, Rf∗ B) X (Lf A, B) − (3.2.3.2) ⏐ ⏐ ⏐ ⏐≃ ≃ ∗ RHom• X (Lf A, B)

−−−− −−→ (3.2.3.1)

RHom• Y (A, Rf∗ B)

see (2.5.10)(b) and (3.2.3.3). (First show the same with all R’s and L’s dropped; then apply (e) and (2.6.5).)

3.3 Δ-Adjoint Functors We now run through the sorites related to adjointness of Δ-functors. Later, we will be constructing numerous functorial maps between multivariate Δ-functors by purely formal (category-theoretic) methods. The results in this section, together with the Proposition in §1.5, will guarantee that the so-constructed maps are in fact Δ-functorial. Let K1 and K2 be Δ-categories with respective translation functors T1 and T2 , and let (f∗ , θ∗ ) : K1 → K2 and (f ∗, θ∗ ) : K2 → K1 be Δ-functors ∼ such that f ∗ is left-adjoint to f∗ . (Recall from §1.5 that θ∗ : f∗ T1 −→ T2 f∗ , ∗ ∗ ∼ ∗ ∗ ∗ and similarly θ : f T2 −→ T1 f .) Let η : 1 → f∗ f , ǫ : f f∗ → 1 be the functorial maps corresponding by adjunction to the identity maps of f ∗ , f∗ respectively. Lemma-Deﬁnition 3.3.1. In the above circumstances, the following conditions are equivalent: (i) η is Δ-functorial. (i)′ ǫ is Δ-functorial. (ii) For all A ∈ K2 and B ∈ K1 , the following natural diagram commutes: θ∗

HomK1 (f ∗A, B) −−−→ HomK1 (T1 f ∗A, T1 B) −−−→ HomK1 (f ∗ T2 A, T1 B) ⏐ ⏐ ⏐ ⏐≃ ≃

HomK2 (A, f∗ B) −−−→ HomK2 (T2 A, T2 f∗ B) −−−→ HomK2 (T2 A, f∗ T1 B) θ∗

When these conditions hold, we say that (f ∗, θ∗ ) and (f∗ , θ∗ ) are Δ-adjoint, or—leaving θ∗ and θ∗ to the reader—that (f ∗, f∗ ) is a Δ-adjoint pair. Proof. (i) ⇒ (ii). Chase a map ξ : f ∗A → B around the diagram in both directions to reduce to showing that the following diagram commutes:

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3 Derived Direct and Inverse Image

T2 A ⏐ ⏐ η(T2 A)

T2 f ξ

T2 η(A)

∗ −−−−−→ T2 f∗ f ∗A −−−− → T2 f∗ B ⏐ ⏐ ⏐ −1 ∗ ⏐ −1 θ∗ (f A) θ∗ (B)

(3.3.1.1)

f∗ f ∗ T2 A −−−∗−−→ f∗ T1 f ∗A −−−−→ f∗ T1 B f∗ T1 ξ

f∗ θ (A)

The ﬁrst square commutes by (i), and the second by functoriality of θ∗ . Conversely, (i) is just commutativity of (3.3.1.1 ) when B := f ∗A and ξ is the identity map. Thus (i) ⇔ (ii); and a similar proof (starting with a map ξ ′ : A → f∗ B) yields (i)′ ⇔ (ii). Q.E.D. Example 3.3.2. Quasi-inverse Δ-equivalences of categories (1.7.2) are Δ-adjoint pairs. Example 3.3.3. The pair (Lf ∗, Rf∗ ) in (3.2.1) is Δ-adjoint. Indeed, in the proof of (3.2.1) the associated η and ǫ were deﬁned to be certain Δ-functorial maps. Example 3.3.4. With reference to (2.6.1)∗ , let K1 := D(A) =: K2 , ﬁx F ∈ D(A), and for any A, B ∈ D(A) set F, f ∗A := A ⊗ =

f∗ B := RHom• (F , B) .

Then this pair (f ∗, f∗ ) is Δ-adjoint. To verify condition (ii) in (3.3.1), consider the following diagram of natural isomorphisms, where H• stands for RHom• and H• stands for RHom• : H• (A ⊗ F , B) = ⏐ ⏐

(2.6.1)∗

− −−−−−−−−−−−−−−−−−−−−−−−−−−→ 1

H• (A, H• (F , B)) ⏐ ⏐

• • H• (A ⊗ F , B[1])[−1] −−→ H• (A, H• (F = ⏐, B[1]))[−1] ←−− H (A, H (F ⏐, B)[1])[−1] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 3 ⏐ ⏐ H• ((A ⊗ F )[1], B[1]) = ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

H• (A[1] ⊗ F , B[1]) −−→ H• (A[1], H• (F , B[1])) ←−− H• (A[1], H• (F , B)[1]) =

Subdiagram 1 commutes because (2.6.1)∗ is Δ-functorial in the last variable; 2 commutes because (2.6.1)∗ is Δ-functorial in the ﬁrst variable; and 3 commutes for obvious reasons. One checks that application of the functor H 0 to this big commutative diagram gives (ii) in (3.3.1). Q.E.D.

3.3 Δ-Adjoint Functors

99

In particular, we have the canonical Δ-functorial maps η ′ : A → RHom• (F , A ⊗ F), =

(3.3.4.1)

F →B. ǫ′ : RHom• (F , B) ⊗ =

Lemma-Deﬁnition 3.3.5. If f∗ : X → Y, g∗ : X → Y are functors with respective left adjoints f ∗ : Y → X, g ∗ : Y → X, then with “Hom” denoting “ functorial morphisms,” the following natural compositions are inverse isomorphisms: ∼ Hom(g ∗, f ∗ ) , Hom(f∗ , g∗ ) −→ Hom(f∗ f ∗, g∗ f ∗ ) −→ Hom(1, g∗ f ∗ ) −→ ∼ Hom(g ∗f∗ , 1) ←− Hom(g ∗f∗ , f ∗f∗ ) ←− Hom(g ∗, f ∗ ) . Hom(f∗ , g∗ ) ←−

Functorial morphisms f∗ → g∗ and g ∗ → f ∗ which correspond under these isomorphisms will be said to be conjugate (the ﬁrst right-conjugate to the second, the second left-conjugate to the ﬁrst). Proof. Exercise, or see [M, p. 100, Theorem 2]. Corollary 3.3.6. Let (f ∗, f∗ ) and (g ∗, g∗ ) be Δ-adjoint pairs of Δ-functors between K1 and K2 . Then a functorial morphism α : f∗ → g∗ is Δ-functorial if and only if so is its conjugate β : g ∗ → f ∗ . In particular, f∗ and g∗ are isomorphic Δ-functors ⇔ so are f ∗ and g ∗ . The ﬁrst assertion follows from (3.3.1) since, for example, α is the composition η via β ǫ f∗ − → g∗ g ∗f∗ −−−→ g∗ f ∗f∗ − → g∗ . That the conjugate of a functorial isomorphism is an isomorphism follows from Exercise (3.3.7)(c) below. Exercises 3.3.7. (a) Maps α : f∗ → g∗ and β : g ∗ → f ∗ are conjugate ⇔ (either of) the following diagrams commute: 1 ⏐ ⏐ η

η

−−−→ g∗ g ∗ ⏐ ⏐via

f∗ f ∗ −−−−→ g∗ f ∗ via α

β

1 ⏐ ǫ⏐

ǫ

←−−− g ∗ g∗ ⏐ ⏐via

α

f ∗f∗ ←−−−− g ∗f∗ via β

(b) The conditions in (a) are equivalent to commutativity, for all X ∈ X, Y ∈ Y of the diagram via α Hom(Y , f∗ X) −−−−→ Hom(Y , g∗ X) ⏐ ⏐ ⏐ ⏐≃ ≃ Hom(f ∗ Y , X) −− −−→ Hom(g ∗ Y , X) via β

(c) Denoting the conjugate of a functorial map α by α′ we have (with the obvious interpretation) 1′ = 1 and (α2 α1 )′ = α′1 α′2 .

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3 Derived Direct and Inverse Image

(d) The conditions in (3.3.1) are equivalent to either one of: ∼ (iii) The functorial map θ∗ : f ∗ T2 −→ T1 f ∗ is left-conjugate to θ −1

∗ f∗ T1−1 = T2−1 T2 f∗ T1−1 −− → T2−1 f∗ T1 T1−1 = T2−1 f∗ .

∼ (iii)′ The functorial map θ∗ : f∗ T1 −→ T2 f∗ is right-conjugate to θ ∗ −1

f ∗ T2−1 = T1−1 T1 f ∗ T2−1 −−−→ T1−1 f ∗ T2 T2−1 = T1−1 f ∗ . The next Proposition, generalizing some of (1.7.2), says that a left adjoint of a Δ-functor can be made into a left Δ-adjoint, in a unique way. Let K1 , K2 be Δ-categories with respective translation functors T1 , T2 , and let (f∗ , θ∗ ) : K1 → K2 be a Δ-functor such that f∗ has a left adjoint f ∗ : K2 → K1 (automatically additive, see ﬁrst footnote in §3.1). Proposition 3.3.8. There exists a unique functorial isomorphism ∼ θ∗ : f ∗ T2 −→ T1 f ∗

such that (i) (f ∗, θ∗ ) is a Δ-functor, and (ii) the Δ-functors (f ∗, θ∗ ) and (f∗ , θ∗ ) are Δ-adjoint. Proof. The functors f ∗ T2 and T1 f ∗ are left-adjoint to T2−1 f∗ and f∗ T1−1 respectively; and since the latter two are isomorphic (in the obvious way via θ∗ ), so are the former two, and one checks that the conjugate isomorphism θ∗ between them is adjoint to the composite map θ −1

η

∗ T2 − → f∗ T1 f ∗ , 9 → T2 f∗ f ∗ −−

i.e., θ∗ is the unique map making the following diagram commute: T2 ⏐ ⏐ η

T2 ⏐ ⏐η

(3.3.8.1)

f∗ f ∗ T2 −−−−→ f∗ T1 f ∗ −−−−→ T2 f∗ f ∗ ∗ f∗ θ

θ∗

If (i) holds, then commutativity of (3.3.8.1) also expresses the condition that η : 1 → f∗ f ∗ be Δ-functorial, i.e., that (ii) hold. Thus no other θ∗ can satisfy (i) and (ii). (So far, the argument is just a variation on (3.3.7)(d).) We still have to show that (i) holds for the θ∗ we have speciﬁed. So let u v w ′ ∗ A −→ B −→ C −→ T2 A be a triangle in K2 . Apply (Δ3) in (1.4) to embed f u ∗ u p ∗ ∗ q ∗ . I claim that into a triangle f ∗A −−f−−→ f B −→ C −→ T1 f A (a) there is a map γ : f ∗ C → C ∗ making the following diagram commute: f ∗u

f ∗v

θ ∗◦f ∗w

f ∗A −−−−→ f ∗B −−−−→ f ∗ C −−−−−→ T1 f ∗A ⏐ ⏐ γ f ∗A −−f−∗−→ f ∗B −−− −→ C ∗ u p

−−− −→ T1 f ∗A q

and that (b) any such γ must be an isomorphism. Given (a) and (b), condition (Δ1)′′ in (1.4) ensures that the top row in the preceding diagram is a triangle, so that (f ∗, θ∗ ) is indeed a Δ-functor. 9

∗

θ ǫ Whence, dually, θ∗−1 is adjoint to T1 ← − T1 f ∗f∗ ←− f ∗ T2 f∗ .

3.4 Adjoint Functors between Monoidal Categories

101

Assertion (a) results by adjunction from the map of triangles A ⏐ ⏐ η

u

−−−−→

B ⏐ ⏐ η

v

−−−−→

w

C −−−−→ ⏐ ⏐ ′ γ

T2 A ⏐ ⏐T η 2

f∗ f ∗A −−−−→ f∗ f ∗B −−−−→ f∗ C ∗ −−−−−→ T2 f∗ f ∗A ∗ f∗ f u

′

f∗ p

θ∗ ◦f∗ q

′′

where γ is given by (Δ3) in (1.4). For (b), consider the commutative diagram (with D ∈ K1 , and obvious maps): Hom(T1 f ∗B, D) −−− −→ Hom(T2 B, f∗ D) ⏐ ⏐ ⏐ ⏐

Hom(T1 f ∗B, D) ⏐ ⏐ Hom(T1 f ∗A, D) ⏐ ⏐ Hom(C ∗ , D) ⏐ ⏐

Hom(f ∗B, D) ⏐ ⏐ Hom(f ∗A, D)

via γ

−−−−→

Hom(T1 f ∗A, D) −−− −→ Hom(T2 A, f∗ D) ⏐ ⏐ ⏐ ⏐ Hom(f ∗ C, D) ⏐ ⏐

−−− −→

Hom(f ∗A, D)

−−− −→

Hom(f ∗B, D) ⏐ ⏐

−−− −→

Hom(C, f∗ D) ⏐ ⏐

Hom(B, f∗ D) ⏐ ⏐ Hom(A, f∗ D)

The left and right columns are exact [H, p. 23, Prop. 1.1, b], hence the map “via γ” is an isomorphism for all D, i.e., γ is an isomorphism. Q.E.D.

3.4 Adjoint Functors between Monoidal Categories This section and the following one introduce some of the formalism arising from a pair of adjoint monoidal functors between closed categories. A simple example of such a pair occurs with respect to a map R → S of commutative rings, namely extension and restriction of scalars on the appropriate module categories. The module functors f ∗ and f∗ associated with a map f : X → Y of ringed spaces form another such pair. The example which mosts interests us is that of the pair (Lf ∗, Rf∗ ) of §3.2. The point is to develop by purely categorical methods a host of relations, expressed by commutative functorial , RHom• , Lf ∗ and Rf∗ . diagrams, among the four operations ⊗ = But even the puriﬁed categorical approach leads quickly to stultifying complexity—at which the exercises (3.5.6) merely hint. Ideally, we would like to have an implementable algorithm for deciding when a functorial diagram built up from the data given in the relevant categorical deﬁnitions (see (3.4.1), (3.4.2), (3.5.1)) commutes; or in other words, to prove a “constructive coherence theorem” for the generic context “monoidal functor between closed categories, together with left adjoint.” (Lewis [Lw] does this, to some extent,

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3 Derived Direct and Inverse Image

without the left adjoint.) Though there exists a substantial body of results on “coherence in categories,” see e.g., [K′ ], [Sv], and their references, it does not yet suﬃce; we will have to be content with subduing individual diagrams as needs dictate. We treat symmetric monoidal categories in this section, leaving the additional “closed” structure to the next. Deﬁnition 3.4.1. A symmetric monoidal category M = (M0 , ⊗, OM , α, λ, ρ, γ) consists of a category M0 , a “product” functor ⊗ : M0 × M0 → M0 , an object OM of M0 , and functorial isomorphisms ∼ α : (A ⊗ B) ⊗ C −→ A ⊗ (B ⊗ C) ∼ λ : OM ⊗ A −→ A

(associativity)

∼ ρ : A ⊗ OM −→ A ∼

γ : A ⊗ B −→ B ⊗ A

(units) (symmetry)

(where A, B, C are objects in M0 ) such that γ ◦ γ = 1 and the following diagrams (3.4.1.1) commute. α

(A ⊗ OM ) ⊗ B −−−−−−−−→ A ⊗ (OM ⊗ B) ρ⊗1

1⊗λ

A⊗B α

α

((A ⊗ B) ⊗ C) ⊗ D −−−→ (A ⊗ B) ⊗ (C ⊗ D) −−−→ A ⊗ (B ⊗ (C ⊗ D)) ⏐ ⏐ ⏐1⊗α ⏐ α⊗1 (A ⊗ (B ⊗ C)) ⊗ D −−−−−−−−−−−−−−−−−−−−−−−−−→ A ⊗ ((B ⊗ C) ⊗ D) α

γ

α

(A ⊗ B) ⊗ C −−−→ A ⊗ (B ⊗ C) −−−→ (B ⊗ C) ⊗ A ⏐ ⏐ ⏐α ⏐ γ⊗1 (B ⊗ A) ⊗ C −−−→ B ⊗ (A ⊗ C) −−−→ B ⊗ (C ⊗ A) α

1⊗γ

γ

OM ⊗ A −−−−−−→ A ⊗ OM λ

ρ

A (3.4.1.1)

3.4 Adjoint Functors between Monoidal Categories

103

Deﬁnition 3.4.2. A symmetric monoidal functor f∗ : X → Y between symmetric monoidal categories X, Y is a functor f∗0 : X0 → Y0 together with two functorial maps f∗ A ⊗ f∗ B −→ f∗ (A ⊗ B)

(3.4.2.1)

OY −→ f∗ OX

(where we have abused notation, as we will henceforth, by omitting the subscript “0 ” and by not distinguishing notationally between ⊗ in X and ⊗ in Y), such that the following natural diagrams (3.4.2.2) commute. f∗ OX ⊗ f∗ A −−→ f∗ (OX ⊗ A) ⏐ ⏐f (λ ) ⏐ ∗ X ⏐ f∗ A

OY ⊗ f∗ A −−→ λY

f∗ A ⊗ f∗ B −−→ f∗ (A ⊗ B) ⏐ ⏐ ⏐f (γ ) ⏐ γY ∗ X

f∗ B ⊗ f∗ A −−→ f∗ (B ⊗ A)

(f∗ A ⊗ f∗ B) ⊗ f∗ C −−−−→ f∗ (A ⊗ B) ⊗ f∗ C −−−−→ f∗ ((A ⊗ B) ⊗ C) ⏐ ⏐ ⏐f (α) ⏐ α ∗

f∗ A ⊗ (f∗ B ⊗ f∗ C) −−−−→ f∗ A ⊗ f∗ (B ⊗ C) −−−−→ f∗ (A ⊗ (B ⊗ C)) (3.4.2.2) (3.4.3). We assume further that the symmetric monoidal functor f∗ has a left adjoint f ∗ : Y → X. In other words we have functorial maps η : 1 → f∗ f ∗

ǫ : f ∗f∗ → 1

such that the composites via η

via ǫ

f∗ −−−−→ f∗ f ∗f∗ −−−−→ f∗

via η

via ǫ

f ∗ −−−−→ f ∗f∗ f ∗ −−−−→ f ∗

are identities, giving rise to a bifunctorial isomorphism ∼ HomX (f ∗ F , G) −→ HomY (F , f∗ G)

(F ∈ Y, G ∈ X). (3.4.3.1)

Example 3.4.4. (a) Let f : X → Y be a map of ringed spaces, X (resp. Y) the category of OX - (resp. OY -)modules with its standard structure of symmetric monoidal category (⊗ having its usual meaning, etc. etc.), f∗ and f ∗ the usual direct and inverse image functors, see (3.1.7). , (b) Let f : X → Y be a ringed-space map, X := D(X), Y := D(Y ), ⊗ := ⊗ = f∗ := Rf∗ , f ∗ := Lf ∗ (see (3.2.1)). To establish symmetric monoidality of, e.g., D(X), one need only work with q-ﬂat complexes, . . . .

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3 Derived Direct and Inverse Image

For (3.4.2.1), use the map γ from (3.2.4.2) and the natural composition OY → f∗ OX → Rf∗ OX . One can then deduce via adjointness that Rf∗ is symmetric monoidal from the fact that Lf ∗ is symmetric monoidal when considered as a functor from Yop to Xop , see (3.2.4). For this property of Lf ∗, one can check the requisite commutativity in (3.4.2.2) after replacing each object A in X by an isomorphic q-ﬂat complex, and recalling that if A is q-ﬂat, then so is f ∗A (see proof of (3.2.3)(ii)); in view of (3.1.3), the checking is thereby reduced to the context of (a) above, where one can use adjointness (see (3.1.9)) to deduce what needs to be known about f ∗ after showing directly that f∗ is symmetric monoidal! For example, to show commutativity of γ

Rf∗ (OX ) ⊗ Rf∗ (A) −−−−→ Rf∗ (OX ⊗ A) = = ⏐ ⏐λ ⏐ X ⏐ OY ⊗ Rf∗ (A) =

−−−−→ λY

Rf∗ (A)

consider the following natural diagram, in which we have written f ∗, f∗ , ⊗ respectively: for Lf ∗, Rf∗ , ⊗ =

f ∗ (f∗ OX ⊗ f∗ A)

f ∗f∗ OX ⊗ f ∗f∗ A −−→ ⏐ 1 ⏐

OX ⊗ A ⏐ ⏐

f ∗ (OY ⊗ f∗ A) −−→

f ∗f∗ A

f ∗ OY ⊗ f ∗f∗ A −−→ OX ⏐ 2 ⏐

⊗ f ∗f∗ A ⏐ ⏐

A

It will be enough to show that the outer border commutes, because it is “adjoint” to the preceding diagram, see (3.4.5.2). Subdiagram 1 commutes by exercise (3.2.5)(a). For commutativity of 2 replace f∗ A by an isomorphic q-ﬂat complex to reduce to showing commutativity of the corresponding diagram in context (a); then reduce via adjointness to checking (easily) that in that context the following natural diagram commutes: (3.1.7)

f∗ (A) −−−−→ f∗ (OX ⊗ A) f∗ (OX ) ⊗ = = ⏐ ⏐ ⏐ ⏐ OY ⊗ f∗ (A) =

−−−−→

f∗ (A)

The rest is evident. Exercise 3.4.4.1. Let R be a commutative ring, Z := Spec(R), T an indeterminate, X := Spec(R[T ]) with its obvious Z-scheme structure, δ : X → Y := X ×Z X the

3.4 Adjoint Functors between Monoidal Categories

105

∼ diagonal map, and σ : Y −→ Y the symmetry isomorphism, i.e., π1 σ = π2 and π2 σ = π1 where π1 and π2 are the canonical projections from Y to X. Show that in the context of (3.4.4)(a) the natural composite OX -module map ∼ δ ∗δ∗ F = (σδ)∗ (σδ)∗ F −→ δ ∗σ ∗σ∗ δ∗ F → δ ∗δ∗ F

is the identity map for all OX -modules F ; but that in the context of (3.4.4)(b) the natural composite D(X)-map ∼ Lδ ∗δ∗ OX = L(σδ)∗ (σδ)∗ OX −→ Lδ ∗σ ∗σ∗ δ∗ OX → Lδ ∗δ∗ OX

is not the identity map unless 2 = 0 in R. (More challenging.) Show: if ι : Z → X is the closed immersion corresponding to the R-homomorphism R[T ] ։ R taking T to 0, then the natural composite D(X)map ∼ Lδ ∗δ∗ ι∗ OZ = L(σδ)∗ (σδ)∗ ι∗ OZ −→ Lδ ∗σ ∗σ∗ δ∗ ι∗ OZ → Lδ ∗δ∗ ι∗ OZ

is an automorphism of order 2, inducing the identity map on homology.

(3.4.5) (Duality principle). From (3.4.2.1) we get, by adjunction, functorial maps f ∗ C ⊗ f ∗ D ←− f ∗ (C ⊗ D) ,

(3.4.5.1)

OX ←− f ∗ OY .

Speciﬁcally, the second of these maps is deﬁned to be adjoint to the map OY → f∗ OX in (3.4.2.1) (i.e., the two maps correspond under the isomorphism (3.4.3.1)); and the ﬁrst adjoint to the composition (3.4.2.1)

η⊗η

C ⊗ D −−−−→ f∗ f ∗ C ⊗ f∗ f ∗ D −−−−−→ f∗ (f ∗ C ⊗ f ∗ D) . It follows that “dually,” (3.4.2.1)

(3.4.5.2): f∗ A ⊗ f∗ B −−−−−→ f∗ (A ⊗ B) is adjoint to the composition A ⊗ B ←−−−− f ∗f∗ A ⊗ f ∗f∗ B ←−−−−− f ∗ (f∗ A ⊗ f∗ B) . ǫ⊗ǫ

(3.4.5.1)

To see this, it suﬃces to note that the following diagram, whose top row composes to the identity, commutes: ǫ⊗ǫ

η⊗η

f∗ A ⊗ f∗ B ←−−−− f∗ f ∗f∗ A ⊗ f∗ f ∗f∗ B ←−−−− ⏐ ⏐ ⏐(3.4.2.1) ⏐ 2 1 (3.4.2.1)

f∗ A ⊗ f∗ B ⏐ ⏐η

f∗ (A ⊗ B) ←−−−− f∗ (f ∗f∗ A ⊗ f ∗f∗ B) ←−−−−− f∗ f ∗ (f∗ A ⊗ f∗ B) ǫ⊗ǫ

(3.4.5.1)

(Subdiagram 1 commutes by functoriality of (3.4.2.1), and 2 commutes by the above deﬁnition of (3.4.5.1).) The maps (3.4.5.1) satisfy compatibility conditions with the associativity, unit, and symmetry isomorphisms in the symmetric monoidal categories X

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3 Derived Direct and Inverse Image

and Y, conditions dual to those expressed by the commutativity of the diagrams (3.4.2.2) (i.e., in (3.4.2.2) replace f∗ by f ∗ , interchange OX and OY , and reverse all arrows). Proofs are left to the reader. The maps (3.4.5.1) do not make f ∗ monoidal, since they point in the wrong direction (and we do not assume in general that they are isomorphisms, as happens to be the case in (3.4.4(a)) and (3.4.4(b)), so we cannot use their inverses). However, to any symmetric monoidal category M = (M0 , ⊗, OM , α, λ, ρ, γ) we can associate the dual symmetric monoidal category op

Mop = (M0 , ⊗op , OM , α, λ, ρ, γ) op

where M0 is the dual category of M0 (same objects; arrows reversed), ⊗op is the functor op

op

⊗op

op

M0 × M0 = (M0 × M0 )op −−−−→ M0 (so that A ⊗op B = A ⊗ B for all objects A, B ∈ M0 ),

∼ α = (αop )−1 = (α−1 )op : (A ⊗ B) ⊗ C −→ A ⊗ (B ⊗ C)

(in Mop 0 )

and similarly for λ, ρ, γ. Then one checks that the functor (f ∗ )op : Yop → Xop together with the maps (3.4.5.1) is indeed symmetric monoidal;10 and it has a left adjoint (f∗ )op : Xop → Yop (which need no longer be monoidal, because, for example, there may be no good map OY → f∗ OX in Yop ). Thus to every pair f∗ , f ∗ as in (3.4.3), we can associate a “dual” such pair (f ∗ )op , (f∗ )op . This gives rise to a duality principle, which we now state rather imprecisely, but whose meaning should be clariﬁed by the illustrations which follow (in connection with projection morphisms). We will be considering numerous diagrams whose vertices are functors build up from the constant functors OX and OY (on X, Y respectively), identity functors, f∗ , f ∗ , and ⊗, and whose arrows are morphisms of functors built up from those which express the “monoidality” of f∗ , and from the adjunction isomorphism (3.4.3.1). (For 10

f ∗ may then be said to be “op-monoidal” or “co-monoidal.”

3.4 Adjoint Functors between Monoidal Categories

107

example the above-mentioned “compatibility conditions” state that certain such diagrams commute.) By interpreting any such diagram in the dual context, we get another such diagram: speciﬁcally, in the original diagram, interchange - OX and OY - the identity functors of X and Y - the adjunction maps η and ǫ - the functors f ∗ and f∗ - the maps in (3.4.2.1) and (3.4.5.1). If the original diagram commutes solely by virtue of the fact that f∗ is a monoidal functor with left adjoint f ∗, then the second diagram must also commute (because (f ∗ )op is a monoidal functor with left adjoint (f∗ )op ). Example 3.4.6 (Projection morphisms). With preceding notation, and F ∈X, G ∈ Y, the bifunctorial projection morphisms p1 = p1 (F , G) : f∗ F ⊗ G −→ f∗ (F ⊗ f ∗ G) p2 = p2 (G, F ) : G ⊗ f∗ F −→ f∗ (f ∗ G ⊗ F ) are the respective compositions 1⊗η

(3.4.2.1)

η⊗1

(3.4.2.1)

f∗ F ⊗ G −−−−→ f∗ F ⊗ f∗ f ∗ G −−−−−→ f∗ (F ⊗ f ∗ G) G ⊗ f∗ F −−−−→ f∗ f ∗ G ⊗ f∗ F −−−−−→ f∗ (f ∗ G ⊗ F ) . Remarks 3.4.6.1. p1 and p2 determine each other via the following commutative diagram, in which γX , γY are the respective symmetry isomorphisms in X, Y: p

f∗ F ⊗ G −−−1−→ f∗ (F ⏐ ⏐ γY

⊗ f ∗ G) ⏐ ⏐f (γ ) ∗ X

∗ G ⊗ f∗ F −−− p −→ f∗ (f G ⊗ F ) 2

The commutativity of this diagram follows from that of (3.4.2.1)

f∗ F ⊗ f∗ f ∗ G −−−−−→ f∗ (F ⏐ ⏐ γY

⊗ f ∗ G) ⏐ ⏐f (γ ) ∗ X

f∗ f ∗ G ⊗ f∗ F −−−−−→ f∗ (f ∗ G ⊗ F ) (3.4.2.1)

which holds as part of the deﬁnition of “symmetric monoidal functor” (see (3.4.2.2)).

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(3.4.6.2). The map p1 (F , G) is adjoint to the composed map (3.4.5.1)

ǫ⊗1

f ∗ (f∗ F ⊗ G) −−−−−→ f ∗f∗ F ⊗ f ∗ G −−−−→ F ⊗ f ∗ G (a map which is dual (3.4.5) to p2 (F , G)): this follows from commutativity of the natural diagram (3.4.2.1)

via η

f ∗ (f∗ F ⊗ G) −−−−→ f ∗ (f∗ F ⊗ f∗ f ∗ G) −−−−−→ f ∗f∗ (F ⊗ f ∗ G) ⏐ ⏐ ⏐ ⏐ǫ ⏐ ⏐ 2 1 (3.4.5.1) (3.4.5.1) f ∗f∗ F ⊗ f ∗ G −−−−→ f ∗f∗ F ⊗ f ∗f∗ f ∗ G −−−−→

F ⊗ f ∗ G.

ǫ⊗ǫ

via η

(Here commutativity of 1 is clear, and that of 2 results from (3.4.5.2).) Similarly p2 (G, F ) is adjoint to the dual of p1 (G, F ). Lemma 3.4.7. The following diagrams commute: η

A⊗B ⏐ ⏐ 1⊗η

(i)

−−−−→

f∗ f ∗ (A ⊗ B) ⏐ ⏐(3.4.5.1)

A ⊗ f∗ f ∗B −−−−→ f∗ (f ∗A ⊗ f ∗B) p2

p

(3.4.2.1)

(ii)

2 A ⊗ OY −−−−−−→ A ⊗ f∗ OX −−−−− −→ f∗ (f ∗A ⊗ OX ) ⏐ ⏐ ⏐ ⏐f (ρ) ρ ∗

A

−−−−−−−−−−−− −− −−−−−−−−−−−−→ η

f∗ f ∗A

p

1 f∗ B ⊗ OY −−−−→ f∗ (B ⊗ f ∗ OY ) ⏐ ⏐ ⏐ ⏐(3.4.5.1) ρ

(iii)

f∗ B

(A ⊗ B) ⊗ f∗ C ⏐ (iv) p2 ⏐

α

−−−−→

←−−−− f∗ (ρ)

f∗ (B ⊗ OX )

A ⊗ (B ⊗ f∗ C)

1⊗p

2 −−−−→

A ⊗ f∗ (f ∗B ⊗ C) ⏐ ⏐p 2

f∗ (f ∗ (A ⊗ B) ⊗ C) −−−−−→ f∗ ((f ∗A ⊗ f ∗B) ⊗ C) −−−−→ f∗ (f ∗A ⊗ (f ∗B ⊗ C)) α

(3.4.5.1)

Proof. (i) The commutativity of this diagram simply states that the ﬁrst map in (3.4.5.1) is adjoint to the composition 1⊗η

η⊗1

(3.4.2.1)

A ⊗ B −−→ A ⊗ f∗ f ∗B −−→ f∗ f ∗A ⊗ f∗ f ∗B −−−−−→ f∗ (f ∗A ⊗ f ∗B) which is so by deﬁnition (see beginning of (3.4.5)).

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109

(ii) We expand the diagram in question as follows: (3.4.2.1)

η⊗1

A ⊗ f∗ OX −−−−→ f∗ f ∗A ⊗ f∗ OX −−−−−→ f∗ (f ∗A ⊗ OX ) ⏐ ⏐f (ρ) ⏐ ⏐ 1 2 (3.4.2.1)⏐ ∗ ⏐ A ⊗ OY ⏐ ⏐ ρ A

−−−−→ f∗ f ∗A ⊗ OY η⊗1

−−−−−→ ρ

3 −−−−−−−−−−−−−−−−−−−−−−−−→ η

f∗ f ∗A

f∗ f ∗A

Subdiagrams 1 and 3 clearly commute; and so does 2 because of the compatibility of (3.4.2.1) and ρ, which can be deduced from the two top diagrams in (3.4.2.2) (the ﬁrst of which expresses the compatibility of (3.4.2.1) and λ) and the bottom diagram in (3.4.1.1). (iii) The diagram expands as 1⊗η

f∗ B ⊗ OY −−−−−−−−−−−−−−−−−−−−−−−−→ f∗ B ⊗ f∗ f ∗ OY ⏐ ⏐ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 f∗ B ⊗ f∗ OX 3 ρ⏐ ⏐ (3.4.2.1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (3.4.2.1) f∗ B

←−−−−− f∗ (B ⊗ OX ) ←−−−−− f∗ (B ⊗ f ∗ OY ) f∗ (ρ)

(3.4.5.1)

Subdiagram 1 commutes by the deﬁnition of the map f ∗ OY → OX in (3.4.5.1), 2 by the compatibility of (3.4.2.1) and ρ (see preceding proof of (ii)), and 3 by functoriality of (3.4.2.1). (iv) An expanded version of this diagram can be obtained by ﬁtting together the following two diagrams (whose maps are the obvious ones): (A ⊗ B) ⊗ f∗ C

(A ⊗ B) ⊗ f∗ C ⏐ ⏐

−−−→

A ⊗ (B ⊗ f∗ C)

1 −−−→ (f∗

2

f∗ f ∗ (A ⊗ B) ⊗ f∗ C −−−→ ⏐ ⏐ 3

f∗ (f ∗ (A ⊗ B) ⊗ C) −−−→

f ∗A

−−−→

A ⊗ (f∗ f ∗B ⊗ f∗ C) ⏐ ⏐a

⊗ f∗ f ∗B) ⊗ f∗ C −−−→ f∗ f ∗A ⊗ (f∗ f ∗B ⊗ f∗ C) b ⏐ ⏐c

f∗ (f ∗A ⊗ f ∗B) ⊗ f∗ C ⏐ ⏐ d f∗ ((f ∗A ⊗ f ∗B) ⊗ C)

110

3 Derived Direct and Inverse Image

b

A ⊗ (f∗ f ∗B ⊗ f∗ C) ⏐ ⏐ a

−−−→

5

A ⊗ f∗ (f ∗B ⊗ C) ⏐ ⏐

(f∗ f ∗A ⊗ f∗ f ∗B) ⊗ f∗ C −−−→ f∗ f ∗A ⊗ (f∗ f ∗B ⊗ f∗ C) −−−→ f∗ f ∗A ⊗ f∗ (f ∗B ⊗ C) ⏐ ⏐ ⏐ ⏐ c ⏐ ⏐ ⏐ f∗ (f ∗A ⊗ f ∗B) ⊗ f∗ C 4 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ d f∗ ((f ∗A ⊗ f ∗B) ⊗ C) −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ f∗ (f ∗A ⊗ (f ∗B ⊗ C))

Subdiagram 1 commutes by functoriality of a; 2 by the deﬁnition of (3.4.5.1); 3 by functoriality of (3.4.2.1); 4 by commutativity of the bottom diagram in (3.4.2.2); and 5 for obvious reasons. Q.E.D. Remarks 3.4.7.1. By duality (3.4.5) we get four other commutative diagrams out of (3.4.7). For example, the dual of (ii) is (3.4.5.1)

(3.4.6.2)

A ⊗ OX ←−−−−− A ⊗ f ∗ OY ←−−−−− f ∗ (f∗ A ⊗ OY ) ⏐ ⏐ ⏐f ∗ (ρ) ⏐ ρ A

←−−−−−−−−−−−−−−−−−−−−− ǫ

f ∗f∗ A

Using the symmetry isomorphism γ, Remark (3.4.6.1), the bottom diagram in (3.4.1.1), etc., we can also transform the commutative diagrams in (3.4.7) into similar ones with p2 (resp. p1 ) replaced by p1 (resp. p2 ), and with ρ replaced by λ.

3.5 Adjoint Functors between Closed Categories The adjoint symmetric functors f∗ , f ∗ , remain as in (3.4.3). Additional structure comes into play when the monoidal categories X and Y are closed, in the following sense. Deﬁnition 3.5.1. A symmetric monoidal closed category (brieﬂy, a closed category) is a symmetric monoidal category M = (M0 , ⊗, OM , α, λ, ρ, γ) as in (3.4.1), together with a functor, called “internal hom,” [−, −] : Mop 0 × M0 → M0

(3.5.1.1)

(where Mop 0 is the dual category of M0 ) and a functorial isomorphism

3.5 Adjoint Functors between Closed Categories

111

∼ π : Hom(A ⊗ B, C) −→ Hom(A, [B, C ]) .

(3.5.1.2)

The notion of closed category reduces myriad relations among, and maps involving, “tensor” and “hom” to the few basic ones appearing in the deﬁnition. (See, e.g., the following exercises (3.5.3).)11 The original treatise on closed categories is [EK], in particular Chap. III, (p. 512 ff ). Some more recent theory can be found starting with [Sv] and its references. Example 3.5.2. (a) Let M0 be the category of modules over a given commutative ring R. Let ⊗ be the usual tensor product, OM := R, and [B, C ] := HomR (B, C). Fill in the rest. (b) M0 is the category of OX -modules on a ringed space X. Let ⊗ be the usual tensor product, OM := OX , and [B, C ] := HomX (B, C) . . . . (c) Let M′0 := K(X) be the homotopy category of complexes in the category M0 of (b). Let ⊗ be the tensor product in (1.5.4), set OM ′ := OX (a complex vanishing in all nonzero degrees), and set [B, C ] := Hom•X (B, C), see (2.4.5), (2.6.7), . . . . (d) M′′0 := D(X), the derived category of M0 in (b), ⊗ := ⊗ (2.5.7), = OM ′′ := OX , [B, C ] := RHom•X (B, C), see (2.6.1)′ , (3.4.4)(b), . . . . Exercises 3.5.3. Let (M, [−, −], π) as above be a closed category. Write (A, B) for HomM0 (A, B). (a) Deﬁne the set-valued functor Γ on M0 to be the usual functor (OM , −). Establish a bifunctorial isomorphism ∼ Γ[A, B ] −→ (A, B).

(b) Let tAB : [A, B ] ⊗ A → B correspond under π to the identity map of [A, B ]. Use tAB and π to obtain a natural map [A, B ] → [A ⊗ C, B ⊗ C ]. (c) Use π, tCA , and tAB (see (b)) to get a natural “internal composition” map c : [A, B ] ⊗ [C, A] → [C, B ]. Prove associativity (up to canonical isomorphism) for this c. (d) Show that the map ℓ = ℓA,B,C : [A, B ] → [C, A], [C, B ]

corresponding under π to internal composition (see (c)) is compatible with ordinary composition in M0 in that the following natural diagram (with Γ as in (a) and “Hom” meaning “set maps”) commutes: Γ[A, B ] ⏐ ⏐ Γ(ℓ)

11

−−− −→

composition

(A, B) −−−−−−−−→ ⏐ ⏐ functoriality of [C,−]

Hom (C, A), (C, B) ⏐ ⏐≃

functoriality Γ [C, A], [C, B ] −−− −→ [C, A], [C, B ] −−−−−−−−−→ Hom Γ[C, A], Γ[C, B ] of Γ

When M0 has direct sums, π gives rise to a distributivity isomorphism ∼ (A′ ⊕ A′′ ) ⊗ B −→ (A′ ⊗ B) ⊕ (A′′ ⊗ B)

whose consequences we will not follow up here—but see [L], [L′ ], [K′ ].

112

3 Derived Direct and Inverse Image

(e) From the sequence of functorial isomorphisms

α π D, [A ⊗ B, C ] −→ D ⊗ (A ⊗ B), C −→ (D ⊗ A) ⊗ B, C π π −→ D ⊗ A, [B, C ] −→ D, A, [B, C ]

deduce a functorial isomorphism

∼ p = pA,B,C : [A ⊗ B, C ] −→

A, [B, C ] .

(Take D := [A ⊗ B, C ].) Referring to (a), show that Γ(p) = π. In example (3.5.2)(d), does this p coincide with the isomorphism in (2.6.1)∗ ? (f) Let uAB : A → [B, A ⊗ B ] correspond under π to the identity map of A ⊗ B. Show that the map pA,B,C in (e) factors as via u ℓA⊗B,C,B [A ⊗ B, C ] −−−−−−→ [B, A ⊗ B ], [B, C ] −−−−−AB −→ A, [B, C ] .

with ℓ as in (d). Let tAB : [A, B ] ⊗ A → B correspond under π to the identity map of [A, B ]. Show that ℓA,B,C factors as p[C,A],C,B via tAC [A, B ] −−−−− → [C, A] ⊗ C, B −−−−−−→ [C, A], [C, B ] .

(g) The preceding exercises make no use of the symmetry isomorphism γ, but this one does. Construct functorial maps [A, B ] ⊗ [C, D] → [B, C ], [A, D] , [A, B ] ⊗ [C, D] → A ⊗ C, B ⊗ D .

using π, c and γ for the ﬁrst (see (c)), π, t and γ for the second (see (b)). (h) Let α : B → A be an M-map. Show that the following diagrams—in which unlabeled maps correspond under π to identity maps—commute for any C : via α

[A, C ] ⊗ B −−−−→ [A, C] ⊗ A ⏐ ⏐ ⏐ ⏐ via α [B, C ] ⊗ B −−−−→

C

via α

[B, C ⊗ A] ←−−−− [A, C ⊗ A] ⏐ ⏐ via α⏐ ⏐ [B, C ⊗ B] ←−−−−

C

Hint. For the ﬁrst diagram, consider the adjoint (via π) diagram, with D arbitrary, Hom([A, C] ⊗ B, D) ←−−−− Hom([A, C] ⊗ A, D) ⏐ ⏐ ⏐ ⏐

Hom([B, C] ⊗ B, D) ←−−−−

Hom(C, D)

Commutativity of the second diagram can be deduced from that of the ﬁrst (and vice-versa), or proved independently.

(3.5.4). Now let us see how f∗ and f ∗ interact with closed structures (assumed given) on X and Y. First we have a functorial map, with A, B ∈ X, f∗ [A, B ] −→ [f∗ A, f∗ B ]

(3.5.4.1)

3.5 Adjoint Functors between Closed Categories

113

corresponding under π (3.5.1.2) to the composed map f tAB f∗ [A, B ] ⊗ f∗ A −−−−−→ f∗ [A, B ] ⊗ A −−−∗−−−→ f∗ B . (3.5.3)(b)

(3.4.2.1)

Conversely (verify!), the functorial map f∗ (A ⊗ B) ←− f∗ A ⊗ f∗ B in (3.4.2.1) corresponds to the composition f uAB f∗ B, f∗ (A ⊗ B) ←−−−−− f∗ [B, A ⊗ B ] ←−∗−−−− f∗ A . (3.5.4.1)

(3.5.3)(f)

There results a functorial composition

via η

f∗ [f ∗A, B ] −−−−−→ [f∗ f ∗A, f∗ B ] −−−−→ [A, f∗ B ] , (3.5.4.1)

(3.4.3)

(3.5.4.2)

from which (verify!) (3.5.4.1) can be recovered as the composition via ǫ

f∗ [A, B ] −−−−→ f∗ [f ∗f∗ A, B ] −−−−−→ [f∗ A, f∗ B ] . (3.4.3)

(3.5.4.2)

The functors C → f ∗ (C ⊗ A) and C → f ∗ C ⊗ f ∗A (from Y to X) both have right adjoints, namely B → [A, f∗ B ] and B → f∗ [f ∗A, B ]. Hence there is a functorial map [A, f∗ B ] ←− f∗ [f ∗A, B ]

(3.5.4.3)

right-conjugate (see (3.3.5)) to the functorial map f ∗ (C ⊗ A) → f ∗ C ⊗ f ∗A in (3.4.5.1). Similarly, there is a functorial map f∗ [B, A] −→ [f∗ B, f∗ A]

(3.5.4.4)

right-conjugate to the adjoint f ∗ C ⊗ B ← f ∗ (C ⊗ f∗ B) of p2 (C, B) (3.4.6). If f ∗ (C ⊗ A) → f ∗ C ⊗ f ∗A—and hence its conjugate (3.5.4.3)—is a functorial isomorphism, then we have the functorial map f ∗ [A, B ] −→ [f ∗A, f ∗B ] which is adjoint to the composition η

(3.5.4.3)−1

[A, B ] −−−−→ [A, f∗ f ∗B ] −−−−−−−→ f∗ [f ∗A, f ∗B ] ;

(3.5.4.5)

114

3 Derived Direct and Inverse Image

and (verify!) (3.5.4.3)−1 is the map adjoint to the composition (3.5.4.5)

via ǫ

f ∗ [A, f∗ B ] −−−−−→ [f ∗A, f ∗f∗ B ] −−−→ [f ∗A, B ] , from which (3.5.4.5) can be recovered as the composition via η

f ∗ [A, B ] −−−−→ f ∗ [A, f∗ f ∗B ] −−−−→ [f ∗A, f ∗B ] . This all holds in the most relevant (for us) cases, see e.g., (3.4.4)(a), (b), and (3.5.2). Does the map in (3.5.4.3) (resp. (3.5.4.4)) coincide with that in (3.5.4.2) (resp. (3.5.4.1))? Of course, but it’s not entirely obvious: it amounts to commutativity of the respective diagrams in (3.5.5) below. (Cf. (3.2.4)(i), but recall that in proving (3.2.4)(i), we used (3.1.10), for whose last assertion, given (3.1.8), (3.5.5) provides a formal proof.)12 Proposition 3.5.5. The following functorial diagrams—where A, B, G ∈ X0 , E, F , C ∈ Y0 , HX , HY stand for HomX0 , HomY0 respectively, and with maps arising naturally from those deﬁned above—commute: HX f ∗ E, [f ∗ F , G] −− −−→ HX f ∗ E ⊗ f ∗ F , G ⏐ ⏐ ⏐(3.4.5.1) ⏐ ≃ HY E, f∗ [f ∗ F , G] HX f ∗ (E ⊗ F ), G ⏐ ⏐ ⏐≃ ⏐ (3.5.4.2) −− −−→ HY E ⊗ F , f∗ G HY E, [F , f∗ G]

−−− HY C, [f∗ B, f∗ A] HY C ⊗ f∗ B, f∗ A ←− ⏐(3.5.4.1) ⏐ ≃⏐ ⏐ ∗ HX f (C ⊗ f∗ B), A HY C, f∗ [B, A] ⏐≃ ⏐ (3.4.6.2)⏐ ⏐ ∗ ∗ HX f C ⊗ B, A ←− −−− HX f C, [B, A] 12

(3.5.5.1)

(3.5.5.2)

Diagram (3.2.4.6) is, in view of (3.1.8), an instance of (3.5.5.1). So is (3.2.4.5); but we don’t know that a priori, because we don’t know that the maps in (3.2.3.2) and (3.5.4.2) coincide until after proving either (3.2.4)(i) or the derived-category analog of (3.1.8), viz. (3.2.4)(ii)—in whose proof (3.5.5) was used.

3.5 Adjoint Functors between Closed Categories

115

The proof will be based on: Lemma 3.5.5.3. The following diagram (with preceding notation) commutes: natural (3.5.4.1) HX A, [B, G] −−−−−→ HY f∗ A, f∗ [B, G] −−−−−−→ HY f∗ A, [f∗ B, f∗ G] ⏐ ⏐ ⏐ ⏐≃ ≃ HX A ⊗ B, G −natural −−−−→ HY f∗ (A ⊗ B), f∗ G −−−−−−→ HY f∗ A ⊗ f∗ B, f∗ G (3.4.2.1)

Proof. Chasing a map ϕ : A → [B, G] around the diagram both clockwise and counterclockwise from upper left to lower right, one comes down to showing commutativity of the following diagram (with t as in (3.5.3(b)): f∗ ϕ ⊗ 1f

B

(3.5.4.1)

f∗ (ϕ ⊗ 1B )

f∗ (tBG )

∗ f∗ A ⊗ f∗ B −−−−−−− → f∗ [B, G] ⊗ f∗ B −−−−−→ [f∗ B, f∗ G] ⊗ f∗ B ⏐ ⏐ ⏐ ⏐tf B,f G ⏐ ⏐ (3.4.2.1) (3.4.2.1) ∗ ∗ f∗ G f∗ (A ⊗ B) −−−−−−−→ f∗ [B, G] ⊗ B −−−−−→

The left square commutes by functoriality, and the right one by the deﬁnition of (3.5.4.1). Q.E.D. ∗ Proof of (3.5.5). Expand (3.5.5.1) to (3.5.5.1.) , shown on the next page, where the map ξ is induced by the map ξ ′ : E ⊗ F → f∗ (f ∗ E ⊗ f ∗ F ) adjoint to f ∗ (E ⊗ F ) → f ∗ E ⊗ f ∗ F , see (3.4.5.1); and the other maps are the obvious ones. The outer border of (3.5.5.1)∗ commutes, by (3.5.5.3) (with A := f ∗ E, B := f ∗ F ). Hence if all the subdiagrams other than (3.5.5.1) commute, then so does (3.5.5.1), as desired. Commutativity of 1 follows from adjointness of f∗ and f ∗ . Commutativity of 2 follows from the deﬁnition (3.5.4.2) of the map f∗ [f ∗ F , G] → [F , f∗ G]. Commutativity of 3 follows from functoriality of π (3.5.1.2). Commutativity of 4 and of 5 result respectively from the following two factorizations of the map ξ ′ : η

E ⊗ F −−−−→

f∗ f ∗ (E ⊗ F )

η⊗η

(3.4.5.1)

−−−−−→ f∗ (f ∗ E ⊗ f ∗ F ) , (3.4.2.1)

E ⊗ F −−−−→ f∗ f ∗ E ⊗ f∗ f ∗ F −−−−−→ f∗ (f ∗ E ⊗ f ∗ F ) . Thus (3.5.5.1) does commute.

HY f∗ f ∗ E, [f∗ f ∗ F , f∗ G] HY f∗ f ∗ E, [f∗ f ∗ F , f∗ G] ⏐ ⏐≃ HY f∗ f ∗ E ⊗ f∗ f ∗ F , f∗ G HY f∗ f ∗ E ⊗ f∗ f ∗ F , f∗ G

116

HX f ∗ E, [f ∗ F , G] −−−→ HY f∗ f ∗ E, f∗ [f ∗ F , G] −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ ⏐ ⏐ 1 2 ∗ ←−−− −−−→ HY E, [F , f∗ G] HY E, f∗ [f ∗ F , G] HX f E, [f ∗ F , G] −−−→ ⏐ ⏐ ⏐ ⏐ ≃ ≃ (3.5.5.1) 3 ∗ ←−−− −−−→ HY E ⊗ F , f∗ G HX f ∗ (E ⊗ F ), G HX f E ⊗ f ∗ F , G −−−→ ⏐ ξ⏐ 4 5 ∗ HX f E ⊗ f ∗ F , G − −−−−−−−−−−−−−−−−−−−−−−−→ HY f∗ (f ∗ E ⊗ f ∗ F ), f∗ G −−−→

(3.5.5.1)*

(3.5.5.2)*

3 Derived Direct and Inverse Image

HX f ∗ C, [B, A] −−−−−−−−−−−−−−−−−−−−−−−−−→ HY C, f∗ [B, A] HY C, f∗ [B, A] ⏐ ⏐ ⏐ ⏐ 2 1 ∗ ∗ HX f C, [B, A] −−−→ HX f C, [f ∗ f∗ B, A] −−−→ HY C, f∗ [f ∗ f∗ B, A] −−−→ HY C, [f∗ B, f∗ A] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≃ ≃ ≃ 4 3 HX f ∗ C ⊗ B, A −−−→ HX f ∗ C ⊗ f ∗ f∗ B, A −−−→ HX f ∗ (C ⊗ f∗ B), A −−−→ HY C ⊗ f∗ B, f∗ A

3.5 Adjoint Functors between Closed Categories

117

Now look at (3.5.5.2)∗ , whose outer border is identical with (3.5.5.2). Subdiagrams 1 and 3 commute by functoriality. Commutativity of 2 comes from the statement immediately following (3.5.4.2). Subdiagram 4 is just (3.5.5.1) with E := C, F := f∗ B, G := A; so it commutes too. Thus (3.5.5.2)∗ commutes. Q.E.D. Exercises 3.5.6. (a) Show that if the natural map f ∗ (C ⊗ A) → f ∗ C ⊗ f ∗A is an isomorphism for all C and A, then (3.5.4.5) corresponds under π (see (3.5.1.2)) to ∼ the natural composite map f ∗ [A, B ] ⊗ f ∗A −→ f ∗ [A, B ] ⊗ A −→ f ∗B . (b) Given a ﬁxed map e : B ′ → B, show that the functorial maps via e

f∗ [B, A] −−−→ f∗ [B ′ , A]

and

via e

f ∗ C ⊗ B ←−−− f ∗ C ⊗ B ′

are conjugate; and then deduce the equality of the maps (3.5.4.1) and (3.5.4.4) from that of (3.5.4.2) and (3.5.4.3). (c) In (3.5.5.i) (i = 1, 2, 3), replace HX (−, −) by f∗ [−, −], and HY (−, −) by [−, −]. Show that the resulting diagrams commute. (For example, reduce to commutativity of (3.5.5.i), by applying the functor HY (D, −) to the diagram in question.) Show that (3.5.5.i) can be recovered from “the resulting diagram” by application of the functor ΓY := HY (OY , −) of (3.5.3)(a). (d) By Yoneda’s principle, commutativity of (3.5.5.1) can be proved by taking E = f∗ [f ∗ F , G] and chasing the identity map of f∗ [f ∗ F , G] around the diagram in both directions. Deduce that commutativity of (3.5.5.1) is equivalent to that of the diagram tF ,f G f ∗ f∗ [f ∗ F , G] ⊗ F −−−−−−→ f ∗ [F , f∗ G] ⊗ F −−−−−∗−−→ f ∗f∗ G (3.5.3)(b) (3.5.4.2) ⏐ ⏐ ⏐ ⏐ǫ (3.4.5.1)

f ∗f∗ [f ∗ F , G] ⊗ f ∗ F

(3.4.3)

−−via −−− → ǫ

[f ∗ F , G] ⊗ f ∗ F

G

−− −→ ∗F ,G tf−

(e) In a closed category X the natural composite functorial map ∼ ∼ Hom(F , G) −→ Hom(F ⊗ OX , G) −→ Hom(F , [OX , G]),

being an isomorphism, takes (when F = G) the identity map of G to an isomorphism ∼ G −→ [OX , G]. Let Y be another closed category, and (f ∗ , f∗ ) be as in (3.4.3). Show that for G ∈ X and E ∈ Y the following natural diagrams commute: f∗ [OX , G] −−−−→ [f∗ OX , f∗ G] ⏐ ⏐ ⏐ ⏐ ≃ f∗ G

−−− −→

[OY , f∗ G]

f ∗ [OY , E ] −−−−→ [f ∗ OY , f ∗ E ] ⏐ ⏐ ⏐ ⏐≃ ≃ f ∗E

−−− −→

[OX , f ∗ E ]

Hint. The ﬁrst diagram is right-conjugate to the dual (3.4.5) of (3.4.7)(iii). For the second diagram, use, e.g., (a) above. (f) With notation as in (e), and πX , πY as in (3.5.1.2), and assuming the functorial map f ∗ (C ⊗ D) → f ∗ C ⊗ f ∗ D in (3.4.5.1) to be a functorial isomorphism, show that πX takes the inverse of the isomorphism f ∗ (G ⊗ B) → f ∗ G ⊗ f ∗ B to the composite map natural

(3.5.4.5)

f ∗ G −−−−−→ f ∗ [B, G ⊗ B] −−−−−−→ [f ∗B, f ∗ (G ⊗ B)], or, equivalently, that the following diagram commutes.

118

3 Derived Direct and Inverse Image (3.4.5.1)−1

[f ∗ B, f ∗ (G ⊗ B)] ←−−−−−−− [f ∗B, f ∗ G ⊗ f ∗ B] ⏐ ⏐via π (3.5.4.5)⏐ ⏐ X f ∗ [B, G ⊗ B]

f ∗G

←−−−−−−−− via πY

(g) With assumptions as in (f), and using the commutative diagram in (f)—or otherwise—show that for any Y-map α : C ⊗ D → E, and αf the composite map (3.4.5.1)−1

f ∗α

f ∗C ⊗ f ∗D −−−−−−−→ f ∗ (C ⊗ D) −−−−→ f ∗E, f it holds that πX α is the composite map f ∗ (π α)

(3.5.4.5)

Y −→ f ∗ [D, E ] −−−−−−→ [f ∗D, f ∗E ]. f ∗C −−−−−

3.6 Adjoint Monoidal Δ-Pseudofunctors We review next the behavior of derived direct and inverse image functors f g vis-`a-vis a pair of ringed-space maps X −→ Y −→ Z. First, relative to the categories of OX - (OY - , OZ -) modules we have the functorial isomorphism (in fact equality) ∼ (gf )∗ −→ g∗ f∗

(3.6.1)∗

and hence, since f ∗ g ∗ is left-adjoint to g∗ f∗ and (f g)∗ is left-adjoint to (gf )∗ there is a unique functorial isomorphism ∼ f ∗ g ∗ −→ (gf )∗

(3.6.1)∗

such that the following natural diagram of functors commutes: 1 ⏐ ⏐

−−−−→

g∗ g ∗

−−−−→ g∗ (f∗ f ∗ g ∗ )

(3.6.2)

(gf )∗ (gf )∗ −− −−→ g∗ f∗ (gf )∗ ←− −−− g∗ f∗ f ∗ g ∗

or, equivalently, such that the “dual” diagram 1 ⏐ ⏐

←−−−−

f ∗f∗

←−−−− f ∗ (g ∗ g∗ f∗ )

(gf )∗ (gf )∗ ←− −−− f ∗ g ∗ (gf )∗ −− −−→ f ∗ g ∗ g∗ f∗

(3.6.2)op

commutes. (This statement follows from (3.3.5), see also (3.3.7)(a)).

3.6 Adjoint Monoidal Δ-Pseudofunctors

119

h

Given a third map Z −→ W , we have the commutative diagram of functorial isomorphisms (actually equalities) (hgf )∗ −−−−→ (hg)∗ f∗ ⏐ ⏐ ⏐ ⏐

(3.6.3)∗

h∗ (gf )∗ −−−−→ h∗ g∗ f∗

from which we deduce formally, via adjunction, a commutative diagram of functorial isomorphisms (hgf )∗ ←−−−− f ∗ (hg)∗ ⏐ ⏐ ⏐ ⏐

(3.6.3)∗

(gf )∗ h∗ ←−−−− f ∗ g ∗ h∗

From these observations we can derive similar ones involving the corresponding derived functors. Indeed, taking U := g −1 V (V open ⊂ Z) in (3.2.3.3), we ﬁnd that f∗ B is g∗ -acyclic for any q-injective B ∈ K(X), whence, by (2.2.7), there is a unique Δ-functorial isomorphism ∼ R(gf )∗ −→ Rg∗ Rf∗

(3.6.4)∗

making the following natural diagram commute: (gf )∗ −− −−→ g∗ f∗ −−−−→ (Rg∗ )f∗ ⏐ ⏐ ⏐ ⏐

(3.6.4.1)

R(gf )∗ −−−−−−−− −−−−−−−→ Rg∗ Rf∗

This allows us to build a diagram analogous to (3.6.3)∗ , with Re∗ in place of e∗ for each map e involved. The resulting derived functor diagram still commutes, as can be seen by reduction (via suitable quasi-isomorphisms) to the case of q-injective complexes in D(X), for which the diagram in question is essentially (3.6.3)∗ . In a parallel fashion, using q-ﬂat instead of q-injective complexes, and recalling that f ∗ transforms q-ﬂat complexes into q-ﬂat complexes (see proof of (3.2.4)(i)), etc., we get a natural Δ-functorial isomorphism ∼ Lf ∗ Lg ∗ −→ L(gf )∗ ,

(3.6.4)∗

and a commutative diagram analogous to (3.6.3)∗, with Le∗ in place of e∗ . By (3.3.5), we also have commutative diagrams like (3.6.2) and (3.6.2)op , with f∗ , f ∗ etc. replaced by their respective derived functors.

120

3 Derived Direct and Inverse Image

It is helpful to conceptualize some of the foregoing, as follows, leading up to (3.6.10). We begin with some standard terminology.13 (3.6.5). Let S be a category. A covariant pseudofunctor # on S assigns to each object X ∈ S a category X# , to each map f : X → Y in S a functor f# : X# → Y# , with f# the identity functor if X = Y and f = 1X , and to f

g

each pair of maps X −→ Y −→ Z in S an isomorphism of functors ∼ cf,g : (gf )# −→ g# f#

such that 1) c1,g = cf,1 = identity, and f

g

h

2) for any triple of maps X −→ Y −→ Z −→ W the following diagram commutes: cf,hg (hgf )# −−−−→ (hg)# f# ⏐ ⏐ ⏐cg,h cgf,h ⏐ (3.6.5.1) h# (gf )# −−−−→ h# g# f# cf,g

Similarly, a contravariant pseudofunctor on S assigns to each X ∈ S a category X# , to each map f : X → Y a functor f # : Y# → X# (with 1# = 1), f

g

and to each map-pair X −→ Y −→ Z a functorial isomorphism df,g : f # g # → (gf )# satisfying d1,g = dg,1 = identity, and such that for each triple of maps f

g

h

X −→ Y −→ Z −→ W the following diagram commutes: df,hg

(hgf )# ←−−−− f # (hg)# ⏐d ⏐ dgf,h ⏐ ⏐ g,h

(3.6.5.2)

(gf )# h# ←−−−− f #g # h# df,g

There is an obvious way of identifying contravariant pseudofunctors on S with pseudofunctors on the dual category Sop . (3.6.6). Given covariant pseudofunctors * and # with X* = X# for all X ∈ S, a morphism of pseudofunctors * → # is a family of morphisms of functors αf : f∗ → f# f

g

(one for each map f in S) such that for any pair of maps X −→ Y −→ Z,

13

Pseudofunctors can also be interpreted as 2-functors.

3.6 Adjoint Monoidal Δ-Pseudofunctors

121

the following diagram commutes: αgf

(gf )∗ −−−−−−−−−−−−−−→ (gf )# ⏐ ⏐ ⏐c cf,g ⏐ f,g g∗ f∗ −− −→ g∗ f# −−− −→ g# f# g− ∗ αf α g

and such that for all X ∈ S, with identity map 1X , α1X : (1X )∗ → (1X )# is the identity automorphism of X* = X# . Morphisms of contravariant pseudofunctors are deﬁned analogously. Suppose we are given a pseudofunctor *, and functors f# : X* → Y* , one for each S-morphism f : X → Y , such that f# is an identity functor whenever f is an identity map, and a family of functorial isomorphisms αf : f∗ → f# . It is left as an exercise to show that then there is a unique ∼ family of isomorphisms of functors cf,g : (gf )# −→ g# f# which together with the family (f# ) constitute a pseudofunctor such that the family (αf ) is an isomorphism of pseudofunctors. (3.6.7). Various reﬁnements of these notions can be made. (a). Assume that each category X# is a Δ-category, that each f# (resp. f # ) is a Δ-functor, and that each cf,g (resp. df,g ) is an isomorphism of Δ-functors. We say then that # is a covariant (resp. contravariant) Δ-pseudofunctor. A morphism of Δ-pseudofunctors is then a family αf as in (3.6.6), with each αf a morphism of Δ-functors. (b). Assume that each category X# is a symmetric monoidal category, see (3.4.1), that each f# is a symmetric monoidal functor (3.4.2), and that each cf,g is a morphism of symmetric monoidal functors [EK, p. 474], i.e., that the following natural diagrams commute (where ⊗ denotes the appropriate product functor, and O the unit; and A, B ∈ X# ): OZ ⏐ ⏐

−−−−→ (gf )# OX ⏐ ⏐

(3.6.7.1)

g# OY −−−−→ g# f# OX

(gf )# A ⊗ (gf )# B −−−−−−−−−−−−−−−−−−−−−→ (gf )# (A ⊗ B) ⏐ ⏐ ⏐ ⏐

(3.6.7.2)

g# f# A ⊗ g# f# B −−−→ g# (f# A ⊗ f# B) −−−→ g# f# (A ⊗ B)

We say then that # is a monoidal pseudofunctor. We say that a contravariant pseudofunctor # is monoidal if for each map f : X → Y in S, the opposite functor (f # )op : (Y# )op → (X# )op is monoidal. In other words, we have functorial maps f # (A ⊗ B) → f #A ⊗ f #B

122

3 Derived Direct and Inverse Image

and a map f # OY → OX satisfying the obvious conditions. A morphism of monoidal pseudofunctors is a family αf as in (3.6.6) such that each αf is a morphism of symmetric monoidal functors (i.e., αf is compatible, in an obvious sense, with the maps (3.4.2.1). (c). If every X# is both a Δ-category and a symmetric monoidal category, and if the multiplication X# × X# → X# is a Δ-functor (see (2.4.3)), then we say that X# is a monoidal Δ-category; and we speak correspondingly of monoidal Δ-pseudofunctors and their morphisms. (d). A pair ( * , * ) with * a pseudofunctor and * a contravariant pseudofunctor on S are said to be adjoint if the following conditions hold: (i) X* = X* for all objects X in S. (ii) For every f : X → Y in S there are bifunctorial isomorphisms ∼ HomX* (f ∗ C, D) −→ HomY* (C, f∗ D) (C ∈ Y* , D ∈ X* ), i.e., the functor f∗ : X* → Y* is right adjoint to f ∗ : Y* → X* . (iii) The resulting functorial diagrams (3.6.2) (or (3.6.2)op ) commute. In the monoidal case, we also require: (iv) The natural maps f∗ (A) ⊗ f∗ (B) → f∗ (A ⊗ B), f ∗ (f∗ A ⊗ f∗ B) → f ∗f∗ A ⊗ f ∗f∗ B → A ⊗ B correspond under the adjunction isomorphism of (ii) above, as do the natural maps f ∗ OY → OX , OY → f∗ OX . In the Δ-case, we also require thatf ∗ and f∗ be Δ-adjoint (3.3.1), i.e., (v) The natural functorial morphisms 1 → f∗ f ∗

and

f ∗f∗ → 1

are both morphisms of Δ-functors. (3.6.8). We add some remarks on existence and uniqueness, some of which are relevant to the subsequent construction and understanding of speciﬁc adjoint pairs of pseudofunctors. (3.6.8.1). If * is a monoidal pseudofunctor on S, and if for each map f : X → Y in S the functor f∗ : X* → Y* has a left adjoint f ∗ , then there is a unique contravariant monoidal pseudofunctor * on S such that X* = X* for all objects X ∈ S, f ∗ is the said left adjoint for each f : X → Y , and the pair ( * , * ) is adjoint. Indeed, condition (iii) in (d) above forces df,g : f ∗ g ∗ → (gf )∗ to be the left conjugate of the given cf,g : (gf )∗ → g∗ f∗ (see beginning of this section, up to (3.6.3)∗ ). Similarly, (iv) imposes a unique monoidal structure on (f ∗ )op : given (ii), we see as in (3.4.5) that (iv) is equivalent to the following dual statement:

3.6 Adjoint Monoidal Δ-Pseudofunctors

123

(iv)′ The natural maps f ∗ (A) ⊗ f ∗ (B) ← f ∗ (A ⊗ B), f∗ (f ∗A ⊗ f ∗B) ← f∗ f ∗A ⊗ f∗ f ∗B ← A ⊗ B correspond under the above adjunction isomorphism (ii), as do the natural maps f∗ OX ← OY ,

OX ← f ∗ OY .

The rest of the proof is left to the reader. (3.6.8.2). If * is a Δ-pseudofunctor on S, and if for each map f : X → Y in S the functor f∗ : X* → Y* has a left adjoint f ∗ , then there is a unique contravariant Δ-pseudofunctor * on S such that X* = X* for all objects X ∈ S, f ∗ is the said left adjoint for each f : X → Y , and the pair ( * , * ) is adjoint. Indeed, by (3.3.8), each f ∗ carries a unique structure of Δ-functor such that f g (v) above holds; and for X −→ Y −→ Z in S, the isomorphism df,g —forced by (iii) to be the conjugate of the given Δ-functorial isomorphism cf,g —is Δ-functorial, by (3.3.6). (3.6.8.3). Here is another form of uniqueness: If ( * , * ) and ( # , * ) are adjoint pairs of monoidal (or Δ-)pseudofunctors, and if for each f : X → Y we deﬁne the morphism αf : f ∗ → f # to be adjoint to the natural morphism 1 → f∗ f # , then the family αf is an isomorphism of monoidal (or Δ-)pseudofunctors. Remarks 3.6.9 (Duality principle II). To each adjoint pair of monoidal pseudofunctors ( * , * ) on S, (3.6.7)(d), associate a dual pair ( # , # ) of monoidal pseudofunctors on the dual category Sop as follows: X# := (X* )op ,

X# := (X* )op

for objects X ∈ Sop , and f # := (f∗ )op : (X* )op → (Y* )op ,

f# := (f ∗ )op : (Y* )op → (X* )op

for each map f : Y → X in Sop (i.e., for each map f : X → Y in S), the ∼ ∼ (gf )# and (gf )# −→ g# f# being the obvious ones. isomorphisms f #g # −→ The monoidal structure on the category X# = X# is deﬁned to be dual to that on X* = X* see (3.4.5), and then each functor f# is monoidal, with left adjoint f # . It follows that: Each diagram built up from the basic data deﬁning adjoint monoidal pairs can be interpreted in the dual context, giving rise to a “dual” diagram, obtained by interchanging * and * and reversing arrows, etc., etc. This somewhat imprecise statement will be illustrated in Ex. (3.7.1.1) and in the proof of Prop. (3.7.2) below.

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3 Derived Direct and Inverse Image

(3.6.10). With the terminology of (3.6.7), and with (3.5.2)(d) in mind, we can formally summarize many preceding results as follows. Scholium. Let S be the category of ringed spaces. For each object X ∈ S, set X* = X* := D(X) (the derived category of the category of OX -modules), , unit OX , and internal hom RHom. a closed Δ-category with product ⊗ = f

g

For X −→ Y −→ Z in S, write f ∗ for Lf ∗ : Y* → X* , f∗ for Rf∗ : X* → Y* ,

df,g for the map (3.6.4)∗ , cf,g for the map (3.6.4)∗ .

This deﬁnes an adjoint pair ( * , * ) of monoidal Δ-pseudofunctors on S. Proof. Essentially everything has already been proved, in (3.4.4)(b) and at the beginning of this §3.6, except for the commutativity of (3.6.7.1) and (3.6.7.2) (with ∗ in place of # ). Commutativity of (3.6.7.1) is left to the reader. To show that (3.6.7.2) commutes, ﬁrst do it in the context of sheaves of modules—with the ordinary direct image functors see (3.1.7)—where it follows easily from deﬁnitions. A formal argument, using (iv) or (iv)′ above (details left to the reader), then yields the commutativity of the corresponding (dual) sheaf diagram with ∗ in place of ∗ , and all arrows reversed. In this latter diagram, we can then replace f ∗ etc. by Lf ∗ , etc., and commutativity is preserved since the resulting derived functor diagram need only be checked when A and B are q-ﬂat complexes, in which case it does not diﬀer essentially from the original sheaf diagram. Finally, the preceding formal (adjunction) argument, applied this time to derived functors, gives us commutativity in (3.6.7.2).

3.7 More Formal Consequences: Projection, Base Change We give some additional consequences, to be used later, of the formalism in §3.6. Again, the introductory remarks in §3.4, suitably modiﬁed, are relevant. We consider an adjoint monoidal pair (* , * ) as in (d) of (3.6.7). Condition (ii) there means that for f : X → Y in S, we have functorial maps ǫ : f ∗f∗ → 1 η : 1 → f∗ f ∗ , such that the resulting compositions η

ǫ

f∗ − → f ∗f∗ f ∗ − → f ∗, are both identities.

η

ǫ

f∗ − → f∗ f ∗f∗ − → f∗

3.7 More Formal Consequences: Projection, Base Change

125

For X ∈ S, the product functor on the monoidal category X* = X* will be denoted by ⊗. For a map f : X → Y in S, the functorial “projection” map pf : G ⊗ f∗ F → f∗ (f ∗ G ⊗ F )

(G ∈ Y*, F ∈ X* )

is deﬁned as in (3.4.6). It is compatible with pseudofunctoriality, in the following sense. f

g

Proposition 3.7.1 For any X − →Y − → Z in S, the following diagram, with F ∈ X* and G ∈ Z* , commutes. pg

G ⊗ g∗ f∗ F −−−−→ ⏐ via cf,g ⏐≃

g∗ (g ∗ G ⊗ f∗ F )

g∗ (pf )

−−−−→ g∗ f∗ (f ∗ g ∗ G ⊗ F ) ⏐ ⏐via d f,g

G ⊗ (gf )∗ F −−p−gf−→ (gf )∗ ((gf )∗ G ⊗ F ) −−c − −→ g∗ f∗ ((gf )∗ G ⊗ F ) f,g

Proof. An expanded form of the diagram is obtained by pasting the ﬁrst of the following diagrams, along its right edge, to the second, along its left edge. (All the arrows have an obvious interpretation.) G ⊗ g∗ f∗ F ⏐ ⏐

G ⊗ (gf )∗ F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 ⏐ ⏐ ⏐

−→

−→

g∗ g ∗ G ⊗ g∗ f∗ F ⏐ ⏐

g∗ g ∗ G ⊗ (gf )∗ F ⏐ ⏐

g∗ g ∗ G ⊗ g∗ f∗ F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

g∗ f∗ f ∗ g ∗ G ⊗ (gf )∗ F −→ g∗ f∗ f ∗ g ∗ G ⊗ g∗ f∗ F ⏐ ⏐ ⏐ ⏐

(gf )∗ (gf )∗ G ⊗ (gf )∗ F −→ g∗ f∗ (gf )∗ G ⊗ (gf )∗ F −→ g∗ f∗ (gf )∗ G ⊗ g∗ f∗ F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 g∗ (f∗ (gf )∗ G ⊗ f∗ F ) ⏐ ⏐ ⏐ ⏐ ⏐ (gf )∗ ((gf )∗ G ⊗ F )

−−−−−−−−−−−−−−−−−−−−−→

g∗ f∗ ((gf )∗ G ⊗ F )

126

3 Derived Direct and Inverse Image g∗ g ∗ G ⊗ g∗ f∗ F ⏐ ⏐

−−−−−→

g∗ (g ∗ G ⊗ f∗ F ) ⏐ ⏐

∗ ∗ g∗ f∗ f ∗ g ∗ G ⏐ ⊗ g∗ f∗ F −−−−−→ g∗ (f∗ f g G ⊗ f∗ F ) ⏐ ∗ g∗ f∗ (gf ) G ⊗ g∗ f∗ F ⏐ ⏐

g∗ (f∗ (gf )∗ G ⊗ f∗ F ) ←−−−−− g∗ (f∗ f ∗ g ∗ G ⊗ f∗ F ) ⏐ ⏐ ⏐ ⏐ g∗ f∗ ((gf )∗ G ⊗ F )

←−−−−−

g∗ f∗ (f ∗ g ∗ G ⊗ F )

Subdiagram 1 commutes because of commutativity of (3.6.2) (see condition (iii) in (3.6.7)(d)), Subdiagram 2 commutes because of the commutativity of (3.6.7.2) (which is part of the deﬁnition of monoidal pseudofunctor); and commutativity of the remaining subdiagrams is clear. The conclusion follows. Exercise 3.7.1.1 The preceding Proposition expresses the compatibility of the projection map with the structure “adjoint pair of monoidal pseudofunctors.” One can ask about similar compatibilities for any of the maps introduced in §3.5. Here are some examples which will be needed later. (Challenge: Establish metaresults of which such examples would be instances.) With notation as in (3.7.1), ̺ as in (3.5.4.1), and βf : f∗ [f ∗ −, −] → [−, f∗ −] as in (3.5.4.2) or (3.5.4.3), show that the following diagrams commute: βgf

(gf )∗ [(gf )∗ G, F ] −−−−−−−−−−−−−−−−−−−−−−−→ [G, (gf )∗ F ] ⏐ ⏐ ⏐via c via cf,g ⏐ and df,g f,g g∗ f∗ [f ∗g ∗ G, F ] −−−−→ g∗ [g ∗ G, f∗ F ] −−−−→ [G, g∗ f∗ F ] g∗ βf

f∗̺g

f∗g∗ [F , F ′ ] −−−−→ ⏐ −1 ⏐ cf,g

f∗ [g∗ F , g∗F ′ ]

βg

̺f

−−−−→ [f∗g∗ F , f∗g∗F ′ ] ⏐ ⏐via c−1 f,g

(gf )∗ [F , F ′ ] −−̺−−→ [(gf )∗ F , (gf )∗F ′ ] −−−−−→ [f∗g∗ F , (gf )∗F ′ ] gf

−1 via cf,g

Deduce from the ﬁrst diagram that with ρf : f ∗ [−, −] → [f ∗ −, f ∗ −] as in (3.5.4.5), the next diagram commutes:

3.7 More Formal Consequences: Projection, Base Change f ∗ρg

f ∗g ∗ [G, G′ ] −−−−→ ⏐ ⏐ df,g ≃

127

ρf

f ∗ [g ∗ G, g ∗G′ ]

−−−−→ [f ∗g ∗ G, f ∗g ∗G′ ] ⏐ ⏐ ≃via df,g

(gf )∗ [G, G′ ] −−ρ−gf−→ [(gf )∗ G, (gf )∗ G′ ] −via −− − −→ [f ∗g ∗ G, (gf )∗ G′ ] d f,g

Hints. Apply (3.6.9) to the diagram in (3.6.7.2), resp. Prop. (3.7.1), and compare the result with the diagram left-conjugate to the ﬁrst, resp. second, one above. The third diagram expands naturally as follows.

∗ ∗

′

∗ ∗

′

∗ ∗

∗

′

∗ ∗

∗

∗

′

∗

∗

∗

′

f g [G, G ] −→ f g [G, g∗ g G ] −→ f g g∗ [g G, g G ] −→ f [g G, g G ] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ∗ ∗ ∗ ′ ∗ ∗ ∗ ∗ ∗ ′ ∗ ∗ ∗ ∗ ′ f → f → f g [G, g f f g G ] g g [g G, f f g G ] [g G, f∗ f g G ] ∗ ∗ ∗ ⏐ ⏐ ∗ ⏐ ⏐ ⏐ ⏐ ∗ ∗

∗

′

f g [G, G ] → f g [G, (gf )∗ (gf ) G ] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗

′

∗

∗

′

∗ ∗

∗ ∗

∗ ∗

∗

′

∗ ∗

∗ ∗

′

f g g∗ f∗ [f g G, f g G ] → f f∗ [f g G, f g G ] ⏐ ⏐ ⏐ ⏐ ∗

∗ ∗

∗ ∗

′

∗ ∗

∗ ∗

′

∗

∗

′

(gf ) (gf )∗ [f g G, f g G ] → [f g G, f g G ] ⏐ ⏐ ⏐ ⏐ ∗

∗

∗

′

(gf ) [G, G ] → (gf ) [G,(gf )∗ (gf ) G ] → (gf ) (gf )∗[(gf ) G,(gf ) G ] → [(gf ) G,(gf ) G ]

In this diagram, all but three subdiagrams clearly commute, and those three are taken care of by (3.6.2), (3.6.2)op , and the ﬁrst diagram above.

Next, we introduce an oft-to-be-encountered “base change” morphism. Proposition 3.7.2 (i) To each commutative square σ in S: g′

X ′ −−−−→ ⏐ ⏐ f ′

X ⏐ ⏐f

Y ′ −−−g−→ Y

there is associated a natural map of functors θ = θσ : g ∗f∗ −→ f∗′ g ′∗ , equal to each of the following four compositions (with h = f g ′ = gf ′ ) : η

(cf ′,g )(c−1 ) g ′,f

ǫ

(a)

g ∗f∗ − → g ∗f∗ g∗′ g ′∗ −−−−−−−−→ g ∗ g∗ f∗′ g ′∗ − → f∗′ g ′∗

(b)

→ f∗′ g ′∗ g ∗f∗ −−−→ f∗′ f ′∗ g ∗f∗ g∗′ g ′∗ −−−−−−−−→ f∗′ h∗ h∗ g ′∗ −

η( )η

(df ′,g )(c−1 ) g ′,f

ǫ

128

3 Derived Direct and Inverse Image (d−1 )(df ′,g ) g ′,f

η

ǫ

(c)

g ∗f∗ − → f∗′ g ′∗ → f∗′ f ′∗ g ∗f∗ −−−−−−−−→ f∗′ g ′∗ f ∗f∗ −

(d)

g ∗f∗ − → g ∗ h∗ h∗f∗ −−−−−−−−→ g ∗ g∗ f∗′ g ′∗ f ∗f∗ −−−→ f∗′ g ′∗

(cf ′,g )(d−1 ) g ′,f

η

ǫ( )ǫ

(ii) Given a pair of commutative squares g′

X ′ −−−−→ ⏐ ⏐ f ′

Y ′ −−−g−→ ⏐ ⏐ h′

X ⏐ ⏐f

Y ⏐ ⏐ h

Z ′ −−−′′−→ Z g

the following resulting diagram commutes: θ

g ′′∗ (hf )∗ −−−−−−−−−−−−−−−−−→ (h′ f ′ )∗ g ′∗ ⏐ ⏐ ⏐cf ′,h′ cf,h ⏐ g ′′∗ h∗ f∗ −−−−→ h′∗ g ∗f∗ −−−−→ h′∗ f∗′ g ′∗ θ

θ

(iii) Given a pair of commutative squares h

X ′′ −−−−→ ⏐ ⏐ g ′′

f

X ′ −−−−→ ⏐ ⏐g

X ⏐ ⏐ ′ g

Y ′′ −−−− → Y ′ −−−− → Y ′ ′ h

f

the following resulting diagram commutes: θ

g∗′′ (f h)∗ ←−−−−−−−−−−−−−−−−− (f ′ h′ )∗ g∗′ ⏐d ′ ′ ⏐ dh,f ⏐ ⏐ h ,f g∗′′ h∗f ∗ ←−−−− h′∗ g∗ f ∗ ←−−−− h′∗ f ′∗ g∗′ θ

θ

Proof. (i) To get convinced that (a), (b) and (c) are the same, contemplate the following commutative diagram, noting that ǫ◦η on the right (resp. bottom) edge is the identity map, and recalling for subdiagrams 1 and 2 the condition (iii) in the deﬁnition (3.6.7)(d) of “adjoint pair.”

3.7 More Formal Consequences: Projection, Base Change

g ∗f∗ ⏐ ⏐ η

−−→

g ∗f∗ g∗′ g ′∗ ⏐ ⏐ η

−−→

129

g ∗ g∗ f∗′ g ′∗ ⏐ ⏐ η

−−→

f∗′ g ′∗ ⏐ ⏐η

f∗′ f ′∗ g ∗f∗ −−→ f∗′ f ′∗ g ∗f∗ g∗′ g ′∗ −−→ f∗′ f ′∗ g ∗ g∗ f∗′ g ′∗ −−→ f∗′ f ′∗ f∗′ g ′∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 ⏐ ⏐ ⏐ ⏐ǫ f∗′ g ′∗ f ∗f∗ −−→ f∗′ g ′∗ f ∗f∗ g∗′ g ′∗ −−→ f∗′ h∗ h∗ g ′∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 ǫ

f∗′ g ′∗

−−→ η

f∗′ g ′∗ g∗′ g ′∗

−−−−−−−−−−−−−−−−−−→ ǫ

f∗′ g ′∗

The equality (c) = (d) is obtained from (a) = (b) by duality (3.6.9). (ii) Consider the expanded diagram (3.7.2.2) on the following page. Recall that the composition ǫ◦η of the adjacent arrows in the middle is the identity. Commutativity of subdiagram 1 is an easy consequence of the commutativity of (3.6.5.1) (axiom for pseudofunctors). Commutativity of the other subdiagrams is straightforward, and the conclusion follows. (iii) is simply the dual of (ii) (see (3.6.9)). Q.E.D. Proposition 3.7.3 (Base change and projection). Let g′

X ′ −−−−→ ⏐ ⏐ f ′

X ⏐ ⏐f

Y ′ −−−−→ Y g

be a commutative S-diagram, P ∈ Y* , Q ∈ X* . Then with θ as in (3.7.2), h = f g ′ = gf ′ , and p the projection map, the following diagram commutes: g ∗ P ⊗ g ∗f∗ Q ⏐ ⏐ 1⊗θ

g ∗ P ⊗ f∗′ g ′∗ Q ⏐ pf ′ ⏐

(3.4.5.1)

←−−−−−

g ∗ (P ⊗ f∗ Q)

g ∗ (pf )

−−−−→

g ∗f∗ (f ∗ P ⊗ Q) ⏐ ⏐ θ

f∗′ g ′∗ (f ∗ P ⊗ Q) ⏐ ⏐(3.4.5.1)

f∗′ (f ′∗ g ∗ P ⊗ g ′∗ Q) −−−−→ f∗′ (h∗ P ⊗ g ′∗ Q) ←−−−− f∗′ (g ′∗ f ∗ P ⊗ g ′∗ Q) df ′,g

dg′,f

Proof. Consider the expanded diagram (3.7.3.1) on the following page (a diagram in which the arrows are self-explanatory). With a bit of patience, one checks that it suﬃces to show its commutativity.

g ′′∗ h∗ f∗ ⏐ ⏐ η

−−→

−−→

g ′′∗ (hf )∗ g∗′ g ′∗ −−−−−−−−−−−−−−−−−−−−−−−−→ g ′′∗ g∗′′ (h′ f ′ )∗ g ′∗ ⏐ ⏐ ⏐ ⏐ 1 g ′′∗ h∗ f∗ g∗′ g ′∗ ⏐ ⏐ η

−−→

′ ′∗ g ′′∗ h∗⏐g ∗ f∗ g ⏐⏐ η ⏐ǫ

−−→

g ′′∗ g∗′′ h′∗ f∗′ g ′∗ ⏐ǫ ⏐

−−→

−−→

130

g ′′∗ (hf )∗ ⏐ ⏐

(h′ f ′ )∗ g ′∗ ⏐ ⏐ h′∗ f∗′ g ′∗ ⏐ǫ ⏐

(3.7.2.2)

g ′′∗ h∗ g∗ g ∗f∗ −−→ g ′′∗ h∗ g∗ g ∗f∗ g∗′ g ′∗ −−→ g ′′∗ h∗ g∗ g ∗ g∗ f∗′ g ′∗ −−→ g ′′∗ g∗′′ h′∗ g ∗ g∗ f∗′ g ′∗ −−→ h′∗ g ∗ g∗ f∗′ g ′∗ ⏐ ⏐ ⏐ ⏐ ǫ

g ′′∗ g∗′′ h′∗ g ∗f∗ −−−−−−−−−−− −− −−−−−−−−−−−−−→

←−−

∗ g ∗P ⊗ g⏐ h∗ g ′∗ Q ⏐

g ∗(P ⊗ f∗ Q) ⏐ ⏐

g ∗f∗ (f ∗P ⊗ Q) ⏐ ⏐

−−→

←−−

g ∗ (P ⊗ f∗ g∗′ g ′∗ Q) ⏐ ⏐

g ∗P ⊗ g ∗⏐g∗ f∗′ g ′∗ Q ←−− ⏐ 4

g ∗ (P ⊗ g⏐∗ f∗′ g ′∗ Q) ⏐

3

g ∗P ⊗ f∗′ g ′∗ Q

−−−−−−−−−−−−−−−−−−−−−−−→

g ∗P ⊗ f∗′ g ′∗ Q

g ∗P ⊗ f∗′ g ′∗ Q

g ∗f∗ g∗′ g ′∗(f ∗P ⊗ Q) ⏐ ⏐

−−→ g ∗ g∗ f∗′ g ′∗(f ∗P ⊗ Q) ⏐ ⏐ 1 ⏐ ⏐ ⏐ ∗ ∗ ′ ′∗ ∗ ′ ′∗ ∗ ′∗ ⏐ −−→ g f∗ (f P ⊗ g∗ g Q) −−→ g f∗ g∗ (g f P ⊗ g Q) ⏐ ⏐ ⏐ ⏐ ⏐ 2

−−→

g ∗P ⊗ g ∗f∗ g∗′ g ′∗ Q ←−− ⏐ ⏐

g ∗ (P ⊗ ⏐h∗ g ′∗ Q) ⏐

η

−−−−−−−−−−− −− −−−−−−−−−−−→ h′∗ g ∗f∗ g∗′ g ′∗

−−−−−−−−−−−−−−−−−−−−−−−→

g ∗ h∗ (h∗P ⊗ g ′∗ Q) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

←−− g ∗ g∗ (g ∗P ⊗ f∗′ g ′∗ Q) −−−−−−−−−−−−−−−−−−−−−−−→ g ∗ g∗ f∗′ (f ′∗ g ∗P ⊗ g ′∗ Q) ⏐ ⏐ ⏐ ⏐ f∗′ (f ′∗ g ∗P ⊗ g ′∗ Q)

−−→

f∗′ g ′∗ (f ∗P ⊗ Q) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ′ ′∗ ∗ f∗ (g f P ⊗ g ′∗ Q) ⏐ ⏐ f∗′ (h∗P ⊗ g ′∗ Q)

(3.7.3.1)

3 Derived Direct and Inverse Image

g ∗P ⊗ g ∗f∗ Q ⏐ ⏐

h′∗ g ∗f∗

3.8 Direct Sums

131

Subdiagram 1 commutes by (3.4.7)(i), subdiagrams 2 and 3 by (3.7.1), and 4 by the last sentence in (3.4.6.2). Commutativity of the other subdiagrams is straightforward to check. Remarks 3.7.3.1 In the case of ringed spaces (3.6.10), the unlabeled arrows in the preceding diagram represent isomorphisms. So if θ is an isomorphism too, then the maps g ∗ (pf ) and pf ′ are isomorphic. For such diagrams we can say then that “projection commutes with base change.” For example, when g is an open immersion, then θ is an isomorphism. That amounts to compatibility of Rf∗ with open immersions, which is also an immediate consequence of (2.4.5.2). For other situations in which θ is an isomorphism, see (3.9.5) and its generalization (3.10.3).

3.8 Direct Sums Proposition 3.8.1 Let X be a ringed space. Arbitrary (small ) direct sums exist in K(X) and in D(X); and the canonical functor Q : K(X) → D(X) preserves them. In both K(X) and D(X), natural maps of the type ⊕α∈A Cα [1] → ⊕α∈A Cα [1] are always isomorphisms—direct sums commute with translation; and any direct sum of triangles is a triangle. Proof. Let (Cα )α∈A (A small) be a family of complexes of OX -modules. The usual direct sum C of the family (Cα )—together with the homotopy classes of the canonical maps Cα → C—is also a direct sum in the category K(X). Since any complex in D(X) is isomorphic to a q-injective one, and since HomD(X) (−, I ) = HomK(X) (−, I ) for any q-injective I, see (2.3.8(v)), it follows that C is also a direct sum in D(X).14 The remaining assertions are easily checked for K(X), where we need only consider standard triangles, see (1.4.3); and they follow for D(X) upon application of Q, see (1.4.4). Q.E.D. Proposition 3.8.2 Let Y be a ringed space, and let (Cα )α∈A be a small family of complexes of OY -modules. Then: (i) For any D ∈ D(Y ), the canonical map is an isomorphism ∼ ⊕α (Cα ⊗ D) −→ (⊕α Cα ) ⊗ D. = =

(ii) For any ringed-space map f : X → Y , the canonical map is an isomorphism ∼ ⊕α Lf ∗ Cα −→ Lf ∗ (⊕α Cα ).

14

A more elementary proof, not using q-injective resolutions, is given in [BN, §1].

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3 Derived Direct and Inverse Image

Proof. Each Cα is isomorphic to a q-ﬂat complex; and any direct sum of q-ﬂat complexes is still q-ﬂat, see §2.5. Hence the assertions reduce to the and f ∗ in corresponding ones for ordinary complexes, with ⊗ in place of ⊗ = ∗ place of Lf . Alternatively, in view of (2.6.1)∗ and (3.2.1) one can use the fact that any functor having a right adjoint respects direct sums. Q.E.D. Proposition 3.8.3 (See [N′ , p. 38, Remark 1.2.2]). Let Y be a ringed space and Cα′ −→ Cα −→ Cα′′ −→ T Cα′ (α ∈ A) a small family of D(Y )-triangles. Then the naturally resulting sequence ⊕α Cα′ −→ ⊕α Cα −→ ⊕α Cα′′ −→ ⊕α T Cα′ ∼ = T (⊕α Cα′ )

(α ∈ A)

is also a D(Y )-triangle. Exercise. Deduce (3.8.2)(i) from (2.5.10)(c). Using, e.g., (2.5.5), prove an analogous generalization of (3.8.2)(ii), i.e., show that if (Cα ) is a (small, directed) inductive system of complexes of OY -modules, then there are natural isomorphisms ∼ limH n Lf ∗ Cα −→ H n Lf ∗ (limCα ) −→ −→ α α

(n ∈ Z).

3.9 Concentrated Scheme-Maps This section contains some reﬁnements of preceding considerations as applied to a map f : X → Y of schemes, see (3.4.4)(b). Except in (3.9.1), which does not involve Rf∗ , we need f to be concentrated (= quasi-compact and quasi-separated). The main result (3.9.4) asserts that under mild restrictions on f or on the OX -complex F , the projection map p : Rf∗ F ⊗ G → Rf∗ (F ⊗ Lf ∗ G) see (3.4.6) = =

is an isomorphism for any OY -complex G having quasi-coherent homology. The results of (3.9.1) and (3.9.2) on good behavior, vis-` a-vis quasi-coherence, of the derived direct and inverse image functors of a concentrated map allow “way-out” reasoning to reduce (3.9.4) essentially to the trivial case G = OY , provided that F and G are bounded above; homological compatibility of Rf∗ and lim (proved in (3.9.3)) then gets rid of the boundedness. −→ Another Proposition, (3.9.5), says that for concentrated f the map θ associated as in (3.7.2) to certain ﬂat base changes is an isomorphism. A stronger result will be given in Theorem (3.10.3), which contains (3.9.4) as well. (But (3.9.4) is used in the proof of (3.10.3)).

3.9 Concentrated Scheme-Maps

133

Proposition (3.9.6) takes note of, among other things, the fact that on a quasi-compact separated scheme, complexes with quasi-coherent homology are D-isomorphic to quasi-coherent complexes. We begin with some notation and terminology relative to any ringed space X, with K(X) and D(X) as in §3.1. As in (1.6)–(1.8), we have various triangulated (i.e., Δ-)subcategories of K(X), denoted K*(X), K*(X) (with “ * ” indicating a boundedness condition—below (* = +), above (* = −), or both above and below (* = b)— and “ ” indicating application of the boundedness condition to the homology of a complex rather than to the complex itself); and we have the corresponding derived categories D*(X), D*(X), which are Δ-subcategories of D(X). For example, K+(X) is the full subcategory of K(X) whose objects are complexes A• of OX -modules such that An = 0 for all n ≤ n0 (A• ) (where n0 (A• ) is some integer depending on A• ); and D−(X) is the full subcategory of D(X) whose objects are complexes A• such that H n (A• ) = 0 for all n ≥ n1 (A• ). The subscript “qc” indicates collections of OX -complexes whose homology sheaves are all quasi-coherent (see (1.9), with A# the category of quasi-coherent OX -modules, which is a plump subcategory of the category of all OX -modules [GD, p. 217, (2.2.2) (iii)]). For example D+ qc (X) is the Δsubcategory of D(X) whose objects are complexes A• such that H n (A• ) is quasi-coherent for all n ∈ Z, and H n (A• ) = 0 for n ≤ n0 (A• ). Proposition 3.9.1 For any scheme-map f : X → Y we have Lf ∗ Dqc (Y ) ⊂ Dqc (X).

Proof. For C ∈ Dqc (Y ) and Cm := τ≤m C (1.10), there exists a q-ﬂat resolution lim Q = Q → C = lim Cm (m ≥ 0) −→ m −→ where for each i, Qm is a bounded-above ﬂat resolution of Cm , see (2.5.5). The resulting maps lim f ∗ Qm − → f ∗Q ← − Lf ∗ Q − → Lf ∗ C −→ are all isomorphisms in D(X) (recall that, as indicated just before (3.1.3), q-ﬂat ⇒ left-f ∗-acyclic, and dualize the last assertion in (2.2.6)); and hence H n (Lf ∗ C) ∼ lim H n (Lf ∗ Cm ) lim H n (f ∗ Qm ) ∼ =− =− → →

(n ∈ Z).

Since lim preserves quasi-coherence, we need only deal with the case where −→ C = Cm ∈ D− qc (Y ); and then way-out reasoning [H, p. 73, (ii) (dualized)] reduces us further to showing that for any quasi-coherent OY -module F and any i ∈ Z, the OX -modules Li f ∗ (F ) := H −i Lf ∗ (F )(i ≥ 0) are also quasicoherent.

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3 Derived Direct and Inverse Image

For this, note that the restriction of a ﬂat resolution of F to an open subset U ⊂ Y is a ﬂat resolution of the restriction F |U , whence formation of Li f ∗ (F ) “commutes” (in an obvious sense) with open immersions on Y ; so we can assume X and Y to be aﬃne, say X = Spec(B), Y = Spec(A), the quasi-coherent OY -module associated to some A-module G; and F = G, and then if G• → G is an A-free resolution of G, it is easily seen (since is an exact functor of A-modules M [GD, p. 198, (1.3.5)], and since M → M = (B ⊗A M ) [ibid., p. 213, (1.7.7)]) that Li f ∗ (F ) is the quasi-coherent f ∗M i , where Hi is the homology Hi := Hi (B ⊗A G• ) = TorA (B, G). OX -module H i Q.E.D. We will use the adjective concentrated as a less cumbersome synonym for quasi-compact and quasi-separated. Elementary properties of concentrated schemes and scheme-maps can be found in [GD, pp. 290 ﬀ ]. In particular, if f : X → Y is a scheme-map with Y concentrated, then X is concentrated iﬀ f is a concentrated map [ibid., p. 295, (6.1.10)]. Proposition 3.9.2 Let f : X → Y be a concentrated map of schemes. Then (3.9.2.1) Rf∗ Dqc (X) ⊂ Dqc (Y ). Moreover, with notation as in §1.10, for all n ∈ Z it holds that Rf∗ Dqc (X)≥n ⊂ Dqc (Y )≥n ;

(3.9.2.2)

and if Y is quasi-compact, then there exists an integer d such that for every n ∈ Z, Rf∗ Dqc (X)≤n ⊂ Dqc (Y )≤n+d . (3.9.2.3)

Proof. The fact that Rf∗ (D(X)≥n ) ⊂ D(Y )≥n is, implicitly, in (2.7.3): any F ∈ D(X)≥n admits the quasi-isomorphism (1.8.1)+ : F → τ +F , and there is a quasi-isomorphism τ +F → I where I is a ﬂasque complex with I m = 0 for all m < n, so that Rf∗ F ∼ = f∗ I ∈ D(Y )≥n . To ﬁnish proving (3.9.2.2), i.e., to show that if I has quasi-coherent homology then so does f∗ I, use the standard spectral sequence Rpf∗ H q (I) ⇒ H • f∗ I (Rpf∗ := H p Rf∗ )

and the fact (proved in [AHK, p. 33, Thm. (5.6)] or [Kf, p. 643, Cor. 11]) that Rpf∗ preserves quasi-coherence of sheaves. Or, reduce to this fact by “way-out” reasoning, see [H, p. 88, Prop. 2.1]. For the rest, we need: Lemma 3.9.2.4 If Y is quasi-compact then there is an integer d such that for any quasi-coherent OX -module F and any i > d, Rif∗ F = 0.

3.9 Concentrated Scheme-Maps

135

Proof. Since Y is covered by ﬁnitely many aﬃne open subschemes Yk and since for each k the restriction Rif∗ F |Yk is the quasi-coherent sheaf associated to the Γ(Yk , OY )-module H i (f −1 (Yk ), F ) [Kf, p. 643, Cor. 11], we need only show that if Y is aﬃne then there is an integer d such that H i (X, F ) = 0 for all i > d. Note that X is now a concentrated scheme. We proceed by induction on the unique integer n = n(X) such that X can be covered by n quasi-compact separated open subschemes, but not by any n − 1 such subschemes. (This integer exists because X is quasi-compact and its aﬃne open subschemes are quasi-compact and separated.) ˇ cohomology If n = 1, i.e., X is separated, then H i (X, F ) is the Cech with respect to a ﬁnite cover X = ∪dj=0 Xj by aﬃne open subschemes, so it vanishes for i > d. Suppose next that X = X1 ∪ X 2 ∪ · · · ∪ Xn

(n = n(X) > 1)

with each Xj a quasi-compact separated open subscheme of X. Since X is quasi-separated therefore Xj ∩X1 is quasi-compact and separated,15 so setting X0 := X2 ∪ · · · ∪ Xn we have n(X0 ) < n and n(X0 ∩ X1 ) < n. The desired conclusion follows then from the inductive hypothesis and from the long exact sequence · · · → H i−1 (X0 ∩ X1 , F ) → H i (X, F ) → H i (X0 , F ) ⊕ H i (X1 , F ) → · · · associated to the obvious short exact sequence of complexes 0 → Γ(X, I• ) → Γ(X0 , I• ) ⊕ Γ(X1 , I• ) → Γ(X0 ∩ X1 , I• ) → 0 where I• is a ﬂasque resolution of F. Q.E.D. Now let F ∈ Dqc (X) and N ∈ Z. Starting with an injective resolution τ≥N F → IN , and using (3.9.2.5)(ii) below (with J the category of boundedbelow injective complexes), we build inductively a commutative ladder α

n · · · −−→ τ≥n F −−→ τ≥n+1 F −−→ · · · −−→ τ≥N F ⏐ ⏐ ⏐ ⏐β ⏐ ⏐ βn n+1

· · · −−→

In

−−→ γn

In+1

−−→ · · · −−→

IN

where for −∞ < n < N , αn is the natural map, βn is a quasi-isomorphism, In+1 is a bounded-below injective (hence, by (2.3.4), q-injective) complex, 15

Quasi-compactness holds by [GD, p. 296, (6.1.12)], where (Uα ) should be a base of the topology.

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3 Derived Direct and Inverse Image

and γn is split-surjective in each degree. Then I := lim In is q-injective ←− [Sp, p. 130, 2.5]; and the natural map lim τ≥n F = F → I is a quasi←− ∼ isomorphism [Sp, p. 134, 3.13]. So we have an isomorphism Rf∗ F −→ f∗ I. It follows from (2.4.5.2) that Rf∗ is compatible with open immersions on Y , and hence if (3.9.2.1) holds whenever Y is quasi-compact (indeed, aﬃne) then it holds always. Assuming Y to be quasi-compact, we argue further as in loc. cit. Since γn is split surjective in each degree m, its kernel Cn is a bounded-below injective complex, and for any aﬃne open U ⊂ Y, γn induces m a surjection Γ(f −1 U , Inm ) ։ Γ(f −1 U , In+1 ) with kernel Γ(f −1 U , Cnm ). The ﬁve-lemma yields that βn induces a quasi-isomorphism to Cn from the kernel An of the surjection αn ; and in D(X), An ∼ = H n (F )[−n]. Thus Cn [n] n is an injective resolution of H (F ), and so if d is the integer in (3.9.2.4) then for any m > n + d, H m Γ(f −1 U , Cn ) ∼ = H m−n f −1 U , H n (F ) ∼ = Γ U , Rm−nf∗ H n (F ) = 0, so that the sequence

Γ(f −1 U , Cnm−1 ) → Γ(f −1 U , Cnm ) → Γ(f −1 U , Cnm+1 ) → Γ(f −1 U , Cnm+2 ) is exact. A Mittag-Leﬄer-like diagram chase ([Sp, p. 126, Lemma], applied to the inverse system of diagrams Γ(f −1 U , Inm−1 ) → Γ(f −1 U , Inm ) → Γ(f −1 U , Inm+1 ) → Γ(f −1 U , Inm+2 ) where n runs through Z and In := IN for all n > N ) shows then that if m ≥ N + d then the natural map H m Γ(U , f∗ I) = H m lim Γ(f −1 U , In ) ←− m → H Γ(f −1 U , IN ) = H m Γ(U , f∗ IN )

is an isomorphism. Sheaﬁfying on Y , we get that for any m ≥ N + d, the natural composition ∼ ∼ Rmf∗ F = H m (Rf∗ F ) −→ H m (f∗ I) −→ H m (f∗ IN ) −→ Rmf∗ (τ≥N F )

is an isomorphism. From (3.9.2.2) we conclude then that Rmf∗ F is quasicoherent, which gives (3.9.2.1) (since N is arbitrary); and furthermore Q.E.D. if τ≥N F ∼ = 0, proving (3.9.2.3). = 0, then τ≥N +d Rf∗ F ∼ Lemma 3.9.2.5 Let A be an abelian category, and let J be a full subcategory of the category C of A-complexes such that (1): a complex B is in J iﬀ B[1] is, and (2): for any map f in J, the cone Cf (§1.3) is in J.

3.9 Concentrated Scheme-Maps

137

(i) Let u : P → C be a map in C with P ∈ J and such that there exists a quasi-isomorphism h : Q → Cu with Q ∈ J. Then u factors as u1 v P − → P1 −→ C where P1 ∈ J, u1 is a quasi-isomorphism, and in each degree m, v m : P m → P1m is a split monomorphism, i.e., has a left inverse. (ii) Let s : C → I be a map in C with I ∈ J and such that there exists s1 t a quasi-isomorphism Cs → J with J ∈ J. Then s factors as C − → I1 − →I m m where I1 ∈ J, s1 is a quasi-isomorphism, and in each degree m, t : I1 → I m is a split epimorphism, i.e., has a right inverse. Proof. (i) We have a diagram in C v

P −−−−→ Cwh [−1] −−−−→ ⏐ ⏐ g

P −−−−→ Cw [−1] −−−−→ ⏐ ⏐ ϕ 1

P −−−−→ u

C

wh

Q −−−−→ P [1] ⏐ ⏐ h

Cu −−−−→ P [1] w

−−−−→ Cu −−−−→ P [1] w

where the bottom row is the standard triangle associated to u, the top two rows are made up of natural maps, ϕ is as in (1.4.3.1), and g is given in degree m by the map g m = 1 ⊕ hm : Cwh [−1]m = P m ⊕ Qm → P m ⊕ Cum = Cw [−1]m . Here all the subdiagrams other than 1 commute, and 1 is homotopycommutative (see (1.4.3.1)). By (Δ2) in §1.4, the rows of the diagram become triangles in K(A). Since h is a quasi-isomorphism, we see, using the exact homology sequences (1.4.5)H of these triangles, that the composed map ϕ ◦g is also a quasi-isomorphism. Since P and Q are in J, so is Cwh [−1]. Thus we can take P1 := Cwh [−1] and u1 := ϕ ◦g. (ii) A proof resembling that of (i) (with arrows reversed) is left to the reader. See also the following exercise (a), or [Sp, p. 132, proof of 3.3]. Q.E.D. Exercises 3.9.2.6 (a) Convince yourself that (i) and (ii) in (3.9.2.5) are dual, i.e., (ii) is essentially the statement about A obtained by replacing A in (i) by its opposite category Aop . (b) (Cf. (1.11.2)(iv).) Let X be a scheme and let AX (resp. Aqc X ) be the category of all OX -modules (resp. quasi-coherent OX -modules). Let φ : AX → Ab be an additive functor satisfying φ(lim ←− In ) = lim ←− φ(In ) for any inverse system (In )n<0 of AX injectives in which all the maps In → In+1 are split surjective. Then dim+ Rφ| D (X) = dim+ Rφ| Aqc . qc

X

(c) Show: for any proper map f : X → Y of noetherian schemes, Rf∗ Dc (X) ⊂ Dc (Y ). Hint. (3.9.2), [H, p. 74, (iii)], [EGA, III, (3.2.1)].

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3 Derived Direct and Inverse Image

(3.9.3). Henceforth, index sets A for inductive systems are assumed to be (small and) ﬁltered: α, β ∈ A ⇒ ∃γ ∈ A with γ ≥ α and γ ≥ β. (More generally, the results will be valid for limits over ﬁltered—or even pseudoﬁltered—categories [GV, pp. 14–15], [M, p. 211].) Lemma 3.9.3.1 Let f : X → Y be a concentrated scheme-map. Fix n ∈ Z, let (Cα , ϕβα )α,β∈A be an inductive system of OX -complexes all of whose homology vanishes in degree < n, and set C := lim Cα . Then we have natural −α→ isomorphisms ∼ lim Rif (C ) −→ Rif∗ (C) −α→ ∗ α

(Rif∗ := H i Rf∗ , i ∈ Z).

Proof.16 In the category of bounded-below OX -complexes D, we can choose ﬂasque resolutions D → F functorially, as follows: for each q ∈ Z, let 0 → Dq → F 0q → F 1q → F 2q → . . . be the (ﬂasque) Godement resolution of Dq [G, p. 167, 4.3], set F pq := 0 if p < 0, and let F be the complex coming from the double complex F pq , i.e., F m := ⊕p+q=m F pq , etc; then F m is ﬂasque, and diagram chasing, or a simple spectral sequence argument, shows that the family of natural maps Dm → F 0m ⊂ F m gives a quasiisomorphism gD : D → F . We will refer to this gD (or simply F ) as the Godement resolution of D. With Cα and n as above, the truncation operator τ≥n as in §1.10, and Fα the Godement resolution of τ≥n Cα , we have an inductive system of quasi-isomorphisms Cα → τ≥n Cα → Fα , and hence a quasi-isomorphism C → F := lim Fα . Each Fα is ﬂasque, hence f∗ -acyclic (2.7.3). By [Kf, p. 641, −→ Cor. 5 and 7], F is a complex of f∗ -acyclic sheaves, and so, being bounded below, F itself is f∗ -acyclic, see (2.7.4) (dualized). The last assertion in (2.2.6) shows then that the (obvious) map in (3.9.3.1) is isomorphic to the natural map lim H i (f∗ Fα ) = H i (lim f∗ Fα ) → H i (f∗ lim Fα ) = H i (f∗ F ), −→ −→ −→ which is an isomorphism since f∗ commutes with lim [Kf, p. 641, Prop. 6]. −→ Q.E.D. Corollary 3.9.3.2 Let f : X → Y be a concentrated scheme-map. With notation as in §1.9, let A# be a plump subcategory of the category AX of OX modules, such that any lim of objects in A# is itself in A# and such that the −→ restriction of Rf∗ to D# (X) is bounded above (§1.11). Let (Cα , ϕβα )α,β∈A be an inductive system of complexes all of whose homology lies in A#, and set C := lim Cα . Then we have natural isomorphisms −α→ ∼ Rif∗ (C) lim Rif∗ (Cα ) −→ −α→

16

Cf. [EGA, III, Chap. 0, p. 36, (11.5.1)].

(Rif∗ := H i Rf∗ , i ∈ Z).

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139

Remarks. (a) If the map f is ﬁnite-dimensional (2.7.6), (e.g., if X is noetherian, of ﬁnite Krull dimension (2.7.6.2)), then all the hypotheses in (3.9.3.2) are satisﬁed when A# = AX . (b) By (3.9.2.3), if Y is quasi-compact then all the hypotheses in (3.9.3.2) are satisﬁed when A# = Aqc , the category of quasi-coherent OX -modules. Even if Y is not quasi-compact, the conclusion of (3.9.3.2) still holds, because Rf∗ and lim “commute” with open immersions on Y (see (2.4.5.2)), so it −→ suﬃces to check over aﬃne open subsets of Y . Proof of (3.9.3.2). By (1.11.2)(ii) we have natural isomorphisms ∼ Rif∗ (D) −→ Rif∗ (τ≥i−d D) D ∈ D# (X), d := dim+ (Rf∗ | D# (X) ) .

Note that C ∈ D# (X) since homology commutes with lim ; and clearly −→ τ≥i−d C = lim τ≥i−d Cα . Fixing i, we conclude by applying (3.9.3.1) to the −→ Q.E.D. inductive system τ≥i−d Cα . Corollary 3.9.3.3 Let (Cβ )β∈B be a small family of complexes in D≥n # (n ﬁxed, see (1.10)) or in D # (X) (A as in (3.9.3.2)). Then the natural map ⊕β Rf∗ Cβ → Rf∗ ⊕β Cβ (see (3.8.1)) is an isomorphism.

Proof. We need only check that the induced homology maps are isomorphisms, which follows from (3.9.3.1) or (3.9.3.2), a direct sum over B being a lim of the family of direct sums over ﬁnite subsets of B. Q.E.D. −→ Corollary 3.9.3.4 Under the hypotheses of (3.9.3.1) or (3.9.3.2), if each Cα is f∗-acyclic then so is C.

Proof. The assertion is that the natural map f∗ C → Rf∗ C is an isomorphism in D(Y ), i.e., that the induced maps H i (f∗ C) → H i (Rf∗ C) are all isomorphisms. By assumption, this holds with Cα in place of C; and since H i and f∗ commute with lim [Kf, p. 641, Prop. 6], it also holds, by (3.9.3.1) −→ or (3.9.3.2), for C. Q.E.D. Corollary 3.9.3.5 With A# as in (3.9.3.2), any complex C of f∗-acyclic A#objects is itself f∗-acyclic. Proof. The complexes · · · → 0 → 0 → C −n → C −n+1 → · · · (n ∈ Z) form an inductive system of f∗ -acyclic complexes (see (2.7.2), dualized), whose lim −→ is C. Conclude by (3.9.3.4). Q.E.D. Proposition 3.9.4 Let f : X → Y be a concentrated scheme-map, and let F ∈ D(X), G ∈ Dqc (Y ). If f is ﬁnite-dimensional (2.7.6), or if F ∈ Dqc (X), then the projection maps p1 : (Rf∗ F ) ⊗ G → Rf∗ (F ⊗ Lf ∗ G) , = = (see (3.4.6)) are isomorphisms.

p2 : G ⊗ Rf∗ F → Rf∗ (Lf ∗ G ⊗ F) = =

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Proof. We treat only p1 (p2 can be handled similarly; or (3.4.6.1) can be applied). The question is local on Y (check directly, or see (3.7.3.1)), so we may assume Y aﬃne. Suppose ﬁrst that both F and G are bounded-above complexes. Then the source and target of p1 are, for ﬁxed F , bounded-above functors of G: this is clear when f is ﬁnite-dimensional, and if F ∈ Dqc (X) then it follows from (3.9.2.3) since F ⊗ Lf ∗ G ∈ Dqc (X), see (3.9.1) and (2.5.8). = By (1.11.3.1), with A# the category of quasi-coherent OY -modules on the aﬃne scheme Y , we reduce the question to where G is a single free OY module G0 , whence Lf ∗ G is isomorphic to the free OX -module f ∗ G0 . After verifying via (3.8.2) and (3.9.3.3) that everything in sight commutes with direct sums, we have a further reduction to the case G = OY . We check then, via (3.2.5)(a) and commutativity of the upper diagrams in (3.4.2.2), that p1 is isomorphic to the identity map of Rf∗ F . Next, drop the assumption that F is bounded above. For any integer i and any triangle in D(X) based on the natural map F → τ≥i F , the vertex Ci (depending, up to isomorphism, only on F ) lies in D
we get isomorphisms ∼ H j Rf∗ (F ⊗ Lf ∗ G) −→ H j Rf∗ (τ≥i F ⊗ Lf ∗ G) = = for all j > i + e + d. Similarly, we have natural isomorphisms ∼ G −→ H j Rf∗ τ≥i F ⊗ G . H j Rf∗ F ⊗ = =

Therefore, to show for any given j that the homology map H j (p1 ) is an isomorphism—which suﬃces, by (1.2.2)—we can replace F by τ≥ j−1−e−d F . Thus we may assume that F is bounded below. Also, as above, we may assume that G is ﬂat, whence so is f ∗ G ∼ = Lf ∗ G. Let Fm be the Godement resolution of τ≤m F (m ∈ Z), see proof of (3.9.3.1), so that the canonical map F = lim τ≤m F → lim Fm −m → −m →

is the Godement resolution of F .

3.9 Concentrated Scheme-Maps

141

By the ﬁrst part of this proof, there is a natural isomorphism H j f∗ Fm ⊗ G ∼ G = H j Rf∗ τ≤m F ⊗ = ∼ −→ H j Rf∗ (τ≤m F ⊗ Lf ∗ G) ∼ = H j Rf∗ (Fm ⊗ f ∗ G) . =

As before, if F ∈ Dqc (X) then (Fm ⊗ f ∗ G) ∼ Lf ∗ G) ∈ Dqc (X). = (τ≤m F ⊗ = Using (3.9.3.2) and—as in the proof of (3.9.3.1)—the commutativity of lim −→ with f∗ , ⊗, and H j , we ﬁnd then that H j (p1 ) factors as the composition of the natural isomorphisms ∼ H j Rf∗ F ⊗ G −→ H j f∗ lim Fm ⊗ G = −→ ∼ lim H j f∗ Fm ⊗ G −→ −→ ∼ lim H j Rf∗ (Fm ⊗ f ∗ G) −→ −→ ∼ ∼ H j Rf∗ (F ⊗ Lf ∗ G) , −→ H j Rf∗ lim (Fm ⊗ f ∗ G) −→ = −→ proving (3.9.3) whenever G is bounded above. Finally, to extend the assertion to any G ∈ Dqc (Y ), use a quasiisomorphism Q → G where Q = lim Qm with Qm ∈ D− qc (Y ) bounded-above −→ and ﬂat, so that Lf ∗ G ∼ = f ∗ Q, see proof of (3.9.1). As in (3.1.2), Rf∗ F = f∗ IF ; and, again, if F ∈ Dqc (X) then IF ⊗ f ∗ Qm ∈ Dqc (X). Applying lim to the −m → system of natural maps H j f IF ⊗ Qm ∼ = H j Rf F ⊗ Qm ∗

∗

=

−→ H Rf∗ (F ⊗ Lf ∗ Qm ) ∼ = H j Rf∗ (IF ⊗ f ∗ Qm ) , = j

maps which we have already seen to be isomorphisms, we ﬁnd, via (3.9.3.2) and commutativity of lim with H j , with ⊗, and with f ∗ , that the maps −→ H j (p1 ) : H j Rf∗ F ⊗ Q −→ H j Rf∗ (F ⊗ Lf ∗ Q) (j ∈ Z) = = are all isomorphisms, whence the conclusion.

Q.E.D.

Remarks 3.9.4.1 The projection map p1 need not be an isomorphism for non-quasi-coherent OY -modules G. For example, let R be a two-dimensional noetherian local ring with maximal ideal m, Y = Spec(R), X = Spec(R)−{m}, f : X → Y the inclusion, F = OY , and G = OX extended by zero (so that G is a ﬂat OY -module). Then the stalk of R1 f∗ (F ) ⊗ G at m is 0, whereas 2 the stalk of R1 f∗ (F ⊗ f ∗ G) = R1 f∗ (OX ) is H 1 (X, OX ) = Hm (R) = 0 (where Hm denotes local cohomology supported at m).

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3 Derived Direct and Inverse Image

Exercises 3.9.4.2 Let X be a ringed space. (a) Show that an OX -module F is ﬂat iﬀ Tori (F , G) := H −i (F ⊗ G) = 0 for all = OX -modules G and all i = 0. (One need only consider i = 1, see proof of (2.7.6.4).) (b) [I, p. 131]. A complex F of OX -modules has finite flat amplitude (or finite tor-dimension) if for some integers d1 ≤ d2 , Tori (F , G) = 0 for all OX -modules G and all i outside the interval [d1 , d2 ]. Show that this condition is equivalent to there ∼ being a D(X)-isomorphism F −→ P with P flat and P i = 0 for all i ∈ / [−d2 , −d1 ]. (See (2.7.6), with f the identity map of X.) (c) [I, p. 249]. Suppose further in (3.9.4) that f has ﬁnite tor-dimension (2.7.6) and that F has ﬁnite ﬂat amplitude (b). Show that then Rf∗ F also has ﬁnite ﬂat amplitude. (d) Show: if X is an aﬃne scheme and if F ∈ Dqc (X) has ﬁnite ﬂat amplitude, then the complex P in (b) may be assumed to be quasi-coherent. (Use (3.9.6) below.) (e) Let f : X → Y be a concentrated scheme-map. Let F ∈ D+(X) and let G ∈ Dqc (Y ) have finite flat amplitude. Then the projection map p1 in (3.9.4) is an isomorphism. Hint. We may assume Y to be aﬃne. Induction on the number of non-zero terms of a bounded ﬂat quasi-coherent complex P ∼ = G (see (d)) reduces the question to where G is a single ﬂat quasi-coherent OY -module. Then by a theorem of Lazard [GD, p. 163, Prop. (6.6.24)], G is a direct limit of ﬁnite-rank free OY -modules, and so (3.9.3.1) gives a reduction to the trivial case G = OY . (f) Let Y be a ringed space. Show that the following conditions on a complex G of OY -modules are equivalent: (i) For some d ∈ Z, Tori (F , G) = 0 for all OY -modules F and all i > d. (ii) The functor E → E ⊗ G (E ∈ D(Y ) is bounded below (1.11.1). = (iii) In D(Y ), G ∼ = P with P bounded-below and q-ﬂat. (iv) In D(Y ), G ∼ = P with P bounded-below, flat, and q-ﬂat. When these conditions hold we say that G has bounded-below flat amplitude. (g) Do exercise (e) assuming only that G has bounded-below ﬂat amplitude. Hint. Assuming G to be bounded-below, ﬂat, and q-ﬂat, show that it suﬃces to apply (e) to each of the complexes . . . → Gn−1 → Gn → 0 → 0 → . . . (n ∈ Z).

The following result will be generalized in (3.10.3). Proposition 3.9.5 Given a commutative square σ of scheme-maps v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−−−→ Y u

suppose that f is concentrated, that u is ﬂat, and that σ is a ﬁber square (i.e., that the associated map X ′ → X ×Y Y ′ is an isomorphism). Then for any F ∈ Dqc (X), the natural composed map (see (3.7.2)(a)) η

θσ (F ) : u∗ Rf∗ F −→ u∗ Rf∗ Rv∗ v ∗ F ǫ

∼ u∗ Ru∗ Rg∗ v ∗ F −→ Rg∗ v ∗ F −→

is an isomorphism.

3.9 Concentrated Scheme-Maps

143

Proof. It should be noted that since u, and hence v, is ﬂat, we have functorial ∼ ∼ isomorphisms Lu∗ −→ u∗ and Lv ∗ −→ v ∗ . (This follows from (2.2.6)(dualized), since the exactness of (e.g.) u∗ implies at once that every OX -complex is u∗-acyclic.) In view of (3.9.2.2) and (3.9.2.3), (1.11.3)(iv) allows us to assume that F is a single quasi-coherent OX -module. It will suﬃce then, by (1.2.2), to show that application of the homology functors H n to θσ (F ) produces (what else?) the “base change” isomorphisms αn (F ) of [AHK, p. 35, Theorem (6.7)]. For this purpose, we need to express θσ in terms of canonical ﬂasque (Godement) resolutions—which we denote by C. In [AHK, p. 28, §3] there is deﬁned a map ϕ : C(F ) → v∗ C(v ∗ F ) (denoted there by θv• (F )) which, as easily checked, makes the following natural diagram commute: F −−−−→ v∗ v ∗ F ⏐ ⏐ ⏐ ⏐ C(F ) −−−−→ v∗ C(v ∗ F ) ϕ

With the deﬁnitions of ǫ and η in §3.2, and the fact that the direct image of a ﬂasque sheaf is still ﬂasque, it is a straightforward exercise to verify that the map θσ (F ) is isomorphic to the derived category map given by the natural composition ϕ

∼ u∗ f∗ C(F ) →−→ u∗ f∗ v∗ C(v ∗ F ) −→ u∗ u∗ g∗ C(v ∗ F ) −→ g∗ C(v ∗ F ).

Now applying H n , and recalling that u is ﬂat, we get a composed map ϕ ∼ ∗ n α′n : u∗ H n f∗ C(F ) −→ u∗ H n f∗ v∗ C(v ∗ F ) −→ u H u∗ g∗ C(v ∗ F ) γ −→ H n g∗ C(v ∗ F ) .

Let’s look more closely at γ. Setting g∗ C(v ∗ F ) = E •, let K n be the kernel of the diﬀerential E n → E n+1 , and let δ : E n−1 → K n be the obvious map. Then γ can be identiﬁed with the map coker(u∗ u∗ δ) = u∗ coker(u∗ δ) → u∗ u∗ coker(δ) → coker(δ) which is adjoint to the natural map γ ′ : H n (u∗ E • ) = coker(u∗ δ) → u∗ coker(δ) = u∗ H n (E • ). Note that coker(u∗ δ) is the sheaf associated to the presheaf (U open in Y ) U → coker δ(u−1 U ) = H n E • (u−1 U )

and that γ ′ is the sheaﬁﬁcation of the natural presheaf map H n E • (u−1 U ) → Γ u−1 U , H n (E • ) .

It is then readily veriﬁed that the adjoint of α′n , viz. the composed map ϕ ∼ H n f∗ C(F ) −→ H n f∗ v∗ C(v ∗ F ) −→ H n u∗ g∗ C(v ∗ F ) γ′ −→ u∗ H n g∗ C(v ∗ F ) ,

is the map β n (f , g, u, v, F ) near the top of p. 34 of [AHK]. But by deﬁnition the adjoint of this β n is αn (F ); thus α′n = αn (F ), and we are done. Q.E.D.

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3 Derived Direct and Inverse Image

Here are two important results about quasi-coherence on quasi-compact separated schemes. Proofs can be found in the indicated references. Proposition 3.9.6 Let X be a quasi-compact separated scheme and Aqc X the category of quasi-coherent OX -modules. Then: (a) [BN, p. 230, Corollary 5.5.] The natural functor D(Aqc X ) → Dqc (X) is an equivalence of categories. (b) [AJL, p. 10, Proposition 1.1.] Every complex in Dqc (X) is D(X)isomorphic to a quasi-coherent q-ﬂat complex.

3.10 Independent Squares; K¨ unneth Isomorphism Throughout this section, (* , * ) will be the adjoint monoidal pair in (3.6.10), but with S restricted to be the category of quasi-separated schemes and concentrated (= quasi-compact and quasi-separated) maps between them [GD, p. 291, (6.1.5) and p. 294, (6.1.9)], and with the further restriction X* = X* = Dqc (X) for all X ∈ S (see (3.9.1), (3.9.2)). Note that any subscheme of a quasi-separated scheme is quasi-separated; and that the category S is closed under ﬁber product. Note also that if X and Y are quasi-separated then any scheme-map f : X → Y is quasi-separated, and further, quasi-compact if X is [GD, p. 295, (6.1.10)]. Accordingly (except in (3.10.1) and the proof of (3.10.2.2), where we need to distinguish between ordinary and derived functors), for any scheme-map α we . These abbreviations write α∗ for Rα∗ , and α∗ for Lα∗. We also write ⊗ for ⊗ = should not be allowed to obscure the fact that we are working throughout with derived categories and derived functors. After discussing some basic maps we deﬁne, in (3.10.2), various notions of independence of commutative S-squares. The main result, (3.10.3), is that all these independence conditions are equivalent.17 This implies, e.g., that the isomorphism in (3.9.5) holds for any tor-independent S-square, as does a certain K¨ unneth isomorphism, which subsumes the projection isomorphisms of (3.9.4). Independent squares are important in Grothendieck duality theory, where they support base-change maps (Remark (3.10.2.1)(c)). An orientation of a commutative S-square σ v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′ −−− −→ Y u

is an ordering of the pair (u, f ). 17 The knowledgeable reader might wish to place this result in the context of the K¨ unneth spectral sequences of [EGA, III, (6.7.5)].

3.10 Independent Squares; K¨ unneth Isomorphism

145

In this section, unless otherwise indicated, all commutative S-squares will be understood to be equipped with the orientation for which the bottom arrow precedes the right vertical one. To such an oriented σ associate the functorial maps θ = θσ : u∗f∗ → g∗ v ∗

(see Proposition (3.7.2))

and θ′ = θσ′ := θσ′ : f ∗u∗ → v∗ g ∗ where σ ′ is σ with its orientation reversed. Setting h := f v = ug, deﬁne the functorial K¨ unneth map η = ησ : u∗ E ⊗ f∗ F → h∗ (g ∗E ⊗ v ∗F ) to be the natural composition

E ∈ Y′ *, F ∈ X*

u∗ E ⊗ f∗ F → h∗ h∗ (u∗ E ⊗ f∗ F ) (3.4.5.1)

−−−−→ h∗ (g ∗ u∗ u∗ E ⊗ v ∗f ∗f∗ F ) → h∗ (g ∗E ⊗ v ∗F ). ∗ (3.6.1)

The map η generalizes (3.4.2.1): let X ′ = Y ′ = X, let v = g be the identity map, let u = f , so that h = f , and see (3.4.5.2) and 1) in (3.6.5). The map η also generalizes the projection maps p1 and p2 in (3.4.6): for p1 , let f be the identity map of X = Y , let g be the identity map of X ′ = Y ′, so that h = v = u, and see (3.4.6.2); and similarly for p2 let u and v be identity maps, . . . Example 3.10.1 Let us see what the above θσ and ησ look like in a concrete situation, when σ is a diagram of aﬃne schemes. The results are hardly surprising, but do need proof. (a) We deal ﬁrst with θ. On S there is a second adjoint pair (⋆ , ⋆ ) such that for each ringed space X, X⋆ = X ⋆ := K(X), the homotopy category of OX complexes, with monoidal structure given by the ordinary tensor product, and such that for each S-map f : X → Y the associated adjoint functors are the standard (sheaf-theoretic) inverse- and direct-image functors, f ⋆ := f ∗ and f⋆ := f∗ . So, as above, for each commutative S-square σ one gets functorial maps θ = θσ : Lu∗ Rf∗ → Rg∗ Lv ∗ , θ = θσ : u∗f∗ → g∗ v ∗ , related as follows.

(3.10.1.0)

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Lemma 3.10.1.1 With Q : K → D as usual, the following natural diagram of functors from K(X) to D(Y ′ ) commutes. α

Lu∗ Rf∗ Q ←−−−− Lu∗ Qf∗ −−−−→ Qu∗f∗ ⏐ ⏐ ⏐Qθ ⏐ θ Rg∗ Lv ∗ Q −−−−→ Rg∗ Qv ∗ ←−− −− Qg∗ v ∗ γ β

Proof. Expand the diagram (all maps being the obvious ones): Lu∗ Rf∗ Q ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

←−−−

Lu∗⏐Qf∗ ⏐

Qu∗f∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

Qu⏐∗f∗ ⏐

−−−→

Rg∗ Lg ∗ Lu∗ Qf∗ −−−→ Rg∗ Lg ∗ Qu∗f∗ ⏐ ⏐

Rg∗ Lg ∗ Lu∗ Rf∗ Q ←−−− Rg∗ Lg ∗ Lu∗ Qf∗ −−−→ Rg∗ Qg ∗ u∗f∗ ←−−− Qg∗ g ∗ u∗f∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐≃ ≃ ≃

∗ ∗ ∗ ∗ Rg∗ Lv ∗ Lf ∗ Rf∗ Q ←−−− Rg∗ Lv ∗⏐Lf ∗ Qf∗ −−−→ Rg∗ Qv f f∗ ←−−− Qg∗ v⏐ f f∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Rg∗ Lv ∗ Qf ∗f∗ −−−→ Rg∗ Qv ∗f ∗f∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

Rg∗ Lv ∗ Q

Rg∗ Lv ∗ Q

Rg∗ Qv ∗

−−−→

←−−−

Qg∗ v ∗

The upper right (resp. lower left) subdiagram commutes by (3.2.1.3) (resp. (3.2.1.2)). Commutativity of the rest is easy to verify. Q.E.D. Next, we make the map θ in (3.10.1.0) more explicit, at least locally. Lemma 3.10.1.2 Let V ←−−−− ⏐ ⏐

S ⏐ ⏐

U ←−−−− R

be a commutative diagram of commutative-ring homomorphisms, let σ as above be the corresponding diagram of aﬃne schemes (Y := Spec(R), etc.), and let θ = θσ : u∗f∗ → g∗ v ∗ be as in (3.10.1.0). For any S-complex E, let θ0 (E) be the natural composition U ⊗R E → V ⊗R E → V ⊗S E, i.e., the U-homomorphism taking 1 ⊗R e to 1 ⊗S e for all e ∈ E n (n ∈ Z).

3.10 Independent Squares; K¨ unneth Isomorphism

147

Then there is a natural commutative diagram of OY ′ -modules −− u∗f∗ E −−→ (U ⊗R E ) ⏐ ⏐ ⏐ ⏐ ) θ(E θ0 (E ) −− −−→ (V ⊗S E ) g∗ v ∗E

where denotes the usual functor from modules to quasi-coherent sheaves [GD, p. 197ff, §1.3], and where the horizontal arrows are isomorphisms. Proof. The horizontal isomorphisms come from [GD, p. 213, (1.7.7)]. To check commutativity, expand the diagram as follows, where in the right hand column, the complexes to which is applied are all regarded as U-complexes, and the maps are sheaﬁﬁcations of natural U-complex homomorphisms: u∗f∗ E ⏐ ⏐

−−−−−−−−−−− −−−−−−−−−−−−→

g∗ v ∗E

g∗ v ∗E

1

(U ⊗R E ) ⏐ ⏐

−−−−−−−−−−− g∗ g ∗ u∗f∗ E −−−−−−−−−−−−→ V ⊗U (U ⊗R E ) ⏐ ⏐ ⏐≃ ⏐ ≃ −−−−→ g∗ v ∗ (S ⊗R E) −−−−→ V ⊗S (S ⊗R E ) g∗ v ∗f ∗f∗ E ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 −−−−→

(V ⊗S E )

Commutativity of subdiagrams 1 and 2 is given by [GD, p. 214, (1.7.9)]. The rest is straightforward. Q.E.D. Under the hypotheses of (3.10.1.2), for any G ∈ Dqc (X) the map θ(G) : Lu∗ Rf∗ G → Rg∗ Lv ∗G can now be described as follows. By (3.9.6)(a), G is D-isomorphic to a quasi-coherent complex, which is E for some S-complex E. Arguing as in (2.5.5) (using that any S-module F is naturally a homomorphic image of the free S-module P0 (F ) with basis F ) one sees that there exists a quasi-isomorphism P → E with P a lim of bounded−→ above complexes of free S-modules. There results a quasi-isomorphism and P, being a lim of bounded-above complexes of free OX -modules, P → E; −→ by P, i.e., one may assume that there is q-ﬂat, as is v ∗P. One can replace E ∼ such that both E and v ∗E are q-ﬂat E exists a D-isomorphism λ : G −→ as well as quasi-coherent. Since f∗ is an exact functor on the category of quasi-coherent OX -modules → Rf∗ E is a D(Y )[GD, p. 214, (1.7.8)], therefore the natural map f∗ E ∗ ∗ ′ isomorphism. Also, the natural map Lv E → v E is a D(Y )-isomorphism.

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and β(E) in (3.10.1.1) are isomorphisms. Moreover, the So the maps α(E) can be identiﬁed as in (3.10.1.2) with θ map θ(E) 0 (E ). The map θ(E) is thereby determined by (3.10.1.1) and (3.10.1.2); and via λ (a “quasi-coherent q-ﬂat resolution”), so is the map θ(G). (b) We turn now to η. With σ, ( * , * ) and (⋆ , ⋆ ) as in (a), and h = f v = gu, one has for OY ′ -complexes E and OX -complexes F the functorial maps η = ησ (E, F ) : Ru∗ E ⊗ Rf∗ F → Rh∗ (Lg ∗E ⊗ Lv ∗F ), = = η = ησ (E, F ) : u∗ E ⊗ f∗ F → h∗ (g ∗E ⊗ v ∗F ), related as follows. Lemma 3.10.1.3 For all E and F as above, the following natural bifunctorial diagram—where appropriate insertions of “ Q” are left to the reader—commutes. Ru∗ E ⊗ Rf∗ F = ⏐ ⏐ η

α′

←−−−−

u∗ E ⊗ f∗ F =

β′

−−−−→

u∗ E ⊗ f∗ F ⏐ ⏐η

Rh∗ (Lg ∗E ⊗ Lv ∗F ) −−−−→ Rh∗ (g ∗E ⊗ v ∗F ) ←−−−− h∗ (g ∗E ⊗ v ∗F ) = Proof. Paste the following two diagrams along their common edge: Ru∗ E ⊗ Rf∗ F = ⏐ ⏐

Rf∗ F ) Rh∗ Lh∗(Ru∗ E ⊗ = ⏐ ⏐ ≃

Rh∗ (Lh∗ Ru∗ E ⊗ Lh∗ Rf∗ F ) = ⏐ ⏐ ≃

α′

←−−−

←−−−

←−−−

u∗ E ⊗ f∗ F = ⏐ ⏐

Rh∗ Lh∗ (u∗ E ⊗ f∗ F ) = ⏐ ⏐≃

Rh∗ (Lh∗ u∗ E ⊗ Lh∗f∗ F ) = ⏐ ⏐≃

Rh∗ (L(g ∗ u∗ )Ru∗ E ⊗ L(v ∗f ∗ )Rf∗ F ) ←−−− Rh∗ (L(g ∗ u∗ )u∗ E ⊗ L(v ∗f ∗ )f∗ F ) = = ⏐ ⏐ ⏐ ⏐≃ ≃ Rh∗ (Lg ∗ Lu∗ Ru∗ E ⊗ Lv ∗ Lf ∗ Rf∗ F ) ←−−− Rh∗ (Lg ∗ Lu∗ u∗ E Lv ∗ Lf ∗f∗ F ) = ⏐⊗ ⏐ = ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ∗ ⏐ Lv ∗f ∗f∗ F ) Rh∗ (Lg u u∗ E ⊗ 3 = ⏐ ⏐ ⏐ ⏐ ⏐ Rh∗ (Lg ∗E ⊗ Lv ∗F ) =

Rh∗ (Lg ∗E ⊗ Lv ∗F ) =

3.10 Independent Squares; K¨ unneth Isomorphism u∗ E ⊗ f∗ F = ⏐ ⏐

Rh∗ Lh∗ (u∗ E ⊗ f∗ F ) ⏐ = ⏐ ⏐ ⏐ ⏐ ≃⏐ ⏐ ⏐ ⏐

Rh∗ (Lh∗ u∗ E ⊗ Lh∗f∗ F ) ⏐ = ⏐ ≃

Rh∗ (Lg ∗ Lu∗ u∗ E ⊗ Lv ∗ Lf ∗f∗ F ) = ⏐ ⏐ Rh∗(Lg ∗ u∗ u∗ E ⊗ Lv ∗f ∗f∗ F ) = ⏐ ⏐ Rh∗ (Lg ∗E ⊗ Lv ∗F ) =

−−→

−−→

5

−−→

6

u∗ E ⊗ f∗ F ⏐ ⏐

149

4

u∗ E ⊗ f∗ F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

Rh∗ Lh∗ (u⏐ ∗ E ⊗ f∗ F ) ⏐

←−−

Rh∗ (h∗ u∗ E ⊗ h∗f∗ F ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐ ⏐ ⏐

←−− h∗ (h∗ u∗ E ⊗ h∗f∗ F ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐ ⏐ ⏐

Rh∗ h∗ (u∗ E ⊗ f∗ F ) ⏐ ⏐

h∗ h∗ (u∗ E ⊗ f∗ F ) ⏐ ⏐≃

−−→ Rh∗(g ∗ u∗ u∗ E ⊗ v ∗f ∗f∗ F ) ←−− h∗(g ∗ u∗ u∗ E ⊗ v ∗f ∗f∗ F ) = = ⏐ ⏐ ⏐ ⏐ −−→ ′ β

Rh∗ (g ∗E ⊗ v ∗F )

←−−

h∗ (g ∗E ⊗ v ∗F )

Commutativity of the unlabeled subdiagrams of the preceding diagrams is pretty clear. Commutativity of subdiagram 3 follows from that of (3.2.1.2), of 4 from (3.2.1.3), of 5 from (3.2.4.1), and of 6 from the dual of the commutative diagram (3.6.4.1) (see the remarks surrounding (3.6.4)∗ ). Lemma (3.10.1.3) results. Q.E.D. Lemma 3.10.1.4 With notation as in (3.10.1.2), for any U-complex E and F) be as above, and let η = η (E, F ) be the any S-complex F let η = ησ (E, 0 0 natural composition

∼ E ⊗R F → V ⊗R (E ⊗R F ) −→ (V ⊗R E)⊗V (V ⊗R F ) → (V ⊗U E)⊗V (V ⊗S F ).

Then there is a natural commutative diagram of OY -modules ⊗ f∗ F u∗ E ⏐ ⏐ η

−− −−→

(E ⊗R F ) ⏐ ⏐ η0

⊗ v ∗F) −− −−→ (V ⊗U E) ⊗V (V ⊗S F ) h∗ (g ∗E

in which the horizontal arrows are isomorphisms.

Proof. The horizontal isomorphisms in the diagram are given by [GD, p. 213, (1.7.7) and p. 202, (1.3.12)(i)].

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3 Derived Direct and Inverse Image

For commutativity, expand the diagram naturally as follows: ⊗ f∗ F u∗ E ⏐ ⏐

−−−−→

⊗ f∗ F) h∗ h∗ (u∗ E ⏐ ⏐

−−−−→

⊗ h∗f∗ F) h∗ (h∗ u∗ E ⏐ ⏐

−−−−→

E ⊗R F ⏐ ⏐

V ⊗R (E ⊗R F ) ⏐ ⏐

(V ⊗R E) ⊗V (V ⊗R F ) ⏐ ⏐

⊗ v ∗f ∗f∗ F) −−−−→ (V ⊗U U ⊗R E) ⊗V (V ⊗S S ⊗R F ) h∗ (g ∗ u∗ u∗ E ⏐ ⏐ ⏐ ⏐ ⊗ v ∗F) h∗ (g ∗E −−−−→ (V ⊗U E) ⊗V (V ⊗S F )

Veriﬁcation of commutativity of the subdiagrams is left as an exercise. (Suggestion: recall (3.1.9), and use [GD, p. 214, (1.7.9)(ii)].) Q.E.D. As in (a), Lemmas (3.10.1.3) and (3.10.1.4) determine (via quasi-coherent q-ﬂat resolutions) the map η(G1 , G2 ) for any G1 ∈ Dqc (Y ′ ) and any G2 ∈ Dqc (X), in terms of the concrete functorial map η0 . Deﬁnition 3.10.2 A commutative oriented S-square v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′ −−− −→ Y u

is said to be • independent if θσ is a functorial isomorphism; • ′ -independent if θσ′ is a functorial isomorphism; • K¨ unneth-independent if ησ is a bifunctorial isomorphism; • tor-independent if σ is a ﬁber square (i.e., the map X ′ → X ×Y Y ′ associated to σ is an isomorphism) and if the following equivalent conditions hold for all pairs of points y ′ ∈ Y ′, x ∈ X such that y := u(y ′ ) = f (x): (i)

OY ,y

Tori

(OY ′, y′ , OX,x ) = 0 for all i > 0.

(ii) There exist an aﬃne open neighborhood Spec(A) of y and aﬃne open sets Spec(A′ ) ⊂ u−1 Spec(A), Spec(B) ⊂ f −1 Spec(A) such that ′ TorA i (A , B) = 0

for all i > 0.

3.10 Independent Squares; K¨ unneth Isomorphism

151

(ii)′ For any aﬃne open neighborhood Spec(A) of y and aﬃne open sets Spec(A′ ) ⊂ u−1 Spec(A), Spec(B) ⊂ f −1 Spec(A), ′ TorA i (A , B) = 0

for all i > 0.

Remarks 3.10.2.1 (a) The conditions of K¨ unneth-independence and torindependence do not depend on an orientation of σ. (b) Condition (ii)′ in (3.10.2) implies condition (ii); and (ii) implies (i) because if p ⊂ A, q ⊂ A′ , and r ⊂ B are the prime ideals corresponding to y, y ′ and x respectively, then there are natural isomorphisms A A ′ ′ ∼ ′ Tori p (A′q , Br ) ∼ = TorA i (Aq , Br ) = Aq ⊗A′ Tori (A , Br ) ∼ A′ ⊗A′ TorA (A′ , B) ⊗B Br . = i q

These isomorphisms also show that, conversely, (i) implies (ii′ ): for if ′ m ⊂ A′ ⊗A B were a prime ideal in the support of TorA i (A , B) and p, q, r ′ ′ were its inverse images in A, A and B respectively, then 0 = TorA i (A , B)m Ap ′ would be a localization of Tori (Aq , Br ) = 0. (c) Let σ, as above, be an independent square; and suppose that the functors f∗ and g∗ have right adjoints f × and g × respectively. Then one can associate to σ a functorial base-change map (for f × rather than f∗ ): βσ : v ∗f × → g × u∗ , θ −1

adjoint to the natural composition g∗ v ∗f × −→ u∗f∗ f × → u∗ . This map plays a crucial role in Grothendieck duality theory on, say, the full subcategory of S whose objects are all the concentrated schemes, in which situation the right adjoints f × and g × exist, see (4.1.1) below. (d) We call an S-map f : X → Y isofaithful if any X* -map α such that f α is a Y* -isomorphism is itself an isomorphism. ∗

For example, if f is an open immersion then f is isofaithful because of the ∼ natural functorial isomorphism G −→ Lf ∗ Rf∗ G (G ∈ D(Y )). Lemma 3.10.2.2 If the S-map f : X → Y is aﬃne [GD, p. 357, (9.1.10)]: for each aﬃne open U ⊂ Y , f −1 U is aﬃne then f is isofaithful.

Proof. In this proof only, f∗ : K(X) → K(Y ) will be the ordinary directimage functor, and Rf∗ : D(X) → D(Y ) its derived functor. From (2.4.5.2) it follows that Rf∗ “commutes” with open immersions, so the question is local, and we may assume that X and Y are aﬃne, let us say X = Spec(B), Y = Spec(A). By (3.9.6)(a), every complex in Dqc (X) is D-isomorphic to a quasicoherent complex. Therefore—and since a D-map α is an isomorphism iﬀ the vertex of a triangle based on α is exact—we need only show: if C is a quasi-coherent OX -complex such that Rf∗ (C) is exact then C is exact.

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3 Derived Direct and Inverse Image

Since the functor f∗ of quasi-coherent OX -modules is exact, therefore, by (3.9.2.3) and the dual of (2.7.4), C is f∗ -acyclic, so f∗ C ∼ = Rf∗ C is exact, i and for all i, f∗ H i C ∼ H f C = 0. = ∗ for some B-complex E, so H i C = (H i E), and when H i E Finally, C = E is regarded as an A-module, f∗ H i C = (H i E) (see [GD, p. 214, (1.7.7.2)]), whence H i E = 0. The desired conclusion results. Q.E.D. The following assertions result at once from commutativity (to be shown) of diagram (3.10) below, for any E ∈ Y′ * and F ∈ X* . unneth independence. • Independence or ′ -independence of σ implies K¨ • If u (resp. f ) is isofaithful then K¨ unneth independence of σ implies independence (resp. ′ -independence). (Take E (resp. F ) to be OY ′ (resp. OX ).) Thus: unneth • If u and f are isofaithful then independence, ′ -independence and K¨ independence are equivalent conditions on σ. This applies, for instance, if the schemes Y ′, Y and X are aﬃne. u∗ (E ⊗ u∗f∗ F ) ⏐ ⏐ via θ u∗ (E ⊗ g∗ v ∗F ) ⏐ ⏐ ≃(3.9.4)

←− −−− (3.9.4)

u∗ E ⊗ f∗ F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐η ⏐ ⏐

−− −−→ (3.9.4)

f∗ (f ∗ u∗ E ⊗ F ) ⏐ ⏐ via θ′ f∗ (v∗ g ∗E ⊗ F ) ⏐ ⏐ (3.9.4)≃

u∗ g∗ (g ∗E ⊗ v ∗F ) −−− −−→ h∗ (g ∗E ⊗ v ∗F ) ←−− −−− f∗ v∗ (g ∗E ⊗ v ∗F ) (3.6.4)∗

(3.6.4)∗

(3.10.2.3)

Proving commutativity of (3.10) is a formal exercise on adjoint monoidal pseudofunctors. For example, in view of the deﬁnition of θσ(F ) in (3.7.2)(c), commutativity of the left half follows from that of the natural diagram u∗ E ⊗ f∗ F ⏐ ⏐ (3.9.4)≃

−−→ 1

u∗ (E ⊗ u∗f∗ F ) ⏐ ⏐ g ∗ u∗f

u∗ (E ⊗ g∗ ⏐ ⏐≃

∗U )

←−−

←−−

u∗ u∗ (u∗ E ⊗ f∗ F ) ⏐ ⏐

u∗ (u∗ u∗ E ⊗ u∗f∗ F ) ⏐ ⏐

u∗ (u∗ u∗ E

⊗ g∗ ⏐ ⏐≃

g ∗ u∗f

∗F )

−−→

−−→ 2

u∗ g∗ g ∗ u∗ (u∗ E ⊗ f∗ F ) ⏐ ⏐

u∗ g∗ g ∗ (u∗ u∗ E ⊗ u∗f∗ F ) ⏐ ⏐

−− → u∗ g∗ (g ∗ u∗ u∗ E ⊗ g ∗ u∗f∗ F ) ⏐ ⏐ ≃

(3.9.4)

u∗ (E ⊗ g∗ v ∗f ∗f∗ U ) ←−− u∗ (u∗ u∗ E ⊗ g∗ v ∗f ∗f∗ F ) −− → u∗ g∗ (g ∗ u∗ u∗ E ⊗ v ∗f ∗f∗ F ) (3.9.4) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ u∗ (E ⊗ g∗ v ∗F )

u∗ (E ⊗ g∗ v ∗F )

−− →

(3.9.4)

u∗ g∗ (g ∗E ⊗ v ∗F )

3.10 Independent Squares; K¨ unneth Isomorphism

153

Commutativity of subsquare 1 is given by 3.4.6.2, and of 2 by (3.4.7)(i). Commutativity of the other subsquares is straightforward to check. Commutativity of the right half of (3.10.3.2) is shown similarly. Theorem 3.10.3 For any ﬁber square of concentrated maps of quasiseparated schemes v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

−→ Y Y ′ −−− u

(σ commutes and the associated map X ′ → Y ′ ×Y X is an isomorphism), the four independence conditions in Deﬁnition (3.10.2) are equivalent. Proof. We ﬁrst prove a special case. Lemma 3.10.3.1 Theorem (3.10.3) holds when all the schemes appearing in σ are aﬃne. Proof. We saw above (just before (3.10)) that the ﬁrst three independence conditions are equivalent. From (3.10.2.2) and (3.10) with F = OX , it follows that if θ(OX ) is an isomorphism then θ′ (E) is an isomorphism for all E, i.e., σ is ′ -independent. Thus it will suﬃce to show that θ(OX ) is an isomorphism if and only if σ is tor-independent. From (3.10.1.2) with E = S, and the assumption that σ is a ﬁber square, one sees that when applied to OX the right column in (3.10.1.1) becomes an isomorphism. As OX is ﬂat and quasi-coherent, the maps α(OX ), β(OX ) and γ(OX ) in (3.10.1.1) are isomorphisms, and hence the left column— which is what we are now denoting by θ(OX )—is an isomorphism iﬀ so is the canonical map ψ : Lu∗f∗ OX → u∗f∗ OX . Since sheaﬁﬁcation is exact and preserves ﬂatness (ﬂatness of a sheaf being guaranteed by ﬂatness of its stalks), using [GD, p. 214, (1.7.7.2)] one ﬁnds that ψ is D(Y ′ )-isomorphic to the sheaﬁﬁcation φ of the natural U -homomorphism φ : U ⊗R P • → U ⊗R S, where U, R and S are as in (3.10.1.2) and P • → S is an R-ﬂat resolution of S. Since φ is a quasi-isomorphism precisely when TorR i (U , S) = 0 for all i > 0, that is, when σ is tor-independent, the desired conclusion results. Q.E.D. The strategy now is to show that: (A) Independence is a local condition, i.e., it holds for σ iﬀ it holds for every induced ﬁber square v

X0′ −−−−→ ⏐ ⏐ g σ0

X0 ⏐ ⏐f

−→ Y0 Y0′ −−− u

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3 Derived Direct and Inverse Image

such that Y0 is an aﬃne open subscheme of Y , and Y0′ , X0 are aﬃne open subschemes of u−1 Y0 , f −1 Y0 respectively. (See ﬁrst paragraph of §3.10.) It follows then from (3.10.3.1) that tor-independence for σ in (3.10.3) implies independence and, by symmetry, ′ -independence. It has already been noted (before (3.10)) that independence or ′ -independence implies K¨ unneth independence. To ﬁnish proving (3.10.3) it will therefore suﬃce to show that: (B) K¨ unneth independence for σ implies the same for any σ0 as above. For then it will follow from (3.10.3.1) that K¨ unneth-independence implies tor-independence. Finally, (A) and (B) result at once from the ﬁrst assertion in (3.10.3.3) and the last assertion in (3.10.3.4) below. Lemma 3.10.3.2 (Independence and concatenation). For each one of the following S-diagrams, assumed commutative, w

v

X ′′ −−−1−→ ⏐ ⏐ σ1 h

v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′′ −−− −→ Y ′ −−− −→ Y u u 1

Z ′ −−−−→ ⏐ ⏐ σ1 g1 v

X ′ −−−−→ ⏐ ⏐ g σ

Z ⏐ ⏐f 1

X ⏐ ⏐f

Y ′ −−− −→ Y u

if σ and σ1 are independent (resp. ′ -independent, K¨ unneth-independent) then so is the rectangle σ0 := σσ1 enclosed by the outer border. Proof. As in (3.7.2)(iii), the following natural diagram commutes for any G ∈ X* : θσ0(G)

(uu1 )∗f∗ G −−−−−−−−−−−−−−−−−−−−→ h∗ (vv1 )∗ G ⏐ ⏐ ⏐≃ ⏐ ≃ u∗1 u∗f∗ G

−−∗−−−→ u1 θσ (G)

u∗1 g∗ v ∗ G

−−−−∗−→ θσ1(v G)

(3.10.3.2.1)

h∗ v1∗ v ∗ G

whence the independence assertion for the ﬁrst of the diagrams in (3.10.3.2). The second is dealt with similarly via (3.7.2)(ii). The assertion for ′ -independence follows by symmetry. (Reﬂection in the appropriate diagonal interchanges independence and ′ -independence.)

3.10 Independent Squares; K¨ unneth Isomorphism

155

K¨ unneth independence for the ﬁrst diagram in (3.10.3.2)—and hence, since K¨ unneth independence does not depend on orientation, for the second diagram too—is treated via commutativity of the following natural diagram (with E ∈ Y′′ * and F ∈ X* ): (uu1 )∗ E ⊗ f∗ F ⏐ ⏐ ≃

u∗ (u1∗ E) ⊗ f∗ F ⏐ ⏐ ησ(u1∗E,F )

ησ0(E,F )

−−−−−−−−−−−−−−−−−−−−−−−−−−→ (uu1 h)∗ (h∗E ⊗ (vv1 )∗F ) ⏐ ⏐≃

∗ ∗ (ug)∗ (g ∗ u1∗ E ⊗ v ∗F ) −− −− → u∗ g∗ (g u1∗ E ⊗ v F ) (3.9.4) →

u∗ (u1 h)∗ (h∗E ⊗ v1∗ v ∗F ) ⏐u η (E,v∗F ) ⏐ ∗ σ1 u∗ (u1∗ E ⊗ g∗ v ∗F )

(3.10.3.2.2)

Commutativity can be veriﬁed, e.g., by using the left half of the commutative diagram (3.10) to reduce the question to commutativity of the natural diagram: (uu1 )∗ E ⏐ ⊗ f∗ F ⏐≃

u∗ (u1∗ E) ⊗ f∗ F ⏐ ⏐ (3.9.4)≃

θσ0

−− −→ (uu1 )∗ (E ⊗⏐(uu1 )∗f∗ F ) −−→ (uu1 )∗ (E ⊗⏐h∗ (vv1 )∗F ) ⏐≃ ⏐ ≃(3.9.4) etc.

(3.9.4)

1

u∗ (u1∗ E ⊗ u∗f∗ F ) −− −→ (3.9.4) ⏐ ⏐ θσ u∗ (u1∗ E ⊗ g∗v ∗F ) −− −→

(3.9.4)

u∗ u1∗ (E ⊗ (uu1 )∗f∗ F ) ⏐ ⏐≃ ∗ u∗ u1∗ (E ⊗ u∗ 1 u f∗ F ) ⏐ ⏐θ σ ∗ u∗ u1∗ (E ⊗ u∗ 1 g∗v F )

θσ0

−−→ 2

−−→ θσ1

u∗ u1∗ h∗ (h∗ E ⊗ v1∗ v ∗F ) ⏐ ≃⏐(3.9.4) u∗ u1∗ (E ⊗ h∗ v1∗ v ∗F )

u∗ u1∗ (E ⊗ h∗ v1∗ v ∗F )

Commutativity of subdiagram 1 follows from (3.7.1), and of subdiagram 2 from (3.7.2)(iii). The rest is straightforward. Q.E.D. Corollary 3.10.3.3 For σ as in (3.10.3): (i) σ is independent if and only if for every diagram as in (3.10.3.2) with Y ′′ aﬃne, u1 : Y ′′ → Y ′ an open immersion and σ1 a ﬁber square, σ0 := σ ◦ σ1 is independent. (i)′ σ is ′ -independent if and only if for every diagram as in (3.10.3.2) with Z aﬃne, f1 : Z → X1 an open immersion and σ1 a ﬁber square, σ0 := σ ◦ σ1 is ′ -independent. Proof. It follows from (1.2.2) that θσ is an isomorphism iﬀ so is u∗1 θσ for all open immersions u1 : Y ′′ → Y ′ with Y ′′ aﬃne.

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3 Derived Direct and Inverse Image

For such a u1 the ﬁber square σ1 is independent (as follows readily from (2.4.5.2)), so the commutative diagram (3.10.3.2.1) shows that u∗1 θσ is isomorphic to θσ0 , and (i) results. Q.E.D. Up to reversal of orientation, (i)′ is the same statement as (i). Lemma 3.10.3.4 (Independence and base change). Given σ as in (3.10.3) let i : U → Y be an open immersion, let i∗σ be the ﬁber square v

U ×Y X ′ =: V ′ −−−1−→ ⏐ ⏐ g1

V := U ×Y X ⏐ ⏐f 1

U ×Y Y ′ =: U ′ −−− −→ U u1

(with obvious maps) and let j : V → X and i′ : U ′ → Y ′ be the projections. Then i∗σ is an S-square, and for any G ∈ Dqc (X) the map θi∗σ (j ∗ G) : u∗1 f1∗ j ∗ G → g1∗ v1∗ j ∗ G is isomorphic to the map i′∗ θσ (G) : i′∗ u∗f∗ G → i′∗g∗ v ∗ G . Moreover, for any E ∈ Dqc (U ′ ) and F ∈ Dqc (X) the map i∗ η i∗σ (E, j ∗F ) : i∗(u1∗ E ⊗ f1∗ j ∗F ) → i∗(u1 g1 )∗(g1∗E ⊗ v1∗ j ∗F ) is isomorphic to the map ησ (i′∗ E, F ) : u∗(i′∗ E) ⊗ f∗ F → (ug)∗(g ∗ i′∗ E ⊗ v ∗F ). Consequently, σ is independent if and only if i∗σ is independent for every open immersion i : U ֒→ Y with U aﬃne; and if σ is K¨ unneth-independent then so is i∗σ for all such i. Proof. That U , U ′, V and V ′ are quasi-separated is given by [GD, p. 294, (6.1.9)(i) and (ii)]; and that u1 , f1 , g1 and v1 are quasi-compact by [GD, p. 291, (6.1.5)(iii)]. By (3.7.2)(iii), the diagrams v

V ′ −−−1−→ ⏐ ⏐ g1 i∗σ

j

V −−−−→ ⏐ ⏐f σ ′ 1

X ⏐ ⏐f

U ′ −−− −→ Y u1−→ U −−− i

j′

V ′ −−−−→ ⏐ ⏐ g1 σ ′′

v

X ′ −−−−→ ⏐ ⏐g σ

X ⏐ ⏐f

U ′ −−−′−→ Y ′ −−− u−→ Y i

which are two decompositions of the same square—call it τ —give rise to a commutative diagram of functorial maps (cf. (3.10.3.2.1)):

3.10 Independent Squares; K¨ unneth Isomorphism u∗ θ

157

θi∗σ (j ∗ G)

′ (G)

σ −−→ u∗1 f1∗ j ∗ G −−−−−−→ g1∗ v1∗ j ∗ G u∗1 i∗f∗ G −−1−− ⏐ ⏐ ⏐≃ ⏐ ≃

θτ (G)

(iu1 )∗f∗ G −−−−−−−−−−−−−−−−−−−−−−→ g1∗ (jv1 )∗ G (ui′ )∗f∗ G −−−−−−−−−−−−−−−−−−−−−−→ g1∗ (vj ′ )∗ G θτ (G) ⏐ ⏐ ⏐≃ ⏐ ≃ i′∗ u∗f∗ G −− −−−→ i′∗ g∗ v ∗ G −−−−− −→ g1∗ j ′∗ v ∗ G ∗ ′∗ i θσ (G)

θσ′′ (v G)

Since i and i′ are open immersions, the maps θσ′ and θσ′′ are isomorphisms (see proof of (3.10.3.3)), and the ﬁrst isomorphism assertion in the Lemma results. A similar argument using (3.10) proves the second isomorphism assertion. The independence consequence for θ then follows from (1.2.2) and the fact that since j is an open immersion therefore F ∼ = j ∗j∗ F for every F ∈ D(V ). The K¨ unneth-independence consequence is proved similarly, with the additional observation that i is isofaithful (see (3.10.2.1)(d)). Q.E.D. Exercises 3.10.4 (Conjugate base change). Let σ be a ﬁber square as in (3.10.3), and assume the schemes in σ are concentrated, so that by (4.1.1) below, f∗ and g∗ have right adjoints f × and g × respectively. (a) Show that the map φσ : v∗ g × → f × u∗ (between functors from Dqc (Y ′ ) to Dqc (X)) corresponding by adjunction to the ∼ natural composition f∗ v∗ g × −→ u∗ g∗ g × → u∗ is right-conjugate to θσ. Deduce that σ is independent iff φσ (or φσ′ ) is an isomorphism. Hint. The ﬁrst assertion is that φσ (E) is the image of the identity map under the sequence of natural isomorphisms ∼ ∼ Hom(v∗ g ×E, v∗ g ×E) −→ Hom(v ∗ v∗ g ×E, g × E) −→ Hom(g∗ v ∗ v∗ g ×E, E) ∼ ∼ Hom(u∗f∗ v∗ g ×E, E) −→ Hom(f∗ v∗ g ×E, u∗ E) −→ ∼ −→ Hom(v∗ g × E, f × u∗ E). −1 (b) Show that when σ is independent the map φ−1 σ —right-conjugate to θσ , see (a)—corresponds to the composition via β

natural

σ v ∗f × u∗ −−−−→ g × u∗ u∗ −−−−→ g ×

with βσ as in (3.10.2.1)(c). (b)′ Show that when σ is independent the map βσ corresponds to the composition natural

via φ−1

σ v ∗ g × u∗ . f × −−−−→ f × u∗ u∗ −−−−→

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3 Derived Direct and Inverse Image

Hint. To deduce (b)′ from (b), use the natural diagram (whose bottom row and right column both compose to the identity): f× ⏐ ⏐

−−−−→

f × u∗ u∗ ⏐ ⏐

φ−1

−−−σ−→ φ−1

v ∗g × u∗ ⏐ ⏐

v∗ v ∗f × −−−−→ v∗ v ∗f × u∗ u∗ −−−σ−→ v∗ v ∗ v ∗g × u∗ ⏐ ⏐ ⏐ ⏐ ⏐v β ⏐ v∗ βσ ∗ σ v∗ g × u∗ −−−−→ v∗ g × u∗ u∗ u∗ −−−−→

v ∗ g × u∗

Similarly, (b)′ ⇒ (b). (c) Show that φσ corresponds to the natural composition ∼ g × −−−−→ g × u× u∗ −→ v ×f × u∗ .

Chapter 4

Abstract Grothendieck Duality for Schemes

In this chapter we review and elaborate on—with proofs and/or references— some basic abstract features of Grothendieck Duality for schemes with Zariski topology, a theory initially developed by Grothendieck [Gr′ ], [H], [C], Deligne [De′ ], and Verdier [V′].1 The principal actor in this Chapter is the twisted inverse image pseudofunctor, described in the Introduction. The basic facts about this pseudofunctor—which may be seen as the main results in these Notes—are existence and ﬂat base change, Theorems (4.8.1) and (4.8.3). The abstract theory begins with Theorem (4.1) (Global Duality), asserting for any map f : X → Y of concentrated schemes the existence of a right adjoint f × for the functor Rf∗ : Dqc (X) → Dqc (Y ). In order to sheaﬁfy this result, or, more generally, to prove tor-independent base change for f × —see (4.4.2) and (4.4.3), we need f to be quasi-proper, a condition which coincides with properness when the schemes involved are noetherian. This condition is discussed in section 4.3. The proofs of (4.4.2) and (4.4.3) are given in sections (4.5) and (4.6). That prepares the ground for the above main results. Section (4.7) is concerned with quasi-perfect ( = quasi-proper plus ﬁnite tor-dimension) maps of concentrated schemes. These maps have a number of especially nice properties with respect to f × . Analogously, section (4.9) deals with perfect ( = ﬁnite tor-dimension) ﬁnite-type separated maps of noetherian schemes. These maps behave nicely with respect to the twisted inverse image. For example, if f : X → Y is a ﬁnite-type separated map of noetherian schemes, and f ! is the associated twisted inverse image functor, perfectness of f is characterized by boundedness of f ! OY plus the existence of a functorial isomorphism ∼ f ! OY ⊗ Lf ∗F −→ f !F F ∈ D+ qc (Y ) . =

1

As regards these Notes, see the Introduction for some comments on “abstract” vis-` a-vis “concrete” duality. Exercise (4.8.12)(b) is an example of the latter. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

159

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4 Abstract Grothendieck Duality for Schemes

This, and other characterizations, are in Theorem (4.9.4). Theorem (4.7.1) contains the corresponding result for the functor f × associated to a quasiperfect map f . In an appendix, section (4.10), we say something about the role of dualizing complexes in duality theory. This is an important topic, but not a central one in these Notes. Throughout, all schemes are assumed to be concentrated, i.e., quasiseparated and quasi-compact.

4.1 Global Duality Fix once and for all a universe U [M, p. 22]. Henceforth, any category is understood to have all its arrows and objects in U. Call a set small if it is a member of U. A small category is one whose arrows—and hence objects— form a small set. Every topological space X is understood to be small; and any sheaf E on X is understood to be such that for every open U ⊂ X, Γ(U , E) is a small set. For any scheme (X, OX ), AX is, as before, the abelian category of OX qc modules and their homomorphisms, and AX is the full abelian subcategory whose objects are all the quasi-coherent OX -modules. Though these two categories are not small, they are well-powered, i.e., for each object E there is a small setJE such that every subobject (or every quotient) of E is isomorphic to a member of JE ; and they have small hom-sets, i.e., for any objects E, F, the set Hom(E, F ) is small. “Global Duality” means: Theorem 4.1 Let X be a concentrated (= quasi-compact, quasi-separated) scheme and f : X → Y a concentrated scheme-map. Then the Δ-functor Rf∗ : Dqc (X) → D(Y ) has a bounded-below right Δ-adjoint. By (1.2.2), (2.4.2), and the description of θ∗ in (3.3.8) (where it may be assumed that θ∗ is the identity, see (2.7.3.2)), the following statement is equivalent to (4.1). Theorem 4.1.1 Let X be a concentrated (quasi-compact, quasi-separated ) scheme and f : X → Y a concentrated scheme-map. Then there is a boundedbelow Δ-functor (f ×, identity): D(Y ) → Dqc (X) and a map of Δ-functors τ : Rf∗ f × → 1 such that for all F ∈ Dqc (X) and G ∈ D(Y ), the composite Δ-functorial map (in the derived category of abelian groups) (3.2.1.0)

• • (F , f ×G) −−−−−→ RHomX (Lf ∗ Rf∗ F , f ×G) RHomX (3.2.3.1)

−−−−−→ RHom•Y (Rf∗ F , Rf∗ f ×G) via τ

−−−−−→ RHom•Y (Rf∗ F , G) is a Δ-functorial isomorphism.

4.1 Global Duality

161

Corollary 4.1.2 When restricted to concentrated schemes, the Dqc -valued pseudofunctor “derived direct image” (see (3.9.2)) has a pseudofunctorial right Δ-adjoint × (see (3.6.7)(d)). Proofs. To get (4.1.2) from (4.1.1), recalling that a map f : X → Y of concentrated schemes is itself concentrated [GD, §6.1, pp. 290ﬀ ], choose for each such f a functor f × right-Δ-adjoint to Rf∗ : Dqc (X) → Dqc (Y ), with f × the identity functor whenever f is an identity map. For another such g : Y → Z, deﬁne df,g : f ×g × → (gf )× to be the functorial map adjoint to the natural composition ∼ R(gf )∗ f × g × −→ Rg∗ Rf∗ f × g × → Rg∗ g × → 1.2

This df,g is an isomorphism, its inverse (gf )× → f ×g × being the map adjoint to the natural composition ∼ Rg∗ Rf∗ (gf )× −→ R(gf )∗ (gf )× → 1.

The veriﬁcation of (4.1.2) is then straightforward (see (3.6.5)). As for (4.1), the classical abstract method was introduced by Verdier in his treatment of duality for locally compact spaces, then adapted to schemes by qc Deligne [De′ ] to show that with j : D(AX ) → Dqc (X) the natural functor, Rf∗ ◦j has a right adjoint. This suﬃces only when f is separated, see (3.9.6). The proof given below (for historical reasons, because of the compactness of Deligne’s original presentation) is just an elaboration of Deligne’s arguments. The reader may prefer to look up in [N] the more modern, lucidly exposed, approach of Neeman, who uses Brown Representability instead of, as below, the Special Adjoint Functor Theorem applied via injective resolutions. This is conceptually more elegant in that it gives a direct criterion for the existence of a right adjoint for a triangulated functor F on any compactly generated triangulated category, such as Dqc (X). In analogy with the “cocontinuity” used in Deligne’s method (see below), the condition on F is that it commute with small direct sums, a condition which follows for F = Rf∗ from (3.9.3.3). The (nontrivial) proof in [N] that Dqc (X) is compactly generated ostensibly requires X to be separated; but essentially the same proof shows that Dqc (X) is compactly generated for any concentrated X, see [BB, §3], and this gives Theorem (4.1) in full generality.3 Proof of (4.1) (when X is separated, see above). 1. First, we review some terminology and basic results about abelian categories. Let A be an abelian category with small direct sums (i.e., every family of objects in A indexed by a small set has a direct sum). Any two 2

This deﬁnition makes the property TRA 1 in [H, p. 207] tautologous. Arguments much like Deligne’s or Neeman’s apply also to noetherian formal schemes, see [AJL′, §4, pp. 42–46] resp. [AJL′, p. 41, 3.5.2] and [AJS, p. 245, Cor. 5.9].

3

162

4 Abstract Grothendieck Duality for Schemes

arrows in A with the same source and target have a coequalizer, namely the cokernel of their diﬀerence [M, p. 70]. Hence A is small-cocomplete, i.e., any functor from a small category into A has a colimit, see [M, p. 113, Cor. 2] (dualized). An additive functor F from A to an abelian category A′ is cocontinuous if F commutes with small colimits, in the sense that if G is any functor from a small category C into A and G, (gc : Gc → G)c∈C is a colimit of G then FG, (Fgc )c∈C is a colimit of FG. It follows from [M, p. 113, Thm. 2] that F is cocontinuous iﬀ it is right-exact and transforms small direct sums in A into small direct sums in A′ . We reserve the symbol lim for denoting direct limits of small directed −→ systems in A, i.e., colimits of functors G : C → A where C is the category associated to a small preordered set in which any two elements have an upper bound [M, p. 11, p. 211]. All such lim’s exist in an abelian category A iﬀ A −→ is small-cocomplete [M, p. 212, Theorem 1]. Similarly, an additive functor F : A → A′ is cocontinuous iﬀ it is right-exact and commutes with all lim’s. −→ 2. An essential ingredient of the proof of Theorem (4.1) is the following consequence of the Special Adjoint Functor Theorem [M, p. 130, Corollary]. (See also [De′, p. 408, Cor. 1]). Proposition 4.1.3 For a concentrated scheme X, an additive functor F from qc AX to an abelian category A′ with small hom-sets has a right adjoint if and (clearly) only if it is cocontinuous. (4.1.3.1). For the Special Adjoint Functor Theorem to be applicable here, qc the category AX —which, as above, is well-powered and has small hom-sets, and which is also small-cocomplete [GD, p. 217, (2.2.2)(iv)]—must have a small set of generators. Recall that an OX -module E on a ringed space X is locally ﬁnitely presentable (lfp for short) if X is covered by open subsets U such that for each U the restriction E|U is isomorphic to the cokernel of a map m n → OU with ﬁnite m and n. Since every quasi-coherent OX -module is the OU lim of its lfp submodules [GD, p. 319, (6.9.9)], the small-generated property −→ follows from the fact that for any scheme X there exists a small set S of lfp OX -modules such that every lfp OX -module is isomorphic to a member of S. Proof. With U ranging over the small set of aﬃne open subschemes of X, and iU : U֒→ X the inclusion, any OX -module E is isomorphic to a submodule of U iU ∗ i∗U E. If E is lfp then so is the OU -module i∗U E, so n for some ﬁnite n [GD, p. 207, (1.4.3)]. that i∗U E is a quotient of OU Thus every lfp E is isomorphic to a subsheaf of a sheaf of the form U iU ∗ EU where for each U , EU ranges over a ﬁxed small set of OU -modules, whence the conclusion. Q.E.D. (For another argument see [Kn, pp. 43–44, proof of Thm. 4.]).

4.1 Global Duality

163

3. The basic idea for proving (4.1) is to show that there is a functorial qc ֒→ AX ) exact AX -sequence (i.e., a ﬁnite resolution of the inclusion AX 0

1

d−1

δ(M ) δ (M ) δ (M ) δ (M ) 0 → M −−−−→ D0(M )−−−−→D1(M ) −−−−→ · · · −−−−→ Dd(M ) → 0 (4.1.4) qc M ∈ AX

qc such that the functors Di : AX → AX (0 ≤ i ≤ d) are additive and cocontinuous, such that for all M , Di(M ) is f∗ -acyclic, and such that the functors f∗ Di are right-exact. Here is one way to do this. Recall the Godement resolution

0 → M → G 0 (M ) → G 1 (M ) → · · · where, with G −2 (M ) := 0, G −1 (M ) := M , and Ki (M ) (i ≥ 0) the cokernel of G i−2 (M ) → G i−1 (M ), the sheaf G i (M ) is deﬁned inductively by

G i (M ) U := Ki (M )x (U open in X). x∈U

One shows by induction on i that all the functors G i and Ki (from AX to itself) are exact. Moreover, for i ≥ 0, G i (M ) is ﬂasque, hence f∗ -acyclic. With d as in (3.9.2.4), the dual version of (2.7.5)(iii) shows that Kd (M ) is f∗ -acyclic. So, setting ⎧ i ⎪ (0 ≤ i < d) ⎨G (M ) i d D (M ) := K (M ) (i = d) ⎪ ⎩ 0 (i > d)

we get a ﬁnite resolution (4.1.4) having all the desired properties except for commutativity of the Di with lim. −→ To get commutativity with lim we use the next Lemma, proved below. −→ Lemma 4.1.5 Let A′ be a small-cocomplete abelian category in which lim −→ preserves exactness of sequences. Then with F the category of additive funcqc tors from AX to A′, there is a functor (−)cts : F → F and a functorial map iD : Dcts → D (D ∈ F) such that: qc ∼ D(M ). (i) For all lfp M ∈ AX , iD(M ) is an isomorphism Dcts(M ) −→ (ii) For any D ∈ F, Dcts commutes with lim. −→ (iii) If D commutes with lim then iD is a functorial isomorphism. −→ (iv) If D is right-exact then so is Dcts . (v) For any exact sequence D′ → D → D′′ in F (i.e., the A′ -sequence qc ′ D (M ) → D(M ) → D′′ (M ) is exact for all M ∈ AX ), the corresponding ′ ′′ sequence Dcts → Dcts → Dcts is exact. qc (vi) When A′ = AX , if D(M ) is f∗ -acyclic for all M ∈ AX then Dcts (M ) qc is f∗ -acyclic for all M ∈ AX ; and if, further, D is exact, then the functor qc → AY is right-exact. f∗ Dcts : AX

164

4 Abstract Grothendieck Duality for Schemes

Indeed, one can apply any such (−)cts for A′ = AX to the just-constructed truncated Godement resolution, to produce a resolution with all the desired properties. (For this, condition (4.1.5)(iii) is needed only when D = identity functor.) From (4.1.4) there results a Δ-functor qc

(D, Identity) : K(AX ) → K(AX ) =: K(X) qc

taking each AX -complex (M, d) to the f∗ -acyclic AX -complex D(M ) with D(M )m := ⊕p+q=m Dq(M p )

(m ∈ Z, 0 ≤ q ≤ d)

and with diﬀerential D(M )m → D(M )m+1 deﬁned on Dq(M p ) (p + q = m) to be Dq (dp ) + (−1)p δ q (M p ). One checks by elementary diagram chasing— or spectral sequences—that the natural K(X)-map δ(M ) : M → D(M ) is a quasi-isomorphism. It follows that the natural maps are D(Y )-isomorphisms qc ∼ ∼ Rf∗ jM , M ∈ K(AX ) (4.1.6) f∗ D(M ) −→ Rf∗ D(M ) ←− Rf∗ δ(M )

the ﬁrst, in view of (3.9.2.4), by the dual version of (2.7.5)(a). Thus we have realized Rf∗ ◦ j (up to isomorphism) at the homotopy level, as the functor C• := f∗ D. Let us ﬁnd a right adjoint at this level. qc

4. Each functor Cq := f∗ Dq : AX → AY (0 ≤ q ≤ d) is right-exact. Also, Cq commutes with lim since both Dq and f∗ do. (For f∗ see [Kf, p. 641, −→ Prop. 6], or imitate the proof on p. 163 of [G]). Thus Cq is cocontinuous, and qc so by (4.1.3), Cq has a right adjoint Cq : AY → AX . There are then functorial maps δs : Cs+1 → Cs right-conjugate to f∗ (δ s ) : s C → Cs+1 , see (3.3.5). qc For each AY -complex (F , d′ ), let C• F be the AX -complex with

(C• F )m := Cq F p (m ∈ Z, 0 ≤ q ≤ d), p−q = m

and whose diﬀerential (C• F )m → (C• F )m+1 is the unique map making the following diagram (with vertical arrows coming from projections) commute for all r, s with r − s = m +1: Cq F p Cq F p = (C• F )m −−−→ (C• F )m+1 = p−q = m p−q = m+1 ⏐ ⏐ ⏐ ⏐ Cs F r−1 ⊕ Cs+1 F r

−−−−−−−−−−−−−−−→ Cs dr−1 +(−1)r+s δs (F r ) ′

Cs F r

qc

There results naturally a Δ-functor (C• , Identity) : K(Y ) → K(AX ).

4.1 Global Duality

165

One checks that, applied componentwise, the adjunction isomorphism qc ∼ p qc (M, Cp N ) −→ HomA (C M, N ) HomAX M ∈ AX , N ∈ AY Y

produces an isomorphism of complexes of abelian groups

• • ∼ • qc (G, C• F ) −→ HomA (C G, F ) HomA Y

(4.1.7)

X

qc for all AX -complexes G and AY -complexes F . 5. The isomorphism (4.1.7) suggests using C• to construct f ×, as follows. qc qc Recall that a complex J ∈ K(AX ) is K-injective iﬀ for each exact G ∈ K(AX ), • the complex HomAqc (G, J) is exact too. The isomorphisms (4.1.6) show that X C• G is exact if G is; so it follows from (4.1.7) that if F is K-injective qc in K(Y ) then C• F is K-injective in K(AX ). Thus if KI (−) ⊂ K(−) is the full subcategory whose objects are all the K-injective complexes, then we qc ). have a Δ-functor (C• , Id) : KI (Y ) → KI (AX Associating a K-injective resolution to each complex in AY leads to a Δ-functor (ρ, θ) : D(Y ) → KI (Y ). In fact (ρ, θ) is an equivalence of Δ-categories, see §1.7. This ρ is bounded below : an AY -complex E such that H i (E) = 0 for all i < n is quasi-isomorphic to its truncation τ≥n E, which is quasi-isomorphic to an injective complex F vanishing in all degrees below n; and such an F is K-injective. Finally, one deﬁnes f × to be the composition of the functors ρ

C

qc

natural

qc

• → KI (Y ) −→ KI (AX ) −−−−−→ D(AX ), D(Y ) −

and checks, via (4.1.6), (4.1.7), (2.3.8.1) and (2.3.8)(v), that (f × , identity) is indeed a bounded-below right Δ-adjoint of Rf∗ ◦ j. (Checking the Δ-details can be tedious. Note that by (2.7.3.2) and (3.3.8), we can at least assume that f × commutes with translation of complexes.) That f × is bounded below results from (3.9.2.3) and the following general fact. Lemma 4.1.8 Let A#, B# be plump subcategories of the abelian categories A, B respectively, let E = D# (A), D#*(A), or D#*(A), see (1.9), and let E′ = ′ D# (B), D#*(B), or D* # (B). If the functor F : E → E has a right adjoint G, then for any n, d ∈ Z: F (E≤n ) ⊂ E′≤n+d ⇐⇒ G(E′≥n ) ⊂ E≥n−d . Proof. Let B ∈ E′≥n . For A = τ≤n−d−1 G(B), the natural α : A → G(B) induces homology isomorphisms in all degrees < n − d, see (1.10). But since F (A) ∈ E′≤n−1 and τ≤n−1 B ∼ = 0, we have by adjointness and by (1.10.1.1): α ∈ HomE A, G(B) ∼ = HomE′ F (A), B ∼ = HomE′ F (A), τ≤n−1 B = 0. Hence H j G(B) = 0 for all j < n − d, i.e., G(B) ∈ E≥n−d .

166

4 Abstract Grothendieck Duality for Schemes

A dual argument gives the opposite implication. Q.E.D. This completes the proof of Theorem (4.1), except for Lemma (4.1.5). Proof of (4.1.5). For constructing (−)cts let S be a small set of lfp OX -modules such that every lfp OX -module is isomorphic to a member of S, see (4.1.3.1). For any qc M ∈ AX let S↓M be the small category whose objects are all the maps s → M (s ∈ S), qc a morphism from α : s → M to β : s′ → M being an AX -map μ : s → s′ with βμ = α. qc Sending each α : s → M in S↓M to its source sα := s, we get a functor sM : S↓M → AX . For any D ∈ F, the additive functor Dcts ∈ F is deﬁned as follows: Dcts (M ) := colim D◦ sM S↓M

qc

(M ∈ AX );

qc

and for any AX -map φ : M → M ′, Dcts (φ) is the A′ -map induced by the map sM → sM ′ given by composition with φ.4 The map iD : Dcts (M ) → D(M ) is the one whose composition with the canonical map D(sα ) = DsM (α) → Dcts (M ) is D(α) : D(sα ) → D(M ) for each object α : sα → M in S↓M . Condition (4.1.5)(i) follows from the observation that when M is lfp, the identity map of M is a ﬁnal object in the category S↓M . To prove (ii) we need: (∗) : For any lfp E and directed system Nσ of quasi-coherent OX -modules the natural map is an isomorphism ∼ lim − → HomOX (E, Nσ ) −→ HomOX (E, lim − → Nσ ). σ σ

(Proof : Since X is concentrated, therefore Γ(X, −) commutes with lim −→ [Kf, p. 641, Prop. 6], so it suﬃces to prove the statement with Hom in place of Hom. Thus the statement is local, and so equivalent to the analogous well-known—and easily veriﬁable—one for modules over rings.) qc Given a small directed system Mγ , (φδγ : Mγ → Mδ )δ≥γ in AX , (∗) shows that each map s → M := lim −→ Mγ with s ∈ S is determined by a unique equivalence class of maps s → Mγ (s ﬁxed, γ variable), where [s → Mγ ′ ] ≡ [s → Mγ ′′ ] if and only if there exists a commutative diagram s ⏐ ⏐

−−−−→ Mγ ′ ⏐ ⏐φ ′ γγ

Mγ ′′ −− −−→ Mγ φ ′′ γγ

φδγ

This is the least equivalence relation such that [s → Mγ ] ≡ [s → Mγ −→ Mδ ] ′ for all δ ≥ γ. Moreover, A -maps f : Dcts (M ) → A correspond naturally to families of maps fα : D(sα ) → A α∈S↓M such that for any OX -homomorphism μ : s′ → sα (s′ ∈ S), fα◦μ = fα ◦ D(μ). Hence an A′ -map g : Dcts (M ) → A corresponds to a family of maps gα : D(sα ) → A indexed by OX -homomorphisms α : s → Mγ with variable s ∈ S and γ, such that for any φ = φδγ (δ ≥ γ), φ gs→Mγ − →Mδ = gs→Mγ

For example, if X is noetherian then Dcts (M ) ∼ = lim −→ D(N ) where N runs through all ﬁnite-type OX -submodules of M .

4

4.1 Global Duality

167

and such that for any OX -homomorphism μ : s′ → sα with s′ ∈ S, gα◦μ = gα ◦ D(μ). One checks that an A′ -map lim −→ Dcts (Mγ ) → A is speciﬁed by a family gα subject to exactly the same conditions, whence the natural map is an isomorphism ∼ lim −→ Dcts (Mγ ) −→ Dcts (M ) = Dcts (lim −→ Mγ ),

proving (ii). Then (iii) results by application of lim −→ to (i), since by [GD, p. 320, (6.9.12)] every qc of lfp O -modules. M ∈ AX is a lim X −→ qc Again, [GD, p. 320, (6.9.12)] allows each M ∈ AX to be represented in the form M = lim −→ (Mλ ) with each Mλ lfp. From (∗) above we get a natural isomorphism Dcts (M ) ∼ = lim −→ D(Mλ ). qc -acyclicity in AX (see [Kf, p. 641, Thm. 8] Since lim preserves both exactness and f ∗ −→ for acyclicity), assertion (v) and the ﬁrst part of (vi) follow. qc

ρ

As for (iv), for any exact AX -sequence (♯) : 0 → M ′ → M − → M ′′ → 0 we must ′ show exactness of the resulting sequence Dcts (M ) → Dcts (M ) → Dcts (M ′′ ) → 0. As in the preceding paragraph, write M = lim −→ (Mλ ) with each Mλqc lfp, and let φλ : Mλ → M be the natural maps. Then (♯) is the lim −→ of the exact AX -sequences (♯)λ : 0 → ker(ρφλ ) → Mλ → im(ρφλ ) → 0. Since Dcts commutes with lim −→ and lim −→ preserves exactness, we can replace (♯) by (♯)λ , i.e., we may assume that M is lfp. ′ ′ ′ ′ ∼ Now write M ′ = lim D(Mμ ). −→′′ −→ (Mμ ) with lfp Mμ , so that as ′above, ′Dcts (M ) = lim ′′ If Mμ is the cokernel of the natural composition Mμ → M → M then Mμ is lfp; ′′ ∼ ′′ ′′ ′′ and since lim ) ∼ = lim −→ Mμ and Dcts (M −→ D(Mμ ). −→ preserves exactness, M =′ lim ′′ Applying lim −→ to the exact sequences D(Mμ ) → D(M ) → D(Mμ ) → 0, we conclude that Dcts is right-exact. Finally, for the last part of (vi), note that if D is exact then since R1f∗ D(M ) = 0 qc for all M ∈ AX (because D(M ) is f∗ -acyclic), therefore f∗ D is exact, and hence by (iv), (f∗ D)cts is right-exact. But since, as above, f∗ commutes with lim −→, there are functorial isomorphisms ∼ ∼ (f∗ D)cts (M ) ∼ = lim −→ f∗ D(Mλ ) = f∗ lim −→ D(Mλ ) = f∗ Dcts (M ), and so f∗ Dcts is right-exact, as asserted.

Q.E.D.

Exercise 4.1.9 (a) In (4.1.1), suppose only that X is noetherian as a topological space (resp. that both X and Y are concentrated). Then the conclusion is valid for any scheme-map f : X → Y . Hint. See the remarks just before the proof of (4.1), resp. [GD, p. 295, (6.1.10(i) and (iii))]). (b) If f : X → Y is a concentrated scheme-map and Y is a ﬁnite union of open subschemes Yi with f −1 Yi concentrated, then the conclusion of Theorem (4.1.1) holds.

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4 Abstract Grothendieck Duality for Schemes

Hint. Arguing as in [AJL′, p. 60, 6.1.1], by induction on the least possible number of Yi , one reduces via [GD, p. 296, (6.1.12), a)⇒c)] to where X itself is concentrated; and then the remarks just before the proof of (4.1) apply. (c) Let f : X ֒→ Y be an open-and-closed immersion of concentrated schemes (i.e., an isomorphism of X onto a union of connected components of Y ). Then the sheaf-functors f∗ and f ∗ are exact, so may also be regarded as derived functors. Establish, for E ∈ D(Y ), F ∈ D(X), natural bifunctorial isomorphisms ∼ ∼ HomD(X) (f∗ E, F ) −→ HomD(X) (f ∗f∗ E, f ∗f ) ←− HomD(Y ) (E, f ∗F ),

whence, with f × as in (b), for F ∈ Dqc (Y ) there is a functorial isomorphism ∼ ξ(F ) : f ×F −→ f ∗F ,

corresponding under the preceding isomorphism (with E = f ×F ) to the natural map f∗ f ×F → F , and with inverse adjoint to the natural map f∗ f ∗F → F = f∗ f ∗F ⊕ g∗ g ∗F where g is the inclusion (Y \ X) ֒→ Y . Verify that for the independent square 1

X −−−−→ ⏐ ⏐ τ 1

X ⏐ ⏐f

X −−−−→ Y f

the associated map θτ : f ∗f∗ → 1∗ 1∗ = 1 is the identity, and hence the functorial base-change map from (3.10.2.1)(c) βτ : 1∗f × = f × → f ∗ = 1×f ∗ is just the above isomorphism ξ. Deduce (or prove directly) that ξ is a pseudofunctorial isomorphism. (Cf. (4.6.8), (4.8.1) and (4.8.7) below.) (d) (Cf. [Kn, p. 43, Thm. 4].) Let f : X → Y be as in Theorem (4.1.1), with Y quasi-compact, and let d be an integer as in (3.9.2.3). Deduce from (4.1.1) a natural bifunctorial isomorphism ∼ HomX A, H −d f × (B) −→ HomY Rdf∗ (A), B

for all quasi-coherent OX -modules A and all OY -modules B. For the smallest such d, i.e., dim+ Rf∗ |Dqc(X) , the quasi-coherent OX -module Df := H −d f × OY is the lowest-degree nonvanishing homology of f × OY . When f is proper, Df is often called a relative dualizing sheaf for f . (But certain features of the duality theory for sheaves do not just come out of the abstract theory—see [Kn], [S].) qc (e) Show that the inclusion AX ֒→ AX has a right inverse. Deduce that every qc qc M ∈ AX admits a monomorphism into an AX -injective OX -module. (f) Show that the functor (−)cts : F → F constructed in the proof of (4.1.5) is rightadjoint to the inclusion into F of the full subcategory of functors that commute with filtered colimits (see [M, p. 212]). Also, the restriction of (−)cts to the full subcategory of right-exact functors is right adjoint to the inclusion of the full subcategory of cocontinuous functors.

4.2 Sheaﬁﬁed Duality—Preliminary Form

169

4.2 Sheaﬁﬁed Duality—Preliminary Form Theorem 4.2 Let f : X → Y , f × and τ be as in Theorem (4.1.1). Then with Hom := HomD(Y ) , for any E ∈ Dqc (Y ), F ∈ Dqc (X) and G ∈ D(Y ), the composite map • Hom E, Rf∗ RHomX (F , f ×G) (3.2.1.0) • −−−−−→ Hom E, Rf∗ RHomX (Lf ∗ Rf∗ F , f ×G) (3.2.3.2) −−−−−→ Hom E, RHom•Y (Rf∗ F , Rf∗ f ×G) via τ −−−−−→ Hom E, RHom•Y (Rf∗ F , G) is an isomorphism.

Proof.5 Using (2.6.2)∗ and (3.2.3), and checking all the requisite commutativities, one shows for ﬁxed F ∈ Dqc (Y ) that the composite duality map (3.2.1.0)

• • Rf∗ RHomX (F , f ×G) −−−−−→ Rf∗ RHomX (Lf ∗ Rf∗ F , f ×G) (3.2.3.2)

−−−−−→ RHom•Y (Rf∗ F , Rf∗ f ×G)

(4.2.1)

via τ

−−−−−→ RHom•Y (Rf∗ F , G) (functorial in G) is right-conjugate (see (3.3.5)) to the functorial (in E) projection map p2 : E ⊗ Rf∗ F → Rf∗ (Lf ∗E ⊗ F ), which, by (3.9.4), is an = = isomorphism when E ∈ Dqc (Y ). Now apply Exercise (3.3.7)(b) (with Y = E and X = G). Q.E.D. ! × For proper maps f : X → Y one writes f instead of f . When Y is − noetherian and f is proper, it holds that Rf∗D− c (X) ⊂ Dc (Y ) (where the subscript c indicates “coherent homology”)—see [H, p. 89, Prop. 2.2] in which, owing to (3.9.2.3) above, it is not necessary to assume that X has ﬁnite + − Krull dimension. So if F ∈ D− c (X) and G ∈ Dqc (Y ), then Rf∗ F ∈ Dc (Y ) and • • + ! ! f G ∈ Dqc (X), whence both Rf∗ RHomX (F , f G) and RHomY (Rf∗ F , G) ′ are in D+ qc (X), see [H, p. 92, 3.3] or [AJL , p. 35, 3.2.4]. One concludes that: Corollary 4.2.2 If f : X → Y is a proper map of noetherian schemes + then for all F ∈ D− c (X) and G ∈ Dqc (Y ), the duality map (4.2.1) is an isomorphism • ∼ Rf∗ RHomX (F , f ! G) −→ RHom•Y (Rf∗ F , G).

One of our goals is to prove this Corollary under considerably weaker hypotheses—see (4.4.2) below. For this purpose we need some facts about pseudo-coherence, reviewed in the following section. 5

Cf. [V, p. 404, Proof of Prop. 3].

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4 Abstract Grothendieck Duality for Schemes

Exercises 4.2.3 Let X be a concentrated scheme. Ex. (4.1.9)(e) says that the incluqc sion AX ֒→ AX has a right adjoint QX , the “quasi-coherator.” (Cf. [I, p. 186, §3].) qc (a) Show that RQX is right-adjoint to the natural functor j: D(AX ) → D(AX ); in other words, RQX = (1X )×. (Cf. [AJL′, p. 49, 5.2.2], where “let” in the second line should be “let j be the”.) In the rest of these exercises, assume all schemes to be quasi-compact and qc ≈ separated, so that by (3.9.6), j induces an equivalence j qc : D(A ) → Dqc . Also, Q denotes the functor j qc ◦ RQ, right-adjoint (from (a)) to the inclusion Dqc ֒→ D; and [−, −] denotes the functor Q ◦ RHom• (−, −) : D × D → Dqc . (b) Redo 3.6.10 with S the category of quasi-compact separated schemes and with X* = X* := Dqc (X). (Recall (2.5.8.1), (3.9.1), (3.9.2); and use the preceding [−, −].) (c) For any scheme-map f : X → Y there are natural functorial isomorphisms ∼ RΓ(X, QX −) −→ RΓ(X, −),

∼ Rf∗ QX −→ QY Rf∗ ,

∼ f × QY −→ f ×.

(d) Deduce from Theorem (4.2) a functorial isomorphism ∼ Rf∗ [F , f ×G]X −→ [Rf∗ F , G]Y

to which application of the functor H0 RΓ(Y , −) produces the adjunction isomorphism ∼ HomDqc (X) (F , f × G) −→ HomD(Y ) (Rf∗ F , G). In particular, if f is an open immersion then there is a functorial isomorphism ∼ G ∈ D(Y ) . f ×G −→ f ∗ [Rf∗ OX , G]Y

(e) Under the conditions of Theorem (4.1.1), show that the map right-conjugate to p1 : Rf∗ E ⊗ F → Rf∗ (E ⊗ Lf ∗F ) (where F ∈ Dqc (Y ) is ﬁxed, and both functors = = of E ∈ Dqc (X) take values in D(Y )) is a functorial isomorphism ∼ [Lf ∗F , f ×G]X −→ f × [F , G]Y G ∈ D(Y ) , (d)

adjoint to the natural composition Rf∗ [Lf ∗ F , f ×G]X −−→ [Rf∗ Lf ∗ F , G]Y → [F , G]Y . (f) Establish a natural commutative diagram, for F ∈ Dqc (Y ), G ∈ D(Y ): × Rf∗ [Lf ∗F ⏐, f G]X ⏐

−−(d) − −→

∗ [Rf∗ Lf ⏐ F , G]Y ⏐

× RHom• Y (F , Rf∗ f G)

−− −−→ via τ

RHom• Y (F , G),

• ∗ Rf∗ RHomX (Lf ∗F , f ×G) −−−−→ RHom• Y (Rf∗ Lf F , G) ⏐ ⏐ ⏐ ⏐ (3.2.3.2)≃

and show that the isomorphism in (e) is adjoint to the map obtained by going from the upper left to the lower right corner of this diagram. (g) Show, via the lower square in (f), or via (3.5.6)(e), or otherwise, that the following natural diagram commutes: Rf∗ f ×G ⏐ ⏐ τ G

(4.2.1)

−−−−−→

−−−− −−→

RHom• Y (Rf∗ OX , G) ⏐ ⏐ RHom• Y (OY , G)

4.3 Pseudo-Coherence and Quasi-Properness

171

In the next three exercises, for a scheme-map h we use the abbreviations h∗ := Rh∗ and h∗ := Lh∗. f g (h) Let X − →Y − → Z be maps of concentrated schemes. Referring to (e), show that for any E, F ∈ Dqc (Z), the following diagram of natural isomorphisms commutes. [(gf )∗E, (gf )×F ]X −−−−→ [f ∗g ∗E, g ×f ×F ]X −−−−→ f × [g ∗E, g ×F ]Y ⏐ ⏐ ⏐ ⏐ (gf )× [E, F ]Z

−−−−−−−−−−−−−−−−−−−−−−−−−−−→ f × g × [E, F ]Z

(i) Let βσ : v∗ g × → f × u∗ be as in (3.10.2.1)(c). Taking into account (3.9.1), show that for any E, F ∈ Dqc (Z) the following diagram commutes. (e)

(3.2.4)

v ∗f × [E, F ]Y ←−−−− v ∗ [f ∗E, f ×F ]X −−−−−→ [v ∗f ∗E, v ∗f ×F ]X ′ ⏐ ⏐ ⏐ ⏐ βσ via (3.6.4)∗ and βσ

g × u∗ [E, F ]Y −−−−−→ g × [u∗E, u∗F ]Y ′ ←−−−− [g ∗ u∗E, g ×u∗F ]X ′ (3.2.4)

(e)

(j) Let φσ : v∗ g × → f × u∗ be as in (3.10.4). Taking into account (3.9.2.1), show that for any E, F ∈ Dqc (Z) the following diagram, with θ′ as near the beginning of §3.10, commutes. v∗ g × [E, F ]Y ′ ⏐ ⏐ φσ

(e)

(3.5.4.1)

←−−−− v∗ [g ∗E, g ×F ]X ′ −−−−−−→ [v∗ g ∗E, v∗ g ×F ]X ⏐ ⏐ via θσ′ and φσ

f × u∗ [E, F ]Y ′ −−−−−−→ f × [u∗E, u∗F ]Y (3.5.4.1)

←−−−− [f ∗ u∗E, f × u∗F ]X (e)

4.3 Pseudo-Coherence and Quasi-Properness (4.3.1). Let us recall brieﬂy some relevant deﬁnitions and results concerning pseudo-coherence. Details can be found in [I], as indicated, or, perhaps more accessibly, in [TT, pp. 283ﬀ, §2].6 Let X be a scheme. A complex F ∈ Db(X) is pseudo-coherent if each x ∈ X has a neighborhood in which F is D-isomorphic to a bounded-above complex of ﬁnite-rank free OX -modules [I, p. 175, 2.2.10]. If X is divisorial, and either separated or noetherian, such an F is (globally) D(X)-isomorphic to a bounded-above complex of ﬁnite-rank locally free OX -modules [ibid., p. 174, Cor. 2.2.9]. If OX is coherent, pseudo-coherence of F means simply that F has coherent homology [ibid., p. 115, Cor. 3.5 b)]. If X is noetherian, 6 Though [I] is written in the language of ringed topoi, the reader who, like me, is uncomfortable with that level of generality, ought with suﬃcient patience to be able to translate whatever’s needed into the language of ringed spaces. A good starting point is 2.2.1 on p. 167 of loc. cit., with examples b) on p. 88 and 2.15 on p. 108 kept in mind.

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4 Abstract Grothendieck Duality for Schemes

pseudo-coherence means that F is D(X)-isomorphic to a bounded complex of coherent OX -modules [ibid., p. 168, Cor. 2.2.2.1]. A scheme-map f : X → Y is pseudo-coherent if it factors locally as f = p ◦i where i : U → Z (U open in X) is a closed immersion such that i∗ OU is pseudo-coherent on Z, and p : Z → Y is smooth [ibid., p. 228, D´ef. 1.2]. Pseudo-coherent maps are locally ﬁnitely-presentable (smooth maps being so by deﬁnition). For example, any smooth map is pseudo-coherent, any regular immersion (= closed immersion corresponding to a quasi-coherent ideal generated locally by a regular sequence) is pseudo-coherent, and any composition of pseudocoherent maps is still pseudo-coherent [ibid., p. 236, Cor. 1.14].7 If f : X → Y is a proper map, and L is an f -ample invertible sheaf, then f is pseudo-coherent if and only if the OY -complex Rf∗ (L⊗−n ) is pseudocoherent for all n ≫ 0. (The proof is indicated below, in (4.3.8)). In particular, a ﬁnite map f : X → Y is pseudo-coherent if and only if f∗ OX is a pseudocoherent OY -module. For noetherian Y , any ﬁnite-type map f : X → Y is pseudo-coherent. Pseudo-coherence persists under tor-independent base change [I, p. 233, Cor. 1.10]. Hence, by descent to the noetherian case [EGA, IV, (11.2.7) and its proof], any ﬂat ﬁnitely-presentable scheme-map is pseudo-coherent. Kiehl’s Finiteness Theorem [Kl, p. 315, Thm. 2.9′ ] (due to Illusie for projective maps [I, p. 236, Thm. 2.2]) generalizes preservation of coherence by higher direct images under proper maps of noetherian schemes: If f : X → Y is a proper pseudo-coherent map of quasi-compact schemes, and if F ∈ Db(X) is pseudo-coherent, then so is Rf∗ F ∈ Db(Y ).8 (4.3.2). For simplicity, we introduced pseudo-coherence only for complexes in Db, but that won’t be enough. So let us recall [I, p. 98, D´ef. 2.3]: Let X be a ringed space, and let n ∈ Z. A complex F ∈ D(X) is said to be n-pseudo-coherent if locally it is D-isomorphic to a bounded-above complex E such that E i is free of ﬁnite rank for all i ≥ n. It is equivalent to say that each x ∈ X has a neighborhood U over which there exists such an E = EU together with a quasi-isomorphism EU → F |U . If OX is coherent, then F ∈ D−(X) is n-pseudo-coherent ⇔ H i (F ) is coherent for all i > n and H n (F ) is of ﬁnite type [I, p. 115, Cor. 3.5 b)]. F is called pseudo-coherent if F is n-pseudo-coherent for all n ∈ Z. For F ∈ Db(X), this deﬁning condition is equivalent to the one given in (4.3.1). Moreover, when X is a quasi-compact separated scheme, then by (3.9.6)(a), [I, p. 173, 2.2.8] shows the same for any F ∈ D(X).

7

In the triangle at the top of [ibid., p. 234], the map X → Z should be labeled h. The theorem actually involves a notion of pseudo-coherence of a complex relative to a map f ; but when f itself is pseudo-coherent, relative pseudo-coherence coincides with pseudo-coherence [I, p. 236, Cor. 1.12].

8

4.3 Pseudo-Coherence and Quasi-Properness

173

(4.3.3). Now the above Finiteness Theorem can be put more precisely (as can be seen from the statement of [Kl, p. 308, Satz 2.8] and the proof of [ibid., p. 310, Thm. 2.9]): For any proper pseudo-coherent map f : X → Y of quasi-compact schemes, there is an integer k such that for any n ∈ Z and any n-pseudo-coherent complex F ∈ Db(X), the complex Rf∗ F is (n + k)-pseudo-coherent. Deﬁnition 4.3.3.1 A map f : X → Y is quasi-proper if Rf∗ takes pseudocoherent OX -complexes to pseudo-coherent OY -complexes. Corollary 4.3.3.2 Proper pseudo-coherent maps are quasi-proper. In particular, ﬂat ﬁnitely-presentable proper maps are quasi-proper. Proof. The question is easily seen to be local on Y , so we may assume that both X and Y are quasi-compact. Let F be a pseudo-coherent OX -complex. It follows from [I, p. 96, Prop. 2.2, b)(ii′ )] that for each n, the truncation τ≥n F ∈ Db(X) (see §1.10) is n-pseudo-coherent, and so there exists an integer k depending only on f such that Rf∗ τ≥n F is (n + k)-pseudo-coherent. Let C ∈ (Dqc )≤n−1 be the summit of a triangle whose base is the natural map F → τ≥n F . With d be as in (3.9.2), application of Rf∗ to this triangle shows that Rf∗ (C) is exact in all degrees ≥ n + d − 1, so the natural map is ∼ τ≥n+d Rf∗ τ≥n F (see (1.4.5), (1.2.2)). Hence an isomorphism τ≥n+d Rf∗ F −→ ′ by [I, p. 96, Prop. 2.2, b)(ii )], τ≥n+d Rf∗ F is (n + d + k)-pseudo-coherent for all n, whence Rf∗ F is pseudo-coherent. Q.E.D. Remark. A projective map is quasi-proper iﬀ it is pseudo-coherent, see the Remark following (4.7.3.3) below. See also Example (4.3.8). As noted above, ﬁnite-type maps of noetherian schemes are pseudocoherent. Using Exercise (4.3.9) below, one concludes that: Corollary 4.3.3.3 If Y is noetherian then a map f : X → Y is proper iﬀ it is ﬁnite-type, separated and quasi-proper. The next two Lemmas are elementary. Lemma 4.3.4 For any scheme-map f : X → Y , if G ∈ D(Y ) is npseudo-coherent then so is Lf ∗ G. This is proved by reduction to the simple case where G is a boundedabove complex of ﬁnite-rank free OY -modules, vanishing in all degrees < n, cf. [I, p. 106, proof of 2.13 and p. 130, 4.19.2]. Lemma 4.3.5 If F ∈ D(X) is n-pseudo-coherent and if the complex G ∈ Dqc (X) is such that H m (G) = 0 for all m < r then H j RHom•X (F , G) is quasi-coherent for all j < r − n. Thus if F is pseudo-coherent then RHom•X (F , G) ∈ Dqc (X).

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4 Abstract Grothendieck Duality for Schemes

Proof. Replacing G by τ +G (1.8.1), we may assume that Gm = 0 for m < r. Also, the question being local, we may assume that F is bounded above and that F i is free of ﬁnite rank for i ≥ n. If F ′ ⊂ F is the bounded free complex which vanishes in degree < n and agrees with F in degree ≥ n, then by (1.4.4) and (1.5.3) we have a triangle (with HX = RHom•X ): HX (F/F ′ , G) → HX (F , G) → HX (F ′, G) → HX (F/F ′ , G)[1] . The complex HX (F/F ′ , G) vanishes in degree ≤ r − n; and so from the exact homology sequence associated (as in (1.4.5)) to the triangle, we get isomorphisms ∼ H j HX (F , G) −→ H j HX (F ′ , G)

(j < r − n).

A simple induction on the number of degrees in which F ′ doesn’t vanish (using [H, p. 70, (1)] to pass from n to n + 1) yields HX (F ′ , G) ∈ Dqc (X), whence the assertion. Q.E.D. There results a generalization of (4.2.2), with a similar proof (given (4.3.3.2) and (4.3.5)): Corollary 4.3.6 If f : X → Y is a quasi-proper concentrated schememap, with X concentrated, then for all pseudo-coherent F ∈ D(X) and all G ∈ D+ qc (Y ), the duality map (4.2.1) is an isomorphism • ∼ (F , f × G) −→ RHom•Y (Rf∗ F , G). Rf∗ RHomX

Here is a fact needed in the proof of Theorem (4.4.1), and elsewhere. Lemma 4.3.7 Let f : X → Y be a ﬁnitely-presentable scheme-map, and let ϕ : A1 → A2 be a map in D+ qc (X). Suppose that for every pseudo-coherent F ∈ D(X), the resulting map Rf∗ RHom•X (F , A1 ) → Rf∗ RHom•X (F , A2 )

(4.3.7.1)

is an isomorphism. Then ϕ is an isomorphism. Proof. There are functorial isomorphisms (see (3.2.3.3), (2.5.10)(b)): ∼ ∼ RΓY Rf∗ RHom•X −→ RΓX RHom•X −→ RHom•X .

Application of the functor H 0 RΓY to (4.3.7.1) gives then, via (2.4.2), an isomorphism ∼ HomD(X) (F , A1 ) −→ HomD(X) (F , A2 ) .

(4.3.7.2)

Let C ∈ D+ qc (X) be the summit of a triangle with base ϕ. The exact homology sequence (1.4.5)H of this triangle shows, in view of (1.2.2), that ϕ is an isomorphism iﬀ H n (C) = 0 for all n ∈ Z.

4.3 Pseudo-Coherence and Quasi-Properness

175

Let us suppose that H n (C) = 0 for all n < m while H m (C) = 0, and derive a contradiction. The whole question being local on Y , we may assume that Y is aﬃne. Since H m (C) is quasi-coherent, there exists then a ﬁnitelypresentable OX -module E together with a non-zero map E → H m (C) [GD, p. 320, (6.9.12)].9 By [EGA, IV, (8.9.1)], there exists a noetherian ring R, a map Y → Spec(R), a ﬁnite-type map X0 → Spec(R), and a coherent OX0 -module E0 , such that, up to isomorphism, X = X0 ⊗R Y and, with w : X → X0 the resulting map, E = w∗E0 = H 0 (Lw∗E0 ). It will be convenient to set F := Lw∗ E0 [−m], so that τ≥m F ∼ = E[−m] (see §1.10). Since X0 is noetherian, therefore E0 is pseudo-coherent, and hence, by (4.3.4), so is F . Now by (1.4.2.1) there is an exact sequence (with Hom := HomD(X) ): ϕ

Hom(F , A1 ) −→ Hom(F , A2 ) −→ Hom(F , C) −→ Hom(F , A1 [1]) −→ Hom(F , A2 [1])

Hom(F [−1], A1 ) −→ ϕ Hom(F [−1], A2 )

where, F and F [−1] being pseudo-coherent, the maps labeled ϕ are isomorphisms, see (4.3.7.2). Thus, 0 = Hom F , C ∼ = Hom τ≥m F , C ∼ = Hom E[−m], C ∼ = Hom E[−m], τ≤m C ∼ = Hom E[−m], (H m (C))[−m]

= 0, a contradiction.

see (1.10.1.2) see (1.10.1.1) see (1.2.3)

Q.E.D.

Example 4.3.8 Let f : X → Y be a proper map of schemes, and let L be an f -ample invertible sheaf [EGA, II, p. 89, D´ef. (4.6.1)]. Then f is pseudo-coherent if and only if the OY -complex Rf∗ (L⊗−n ) is pseudo-coherent for all n ≫ 0. Proof. If f is pseudo-coherent then Rf∗ (L⊗−n ) is pseudo-coherent, by the Finiteness Theorem (4.3.3) (in fact—since f is projective locally on Y [EGA, II, p. 104, Thm. (5.5.3)]—by [I, p. 236, Thm. 2.2 and Cor. 1.12]). We ﬁrst illustrate the converse by treating the special case where f is ﬁnite and f∗ OX is a pseudo-coherent OY -module. To check that f is pseudo-coherent, we may assume that Y —and hence X—is aﬃne, so that for some r > 0, f factors as f = pi with p : ArY → Y the (smooth) projection and i : X ֒→ ArY a closed immersion; and we need to show that i∗ OX is pseudo-coherent. In algebraic terms, we have a ﬁnite ring-homomorphism A → B = A[t1 , . . . , tr ], such that the A-module B is resolvable by a complex E• of ﬁnite-type free A-modules [I, p. 160, Prop. 1.1]. Let T := (T1 , . . . , Tr ) be a sequence of indeterminates, and 9

Recall that ﬁnitely-presentable maps are quasi-compact and quasi-separated, by deﬁnition [GD, p. 305, (6.3.7)], so that X is quasi-compact and quasi-separated.

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4 Abstract Grothendieck Duality for Schemes

let ϕ : B[T ] = B[T1 , . . . , Tr ] → B be the unique B-homomorphism such that ϕ(Tk ) = tk (1 ≤ k ≤ r). Then B is resolved as a B[T ]-module by the Koszul complex K• on (T1 − t1 , . . . , Tr − tr ). Since the A[T ]-module B[T ] is resolved by E• ⊗A A[T ], therefore the free B[T ]-modules Kj can be resolved by ﬁnite-type free A[T ]-modules, whence so can B, giving the desired pseudo-coherence of i∗ OX . Now let us treat (sketchily) the general case. Assuming, as we may, that Y is aﬃne, we have for some r > 0, a factorization f = pi where p : PrY → Y is the (smooth) projection and i : X ֒→ PrY is a closed immersion [EGA, II, p. 104, (5.5.4)(ii)]. With γ : X → X ×Y PrY = PrX the graph of i, there is a natural diagram γ

F

r X −−−−→ PX −−−−→ ⏐ ⏐ q

r PY ⏐ ⏐p

X −−−−→ Y f

and it needs to be shown that i∗ OX = RF∗ (γ∗ OX ) is pseudo-coherent. Note that since γ is a regular immersion [Bt, p. 429, Prop. 1.10], therefore γ∗ OX is pseudo-coherent. So it’s enough to show that F is quasi-proper. By [EGA, II, p. 91, (4.6.13)(iii)], L := q ∗L is F -ample; and for n ≫ 0, say n ≥ m, ∼ RF∗ (L⊗−n ) = RF∗ q ∗(L⊗−n ) = p∗ Rf∗ (L⊗−n ) (3.9.5)

is pseudo-coherent (4.3.4). Imitating the proof of [I, p. 238, Thm. 2.2.2], we can then reduce the problem to showing that RF∗ (E ′ ) is pseudo-coherent for any bounded OX -complex E ′ whose component in each degree is a ﬁnite direct sum of sheaves of the form L⊗−n ; and this is easily done by induction on the number of nonzero components of E ′ . Q.E.D.

Exercises 4.3.9 (a) (Curve selection.) Let Z be a noetherian scheme, Z ⊂ Z a dense open subset, and W := Z \ Z. Show that for each closed point w ∈ W there is an integral one-dimensional subscheme C ⊂ Z such that w is an isolated point of C ∩ W . Hint. Use the local nullstellensatz : in any noetherian local ring A with dim A ≥ 1, the intersection of all those prime ideals p such that dim A/p = 1 is the nilradical of A. (For this, note that the maximal ideal is contained in the union of all the height one primes, so that when dim A > 1 there must be inﬁnitely many height one primes; and deduce that if q ⊂ A is a prime ideal with dim A/q > 1 and a ∈ / q then there exists a prime ideal q ′ = m such that q ′ q and a ∈ / q ′.) (b) Prove that if f : X → Y is a ﬁnite-type separated map of noetherian schemes such that f∗ (OX /I) is coherent for every coherent OX -ideal I, then f is proper. In particular, if f is quasi-proper then f is proper. Outline. If not, let Z ⊂ X be a closed subscheme of Z minimal among those for which the restriction of f is not proper. Then Z is integral [EGA, II, p. 101, 5.4.5]. Let f¯: Z → Y be a compactification of f |Z , see [C′ ], [Lt], [Vj], that is, f = f¯v with f¯ proper and v : Z ֒→ Z an open immersion. If dim Z > 1 then by (a) there is a curve on Z for which the restriction of f is not proper, contradiction. So the problem is reduced to where X is integral, of dimension 1. Then if dim Y = 0, and f is not proper, we may assume that Y = Spec(k), k a ﬁeld, whence X is aﬃne, and f∗ OX is not coherent. If dim(Y ) = 1 and f¯: X → Y is a compactiﬁcation of f, then the map f¯ is ﬁnite; and if u : X ֒→ X is the inclusion, u∗ OX is coherent, whence, by [EGA, IV, p. 117, (5.10.10)(ii)], X = X.

4.4 Sheaﬁﬁed Duality, Base Change

177

4.4 Sheaﬁﬁed Duality, Base Change Unless otherwise indicated, all schemes—and hence all scheme-maps—are assumed henceforth to be concentrated. All proper and quasi-proper maps are assumed to be ﬁnitely presentable. As in §4.3, a scheme-map f : X → Y is called quasi-proper if Rf∗ takes pseudo-coherent OX -complexes to pseudo-coherent OY -complexes. For example, when Y is noetherian and f is of ﬁnite type and separated then f is quasi-proper iﬀ it is proper, see (4.3.3.3). We will need the nontrivial fact that quasi-properness of maps is preserved under tor-independent base change [LN, Prop. 4.4]. The following abbreviations will be used, for a scheme-map h or a scheme Z: h∗ := Rh∗ , RHom•Z

HZ := , ⊗Z := ⊗ Z, =

h∗ := Lh∗ , HZ := RHom•Z , ΓZ (−) := RΓ(Z, −).

Recall the characterizations of independent ﬁber square (3.10.3), of ﬁnite tor-dimension map (2.7.6), and of the “dualizing pair” (f ×, τ ) in (4.1.1). We write f ! for f × when f is quasi-proper. Recall also the natural map (3.5.4.1) = (3.5.4.4) (see (3.5.2)(d)) associated to any ringed-space map f : X → Y , ν : f∗ HX (F , H) → HY (f∗ F , f∗ H) F , H ∈ D(X) . (4.4.0)

The composition (3.2.3.2)◦ (3.2.1.0) in (4.2.1) is an instance of this map. (See the line immediately following (3.5.4.2).) Theorem 4.4.1 Suppose one has an independent ﬁber square v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′ −−− −→ Y u

with f (hence g) quasi-proper and u of ﬁnite tor-dimension. Then for any F ′ ∈ Dqc (X ′ ) and G ∈ D+ qc (Y ), the composition ν

g∗ HX ′ (F ′, v ∗f ! G) −−−−−→ HY ′ (g∗ F ′, g∗ v ∗f ! G) −−− −−→ HY ′ (g∗ F ′, u∗f∗ f ! G) −−− −→ HY ′ (g∗ F ′, u∗ G) τ (3.10.3)

is an isomorphism.

If u and v are identity maps then so is the map labeled (3.10.3), and the resulting composition (with F := F ′ ) ν

τ

→ HY (f∗ F , f∗ f ! G) − → HY (f∗ F , G) δ(F , G) : f∗ HX (F , f ! G) − is just the duality map (4.2.1), whence the following generalization of (4.3.6):

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4 Abstract Grothendieck Duality for Schemes

Corollary 4.4.2 (Duality). Let f : X → Y be quasi-proper. Then for any F ∈ Dqc (X) and G ∈ D+ qc (Y ), the duality map δ(F , G) is an isomorphism. Moreover: Corollary 4.4.3 (Base Change). In (4.4.1), the functorial map adjoint to the composition g∗ v ∗f ! G −−− −−→ u∗f∗ f ! G −−−∗−→ u∗ G, (3.10.3)

u τ

is an isomorphism

∼ β(G) = βσ (G) : v ∗f ! G −→ g ! u∗ G

G ∈ D+ qc (Y ) .

To deduce (4.4.3) from (4.4.1), let F ′ ∈ Dqc (X ′ ) and consider the next diagram, whose commutativity follows from the deﬁnition of β = β(G): β

g∗ HX ′ (F ′, v ∗f ! G) −−−−→ g∗ HX ′ (F ′, g ! u∗ G) ⏐ ⏐ ⏐ν ⏐ ν β

HY ′ (g∗ F ′, g∗ v ∗f ! G) −−−−→ HY ′ (g∗ F ′, g∗ g ! u∗ G) ⏐ ⏐ ⏐ ⏐τ (3.10.3)≃

HY ′ (g∗ F ′, u∗f∗ f ! G) −−− −→ τ

(4.4.3.1)

HY ′ (g∗ F ′, u∗ G)

By (4.4.1), τ ◦(3.10.3)◦ ν is an isomorphism; and by (4.4.2) (a special case of (4.4.1)), the right column is an isomorphism too. (Note that by (2.7.5)(d) ′ and (3.9.1), u∗ G ∈ D+ qc (Y ).) It follows that the top row is an isomorphism, and applying the functor H0 ΓY ′ we get as in (4.3.7.2) an isomorphism via β

HomD(X ′ ) (F ′, v ∗f ! G) −−−→ HomD(X ′ ) (F ′, g ! u∗ G); and since this holds for any F ′ ∈ Dqc (X ′ ), in particular for F ′ = v ∗f ! G and F ′ = g ! u∗ G, it follows that β itself is an isomorphism. Q.E.D. Remarks 4.4.4 (a) Conversely, the commutativity of (4.4.3.1) shows that (4.4.2) and (4.4.3) together imply (4.4.1). (b) An example of Neeman [N, p. 233, 6.5], with f the unique map Spec(Z[T ]/(T 2 )) → Spec(Z) (T an indeterminate), shows that (4.4.2) and (4.4.3) can fail when G is not bounded below. (c) In (4.4.1), tordim v ≤ tordim u < ∞. To see this, let x′ ∈ X ′, x = v(x′ ), y ′ = g(x′ ), y = u(y ′ ) = f (x), A = OY ,y , ′ A = OY ′, y′ , B = OX,x , and B ′ = OX ′, x′ . By (2.7.6.4), the A-module A′ has a ﬂat resolution P• of length d := tordim u < ∞; and so by (i) in (3.10.2), P• ⊗A B is a ﬂat resolution of the B-module B ∗ = A′ ⊗A B. Since B ′ is a

4.6 Steps in the Proof

179

localization of B ∗ , it holds for any B-module M that B ∗ ′ ′ TorB j (B , M ) = B ⊗B ∗ Torj (B , M ) = 0

(j > d);

and it follows then from (2.7.6.4) that tordim v ≤ d. (d) By deﬁnition, β is the unique functorial map making the following diagram commute: g∗ β

g∗ v ∗f ! −−−−→ g∗ g ! u∗ ⏐ ⏐ ⏐ ⏐τg (3.10.3)≃ u∗f∗ f ! −−u− → ∗− τ

u∗

f

This diagram generalizes [H, p. 207, TRA 4.]

4.5 Proof of Duality and Base Change: Outline In describing the organization of the proof of (4.4.1), we will attach symbols to labels of the form (4.4.x) to refer to special cases of (4.4.x): (4.4.1)∗pc := (4.4.1) with F ′ = v ∗F , where F ∈ D(X) is pseudo-coherent. (4.4.2)pc := Corollary (4.3.6) := (4.4.1)∗pc with u = v = identity. (4.4.3)o := (4.4.3) with the map u an open immersion. (4.4.3)af := (4.4.3) with the map u aﬃne. Having already proved (4.4.2)pc , our strategy is to prove the chain of implications (4.4.2)pc ⇔ (4.4.1)∗pc ⇒ (4.4.3)o + (4.4.3)af ⇒ (4.4.3) ⇒ (4.4.3)o ⇔ (4.4.2). By (4.4.4)(a), then, (4.4.1) results.

Remark 4.5.1 For arbitrary ﬁnitely-presentable f , assertions (4.4.1)–(4.4.3) are meaningful—though not necessarily true—with (f ×, g × ) in place of (f !, g ! ). As will be apparent from the following proofs, the equivalence (4.4.1) ⇔ (4.4.2) + (4.4.3) holds in this generality, as do the preceding implications except for (4.4.2)pc ⇒ (4.4.1)∗pc .

4.6 Steps in the Proof I. Proof of (4.4.2)pc This has already been done (Corollary (4.3.6)). II. (4.4.2)pc ⇔ (4.4.1)∗pc The implication ⇐ is trivial. The implication ⇒ follows at once from:

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4 Abstract Grothendieck Duality for Schemes

Lemma 4.6.4 With the assumptions of (4.4.1)∗pc , and δ the duality map in (4.4.2), there is a natural commutative D(Y ′ )-diagram u∗f∗ HX (F , f ! G) ⏐ ⏐ ≃

u∗δ

−−−−−→

u∗ HY (f∗ F , G) ⏐ ⏐≃

g∗ HX ′ (v ∗F , v ∗f ! G) −−−−−∗→ HY ′ (g∗ v ∗F , u∗ G) (4.4.1)pc

in which the vertical arrows are isomorphisms. Commutativity in (4.6.4) is derived from the following relation—to be proved below—among the canonical maps ν, θ (3.7.2), and ρ (3.5.4.5): Lemma 4.6.5 For any commutative diagram of ringed-space maps v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

(4.6.5.1)

Y ′ −−− −→ Y u

and F ∈ Dqc (X), H ∈ D(X), the following diagram commutes: u∗f∗ HX (F , H) ⏐ ⏐ θ

ν

−−−−−−−−−−−−−−−−−−−−−−−−→

g∗ v ∗ HX (F , H) ⏐ ⏐ ρ

u∗ HY (f∗ F , f∗ H) ⏐ ⏐ρ

HY ′ (u∗f∗ F , u∗f∗ H) ⏐ ⏐(1,θ)

g∗ HX ′ (v ∗F , v ∗H) −−→ HY ′ (g∗ v ∗F , g∗ v ∗H) −−−→ HY ′ (u∗f∗ F , g∗ v ∗H) ν (θ,1)

Indeed, if (4.6.5.1) is an independent ﬁber square of scheme-maps, so that by (3.10.3), θ(F ) : u∗f∗ F → g∗ v ∗F is an isomorphism. If G ∈ D(Y ) and H := f × G, so that there is a natural map f∗ H → G (see (4.1.1)), then we get (a generalization of) commutativity in (4.6.4) by gluing the D(X ′ )-diagram in (4.6.5) and the following natural commutative diagram along the common column: u∗ HY (f∗ F , f∗ H) ⏐ ⏐ ρ

HY ′ (u∗f∗ F , u∗f∗ H) ⏐ ⏐ (1,θ)

−−−−−−−−−−−−−−−−−−−−−−−−−−→

u∗ HY (f∗ F , G) ⏐ ⏐ρ

HY ′ (u∗f∗ F , u∗f∗ H) −→ HY ′ (u∗f∗ F , u∗ G) ⏐ ⏐ ⏐ ⏐ ≃(θ −1,1) (θ −1,1)≃

HY ′ (u∗f∗ F , g∗ v ∗H) −−−1 − −− → HY ′ (g∗ v ∗F , u∗f∗ H) −→ HY ′ (g∗ v ∗F , u∗ G) −1 (θ

,θ

)

4.6 Steps in the Proof

181

Here is where we need f to be quasi-proper: since F is, by assumption, pseudo-coherent, therefore f∗ F is pseudo-coherent. In view of (4.4.4)(c), the following Proposition gives then the isomorphism assertion in (4.6.4). Proposition 4.6.6 Let u : Y ′ → Y be any scheme-map of ﬁnite tordimension, and let H ∈ D+(Y ). Then there is an integer e such that for all m ∈ Z and all m-pseudo-coherent C ∈ D(Y ), the map ρu : u∗ HY (C, H) → HY ′ (u∗ C, u∗H) induces homology isomorphisms in all degrees ≤ e − m. In particular, if C is pseudo-coherent then ρu is an isomorphism. Proof. The question is local on Y , because if i : U → Y is an open immersion, U ′ := U ×Y Y ′, and w : U ′ → U , j : U ′ → Y ′ are the projections (so that j is an open immersion), then j ∗ρu ∼ = ρw —more precisely, the following natural diagram commutes for any F , G ∈ D(Y ): j∗ρu

j ∗ u∗ HY (F , G) −−−−→ ⏐ ⏐≃ w∗ i∗ HY (F , G) ⏐ ⏐ w∗ρi ≃

j ∗ HY ′ (u∗f, u∗ G) ⏐ ⏐ ≃ρj

HU ′ (j ∗ u∗f, j ∗ u∗ G) ⏐ ⏐ ≃

w∗ HU (i∗F, i∗ G) −−− −→ HU ′ (w∗ i∗F, w∗ i∗ G) ρw Here ρi and ρj are isomorphisms by the last assertion in (4.6.7) (whose proof does not depend on (4.6.6)); and commutativity follows from (3.7.1.1). So by [I, p. 98, 2.3] we may assume there is a D(Y )-map E → C with E strictly perfect (i.e., E is a bounded complex of ﬁnite-rank locally free OY ∼ τ≥m+1 C. modules), such that the induced map is an isomorphism τ≥m+1 E −→ The contravariant Δ-functors Φ1 (C) := u∗ HY (C, H),

Φ2 (C) := HY ′ (u∗ C, u∗H)

are both bounded below (1.11.1), and so arguing as in the proof of (4.3.3.2) we ﬁnd that there is an integer e such that for i = 1, 2, the natural maps ∼ τ≤e−m Φi (E) ← τ≤e−m Φi (τm+1 E) −→ τ≤e−m Φi (τm+1 C) → τ≤e−m Φi (C)

are isomorphisms. Thus it will be more than enough to prove: Proposition 4.6.7 Let u : Y ′ → Y be a scheme-map, let E be a boundedabove complex of ﬁnite-rank locally free OY -modules, and let H ∈ D+(Y ).

182

4 Abstract Grothendieck Duality for Schemes

If E is strictly perfect or if u has ﬁnite tor-dimension then the map ρ : u∗ HY (E, H) → HY ′ (u∗E, u∗H) is an isomorphism. The same holds for any E, H ∈ D(Y ) if u is an open immersion. Except for the proofs of (4.6.5) and (4.6.7), which are postponed to the end of this section 4.6, the proof of (4.6.4)—and hence of the implication (4.4.2)pc ⇒ (4.4.1)∗pc — is now complete. III. (4.4.1)∗pc ⇒ (4.4.3)o + (4.4.3)af

Let β = β(G) be as in (4.4.3). When u, hence v, is an open immersion or aﬃne, then v is isofaithful ((3.10.2.1)(d) or (3.10.2.2)), so that for β to be an isomorphism it suﬃces that v∗ β be an isomorphism. Let F ∈ D(X) be pseudo-coherent. From (4.4.3.1) with F ′ = v ∗F and with ! replaced by ×, one derives the following commutative diagram: f∗ HX (F , v∗ v ∗f × G) ⏐ ⏐ (3.2.3.2)−1≃

via v∗ β

−−−−−−→

via β

f∗ HX (F , v∗ g × u∗ G) ⏐ ⏐ ≃(3.2.3.2)−1

f∗ v∗ HX ′ (v ∗F , v ∗f × G) −−−−−−→ f∗ v∗ HX ′ (v ∗F , g × u∗ G) ⏐ ⏐ ⏐ ⏐≃ ≃

u∗ g∗ HX ′ (v ∗F , v ∗f × G) −−−−−−→ u∗ g∗ HX ′ (v ∗F , g × u∗ G) via β ⏐ ⏐ ≃u∗ δ

u∗ g∗ HX ′ (v ∗F , v ∗f × G) −−− −−−→ ∗ u∗ (4.4.1)pc

u∗ HY ′ (g∗ v ∗F , u∗ G)

The bottom row is an isomorphism by assumption, as is the right column, by the special case (4.4.2)pc of (4.4.1)∗pc . Thus the top row is an isomorphism, and hence, by (4.3.7), so is v∗ β. IV. (4.4.3)o + (4.4.3)af ⇒ (4.4.3)

The essence of what follows is contained in the four lines preceding “CASE 1” on p. 401 of [V]. Denote the independent square in (4.4.1) by σ, and the corresponding functorial map v ∗f × → g × u∗ by βσ (cf. (4.4.3), without assuming f and g to be quasi-proper). Let us ﬁrst record the following elementary transitivity properties of βσ .

4.6 Steps in the Proof

183

Proposition 4.6.8 For any commutative diagram w

v

X ′′ −−−1−→ ⏐ ⏐ σ1 h

v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Z ′ −−−−→ ⏐ ⏐ σ1 g1 v

X ′ −−−−→ ⏐ ⏐ g σ

or

Y ′′ −−− −→ Y ′ −−− −→ Y u1 u

Z ⏐ ⏐f 1

X ⏐ ⏐f

Y ′ −−− −→ Y u

where both σ and σ1 are independent squares—whence so is the composed square σ0 := σσ1 see (3.10.3.2)—the following resulting diagrams of functorial maps commute: βσ0

(vv1 )∗f × −−−−−−−−−−−−−−−−→ h× (uu1 )∗ ⏐ ⏐ ⏐≃ ⏐ ≃ −→ v1∗ g × u∗ −−−−→ h× u∗1 u∗ v1∗ v ∗f × −−− ∗ βσ1

v1 βσ

βσ0

w∗ (f f1 )× −−−−−−−−−−−−−−−−→ (gg1 )× u∗ ⏐ ⏐ ⏐≃ ⏐ ≃ w∗f1× f × −−−−→ g1× v ∗f × −−× −−→ g1× g × u∗ βσ1

g1 βσ

Proof. (Sketch.) Using the deﬁnition of β, one reduces mechanically to proving the transitivity properties for θ in (3.7.2), (ii) and (iii). Q.E.D. Assuming (4.4.3)o , we ﬁrst reduce (4.4.3) to the case where Y is aﬃne. Let (μi : Yi → Y )i∈I be an open covering of Y with each Yi aﬃne. Consider the diagrams, with σ as in (4.4.1), ν

v

Xi′ −−−i−→ Xi −−−i−→ ⏐ ⏐ ⏐ gi τi σi fi ⏐

X ⏐ ⏐f

Yi′ −−− −→ Y −→ Yi −−− μi ui

ν′

Xi′ −−−i−→ ⏐ ⏐ gi τi′

v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Yi′ −−− −→ Y ′ −−− −→ Y u μ′ i

184

4 Abstract Grothendieck Duality for Schemes

where Yi′ := Y ′ ×Y Yi , ui and μ′i are the projections, and all the squares are ﬁber squares. The composed squares τi σi and στi′ are identical. The squares τi and τi′ are independent because μi and μ′i are open immersions; and by (4.4.3)o , βτi and βτi′ are isomorphisms. Furthermore, since f is quasi-proper therefore so are the maps fi . The map ui , which agrees over Yi with u, has ﬁnite tor-dimension. By (3.10.3.4), the square σi ∼ = μ∗i σ is independent. Thus if (4.4.3) holds whenever Y is aﬃne, then βσi is an isomorphism, and (4.6.8) shows that so are βστi′ (= βτi σi ) ∗ and νi′ βσ . Since (νi′ : Xi′ → X ′ )i∈I is an open covering of X ′, and since isomorphism can be checked locally (see (1.2.2)), it follows that βσ is an isomorphism, whence the asserted reduction. Next, again assuming (4.4.3)o , we reduce (4.4.3) with aﬃne Y to where ′ Y too is aﬃne. That will complete the proof, since when both Y and Y ′ are aﬃne then so is u, and (4.4.3)af applies. Let (νj : Yj′ → Y ′ )j∈J be an open covering of Y ′ with each Yj′ aﬃne. Consider the diagram, with aﬃne Y and σ as in (4.4.1), vj

Xj′ −−−−→ ⏐ ⏐ gj σj

v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

−→ Y ′ −−− −→ Y Yj′ −−− νj u

where σj is a ﬁber square, hence independent. By (4.4.3)o, βσj is an isomorphism. If (4.4.3) holds for independent squares whose bottom corners are aﬃne, then βσσj is an isomorphism; and so by (4.6.8), vj∗ βσ is also an isomorphism. As before, then, βσ is an isomorphism, and we have the desired reduction. Q.E.D. o V. (4.4.3) ⇒ (4.4.3) ⇔ (4.4.2) The ﬁrst implication is trivial. The implication (4.4.2) ⇒ (4.4.3)o is contained in what we have already done, but it’s more direct than that, as we’ll see. Incidentally, the following argument does not need f to be quasiproper. Let us ﬁrst deduce (4.4.2) from (4.4.3)o . As in (4.6.4), via (4.6.5), there is for any F ∈ D(X), G ∈ D(Y ) a commutative diagram u∗f∗ HX (F , f × G) ⏐ ⏐

u∗δ

−−−−→

u∗ HY (f∗ F , G) ⏐ ⏐

(4.6.9)

g∗ HX ′ (v ∗F , v ∗f × G) −−−−→ HY ′ (g∗ v ∗F , u∗ G) (4.4.1)

When u (hence v) is an open immersion, then the vertical arrows in this diagram are isomorphisms. Indeed, these arrows are combinations of ρ and θ, ρ being an isomorphism by (4.6.7), and θ(L) : u∗f∗ L → g∗ v ∗ L being an isomorphism for any L ∈ D(X), as follows easily from (2.4.5.2) after L is

4.6 Steps in the Proof

185

replaced by a q-injective resolution. Furthermore, the functor ΓY ′ := RΓ(Y ′, −) transforms the bottom row of (4.6.9) into an isomorphism. This follows from commutativity of the next diagram, obtained via Exercise (3.2.5)(f) by application of ΓY ′ to the commutative diagram (4.4.3.1), and where, under the present assumption of (4.4.3)o , β is an isomorphism: HX ′ (F ′, v ∗f × G) ΓY ′ (4.4.1)

HY ′ (g∗ F ′, u∗ G)

(4.6.10)

via β

(4.1.1)

HX ′ (F ′, g × u∗ G)

We conclude that ΓY ′ u∗ δ is an isomorphism whenever u : Y ′ → Y is an open immersion; and then (4.4.2) results from: Lemma 4.6.11 Let φ : G1 → G2 be a map in D(Y ). Then φ is an isomorphism iﬀ for every open immersion u : Y ′ ֒→ Y with Y ′ aﬃne, the map ΓY ′ u∗(φ) : ΓY ′ u∗(G1 ) → ΓY ′ u∗(G2 ) is an isomorphism. Proof. Write ΓY ′ for the sheaf-functor Γ(Y ′ , −). We may assume that G1 and G2 are q-injective and that φ is actually a map of complexes, see (2.3.8)(v), so that ΓY ′ u∗(φ) is the map ΓY ′ (φ) : ΓY ′ (G1 ) → ΓY ′ (G2 ). If ΓY ′ u∗(φ) is an isomorphism, then the homology maps Hp ΓY ′ (φ) : Hp ΓY ′ (G1 ) → Hp ΓY ′ (G2 )

(p ∈ Z)

are all isomorphisms; and since H p (Gi ) is the sheaf associated to the presheaf Y ′ → Hp ΓY ′ (Gi ) (i = 1, 2), it follows for every p ∈ Z that the map H p (φ) : H p (G1 ) → H p (G2 ) is an isomorphism, so that by (1.2.2), φ is an isomorphism. The converse is obvious. Q.E.D. Conversely, if (4.4.2) holds, then the top row—and hence the bottom row— in (4.6.9) is an isomorphism. We deduce from (4.6.10) that via β

HX ′ (F ′, v ∗f × G) −−−−→ HX ′ (F ′, g × u∗ G) is an isomorphism for all F ′ , whence (taking homology, see (2.4.2)) that

186

4 Abstract Grothendieck Duality for Schemes via β

HomD(X ′ ) (F ′ , v ∗f × G) −−−−→ HomD(X ′ ) (F ′ , g × u∗ G) is an isomorphism for all F ′ , so that β itself is an isomorphism.

Q.E.D.

It remains to prove (4.6.5) and (4.6.7). Proof of (4.6.5). One veriﬁes, using the deﬁnitions of ν, of θ (via (3.7.2)(a)) and of ρ, and the line following (3.5.4.2), that in the big diagram on the following page—with natural maps, and in which α denotes the map (3.5.4.2) = (3.5.4.3) (of which the isomorphism (3.2.3.2) is an instance, see (3.2.4)(i))—the outer border is adjoint to the diagram in (4.6.5). Therefore it will suﬃce to show that all the subdiagrams in the big diagram commute. For the unnumbered subdiagrams commutativity is clear. Commutativity 1 follows from the deﬁnition of ρ; of 2 from the deﬁnition of θ of via (3.7.2)(a); of 3 from (3.7.1.1) (with β replaced by α, etc.); and of 4 from the deﬁnition of θ via (3.7.2)(c). Q.E.D. Proof of (4.6.7). For this proof, we drop the abbreviations introduced at the beginning of §4.4. Thus u∗ and u∗ will now denote the usual sheaffunctors, and Ru∗ , Lu∗ their respective derived functors. Similarly, H will denote the functor Hom• of complexes, and RHom• its derived functor. We need to understand ρ more concretely, and to that end we will establish commutativity of the following diagram of natural maps, for any complexes E, H of OY -modules: Lu∗ HY (E, H) ⏐ ⏐ a

b

−−−−→

u∗ HY (E, H) ⏐ ⏐ρ 0

HY ′ (u∗E, u∗H) ⏐ ⏐c

Lu∗ RHY (E, H) ⏐ ⏐ ⏐ ⏐ ⏐ ρ⏐ ⏐ ⏐ ⏐

RHY ′ (u∗E, u∗H) ⏐ ⏐ d

RHY ′ (Lu∗E, Lu∗H) −−−e−→ RHY ′ (Lu∗E, u∗H) Here ρ0 is adjoint to the natural composite map of complexes ξ : HY (E, H) → HY (E, u∗ u∗H) −− −−→ u∗ HY ′ (u∗E, u∗H). (3.1.6)

This ξ is such that for any open U ⊂ Y , Γ(U , ξ) is the map

Homf −1 U (u∗E i , u∗H i+n ) HomU (E i , H i+n ) → i∈Z

arising from the functoriality of u∗ .

i∈Z

(4.6.7.1)

−−→

f∗ HX ( f ∗f∗ E, H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

1

ρ

f∗ HX ( f ∗f∗ E, H) ⏐ ⏐

∗ f∗ HX ( f ∗f ∗ E, v∗ v H) ⏐ α⏐≃

f∗ v∗ v ∗ HX ( E, H) ⏐ ⏐ ≃

−−→ f∗ v∗ v ∗ HX ( f ∗f∗ E, H) −−→ f∗ v∗ HX ′ ( v ∗f ∗f∗ E, v ∗H) ⏐ ⏐ ⏐≃ ⏐ ≃

u∗ g∗ HX ′ ( v ∗ E, v ∗H) ⏐ ⏐

−−−−−−−−−−−−−−−−−−−−−−−−→ u∗ g∗ HX ′ ( v ∗f ∗f∗ E, v ∗H) ⏐ ⏐≃ 4

u∗ g∗ v ∗ HX ( E, H) ⏐ ⏐ ρ

−−→ u∗ g∗ v ∗ HX ( f ∗f∗ E, H) −− → u∗ g∗ HX ′ ( v ∗f ∗f∗ E, v ∗H) ρ

α

−−→

−− → α

3

HY ( f∗ E, f∗ H) ⏐ ⏐

HY ( f∗ E, f∗ v∗ v ∗H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐ ⏐ ⏐

HY ( f∗ E, u∗ g∗ v ∗H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ α−1 ⏐ ≃ ⏐ ⏐ ⏐

2

(1,θ)

←−−

HY ( f∗ E, f∗ H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

4.6 Steps in the Proof

f∗ HX ( E, H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

HY ( f∗ E, u∗ u∗f∗ H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≃ ⏐ α−1 ⏐ ⏐ ⏐

u∗ g∗ HX ′ ( g ∗ g∗ v ∗ E, v ∗H) −−−−−−−−−−−−−−−−−−−−−−−−→ u∗ g∗ HX ′ ( g ∗ u∗f∗ E, v ∗H) −−→ u∗ HY ′ ( u∗f∗ E, g∗ v ∗H) ←−− u∗ HY ′ ( u∗f∗ E, u∗f∗ H) (1,θ) (θ,1) ⏐(θ,1) ⏐

u∗ g∗ HX ′ ( g ∗ g∗ v ∗ E, v ∗H) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ u∗ HY ′ ( g∗ v ∗ E, g∗ v ∗H)

187

188

4 Abstract Grothendieck Duality for Schemes

Commutativity of (4.6.7.1) is equivalent to commutativity of the following “adjoint” diagram:10 HY (E, H) ⏐ ⏐

−−−−→

RHY (E, H) ⏐ ⏐

RHY (E, Ru∗ Lu∗H) ⏐ ⏐ (3.2.3.2)−1

Ru∗ u∗ HY (E, H) ⏐ ⏐Ru (ρ ) ∗ 0

Ru∗ HY ′ (u∗E, u∗H) ⏐ ⏐

Ru∗ RHY ′ (u∗E, u∗H) ⏐ ⏐

Ru∗ RHY ′ (Lu∗E, Lu∗H) −−−−→ Ru∗ RHY ′ (Lu∗E, u∗H) But in this diagram the two maps obtained by going around from the top left to the bottom right clockwise and counterclockwise respectively, are both equal to the natural composition −1

HY (E, H)−→ HY (E, u∗ u∗H) (3.1.5) −−−−→ u∗ HY ′ (u∗E, u∗H) ∗ ∗ −→ Ru∗ HY ′ (u E, u H) −→ Ru∗ RHY ′ (u∗E, u∗H) −→ Ru∗ RHY ′ (Lu∗E, u∗H), as shown by the commutativity of the following two diagrams. (In the ﬁrst, the top three horizontal arrows come from the natural functorial composition 1 → u∗ u∗ → Ru∗ u∗ ; and the right column is Ru∗ (ρ0 ).) HY (E, H) ⏐ ⏐

HY (E, u∗ u∗H) ⏐ ⏐

−−−−→

−−−−→

Ru∗ u∗ HY (E, H) ⏐ ⏐

Ru∗ u∗ HY (E, u∗ u∗H) ⏐ ⏐

u∗ HY ′ (u∗E, u∗H) −−−−→ Ru∗ u∗ u∗ HY ′ (u∗E, u∗H) ⏐ ⏐ 1

u∗ HY ′ (u∗E, u∗H) −−−−→

10

Ru∗ HY ′ (u∗E, u∗H)

Recall that by (3.2.4)(i), the map (3.2.3.2) is an instance of the map (3.5.4.3).

4.6 Steps in the Proof

HY (E, H) ⏐ ⏐

RHY (E, H) ⏐ ⏐

RHY (E, Ru∗ Lu∗H) ⏐ ⏐

189

−−−→

−−−→ 2 −−−→

HY (E, u∗ u∗H) ⏐ ⏐

RHY (E, u∗ u∗H) ⏐ ⏐

−− −→ 3

RHY (E, Ru∗ u∗H) ⏐ ⏐

u∗ HY ′ (u∗E, u∗H) ⏐ ⏐

Ru∗ HY ′ (u∗E, u∗H) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

Ru∗ RHY ′ (Lu∗E, Lu∗H) −−−→ Ru∗ RHY ′ (Lu∗E, u∗H) ←−−− Ru∗ RHY ′ (u∗E, u∗H)

Commutativity of subdiagram 1 follows from the natural functorial composition u∗ → u∗ u∗ u∗ → u∗ being the identity. Commutativity of 2 follows from that of (3.2.1.3). Commutativity of 3 follows from that of the diagram immediately following (3.2.3.2). Thus (4.6.7.1) does indeed commute. Proceeding now with the proof of (4.6.7), suppose that E is a boundedabove complex of ﬁnite-rank locally freeOY -modules, and that H ∈ D+(Y ). To show that ρ is an isomorphism, we may assume that H is a complex of u∗-acyclic OY -modules, bounded below if u has ﬁnite tor-dimension, see (2.7.5)(vi). Then in (4.6.7.1), d and e are isomorphisms; and HY (E, H) is also a complex of u∗-acyclic OY -modules (the question being local on Y ), so that b too is an isomorphism, see (2.7.5)(a). That ρ0 is an isomorphism follows from the fact that (exercise) its stalk at y ′ ∈ Y ′ is—with y := u(y ′ ), R′ := OY ′, y′ and R := OY,y —the natural map R′ ⊗R HomR (Ey , Hy ) → HomR′ (R′ ⊗R Ey , R′ ⊗R Hy ). It remains to be shown that a and c are isomorphisms. For a, it suﬃces that if H → I is a quasi-isomorphism with I injective and bounded-below, then the resulting map HY (E, H) → HY (E, I) be an isomorphism. Since HY is a Δ-functor, and by the footnote under (1.5.1), it is equivalent to show that if C is the summit of a triangle whose base is H → I (so that C is exact), then HY (E, C) is exact. For any n ∈ Z, to show that H n HY (E, C) = 0 we may assume that E = 0, let m0 = m0 (E) be the least integer such that E m = 0 for all m > m0 , and argue by induction on m0 , as follows. If m0 ≪ 0, then HY (E, C) vanishes in degree n, so the assertion is obvious. Proceeding inductively, set i = m0 (E), and let E
190

4 Abstract Grothendieck Duality for Schemes

in which the ﬁrst term vanishes by the inductive hypothesis, and the last term vanishes because E i is locally free of ﬁnite rank and C is exact. Hence H n HY (E, C) also vanishes, as desired. Thus a is indeed an isomorphism. Similarly c is an isomorphism. Hence, ﬁnally, so is ρ. For the last assertion in (4.6.7), suppose u is an open immersion. It is left as an exercise to show that now ρ0 is just the obvious restriction map. To show that ρ is an isomorphism we may assume that H—and hence u∗H—is q-injective, see (2.4.5.2). Clearly, then, all the maps in (4.6.7.1) other than ρ are isomorphisms, whence so is ρ. Q.E.D.

4.7 Quasi-Perfect Maps Again, all schemes are assumed to be concentrated. In this section, for a scheme-map f : X → Y the functor f × will be as in (4.1.1), but restricted to Dqc (Y ); in other words, f × is always to be regarded as a functor from Dqc (Y ) to Dqc (X). Quasi-perfect maps are scheme-maps f : X → Y characterized by any one of several nice properties preserved by tor-independent base change (see (4.7.3.1)). Among those properties are the following, the ﬁrst two by (4.7.1), and the next two by (4.7.4) and (4.7.6)(d): • f × commutes with small direct sum in Dqc (i.e., direct sum of any family indexed by a small set, see §4.1). • For all F ∈ Dqc (Y ) the natural map is an isomorphism ∼ Lf ∗F −→ f ×F . χF : f × OY ⊗ =

• f × is a bounded functor, and it satisﬁes universal tor-independent base change, that is, for any independent square as in (4.4.1), and for any G ∈ Dqc (Y )—not necessarily in D+ qc (Y )—the base-change map β(G) in (4.4.3) is an isomorphism. • f × is a bounded functor, and these two conditions hold: (i) For all F ∈ Dqc (X) the duality map (4.2.1) is an isomorphism ∼ Rf∗ RHom• (F , f × OY ) −→ RHom•Y (Rf∗ F , OY ).

(ii) If (Fα ) is a small directed system of ﬂat quasi-coherent OY -modules then for any n ∈ Z the natural map is an isomorphism ∼ lim H n (f ×Fα ) −→ H n (f × lim Fα ). −α→ −α→

It follows that quasi-perfection of f implies the following; and in fact when Y is separated the converse is true, see (4.7.4): • f × is a bounded functor, and the above natural map χF is an isomorphism whenever F is a ﬂat quasi-coherent OY -module.

4.7 Quasi-Perfect Maps

191

Further, though we won’t prove it here, the main result Theorem 1.2 in [LN] is the equivalence of the following conditions: (i) f is quasi-perfect. (ii) f is quasi-proper (4.3.3.1) and has ﬁnite tor-dimension. (iii) f is quasi-proper and the functor f × is bounded. We call a scheme-map f perfect if f is pseudo-coherent and of ﬁnite tordimension. (For pseudo-coherent f, being of ﬁnite tor-dimension is equivalent to boundedness of f ×, see [LN, Thm. 1.2]). For example, since ﬁnite-type maps of noetherian schemes are always pseudo-coherent, the foregoing and (4.3.9) show that a separated such map is quasi-perfect if and only if it is proper and perfect. Perfect maps of noetherian schemes will be treated in §4.9. Before proceeding, we review a few basic facts about perfect complexes. A complex in E ∈ D(X) (X a scheme) is said to be perfect if it is locally D-isomorphic to a strictly perfect complex, i.e., a bounded complex of ﬁniterank free OX -modules. More precisely, E is said to have perfect amplitude in [a, b] (a ≤ b ∈ Z) if locally on X, E is D-isomorphic to a strictly perfect complex vanishing in all degrees which are < a or > b. Thus E is perfect iﬀ it has perfect amplitude in some interval [a, b]. By [I, p. 134, 5.8], this condition is equivalent to E being pseudo-coherent and also having ﬂat amplitude in [a, b] (i.e., being globally D-isomorphic to a ﬂat complex vanishing in all degrees < a and > b). So E is perfect iﬀ it is pseudo-coherent and of ﬁnite tor-dimension (that is, D-isomorphic to a bounded ﬂat complex, see (3.9.4.2)(b)). Proposition 4.7.1 (Neeman). For any scheme-map f : X → Y , the following conditions, with f × as in (4.1.1), are equivalent: (i) f × respects direct sums (see (3.8.1)) in Dqc , i.e., for any small Dqc (Y )family (Fα ) the natural map is an isomorphism ∼ f ×(⊕ Fα ). ⊕ f ×Fα −→ α

α

(ii) The functor Rf∗ takes perfect complexes to perfect complexes. (iii) The functor f × has a right adjoint. (iv) For all F ∈ Dqc (Y ), the map adjoint to ∼ Rf∗ (f ×OY ⊗ Lf ∗F ) −→ Rf∗ f ×OY ⊗ F −−−→ F = = (3.9.4)

is an isomorphism

via τ

∼ f × OY ⊗ Lf ∗F −→ f ×F . =

Proof. (i) ⇔ (ii): [N, p. 215, Prop. 2.5 and Cor. 2.3; and p. 224, Thm. 5.1 (where every s ∈ S is implicitly assumed to be compact)]. (i) ⇒ (iii): [N, p. 215, Prop. 2.5; p. 207, lines 12–13; and p. 223, Thm. 4.1]. (iii) ⇒ (i): simple. (i) ⇒ (iv) ⇒ (i): For the ﬁrst ⇒ see [N, p. 226, Thm. 5.4]. The second implication follows from (3.8.2).

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4 Abstract Grothendieck Duality for Schemes

Strictly speaking, the referenced results in [N] are proved for separated schemes; but in view of [BB, p. 9, Thm. 3.1.1] one readily veriﬁes that the proofs are valid for any concentrated scheme. Q.E.D. Deﬁnition 4.7.2 A map f : X → Y is quasi-perfect if it satisﬁes the conditions in (4.7.1). Remark. The fact, mentioned above, that quasi-perfect maps are quasiproper results from (4.7.1)(ii) and [LN, Cor. 4.3.2], which says that f is quasiproper if and (clearly) only if Rf∗ takes perfect complexes to pseudo-coherent complexes. Example 4.7.3 (a) Any quasi-proper scheme-map f of ﬁnite tordimension—so by (4.3.3.2), any proper perfect map, in particular, any ﬂat ﬁnitely-presentable proper map—is quasi-perfect. Indeed Rf∗ preserves both pseudo-coherence of complexes and—by [I, p. 250, 3.7.2] (a consequence of (3.9.4) above)—ﬁnite tor-dimensionality of complexes; so (4.7.1)(ii) holds. (b) Let f : X → Y be a scheme-map with X divisorial, i.e., X has an ample family (Li )i∈I of invertible OX -modules [I, p. 171, D´efn. 2.2.5]. Then [N, p. 211, Example 1.11 and p. 224, Theorem 5.1] imply that f is quasi-perfect ⇔ for each i ∈ I, there is an integer ni such that the ) is perfect for all n ≥ ni . OY -complex Rf∗ (L⊗−n i (c) (Cf. (4.3.8).) Let f be quasi-projective and let L be an f -ample invertible OX -module. Then: f is quasi-perfect ⇔ the OY -complex Rf∗ (L⊗−n ) is perfect for all n ≫ 0 ⇒ f is perfect. Indeed, condition (4.7.1)(ii), together with the compatibility of Rf∗ and open base change, implies that quasi-perfection is a property of f which is local on Y , and the same holds for perfection of Rf∗ (L⊗−n ); so for the ⇔ we may assume Y aﬃne, and apply (b). The ⇒ is given by (4.7.3.3) below. (d) For a ﬁnite map f : X → Y the following are equivalent: (i) f is quasi-perfect. (ii) f is perfect. (iii) The complex f∗ OX ∼ = Rf∗ OX is perfect. Indeed, the implication (i) ⇒ (iii) is given by (4.7.1)(ii). If (iii) holds then f has ﬁnite tor-dimension (see (2.7.6.4)), and as in the ﬁrst part of the proof of (4.3.8), f is pseudo-coherent; thus f is perfect. The implication (ii) ⇒ (i) is given by (a). Proposition 4.7.3.1 For any independent square of scheme-maps, v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

4.7 Quasi-Perfect Maps

193

(i) if f is quasi-perfect then so is g; and (ii) if the (bounded-below) functor f × : Dqc (Y ) → Dqc (X) is bounded above, then so is g × : Dqc (Y ′ ) → Dqc (X ′ ). Hence, if (Yi )i∈I is an open cover of Y then (iii) f is quasi-perfect ⇔ for all i, the same is true of the induced map f −1 Yi → Yi ; and (iv) if f is quasi-proper then f × is bounded above ⇔ for all i, the same is true of the induced map f −1 Yi → Yi . Proof. To begin with, (iii) follows easily from (i) and (4.7.1)(ii); and (iv) follows from (ii) and (4.4.3). In the rest of this proof, quasi-perfection is characterized by (4.7.1)(i). Suppose ﬁrst that Y ′ is separated. We induct on q = q(Y ′ ), the least number of aﬃne open subschemes needed to cover Y ′. If q = 1 then the map u is aﬃne, whence so is v [GD, p. 358, (9.1.16), (v) and (iii)]; so to prove (i) (resp. (ii)) it suﬃces, by (3.10.2.2), to show that for any small Dqc (Y ′ )-family (Fα ) the natural map is an isomorphism ⊕ Rv∗ g ×Fα α

(3.9.3.3)

∼ =

∼ Rv∗ ⊕ g ×Fα −→ Rv∗ g ×(⊕ Fα ) α

α

(resp.—since every G ∈ Dqc (X ′ ) is isomorphic to a quasi-coherent, hence v∗ -acyclic, OX ′ -complex G′, see (2.7.5)(a), so that H n (Rv∗ G) ∼ = H n (G′ ) = 0 = v∗ H n (G′ ) = 0 =⇒ H n (G) ∼ = H n (v∗ G′ ) ∼ —that Rv∗ g × : Dqc (Y ′ ) → Dqc (X) is bounded). Since Ru∗ is bounded (see (3.9.2.3)), the second of these facts results from the natural isomorphism ∼ Rv∗ g × −→ f × Ru∗ of (3.10.4). The ﬁrst results from the (easily-checked) commutativity of ⊕ Rv∗ g ×Fα −−− −−→ Rv∗ ⊕ g ×Fα −−−−→ Rv∗ g ×(⊕ Fα ) α α (3.9.3.3) ⏐α ⏐ ⏐ ⏐ ≃(3.10.4) (3.10.4)≃ −−→ f × Ru∗ (⊕ Fα ) ⊕ f × Ru∗ Fα −− −−→ f × ⊕ Ru∗ Fα −−− α α α (3.9.3.3)

Suppose q > 1, so Y ′ = Y1′ ∪ Y2′ with Yi′ open in Y ′, q(Y1′ ) = q − 1, and ′ ′ ) ≤ q − 1. (Y ′ being separated, the := Y1′ ∩ Y2′ , so that q(Y12 q(Y2′ ) = 1. Set Y12 ′ intersection of aﬃne subschemes of Y is aﬃne). We have the commutative diagram of immersions w1 ′ Y12 −−−− → Y1′ ⏐ ⏐ ⏐u ⏐ w2 1 ′ Y2′ −−− u−→ Y 2

194

4 Abstract Grothendieck Duality for Schemes

With u12 := u1 w1 = u2 w2 there is, for any F ∈ D(Y ′ ), a natural triangle F → Ru1∗ u∗1 F ⊕ Ru2∗ u∗2 F → Ru12∗ u∗12 F → F [1]

(4.7.3.2)

obtained by applying the standard exact sequence—holding for any injective (or even ﬂasque) OY ′ -module G— ∗ 0 → G → u1∗ u1∗ G ⊕ u2∗ u2∗ G → u12∗ u12 G→0

to an injective q-injective resolution of F (see paragraph around (1.4.4.2)). The inductive hypothesis applied to the natural composite independent square (see (3.10.3.2)), with i = 1, 2, 12, v

Xi′ −−−i−→ ⏐ ⏐ gi

v

X ′ −−−−→ ⏐ ⏐g

X ⏐ ⏐f

′ Yi′ −−− −−− u−→ Y u−→ Y i

gives that

gi×

is bounded. Since Rvi∗ is bounded (3.9.2.3), therefore so is g × Rui∗ u∗i

∼ =

(3.10.4)

Rvi∗ gi× u∗i .

Hence, application of the Δ-functor g × to the triangle (4.7.3.2) shows that g × is bounded above, proving (ii). As for (i), in view of (Δ3)∗ of §1.4 it similarly suﬃces to show (left as an exercise) that the following natural diagram—whose columns are triangles (see (3.8.3)), and where the two middle arrows are isomorphisms by (3.9.3.3), by the inductive hypothesis, and by (3.8.2)(ii) (for the trivial case of an open immersion)—commutes: −−→ g × ⊕ Fα ⊕ g ×Fα α α ⏐ ⏐ ⏐ ⏐ × ∗ × ∗ × ∗ ⊕ Rv1∗ g1 u1 Fα ⊕ Rv2∗ g2 u2 Fα − −→ Rv1∗ g1 u1 ⊕ Fα ⊕ Rv2∗ g2× u∗2 ⊕ Fα α α α ⏐ ⏐ ⏐ ⏐ × ∗ ⊕ Rv12∗ g12 u12 Fα α ⏐ ⏐

⊕ g ×Fα [1] α

− −→ −−→

× ∗ Rv12∗ g12 u12 ⊕ Fα ⏐ α ⏐ g × ⊕ Fα [1] α

Having thus settled the separated case, we can proceed similarly for arbitrary concentrated Y ′, with q(Y ′ ) the least number of separated open subschemes needed to cover Y ′ . Q.E.D.

4.7 Quasi-Perfect Maps

195

Proposition 4.7.3.3 Let f : X → Y be a locally embeddable scheme-map, i.e., every y ∈ Y has an open neighborhood V over which the induced map p i → Z − → V where i is a closed immersion f −1 V → V factors as f −1 V − and p is smooth. (For instance, any quasi-projective f satisﬁes this condition [EGA, II, (5.3.3)].) If f is quasi-perfect then f is perfect. Proof. (i) By (4.7.3.1)(iii), quasi-perfection is local over Y , and the same clearly holds for perfection; so we may as well assume that X = f −1 V . Then by [I, p. 252, Prop. 4.4] it suﬃces to show that the complex i∗ OX is perfect, or, more generally, that the map i is quasi-perfect. But i factors as g γ X − → X ×Y Z − → Z where γ is the graph of i and g is the projection. The map γ is a local complete intersection [EGA, IV, (17.12.3)], so the complex γ∗ OX is perfect, and by Example (4.7.3)(d) (or otherwise) γ is quasi-perfect. Also, g arises from f by ﬂat base change, so by (4.7.3.1)(i), g is quasi-perfect. Hence i = gγ is quasi-perfect, as desired. Q.E.D. Remark. Using the analog of (4.7.3.1)(i) with “quasi-proper” in place of “quasi-perfect” [LN, Prop. 4.4], one shows similarly for locally embeddable f that f quasi-proper ⇒ f pseudo-coherent. The converse holds when f is also proper, see (4.3.3.2). Thus, e.g., a projective map is quasi-proper if and only if it is pseudo-coherent. Exercises 4.7.3.4 For a scheme-map f : X → Y and for E, F ∈ Dqc (Y ), let χE,F : f ×E ⊗ Lf ∗F −−−−→ f ×(E ⊗ F ). = = be the map adjoint to ∼ Rf∗ (f ×E ⊗ Lf ∗F ) −→ Rf∗ f ×E ⊗ F −− −−→ E ⊗ F. = = = via τ (3.9.4)

In particular, χOY ,F is the map in (4.7.1)(iv). (a) Show that for any E, F , G ∈ Dqc (Y ), the following diagram commutes. ∗ × f ×E ⊗ (Lf⏐ F⊗ Lf ∗G) −via −−(3.2.4) − Lf ∗(F ⊗ G) −χ−−−−→ f × (E ⊗ (F ⊗ G)) −−→ f E ⊗ = = = = =⏐ = E,F ⊗ =G ⏐≃ ⏐ ≃

F) ⊗ G) (f ×E ⊗ Lf ∗F ) ⊗ Lf ∗G −χ−−−− → f ×(E ⊗ F) ⊗ Lf ∗G −χ−−−−→ f × ((E ⊗ = = = = = = ⊗1 E⊗ =F,G

E,F =

Taking E = OY , deduce that f is quasi-perfect if and only if χF,G is an isomorphism for all F and G. (For this one needs that for any f the map deﬁned in (4.7.1)(iv) is an isomorphism ∼ f ×OY ⊗ Lf ∗ OY −→ f × OY , = ×

∗

∼

(#) ×

∼

×

since, e.g., it factors naturally as f OY ⊗ Lf OY −→ f OY ⊗ OX −→ f OY . = = In fact (#) obtains with any perfect complex in place of OY : see [N, pp. 227–228 and p. 213]. Cf. also (4.7.5) below.) Hint. Using 3.4.7(iv), show that the adjoint of the preceding diagram commutes.

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4 Abstract Grothendieck Duality for Schemes

(b) Show that, with 1 the identity map of Y , the map χE,F : E ⊗ F = 1×E ⊗ 1∗F → E ⊗ F = = is the identity map. (c) (Compatibility of χ and base change.) In this exercise, v ∗ is an abbreviation for Lv ∗, and u∗, f ∗ and g ∗ are analogously understood. Also, ⊗ stands for ⊗ . = For any independent square v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

show that the following diagram, in which β comes from (4.4.3), and the unlabeled isomorphisms are the natural ones, commutes: β(E)⊗1

v ∗f ×E ⊗ v ∗f ∗F −−−−−→ g × u∗E ⊗ v ∗f ∗F ⏐ ⏐ ⏐≃ ⏐ ≃ v ∗ (f ×E ⊗ f ∗F ) ⏐ ⏐ ⏐ ⏐ ⏐ v∗χE,F ⏐ ⏐ ⏐ ⏐ v ∗f ×(E ⊗ F )

g × u∗E ⊗ g ∗ u∗F ⏐ ⏐χ ∗ ∗ u E,u F

−−−−−−→ β(E⊗F )

g × (u∗E ⊗ u∗F ) ⏐ ⏐≃ g × u∗(E ⊗ F )

Hint. It suﬃces to check commutativity of the following natural diagram, whose outer border is adjoint to that of the one in question. β(E)

g∗ (v ∗f ×E ⊗ v ∗f ∗F ) −−−−−−−−−−−−−−−−−−−−−−−−→

g∗ (v ∗f ×E ⊗ v ∗f ∗F ) −− → g∗ (v ∗f ×E ⊗ g ∗ u∗F ) ⏐ ⏐ ⏐ ⏐ p ⏐ ⏐ ⏐ ⏐ g∗ v ∗f ×E ⊗ u∗F ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≃⏐ u∗f∗ f ×E ⊗ u∗F ⏐ cf. (3.7.3) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ u∗ (f∗ f × ⏐E ⊗ F ) ⏐ ⏐ ⏐ p g∗ v ∗ (f ×E ⊗ f ∗F ) −−→ ⏐ ⏐ g∗ v ∗f × (E ⊗ F )

−− →

u∗f∗ (f × E ⊗ f ∗F ) ⏐ ⏐ u∗f∗ f × (E ⊗ F )

β(E)

g∗ (g × u∗E ⊗ v ∗f ∗F ) ⏐ ⏐≃

−−−→ g∗ (g × u∗E ⊗ g ∗ u∗F ) ⏐ ⏐p β(E)

−−−→

g∗ g × u∗E ⊗ u∗F ⏐ ⏐

−−→

u∗E ⊗ u∗F ⏐ ⏐≃

−−→

u∗ (E ⊗ F )

−−→

u∗ (E ⊗ F )

4.7 Quasi-Perfect Maps

197

(d) (Transitivity of χ). If g : Y → Z is a second scheme-map then the following natural diagram is commutative: f ×g ×E ⊗ Lf ∗ Lg ∗ F −−−−→ f ×(g ×E ⊗ Lg ∗ F ) −−−−→ f × g × (E ⊗ F) =⏐ = = ⏐ ≃ L(gf )∗F −−−−→ (gf )× E ⊗ =

(gf )× (E ⊗ F) =

−−− −→

f ×g ×(E ⊗ F) =

Hint. Using (3.7.1), show that the adjoint diagram commutes. (e) Show that χE,F corresponds via (2.6.1)′ to the composite map f ×E −− −−→ f × RHom• (F , E ⊗ F) − −− −−→ f × [F , E ⊗ F ]Y natural = = (4.2.3)(c)

− −− −−→ [Lf ∗F , f ×(E ⊗ F )]X = (4.2.3)(e)

−natural −−−−→ RHom• (Lf ∗F , f ×(E ⊗ F )). = (f) With notation as in (4.2.3)(e), and E, F , G ∈ Dqc (Y ), establish a natural commutative functorial diagram

f ×F ⊗ Lf ∗ [E, G]Y = ⏐ ⏐

χ

−−−→

[E, G]Y f× F ⊗ =

−−−→

f × [E, F ⊗ G]Y = ⏐≃ ⏐

f ×F ⊗ [Lf ∗E, Lf ∗G]X −−−→ [Lf ∗E, f × F ⊗ Lf ∗G]X −−−→ [Lf ∗E, f × (F ⊗ G)]X = = = via χ

We adopt again the notations introduced at the beginning of §4.4. Apropos of the next theorem, recall from the beginning of §4.7 that f quasi-perfect =⇒ f × bounded. Theorem 4.7.4 Let

v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

be an independent square of scheme-maps, with f quasi-perfect. Then for all E ∈ Dqc (Y ) the base-change map of (4.4.3)—with × in place of ! —is an isomorphism ∼ β(E) : v ∗f ×E −→ g × u∗E. The same holds, with no assumption on f , whenever u is ﬁnite and perfect. Conversely, the following conditions on a scheme-map f : X → Y are equivalent; and if Y is separated and f × bounded above, they imply that f is quasi-perfect: (i) For any ﬂat aﬃne universally bicontinuous map u : Y ′ → Y , (i.e., for any Y ′′ → Y the resulting projection Y ′ ×Y Y ′′ → Y ′′ is a homeomorphism onto its image [GD, p. 249, D´efn. (3.8.1)]) the base-change map associated to

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4 Abstract Grothendieck Duality for Schemes

the independent ﬁber square v

Y ′ ×Y X = X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

∼ is an isomorphism β(OY ) : v ∗f × OY −→ g × u∗ OY . (ii) The map in (4.7.1)(iv) is an isomorphism ∼ χF : f × OY ⊗ Lf ∗F −→ f ×F =

whenever F is a ﬂat quasi-coherent OY -module. Proof. For the ﬁrst assertion, using (4.7.3.1)(i) we reduce as in IV of §4.6 to where u, hence v, is an open immersion or aﬃne, so that v is isofaithful ((3.10.2.1)(d) or (3.10.2.2)), and for β to be an isomorphism it suﬃces that v∗ β be an isomorphism. For this purpose it will clearly suﬃce that the following diagram—in which O′ := OY ′ , φ is the isomorphism in (3.10.4), θ′ is as in (3.10.2) (see (3.10.3)), χ := χE,u∗ O′ is as in (4.7.3.4)(a), q is the natural composite isomorphism f ×E ⊗ v∗ g ∗ O′ −− −−→ v∗ (v ∗f ×E ⊗ g ∗ O′ ) − −→ v∗ v ∗f ×E (3.9.4)

and r is the natural composite isomorphism

−−→ u∗ (u∗E ⊗ O′ ) − −→ u∗ u∗E, E ⊗ u∗ O′ −− (3.9.4)

—is commutative:

f ×E ⊗ v∗ g ∗ O′ −− −q−→ ⏐ 1⊗θ ′ ⏐≃

v∗ v ∗f ×E

−−−−→ v∗ g × u∗E v∗ β(E) ⏐ ⏐ ≃φ

− −→ f ×(E ⊗ u∗ O′ ) −− −−→ f × u∗ u∗E f ×E ⊗ f ∗ u∗ O′ −− χ f ×r

(4.7.4.1)

Since χ is an isomorphism whenever u∗ O′ is perfect (see the end of (4.7.3.4)(a)), and since ﬁnite maps are isofaithful (3.10.2.2), commutativity of (4.1) also implies the theorem’s assertion about ﬁnite perfect u. Now, commutativity of (4.1) results from commutativity of the following diagram (4.1)∗, where q ′ is the composite isomorphism −−→ u∗ (u∗f∗ f ×E ⊗ O′ ) − f∗ f ×E ⊗ u∗ O′ −− −→ u∗ u∗f∗ f ×E (3.9.4)

4.7 Quasi-Perfect Maps

199

and t and t′ are the natural maps, a diagram whose outer border, with the isomorphism (3.4.9) replaced by its inverse, is adjoint to (4.1): f∗ q

f∗ v∗ β

1

u∗ g∗ v ∗f ×E −−−−→ u∗ g∗ g × u∗E ⏐ ⏐ ′ ⏐ u∗ θ ⏐≃ u∗ t 2

f∗ (f ×E ⊗ v∗ g ∗ O′ ) −−−−→ f∗ v∗ v ∗f ×E −−−−→ f∗ v∗ g × u∗E ⏐ f∗ (1⊗θ ′ )⏐≃

f∗ (f ×E ⊗ f ∗ u∗ O′ ) ⏐ (3.9.4)⏐≃ f∗ f ×E ⊗ u∗ O′

u∗ g∗ β

−− −− → u∗ u∗f∗ f ×E −−∗−−→ ′ q

u u∗ t

(4.7.4.1)∗

u∗ u∗E

Subdiagram 2 commutes by the very deﬁnition of β. Expand subdiagram 1 as follows, with an arbitrary F ∈ D(X) in place of f ×E, with unlabeled maps being the natural ones, and with p denoting projection maps from (3.4.6) or (3.9.4): (3.4.2.1)

f∗ (F ⊗ v∗ g ∗ O ′ ) −−→ f∗ (v∗ v ∗F ⊗ v∗ g ∗ O ′ ) −−−−→ f∗ v∗ (v ∗F ⊗ g ∗ O ′ ) −− → f∗ v∗ v ∗F ⏐ ⏐ θ ′⏐ θ′ ⏐

f∗ (F ⊗ f ∗ u∗ O ′ ) −−→ f∗ (v∗ v ∗F ⊗ f ∗ u∗ O ′ ) ⏐ ⏐ p⏐ p⏐ f∗ F ⊗ u∗ O

′

f∗ F ⊗ u∗ O ′

−−→

5

−−→

∗

f∗ v∗ v F ⊗ u∗ O

3

′

u∗ g∗ v ∗F ⊗ u∗ O ′ ⏐ θ⏐

u∗ u∗f∗ F ⊗ u∗ O ′

(3.4.2.1)

−−−−→

−−−−→

(3.4.2.1)

→ u∗ g∗ v ∗F u∗ g∗ (v ∗F ⊗ g ∗ O ′ ) −− ⏐ ⏐ ⏐ ⏐ ⏐ u∗ p 4 ⏐ ⏐ ⏐ ⏐

u∗ (g∗ v ∗F ⊗ O ′ ) ⏐ ⏐θ

u∗ (u∗f∗ F ⊗ O ′ )

−− → u∗ g∗ v ∗F ⏐ ⏐θ

−− → u∗ u∗f∗ F

Commutativity of the unlabeled subdiagrams is clear. That of 5 follows from the deﬁnition (3.7.2)(a) of θ; and that of 4 follows from (3.4.7)(iii). Subdiagram 3 expands as follows: f∗ (v∗ v ∗F ⊗ v∗ g ∗ O′ ) ⏐ ⏐ ⏐ ⏐ ⏐ 6 θ ′⏐ ⏐ ⏐ ⏐ ⏐

f∗ (v∗ v ∗F ⊗ f ∗ u∗ O′ ) ⏐ p⏐ f∗ v∗ v ∗F ⊗ u∗ O′

(3.4.2.1)

f∗ (v∗ v ∗F ⊗ v∗ g ∗ O′ ) −−−−→ f∗ v∗ (v ∗F ⊗ g ∗ O′ ) ⏐(3.4.2.1) ⏐ f∗ v∗ v ∗F ⊗ f∗ v∗ g ∗ O′ 7 (3.4.2.1)

u∗ g∗ v ∗F ⊗ u∗ g∗ g ∗ O′ −−−−→ u∗ g∗ (v ∗F ⊗ g ∗ O′ ) ⏐ ⏐u∗ p 8 ⏐ ⏐ u∗ g∗ v ∗F ⊗ u∗ O′

−−−−→ u∗ (g∗ v ∗F ⊗ O′ )

(3.4.2.1)

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4 Abstract Grothendieck Duality for Schemes

For commutativity of subdiagram 8 , replace p by its deﬁnition (3.4.6), and apply commutativity of (3.6.7.2). Commutativity of 7 also follows from that of (3.6.7.2). Finally, subdiagram 6 expands as follows: θ′

f∗ (v∗ v ∗F ⊗ f ∗ u∗ O′ ) −−−−−−−−−−−−−−−−−−−−−−−−−→f∗ (v∗ v ∗F ⏐ (3.4.2.1)⏐ θ′

f∗ v∗ v ∗F ⊗ f∗ f ∗ u∗ O′ −−−−−−−−−−−−−−−−−−−−−−−−−→f∗ v∗ v ∗F ⏐ 9 ⏐ f∗ v∗ v ∗F ⊗ u∗ O′

f∗ v∗ v ∗F ⊗ u∗ O′

⊗ v∗ g ∗ O ′ ) ⏐(3.4.2.1) ⏐

⊗ f∗ v∗ g ∗ O′

−−→ f∗ v∗ v ∗F ⊗ u∗ g∗ g ∗ O′ −−→ u∗ g∗ v ∗F ⊗ u∗ g∗ g ∗ O′ ⏐ ⏐ u∗ g∗ v ∗F ⊗ u∗ O′

Commutativity of 9 is an easy consequence of the deﬁnition (3.7.2)(a) of θ ′ ; and that of the other two subdiagrams is clear. It is thus established that (4.1)∗ commutes. We show next that (i) ⇔ (ii). Assume (i). Let F be a ﬂat quasi-coherent OY -module. Let F be the OY algebra OY ⊕ F with F 2 = 0 (i.e., the symmetric algebra on F, modulo everything of degree ≥ 2), and let u : Y ′ → Y be an aﬃne scheme-map such that u∗ OY ′ = F (see [GD, p. 355, (9.1.4) and p. 370, (9.4.4)]). This u is a ﬂat aﬃne universally bicontinuous map. With E = OY , all the maps in the commutative diagram (4.7.4.1) other than χ = χOY ⊕ χF are isomorphisms, and so χ must be an isomorphism too. But χOY is an isomorphism (exercise), so χF is an isomorphism, i.e., (ii) holds. Conversely, if u is any ﬂat aﬃne map and (ii) holds for the ﬂat quasicoherent OY -module F = u∗ OY ′ then (4.7.4.1) with E = OY shows that v∗ β(OY ) is an isomorphism, whence, v being aﬃne, so is β(OY ), see (3.10.2.2). Finally, assuming (ii) and that Y is separated and f × bounded-above, let us deduce that the map χE : f × OY ⊗ Lf ∗E → f ×E is an isomorphism = for all E ∈ Dqc (Y ), so that f is quasi-perfect (see 4.7.1(iv)). Since Y is separated, we can replace E by a D-isomorphic q-ﬂat quasicoherent complex, which is a lim of bounded-above ﬂat complexes, see [AJL, −→ p. 10, (1.1)] and its proof. Since the functors f × OY ⊗ Lf ∗ (−) and f ×(−) = are both bounded-above, we may assume that E is bounded-below: for each n ∈ Z, if E ′ is obtained by replacing all suﬃciently-negative-degree components of E by (0) then χE and χE ′ induce identical homology maps in degree n, and (1.2.2) can be applied. Similarly, since f × is bounded below, and Lf ∗E = f ∗E when E is a lim of bounded-above ﬂat complexes, we can −→ reduce further to where E is bounded, ﬂat, and quasi-coherent. Now an induction on the number of nonvanishing components of E (using the triangle [H, p. 70, (1)]) gives the desired conclusion. Q.E.D. For more along these lines see exercise 4.7.6(f) below.

4.7 Quasi-Perfect Maps

201

Proposition 4.7.5 If f : X → Y is quasi-proper and F ∈ Dqc (Y ) has ﬁnite tor-dimension then for all E ∈ Dqc (Y ) the map χE,F of (4.7.3.4) is an isomorphism ∼ f ×E ⊗ Lf ∗F −→ f × (E ⊗ F ). = = Proof. If U ֒→ Y is an open immersion, then by [LN, Prop. 4.4], the projection X ×Y U → U is quasi-proper. Together with (4.4.3) and (4.7.3.4)(c), this implies that the assertion in (4.7.5) is local on Y , so we may assume that Y is aﬃne. We can then replace F by a D-isomorphic bounded-above quasi-coherent complex—see (3.9.6)(a)—which by [H, p. 42, 4.6.1)] (dualized) may be assumed ﬂat. Since F has ﬁnite tor-dimension, an application of [I, p. 131, 5.1.1] to a suitable D-isomorphic truncation of F allows one to assume further that F is bounded. Then an induction on the number of nonvanishing components of F (using the triangle [H, p. 70, (1)]) reduces the problem to where F is a single ﬂat quasi-coherent OY -module. As in the proof of (4.7.4) ((i) ⇔ (ii)), let u : Y ′ → Y be an aﬃne schememap such that u∗ OY ′ = OY ⊕ F. The map u is ﬂat, so u and f are two sides of an independent square, and by (4.4.3) the corresponding base-change map β(E) in the commutative diagram (4.7.4.1) is an isomorphism. One Q.E.D. concludes as before that χE,F is an isomorphism. Exercises 4.7.6 (a). Let f : X → Y be a quasi-perfect scheme-map. Assume that X is divisorial—i.e., X has an ample family of invertible OX -modules—so that by [I, p. 173, 2.2.8 b)] every pseudo-coherent OX -complex is D-isomorphic to a bounded above complex of ﬁnite-rank locally free OX -modules. Show that an OX -complex F is pseudo-coherent iﬀ for every n ∈ Z there is a triangle P → F → R → P [1] with P perfect and R ∈ (Dqc )
α

Hint. Write f∗ for Rf∗ , HX for RHom• X , etc. The triangulated category Dqc (X) ≡ D(Aqc (X)) is generated by perfect complexes (see [N, pp. 215–216], or [LN, Thm. 4.2]), so a Dqc -map ϕ : A1 → A2 is an isomorphism if and only if the induced map Hom(E, A1 ) → Hom(E, A2 ) is an isomorphism for all perfect E ∈ D(X). In the following natural diagram, easily seen to commute, f∗ HX (E,⊕f ×Gα ) −−−→ f∗ HX E, f × (⊕ Gα ) −−− −→ HY (f∗ E, ⊕ Gα ) (4.3.6) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ × × f∗ ⊕ HX (E, f Gα ) ←−− −−− ⊕f∗ HX (E, f Gα ) −−− −→ ⊕ HY (f∗ E, Gα ) (3.9.3.1)

(4.3.6)

the left and right vertical arrows are isomorphisms whenever E is pseudo-coherent.

11

Cf. [V′, p. 396, Lemma 1], where the necessary uniform lower bound on the Gα is omitted.

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4 Abstract Grothendieck Duality for Schemes

(The question being local on X, one can, as in the proof of (4.3.5), replace E by a bounded ﬁnite-rank free complex E ′ and then, using the triangle [H, p. 70, (1)], proceed by induction on the number of degrees in which E ′ doesn’t vanish.) Finally, apply the functor H0 RΓ(Y , −). (c) Deduce from (b) that a quasi-proper scheme-map f with f × bounded above is quasi-perfect. (This is part of [LN, Thm. 1.2.]) (d) Let f : X → Y be a scheme-map. Show that if f is quasi-perfect then the following two conditions hold, and that the converse is true when f × is bounded. (Apropos, recall again from the beginning of this section that if f is quasi-perfect then f × is bounded.) (i) If u : Y ′ → Y is an open immersion, and v : f −1 U → X, g : f −1 U → U are the obvious induced maps, then the base-change map is an isomorphism ∼ β(OY ) : v ∗f × OY −→ g × u∗ OY .

Equivalently (see subsection V in §4.6), for all F ∈ Dqc (X) the duality map δ(F , OY ) deﬁned as in (4.4.2) is an isomorphism × ∼ • Rf∗ RHom• X (F , f OY ) −→ RHomY (Rf∗ F , OY )

(ii) If (Fα ) is a small ﬁltered direct system of ﬂat quasi-coherent OY -modules then for all n ∈ Z the natural map is an isomorphism n × ∼ n × lim − → Fα ). − → H (f Fα ) −→ H (f lim α α

Hint. Use (4.7.3.4)(c) and Lazard’s theorem that over a commutative ring A any ﬂat module is a lim −→ of ﬁnite-rank free A-modules [GD, p. 163, (6.6.24)] to show that (i) and (ii) imply condition (ii) in (4.7.4). (e) (i) (Neeman). Using, e.g., (i) in (d) (with F = OX ), show that if f : X → Y is quasi-perfect then the OY -complex Rf∗ f ×OY is perfect; and deduce that for any perfect OY -complex E, Rf∗ f ×E is perfect. (ii) (cf. [I, p. 257, 4.8]). Let f : X → Y be a concentrated quasi-proper map of quasi-compact schemes. Then for any f -perfect OX -complex E, Rf∗ E is a perfect OY -complex. f u (f) Let U − →X− → Y be scheme-maps, with f quasi-proper, and let E ∈ Dqc (Y ). Show that the following are equivalent. (i) The functor Lu∗f ×(E ⊗ F ) (F ∈ Dqc (Y )) is bounded above. = (ii) Lu∗f ×E ∈ D−(X)), and the map (see exercise (4.7.3.4) above) Lu∗ χE,F : Lu∗f ×E ⊗ L(f u)∗F → Lu∗f ×(E ⊗ F ), = = is an isomorphism for all F ∈ Dqc (Y ). F ) (F ∈ Dqc (Y )) respects (iii) Lu∗f ×E ∈ D−(X)), and the functor Lu∗f ×(E ⊗ = direct sums (cf. (4.7.1)(i)). Moreover, if u has ﬁnite tor-dimension, then the following are equivalent. (i)′ The functor Lu∗f ×(E ⊗ F ) (F ∈ Dqc (Y )) is bounded. = (ii)′ The complex Lu∗f ×E has ﬁnite ﬂat f u-amplitude (2.7.6), and Lu∗ χE,F is an isomorphism for all F ∈ Dqc (Y ).

4.8 Two Fundamental Theorems

203

(iii)′ Lu∗f ×E has ﬁnite ﬂat f u-amplitude, and the functor Lu∗f ×(E ⊗ F) = (F ∈ Dqc (Y )) respects direct sums. Hint. Given (i), one sees as in exercise (c) above that the functor Lu∗f ×(E ⊗ F ) respects direct sums; and then arguing as in [N, p. 226, Thm. 5.4], one = see that Lu∗ χE,F in (ii) is an isomorphism. It follows then from [I, p. 242, 3.3(iv)], and the fact that if V ⊂ Y is open then any quasi-coherent OV -module M is the restriction of a quasi-coherent OY -module, that if (i)′ holds then Lu∗f ×E has ﬁnite ﬂat f u-amplitude.

4.8 Two Fundamental Theorems Up to now we have dealt with the pseudofunctor × (see (4.1.1)) for quite general maps—it cost nothing to do so. But for non-proper maps this pseudofunctor may still be of limited interest (see [De′, p. 416, line 3]). As indicated in the Introduction to these notes, Grothendieck Duality is ! fundamentally concerned with a D+ qc -valued pseudofunctor over the category of say, separated ﬁnite-type maps of noetherian schemes, agreeing with × on proper maps, but, unlike × (see (4.2.3)(d)), agreeing with the usual pseudofunctor * on open immersions (more generally, on separated ´etale maps see [EGA, IV, §§17.3, 17.6]), and compatible in a suitable sense with ﬂat base change. The existence and uniqueness, up to isomorphism, of this remarkable pseudofunctor is given by Theorem (4.8.1), and its behavior vis-` a-vis ﬂat base change is described in Theorem (4.8.3). The proof of (4.8.1) presented here is based on a formal method of Deligne for pasting pseudofunctors (see Proposition (4.8.4)), and on the compactiﬁcation theorem of Nagata, that any ﬁnite-type separable map of noetherian schemes factors as an open immersion followed by a proper map (see [Lt], [C′], [Vj]). The proof of (4.8.3) is based on a formal pasting procedure for base-change setups (see (4.8.2), (4.8.5)). There are other pasting techniques, due to Nayak [Nk], to establish the two basic theorems, (4.8.1) and (4.8.3).12 As mentioned in the Introduction, Nayak’s methods avoid using Nagata’s theorem, and so apply in contexts where Nagata’s theorem may not hold. For example, the results in [Nk, §7.1] are generalizations of (4.8.1) and (4.8.3) to the case of noetherian formal schemes (except for “thickening” as in (4.8.11) below, which allows ﬂat basechange isomorphisms for admissible squares (4.8.3.0) rather than just ﬁber squares, see Exercise (4.8.12)(d).) All commutative squares will be considered to be oriented, as in §3.10. The ﬁrst main result deﬁnes (up to isomorphism) the twisted inverse image pseudofunctor. 12

[Nk, §7.5] discusses the relation between Nayak’s methods and Deligne’s. On the other hand, in [Nk′ ] Nayak extends Nagata compactiﬁcation—and hence Theorems (4.8.1) and (4.8.3)—to separated maps which are essentially of ﬁnite type.

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4 Abstract Grothendieck Duality for Schemes

Theorem 4.8.1 On the category Sf of ﬁnite-type separated maps of ! noetherian schemes, there is a D+ qc -valued pseudofunctor that is uniquely determined up to isomorphism by the following three properties: (i) The pseudofunctor ! restricts on the subcategory of proper maps to a right adjoint of the derived direct-image pseudofunctor, see (3.6.7)(d). (ii) The pseudofunctor ! restricts on the subcategory of ´etale maps to the usual inverse-image pseudofunctor * . (iii) For any ﬁber square in Sf : v

• −−−−→ ⏐ ⏐ g σ

• ⏐ ⏐f

(f , g proper; u, v ´etale),

• −−− −→ • u

the base-change map βσ of (4.4.3) is the natural composite isomorphism ∼ ∼ v ∗f ! = v !f ! −→ (f v)! = (ug)! −→ g ! u! = g ! u∗ .

Remark 4.8.1.1 It follows that when f is both ´etale and proper (hence by [EGA, III, 4.4.11], ﬁnite), then the natural map f∗ f ∗ = f∗ f ! → 1 is precisely—not just up to isomorphism—the standard trace map, see Exercise (4.8.12)(b)(vii). For subsequent considerations, involving base-change isomorphisms and their properties, the following deﬁnition will be convenient to have. Deﬁnition 4.8.2 A base-change setup B S, P, F, !, *, (βσ )σ∈ consists of the following data (a)–(d), subject to conditions (1)–(3): (a) Subcategories P and F of a category S, each containing every object of S. (b) Contravariant pseudofunctors ! on P and * on F such that for all objects X ∈ S, the categories X! and X* coincide (see §3.6.5). (c) A class of (oriented) commutative S-squares, the distinguished squares, each member of which has the form v

• −−−−→ ⏐ ⏐ g σ

• ⏐ ⏐f

(f , g ∈ P; u, v ∈ F)

• −−− −→ • u

(where u precedes f in the orientation of σ, see §3.10). (d) For each distinguished σ as in (c), an isomorphism of functors ∼ g ! u∗ . βσ : v ∗f ! −→

4.8 Two Fundamental Theorems

205

(1) If two commutative S-squares v

• −−−−→ ⏐ ⏐ g σ

v

• −−−1−→ ⏐ ⏐ g1 σ1

• ⏐ ⏐f

• −−− −→ • u

• ⏐ ⏐f 1

• −−− −→ • u 1

are isomorphic, i.e., there exists a commutative cube with front and rear faces σ and σ1 respectively, and i, i1 , j, j1 isomorphisms: v1

•

•

j1

j v

•

• f1

g1

g

f

•

•

u1 i

i1

•

•

u

then σ is distinguished ⇔ σ1 is distinguished. (2) For every P-map f , the square 1

• −−−−→ ⏐ ⏐ σ f

• ⏐ ⏐f

• −−−−→ • 1

is distinguished, and βσ : f ! → f ! is the identity map. ′

(2)

For every F-map u, the square u

• −−−−→ ⏐ ⏐ 1 σ

• ⏐ ⏐ 1

• −−− −→ • u

is distinguished, and βσ : u∗ → u∗ is the identity map. (3) (Horizontal and vertical transitivity.) If the square σ0 = σ2 ◦σ1 (with g resp. v deleted)

206

4 Abstract Grothendieck Duality for Schemes w

v

v

• −−−1−→ • −−−2−→ ⏐ ⏐ ⏐ σ1 g⏐ σ2 h

• ⏐ ⏐f

• −−−−→ ⏐ ⏐ σ1 g1 v

• −−−−→ ⏐ ⏐ σ2 g2

resp.

• −−− −→ • −−− −→ • u1 u2

• ⏐ ⏐f 1

• ⏐ ⏐f 2

• −−− −→ • u

as well as its constituents σ2 and σ1 are all distinguished, then the corresponding natural diagram of functorial maps commutes: βσ0

(v2 v1 )∗f ! −−−−−−−−−−−−−−−−→ h! (u2 u1 )∗ ⏐ ⏐ ⏐≃ ⏐ ≃ v1∗ v2∗f !

−−∗−−→ v1∗ g ! u∗2 −−−−→ βσ1

v1 βσ2

h! u∗1 u∗2

resp. βσ0

(g2 g1 )! u∗ ←−−−−−−−−−−−−−−−− w∗ (f2 f1 )! ⏐ ⏐ ⏐≃ ⏐ ≃ g1! g2! u∗

←−!−−− g1! v ∗f2! ←−−−− βσ1

g1βσ2

w∗f1! f2!

Remarks 4.8.2.1 (a) Let u and v be S-isomorphisms. If f and g are S-maps such that f v = ug is in P, then the squares v

• −−−−→ ⏐ ⏐ g σ

• ⏐ ⏐f

1

• −−−−→ ⏐ ⏐ ug σ ˜

and

• −−− −→ • u

• ⏐ ⏐f v

• −−−−→ • 1

are isomorphic, so that by (1) and (2), σ is distinguished—which entails that u and v are in F and that f and g are in P. In particular, v

• −−−−→ ⏐ ⏐ v

• ⏐ ⏐ −1 v

• −−− −→ • −1 v

is distinguished, and consequently every S-isomorphism lies in P ∩ F (whence f v ∈ P ⇐⇒ f ∈ P, and ug ∈ P ⇐⇒ g ∈ P).

4.8 Two Fundamental Theorems

207

Similarly, if f and g are S-isomorphisms, and u and v are any F-maps such that f v = ug, then σ is distinguished. (b) That the isomorphism βσ in (2) is idempotent, hence the identity, actually follows from (3), with ui = vi = 1 (resp. fi = gi = 1). (c) To each base-change setup B = B S, P, F, !, *, (βσ )σ∈ is associated a dual setup B op := B S, F, P, *, !, (βσ′ := βσ−1 )σ′ ∈′ , where σ ′ is the transpose of σ (i.e., σ with its orientation reversed, or, visually, the reﬂection of σ in its upper-left to lower-right diagonal), and ′ consists of all transposes of squares in . Example 4.8.2.2 Let S be a category, take P = F = S, let ! = * be a contravariant pseudofunctor on S, let all commutative squares in S be distinguished, and for any such square σ, let ∼ ∼ βσ : v ∗f ∗ −→ (f v)∗ = (ug)∗ −→ g ∗ u∗

be the isomorphism naturally associated with the pseudofunctor * . Then (4.8.2)(1) holds trivially, and (2), (2)′ , (3) follow readily from the deﬁnition of “pseudofunctor.” We will denote such a base-change setup by B S, * .

Example 4.8.2.3 Let S be a subcategory of the category of quasi-compact separated schemes, P ⊂ S the subcategory of quasi-proper maps, and F ⊂ S the subcategory of ﬁnite-tor-dimension maps. On P there is the D+ qc -valued -valued pseudofuncpseudofunctor × (see (4.1.2)); and on F there is the D+ qc tor * with u∗ := Lu∗ for any F-map u. Let be the class of independent ﬁber squares of the form speciﬁed in 4.8.2(c). For σ ∈ , let βσ : v ∗f × → g × u∗ be the corresponding base-change isomorphism from (4.4.3). Conditions (1), (2) and (2)′ in (4.8.2) are then easily veriﬁed; and as in (4.6.8), (3) follows formally from (3.7.2), (ii) and (iii). So we have a base change setup B S, P, F, × , *, (βσ )σ∈ . Example 4.8.2.4 As a special case, we have the base-change setup B Sf , P, E, × , *, (βσ )σ∈ with Sf as in (4.8.1), P ⊂ Sf the subcategory of proper maps, E ⊂ Sf the subcategory of ´etale maps, and × , * , , βσ as in the preceding example (4.8.2.3) (with F replaced by E). To prove (4.8.1), we will need to way to enlarge show that there is a unique the preceding setup to a setup B Sf , P, E, × , *, (βσ′ )σ∈′ where ′ consists of all commutative Sf -squares

208

4 Abstract Grothendieck Duality for Schemes v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′ −−− −→ Y u

with f , g proper and u, v ´etale. This, and more, will be done in (4.8.11). Meanwhile, we’ll refer to this unique enlarged setup as Example (4.8.2.4)′ . Notation-Deﬁnition (4.8.3.0). A category S having been given, for S-maps v, f , g, u with f v = ug, σv,f, g,u is the commutative square v

• −−−−→ ⏐ ⏐ g

• ⏐ ⏐f

• −−− −→ • u

In the category of schemes, such a σv,f, g,u: v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

is an admissible square if u is ﬂat, f is ﬁnitely presentable, and in the associated diagram q

i

X ′ −−−−→ X ×Y Y ′ −−−1−→ ⏐ ⏐ q2 Y′

X ⏐ ⏐f

−−− −→ Y u

where q1 , q2 are the projections, q1 i = v and q2 i = g, the map i is ´etale. (Note that then g = q2 i is ﬁnitely presentable, and v = q1 i is ﬂat, so that Lv ∗ = v ∗.) Theorem 4.8.3 Let S be the category of separated maps of noetherian schemes, let Sf ⊂ S and ! be as in (4.8.1), let F ⊂ S be the subcategory pseudofunctor of ﬂat maps, and let * be the usual D+ qc -valued inverse-image on F. Then there is a unique base-change setup B S, Sf , F, !, *, (βσ )σ∈ with the class of admissible S-squares, such that the following conditions hold for any admissible S-square σ = σv,f, g,u :

4.8 Two Fundamental Theorems

209

(i) If σ is a ﬁber square with f proper then βσ is the base-change isomorphism in (4.4.3). (ii) If f —and hence g—is ´etale, so that f ! = f ∗ and g ! = g ∗, then βσ is ∼ the natural isomorphism v ∗f ∗ −→ g ∗ u∗ . (iii) If u—and hence v—is ´etale, so that u∗ = u! and v ∗ = v !, then βσ is ∼ the natural isomorphism v !f ! −→ g ! u! . Remarks 4.8.3.1 (a) Since ´etale maps are unramiﬁed [EGA, IV, (17.6.2)], therefore by [EGA, IV, (17.3.3)(iii) and (17.3.4)], every commutative Sf -square σv,f, g,u with u and v ﬂat and such that either f and g or u and v are ´etale is admissible. (b) Uniqueness in (4.8.3) is implied by (i), (ii) and vertical transitivity as in (4.8.2)(3), because if σv,f, g,u is admissible, then, by Nagata’s theorem, f = f2 f1 with f2 proper and f1 an open immersion, whence σ decomposes as in the second diagram in (4.8.2)(3), with σ1 having v, w ﬂat and f1 , g1 ´etale, and with σ2 an admissible ﬁber square. (c) As for existence, the preceding suggests deﬁning βσ via a choice of such factorizations, one for each f, then showing that the deﬁnition does not depend on the choice, and that (i)–(iii) in (4.8.3) are satisﬁed. This purely formal procedure is straightforward in principle but, as will emerge, lengthy in practice. In view of Nagata’s compactiﬁcation theorem, it is readily veriﬁed that the existence of the pseudofunctor ! in Theorem (4.8.1) results from the next Proposition (4.8.4) on the pasting of pseudofunctors, as applied to the basechange setup (4.8.2.4)′ . Proposition 4.8.4 ([De, p. 318, Prop. 3.3.4]) Let there be given a basechange setup B = B S, P, E, × , *, (βσ )σ∈ such that: (a) the ﬁber product in P of any two P-maps with the same target exists, and is a ﬁber product in S of the same two maps; (b) every map f ∈ S has a “compactiﬁcation,” i.e., a factorization f = f¯i with f¯ ∈ P and i ∈ E; and (c) consists of all of the commutative S-squares σv,f, g,u for which f , g ∈ P and u, v ∈ E. Then there exists a contravariant pseudofunctor ! on S, uniquely determined up to isomorphism by the properties that X! = X× = X* for all X ∈ S and ∼ × that there exist isomorphisms of pseudofunctors (see (3.6.6)) αP : ! |P −→ ∼ and αE : ! |E −→ * such that for any σ = σv,f, g,u ∈ , βσ is the natural composition (with ﬁrst and last isomorphisms coming from αP and αE ) : ∼ ∼ −−→ (f v)! = (ug)! −− −−→ g ! u! −→ v ∗f × −→ v ! f ! −− g × u∗. In other words, B S, ! (see (4.8.2.2)) extends B, via αP and αE. In fact there is a ! such that, furthermore, ! |E = * and αE is the identity isomorphism.

210

4 Abstract Grothendieck Duality for Schemes

Remark. Uniqueness (up to isomorphism) in (4.8.1) also results from (4.8.4), as follows. Let P ⊂ Sf , E ⊂ Sf and ′ be as in (4.8.2.4). If the pseudofunctor ! satisﬁes the conditions in (4.8.1) then there is a natural pseudofunctorial ∼ × |P (since both ! |P and × |P have the same pseudoisomorphism αP : ! |P −→ functorial left adjoint). For any σv,f, g,u ∈ ′ let βσ′′ be the natural composite isomorphism v ∗f × −−∗ −− → v ∗f ! = v !f ! − −→ g ! u! = g ! u∗ −−α −−→ g × u∗. −1 P v αP

This gives a setup B ′′ = B Sf , P, E, × , *, (βσ′′ )σ∈′ . (Check directly, or see Exercise (4.8.12)(a).) When σ is a ﬁber square then, one checks, βσ′′ is the base change map of (4.4.3). Thus B ′′ is the unique enlargement (4.8.2.4)′ of the setup (4.8.2.4), so that the uniqueness assertion in (4.8.4) gives the uniqueness in (4.8.1). Proof of (4.8.4). (Outline: more details are in [De, pp. 304–318].13 ) If the pseudofunctor ! exists then to each compactiﬁcation f = f¯i there is × ∼ i∗f¯ ; and for a composed map f1 f2 naturally associated an isomorphism f ! −→ and compactiﬁcations f1 = f¯1 i1 , f2 = f¯2 i2 , i1 f¯2 = g¯j, with σ := σj, g¯,f¯2,i1 , the ∼ canonical isomorphism f2! f1! −→ (f1 f2 )! factors naturally as ×

×

∼ −→ i2∗ j ∗ g¯×f¯1× (f¯2 i2 )! (f¯1 i1 )! −→ i2∗ f¯2 i1∗f¯1 − −1 βσ

∼ ∼ −→ (ji2 )∗ (f¯1 g¯)× −→ (f¯1 g¯ji2 )! = (f¯1 i1 f¯2 i2 )! .

(4.8.4.1)

If !! is another pseudofunctor with the same property as ! then for each compactiﬁcation f = f¯i we have a natural composite functorial isomorphism !

×

∼ ∼ f ! = (f¯i)! −→ i! f¯ −→ i∗f¯

!!

∼ ∼ −→ i!! f¯ −→ (f¯i)!! = f !! .

(4.8.4.2)

One must show that (4.8.4.2) depends only on the S-map f : X → Y , not on any particular compactiﬁcation. Then it is a simple exercise to check via (4.8.4.1) that these isomorphisms, for variable f, constitute an isomorphism of pseudofunctors, giving uniqueness of ! (up to a pseudofunctorial isomorphism—itself unique if we require compatibility with αP and αE ). For comparing (4.8.4.2) relative to various compactiﬁcations of f, is f¯s (is , f¯s ) := X − →Y , → Xs −

let [(i1 , f¯1 ), (i2 , f¯2 )] be the natural composite isomorphism × × ∼ ! ! ∼ ! ! ∼ ∼ i∗2f¯2 −→ i∗1f¯1 . f ! −→ i1 f¯1 −→ i2 f¯2 −→

13

Where there are a few minor misprints (for example, (3.2.4.∗) should be (3.2.5.∗)), and omissions of symbols.

4.8 Two Fundamental Theorems

211

Noting that the compactiﬁcations of f are the objects of a category C in which a morphism (i1 , f¯1 ) → (i2 , f¯2 ) is a P-map g : X1 → X2 such that gi1 = i2 and f¯2 g = f¯1 , one shows the following identity, transitivity and normalization properties (sketch the diagrams!): (i) [(i1 , f¯1 ), (i1 , f¯1 )] = identity. (ii) [(i1 , f¯1 ), (i2 , f¯2 )]◦[(i2 , f¯2 ), (i3 , f¯3 )] = [(i1 , f¯1 ), (i3 , f¯3 )]. (iii) For any g : (i1 , f¯1 ) → (i2 , f¯2 ), and σ := σi1, g,1,i2 , the isomorphism × × ∼ × ∼ [(i2 , f¯2 ), (i1 , f¯1 )] factors naturally as i∗1f¯1 −→ i∗2 f¯2 . i∗1g ×f¯2 −→ βσ

Making use of condition (4.8.4)(a), Deligne shows in [De, p. 308, 3.2.6(ii)] that the opposite category C op is ﬁltered (see [M, p. 211]).14 It follows that the independence veriﬁcation for (4.8.4.2) need only be done for a pair of compactiﬁcations of which one maps to the other. This is now a straightforward exercise, using isomorphisms of the form [(i1 , f¯1 ), (i2 , f¯2 )]. To prove existence of ! Deligne constructs, for each map f , a family of × ∼ × functorial isomorphisms [(i1 , f¯1 ), (i2 , f¯2 )] : i∗2f¯2 −→ i∗1f¯1 , indexed by pairs of compactiﬁcations of f, and satisfying (i)–(iii) [De, p. 313, 3.3.2.1]. (There is a pretty obvious such isomorphism when (i1 , f¯1 ) maps to (i2 , f¯2 ); and the rest follows from the fact that C op is ﬁltered.) He then makes an arbitrary choice of a compactiﬁcation f = f¯i, and sets f ! := i∗f¯×. Thus for any compactiﬁcation f = f¯• i• one has an isomorphism × ∼ [(i• , f¯• ), (i, f¯)] : f ! = i∗f¯× −→ i∗•f¯• .

(4.8.4.3)

∼ For f ∈ E, taking f¯• = 1, i• = f , one gets f ! −→ f ∗, giving αE at the functorial—but not yet the pseudofunctorial—level. Analogous remarks lead to αP . Substituting isomorphisms as in (4.8.4.3) at each of the three appropriate ∼ places in (4.8.4.1), one gets a deﬁnition of df1,f2 : f2! f1! −→ (f1 f2 )! , provided it is ﬁrst shown that the result of this substitution does not depend on the choice of g¯ and j. As before, since C op is ﬁltered it suﬃces to show that (4.8.4.1) (as here modiﬁed) is unaltered by the substitution for (j, g¯) ¯ with of a compactiﬁcation (j1 , g¯1 ) of i1 f¯2 such that there exists a P-map h ¯ ¯ j = j1 h and g¯h = g¯1 . This is done in [De, pp. 314–316]. Finally, a brief check [De, p. 317, 3.3.2.4] ensures that this d endows ! , αP and αE with all the desired pseudofunctorial properties. The last assertion in (4.8.4) simply reﬂects the possibility in the above deﬁnition of ! of making the obvious choice f¯ = 1, i = f whenever f ∈ E. Q.E.D.

The proof of (4.8.3) will be based on the following pasting result for basechange setups.15 14 In that proof take K to be the inverse image of the diagonal under the map (r, s) : Y 1 → Y 2 ×X Y 2 . 15 This result should be compared with [Nk, p. 205, Thm. 2.3.2].

212

4 Abstract Grothendieck Duality for Schemes

Proposition 4.8.5 With notation and assumptions as in (4.8.4), let S be a category containing S as a subcategory. Let B ′ := B S, E, F, *, # , (βσ′ )σ∈′ , B ′′ := B S, P, F, × , # , (βσ′′ )σ∈′′ ,

be base-change setups with ′ (resp. ′′ ) the class of S-ﬁber squares σv,f, g,u such that f , g ∈ E (resp. P) and u, v ∈ F. Assume that for any f ∈ E (resp. P) and u ∈ F, such a σv,f, g,u exists. Then there is at most one base-change setup # B := B S, S, F, !, , (β¯σ )σ∈

which extends—in the obvious sense, via αP and αE —both B ′ and B ′′, and with the class of S-ﬁber squares σv,f, g,u such that f , g ∈ S and u, v ∈ F. Such a B exists if and only if for any S-cube with i, i1 , j, j1 ∈ E, f , f1 , g, g1 ∈ P, and u, u1 , v, v1 ∈ F, and in which all the faces are distinguished (for the appropriate one of B, B ′, or B ′′ ) : v1

•

•

j1

j v

•

• f1

g1

g

f

•

•

u1 i

i1

•

u

•

the following diagram commutes: β′

β ′′

v1# j ∗f × −−−−→ j1∗ v #f × −−−−→ j1∗ g × u# ⏐ ⏐ ⏐β ⏐ β

(4.8.5.1)

∗ → g1× i∗1 u# v1# f1× i∗ −−−′′−→ g1× u# 1 i −−−− ′ β

β

Remark 4.8.5.2 The existence part of Theorem (4.8.3), weakened by substituting for the class of ﬁber squares σv,f, g,u with u, v ﬂat and f , g ﬁnitely presentable, and by leaving aside conditions (4.8.3)(iii), results from an application of (4.8.5) to the following base-change setups B ′ and B ′′. For B ′, let S be the category of separated maps of noetherian schemes; F the subcategory of ﬂat maps, with # = *, the usual D+ qc -valued inverseimage pseudofunctor; E ⊂ F the subcategory of ´etale maps, with the same

4.8 Two Fundamental Theorems

213

inverse-image pseudofunctor *; ′ the class of all S-ﬁber squares σv,f, g,u with ∼ g ∗ u∗ the natural isomorphism. (This f , g ´etale and u, v ﬂat; and βσ : v ∗f ∗ −→ ∗ is just a “subsetup” of B(S, L− ), see (4.8.2.2).) For B ′′, let S and (F,# ) be the same as for B ′; let P be the subcategory × (see (4.1.2)); ′′ the of proper maps, with the D+ qc -valued pseudofunctor class of those S-ﬁber squares σv,f, g,u with f and g proper, u and v ﬂat; and βσ (σ ∈ ′ ) the base-change isomorphism from (4.4.3). In this situation, commutativity of (4.8.5.1) is easily checked, via “horizontal transitivity” in Example (4.8.2.3). In (4.8.6)–(4.8.11), the resulting base-change setup B will be extended to where consists of all admissible S-squares. Proof of (4.8.5). Fiber products being unique up to isomorphism, it follows from (4.8.2.1)(a) and the assumption in (4.8.5) that any S-ﬁber square σv,f, g,u with f ∈ E (resp. P) and u ∈ F is in ′ (resp. ′′ ). It is then straightforward to see via (4.8.4)(b) that any σ ∈ is a vertical composite σ2 ◦σ1 with σ1 ∈ ′ and σ2 ∈ ′′ : v

σ =

• −−−−→ ⏐ ⏐ σ1 j w

• −−−−→ ⏐ ⏐ σ2 g ¯

• ⏐ ⏐ i

• ⏐ ⏐¯ f

(4.8.5.3)

• −−− −→ •, u

and to check that if B exists then β¯σ has to be the natural composition ! ∼ × ∼ × ∼ × ∼ v #(f¯i)! −→ v #i!f¯ −→ v #i!f¯ −→ v #i∗f¯ −→ j ∗w#f¯ α α ′ P

E

β

∼ ∼ ∼ ∼ −→ j ∗ g¯× u# −→ j ! g¯× u# −→ j ! g¯! u# −→ (¯ g j)! u#, −1 −1 ′′ β

αE

αP

whence the uniqueness of B (if it exists). Expanding the two instances of β in (4.8.5.1) according to the description of βσ in (4.8.4), one ﬁnds then that (4.8.5.1) commutes. (The commutativity amounts to two ways ∼ of expanding β¯ : v1# (f j)! = v1# (if1 )! −→ (gj1 )! u# = (i1 g1 )! u# according to vertical transitivity (4.8.2)(3).) To prove the existence of B, we ﬁrst show that the above expression for β¯σ depends only on σ. whose objects are F-maps, the For this purpose, consider the category S ′ morphisms from an F-map v : X → X to an F-map u : Y ′ → Y being the ﬁbre squares σv,f, g,u ∈ , with the obvious deﬁnition of composition. ⊂S (resp. P ⊂ S) to be the one having the same Deﬁne the subcategory E but with morphisms σv,f, g,u ∈ such that f , g ∈ E (resp. P). objects as S,

214

4 Abstract Grothendieck Duality for Schemes

The above decomposition σ = σ2 ◦ σ1 signiﬁes that every S-morphism has an (E, P)-compactiﬁcation, i.e., it factors as an E-morphism followed by a P-morphism. ⊂ S. It is left as an exercise to deduce from (4.8.4)(a) its analogue for P It follows then, as in the proof of (4.8.4), that it will be enough to show that two diﬀerent compactiﬁcations of σ ∈ give the same β¯σ when one of them maps to the other, via P—cf. the deﬁnition of morphisms of compactiﬁcations which appears in the proof of (4.8.4). Let the target compactiﬁcation be given by factorizations f = f¯i, g = g¯j (see (4.8.5.3)); let the source compactiﬁcation be given similarly by factorizations f = f¯1 i1 , g = g¯1 j1 . Then the map of compactiﬁcations is given by P-maps p and q ﬁtting into commutative cubes (with a common face), whose horizontal arrows are F-maps: v

•

•

j1

•

•

1

1

q

1

•

v

•

•

•

w f¯

g ¯

i

w

p

1

j

•

•

q

p

•

¯ f 1 u

•

1

•

g ¯1

i1 w

w1

•

•

•

u

The ﬁrst cube entails, via (4.8.5.1), a commutative diagram β′

β ′′

v # i∗1 p× −−−−→ j1∗w1#p× −−−−→ j1∗ q × w# ⏐ ⏐ ⏐β ⏐ β v # i∗

v # i∗

−−−− → ′ β

(4.8.5.4)

j ∗w#

Vertical transitivity (4.8.2)(3) for the setup B S, P, F, × , # , (βσ′′ )σ∈′′ , applied to the composite diagram consisting of the rear and bottom faces of the second cube, yields a commutative diagram g¯1× u# ⏐ ⏐

−−−−−−−−−−−−−−−−→

×

q × g¯× u# ←−−−− q × w#f¯

×

w1#f¯1 ⏐ ⏐

(4.8.5.5) ×

←−−−− w1# p×f¯

4.8 Two Fundamental Theorems

215

Now, by the deﬁnition of β¯σ with respect to a given compactiﬁcation, the present problem is to show commutativity of the outer border of the following diagram, in which the maps are the obvious isomorphisms. (Recall that i◦ 1 = i = pi1 , f¯1 = f¯p, j ◦1 = j = qj1 , wq = pw1 and g¯1 = g¯q.) v #f ! ⏐ ⏐

!

v # i!f¯ ⏐ ⏐

1

×

v # i∗f¯ ⏐ ⏐

−−−−−−−−−−−−−−−−−−→

×

j ∗ w#f¯ ⏐ ⏐

!

v # i!1 p!f¯ ⏐ ⏐

×

−−−−−−−−−−−−−−−−−−→ v # i∗1 p×f¯ ⏐ ⏐ 3 ×

−−−−→ j1∗ q × w#f¯ ⏐ ⏐

×

−−−−→ j1∗ w1# p×f¯ 4

v #f ! ⏐ ⏐

! −−−−→ v # i!1f¯1 ⏐ ⏐ 2

×

−−−−→ v # i∗1f¯1 ⏐ ⏐

×

−−−−→ j1∗ w1#f¯1 ⏐ ⏐

j ∗ g¯× u# −−−−→ j1∗ q × g¯× u# −−−−−−−−−−−−−−−−−−→ j1∗g¯1× u# ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 5 j ! g¯! u# −−−−→ ⏐ ⏐ g ! u#

j1! q ! g¯! u#

−−−−−−−−−−−−−−−−−−→

j1! g¯1! u# ⏐ ⏐ g ! u#

Subdiagram 1 commutes by (4.8.4) (for v := i1 , f := p, u := i and g := 1), 3 by (4.8.5.4), and 4 by (4.8.5.5). Subdiagrams 2 and 5 commute because the isomorphism αP is pseudofunctorial. Commutativity of the remaining subdiagrams is clear. Thus the entire diagram does commute, and so β¯σ depends only on σ. It remains to check conditions (1)–(3) in (4.8.2), of which only “vertical transitivity for β¯σ” is not straightforward enough to be left to the reader. So we need to consider a commutative diagram, with f¯t , g¯t ∈ P and it , jt ∈ E (t = 1, 2), w, x, y, z, u ∈ F, and in which all the squares are ﬁber squares:

216

4 Abstract Grothendieck Duality for Schemes w

• −−−−→ ⏐ ⏐ j1 x

X ′ −−−−→ ⏐ ⏐ g1 y

• −−−−→ ⏐ ⏐ j2 z

Z ′ −−−−→ ⏐ ⏐ g2 u

• ⏐ ⏐i 1

X ⏐ ⏐f 1 • ⏐ ⏐i 2

Z ⏐ ⏐f 2

• −−−−→ •

Let i2 f1 = f i with f : Y → Z ∈ P and i : X → Y ∈ E. Let g : Z ′ ×Z Y → Z ′ and v : Z ′ ×Z Y → Y be the projections, so that g ∈ P and v ∈ F. Then there is a unique E-map j : X ′ → Z ′ ×Z Y such that gj = j2 g1 and vj = ix. One sees then that in the cube x

X′

X

j i v

′

Z ×Z Y

Y

g1

f1

g

f

•

•

y

j2

Z′

i2

z

Z

the top and bottom faces are B ′ -distinguished, the front and back faces are B ′′ -distinguished, and the other two faces are B-distinguished.

4.8 Two Fundamental Theorems

217

Now vertical transitivity amounts to commutativity of the diagram w# (ii1 )∗ (f2 f )× −−−−→ w# i∗1 i∗f ×f2× −−−−→ w# i∗1 f1× i∗2 f2× ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 j1∗ x# i∗f ×f2× −−−−→ j1∗ x#f1× i∗2 f2× ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (jj1 )∗ v # (f2 f )× −−−−→ j1∗ j ∗ v #f ×f2× 3 j1∗ g1× y # i∗2 f2× ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 j1∗ j ∗ g × z #f2× −−−−→ j1∗ g1× j2∗ z #f2× ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

(jj1 )∗ (g2 g)× u# −−−−→ j1∗ j ∗ g × g2× u# −−−−→ j1∗ g1× j2∗ g2× u#

Subsquares 1 and 2 commute by vertical transitivity for B ′′. Commutativity of 3 is the instance of (4.8.5.1) corresponding to the preceding cube. Commutativity of the remaining two subsquares is obvious. This completes the proof of Proposition (4.8.5). Q.E.D. As previously noted, to ﬁnish the proof of (4.8.1) we need to enlarge the setup (4.8.2.4) to (4.8.2.4)′. Similarly, to ﬁnish the proof of (4.8.3) we need of the setup B at the end to show that there exists a unique enlargement B of (4.8.5.2) such that all admissible S-squares are B-distinguished. In addi tion, we need to check that (4.8.3)(ii) and (iii) hold for this B. All this will be done in (4.8.11), after the supporting formal details are developed in (4.8.6)–(4.8.10). Deﬁnition 4.8.6 For a base-change setup B S, P, F, !, *, (βσ )σ∈ a subcategory E ⊂ S is special if for any maps i : X → Y in E, g : X ′ → X in P, and v : X ′ → X in F, the squares 1

X ′ −−−−→ ⏐ ⏐ g

X′ ⏐ ⏐ig

X −−−−→ Y i

v

X ′ −−−−→ ⏐ ⏐ 1

X ⏐ ⏐ i

X ′ −−−−→ Y iv

are distinguished. Remarks 4.8.6.1 (a) If E is special then E ⊂ P ∩ F. (b) If E is special for B, then E is also special for the dual of B (see (4.8.2.1)(c)).

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4 Abstract Grothendieck Duality for Schemes

Example 4.8.6.2 For (4.8.2.4), or for B ′, B ′′ or B in (4.8.5.2), the category E whose maps are all the open-and-closed immersions of noetherian schemes is special. Indeed, since i is a monomorphism, both squares in (4.8.6) are ﬁber squares. After ﬁxing a special subcategory E, we will call its maps special. For any special map i : X → Y , ∼ βi : i! −→ i∗

(4.8.7.0)

is deﬁned to be the isomorphism βτ associated to the distinguished square 1

X −−−−→ ⏐ ⏐ 1 τ

X ⏐ ⏐ i

X −−−−→ Y i

Proposition 4.8.7 Let B S, P, F, !, *, (βσ )σ∈ be a base-change setup and E a special subcategory. Then the restrictions of the pseudofunctors ! and * to E are naturally isomorphic. Proof. The family of isomorphisms βi (i ∈ E) of (4.8.7.0) is pseudofunctorial (see (3.6.6)): if i : X → Y and j : Y → Z are in E, apply (3) and (2) of (4.8.2) to 1 1 X −−−−→ X −−−−→ X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ i i ⏐ ⏐ 1 1⏐ Y −−−−→ Y ⏐ ⏐ ⏐ ⏐ ⏐j ⏐ ⏐ 1 X −−−−→ Y −−−−→ Z i

j

to see that the left and right halves of the following diagram commute: (ji)! ⏐ ⏐ ≃

βji

(ji)! −−−−→ (ji)∗ ⏐ ⏐ ⏐≃ ⏐ ≃

i! j ! −−− −→ i! j ∗ −−−−→ i∗ j ∗ ! i βj

βi

Q.E.D. Proposition 4.8.8 Let B S, P, F, !, *, (βσ )σ∈ be a base-change setup, E a special subcategory, and βi (i ∈ E) as in (4.8.7.0). Then :

4.8 Two Fundamental Theorems

219

(i) For each distinguished square v

• −−−−→ ⏐ ⏐ g σ

• ⏐ ⏐f

• −−− −→ • u

with f and g in E, the following diagram commutes : βσ

v ∗f ! −−−−−−−−−−−−−−−−−−−−−−→ g ! u∗ ⏐ ⏐ ⏐β ⏐ v∗βf g v ∗f ∗ −− −−→ (f v)∗ = (ug)∗ ←− −−− g ∗ u∗

(ii) For each distinguished square

v

• −−−−→ ⏐ ⏐ g σ

• ⏐ ⏐f

• −−− −→ • u

with u and v in E, the following diagram commutes : βσ

v ∗f ! −−−−−−−−−−−−−−−−−−−−−−→ g ! u∗ ⏐ ! ⏐ βv ⏐ ⏐g βu −−→ g ! u! −−− (f v)! = (ug)! −− v ! f ! ←−

Proof. Deﬁnition (4.8.6) shows that the following composite square ρ is distinguished, as are its constituents: 1

v

• −−−−→ • −−−−→ ⏐ ⏐ ⏐ ⏐ g σ 1

• ⏐ ⏐f

• −−−g−→ • −−− −→ • u

so horizontal transitivity (4.8.2)(3) gives a commutative diagram βσ

v ∗f ! −−−−→ g ! u∗ ⏐ ⏐ ⏐β ⏐ βρ g

(ug)∗ ←− −−− g ∗ u∗

(4.8.8.1)

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4 Abstract Grothendieck Duality for Schemes

Also, the following decomposition of ρ v

• −−−−→ ⏐ ⏐ 1

1

• −−−−→ ⏐ ⏐ 1

• ⏐ ⏐f

• −−−v−→ • −−−−→ • f

yields—via (2)′ and (3) of (4.8.2)—the commutative diagram v ∗f ! ⏐ ⏐β ρ

v ∗f ! ⏐ ⏐ v∗βf

v ∗f ∗ −− −−→ (f v)∗

(4.8.8.2)

(ug)∗

Pasting (4.8.8.1) and (4.8.8.2) along their common edge, we get (i). Assertion (ii) is just (i) for the dual setup (see (4.8.2.1)(c)). Q.E.D. (4.8.9) We will now see how to enlarge certain base-change setups. Consider a category S in which for any maps X → Y and Y ′ → Y a ﬁber product X ×Y Y ′ exists. A square σv,f, g,u in S: v

X ′ −−−−→ ⏐ ⏐ g Y

′

X ⏐ ⏐f

(4.8.9.1)

−−− −→ Y u

is, as usual, called a ﬁber square if the corresponding map X ′ → X ×Y Y ′ is an isomorphism. Let B := B S, P, F, !, *, (βσ )σ∈ be a base-change setup, and E a special subcategory, see (4.8.6). We make the following assumptions, in addition to those in (4.8.2). (4) In the following S-diagrams, suppose that u1 ∈ F (resp. f1 ∈ P). w

v1

v2

• −−−−→ • −−−−→ ⏐ ⏐ ⏐ ⏐ h σ1 g σ2

• ⏐ ⏐f

• −−− −→ • −−− −→ • u1 u2

• −−−−→ ⏐ ⏐ σ1 g1 v

• −−−−→ ⏐ ⏐ g2 σ2

• ⏐ ⏐f 1

• ⏐ ⏐f 2

• −−− −→ • u

In either diagram, if σ2 is a ﬁber square and the composed square σ2 σ1 is in , then σ1 ∈ .

4.8 Two Fundamental Theorems

221

(5) For any ﬁber square (4.8.9.1) in , if u (resp. f ) is special (i.e., lies in E) then so is v (resp. g). (6) If the square (4.8.9.1) is in then so is any ﬁber square with the same u and f, X ′′ −−−−→ ⏐ ⏐

X ⏐ ⏐f

Y ′ −−− −→ Y u

and furthermore, the resulting map X ′ → X ′′ is special. Example 4.8.9.2 Conditions (4)–(6) are easily seen to be satisﬁed in any of the situations in Example (4.8.6.2), where all distinguished squares are ﬁber squares. Remark 4.8.9.3 Let μ : X ′ → X ′′ be an isomorphism and consider the following ﬁber squares, the ﬁrst of which is, by (4.8.2)(2), distinguished: 1

X ′′ −−−−→ ⏐ ⏐ 1

μ

X ′ −−−−→ ⏐ ⏐ μ

X ′′ ⏐ ⏐ 1

X ′′ ⏐ ⏐ 1

X ′′ −−−1−→ X ′′

X ′′ −−−1−→ X ′′

From (6) it follows that μ is special. Thus every isomorphism is special. Proposition 4.8.10 Under the preceding assumptions, there is a unique ′ base-change setup B ′ = BE = B S, P, F, !, *, (βσ′ )σ∈′ such that: (i) A commutative square v

X ′ −−−−→ ⏐ ⏐ g

X ⏐ ⏐f

Y ′ −−− −→ Y u

is in ′ if and only if there is a ﬁber square in X ′′ −−−−→ ⏐ ⏐

X ⏐ ⏐f

Y ′ −−− −→ Y u

such that the resulting map X ′ → X ′′ is special. So by (4.8.9)(6) and (4.8.9.3), ⊆ ′ ; and by (4.8.2)(1), every ﬁber square in ′ is in . (ii) For every σ ∈ ⊆ ′ it holds that βσ = βσ′ .

222

4 Abstract Grothendieck Duality for Schemes

Proof. For uniqueness, suppose that B ′ satisﬁes (i) (which determines ′ ) and (ii). We note ﬁrst that if i : X → Y is a special map, then by (i), the square τ ′ in the following diagram is in ′ , as are the squares τ (by (4.8.6)) and τ ′ τ (by (4.8.2)(2)′ ): 1

X −−−−→ ⏐ ⏐ 1 τ

i

X −−−−→ ⏐ ⏐ i τ ′

Y ⏐ ⏐ 1

X −−−−→ Y −−−−→ Y i

1

It follows then from (4.8.2)(3) and (4.8.2)(2)′ that (ii)

βτ′ ′ = (βτ′ )−1 = (βτ )−1 Now, any σ ∈ ′ :

v

X ′ −−−−→ ⏐ ⏐ g σ

(4.8.7.0)

=

(βi )−1 .

X ⏐ ⏐f

Y ′ −−− −→ Y u

can, according to (i), be decomposed as i

X ′ −−−−→ ⏐ ⏐ i σ1

X ′′ −−−−→ 1 ⏐ ⏐ h σ2

w

X ′′ −−−−→ ⏐ ⏐ 1 X ′′ ⏐ ⏐ h

σ3

X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐f ⏐ ⏐ ⏐

(4.8.10.1)

−→ Y Y ′ −−−1−→ Y ′ −−− u

with σ3 ∈ a ﬁber square (so that h ∈ P), and i special. The ﬁber square σ2 is in , by (4.8.2)(2); and by (i), σ1 and σ2 σ1 ∈ ′ . We saw above that βσ′1 = (βi )−1 ; and the maps βσ′k (k = 2, 3) are determined by (ii). Hence βσ′ 2 σ1 is determined, and then so is βσ′ (see (4.8.2)(3)). Thus B ′ is unique. For the existence, let ′ be the class of all squares v

X ′ −−−−→ ⏐ ⏐ g σ

X ⏐ ⏐f

Y ′ −−− −→ Y u

satisfying (i), that is, decomposing as in (4.8.10.1)—where i ∈ P ∩ F (see (4.8.6.1)), h ∈ P and w ∈ F, so that f , g ∈ P and u, v ∈ F, as required of distinguished squares.

4.8 Two Fundamental Theorems

223

To such a decomposition we associate the natural composite map ∼ ∼ ∼ i! h! u∗ −→ g ! u∗. v ∗f ! −→ i∗ w∗f ! −−∗−−→ i∗ h! u∗ −→ −1 i βσ3

(4.8.10.2)

βi

We will deﬁne βσ′ for B ′ to be (4.8), but ﬁrst we need to show it independent of the chosen decomposition. Suppose then that we have another decomposition with (X ′′, i, h, w) ∼ X1′′ replaced by (X1′′, i1 , h1 , w1 ), i.e., there is an isomorphism μ : X ′′ −→ such that i1 = μi, h1 = hμ−1 , w1 = wμ−1 . For the special map μ (see (4.8.9.3)), we have the isomorphism βμ of (4.8.7.0). We have also the isomorphism βρ associated to the square 1

X ′ −−−−→ ⏐ ⏐ i ρ

X′ ⏐ ⏐i 1

−→ X1′′ X ′′ −−− μ

which is in by (4.8.2.1)(a). We want to show that the following diagram of natural maps (with outside columns as in (4.8)) commutes: v ∗f ! ⏐ ⏐

v ∗f ! ⏐ ⏐

i∗ w∗f ! −−−−→ i∗μ∗ w1∗f ! −−−−−−−−−−−−−−−−−−→ i∗1 w1∗f ! ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ∗ ∗ ∗ ! ∗ ∗ ∗ ! ⏐ i μ h1 u i μ h1 u −−−−→ i1 h!1 u∗ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐i∗β −1 ⏐ ⏐ μ ⏐ ⏐ ⏐ ⏐ ⏐ βi−1 i∗ h! u∗ −−−−→ i∗μ! h!1 u∗ 2 βi−1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ −1 ⏐ ⏐ −1 ⏐ βi βi 1

i!βμ

βρ−1

i! h! u∗ −−−−→ i! μ! h!1 u∗ −−−−→ i! μ∗ h!1 u∗ −−−−→ i!1 h!1 u∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 3 g ! u∗

−−−−→ i!1 h!1 u∗

−−−−−−−−−−−−−−−−−−→

g ! u∗

Commutativity of 2 (resp. 3 ) follows from (4.8.8)(i) (resp. (4.8.8)(ii)) applied to ρ.

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4 Abstract Grothendieck Duality for Schemes

Commutativity of 1 follows from (4.8.2)(3) and (4.8.8)(ii), applied respectively to the following ﬁber squares σ3 = σ3′ σ ′ and σ ′ (σ ′ being distinguished, by (4.8.2.1)(a)): w1 μ X ′′ −−−−→ X1′′ −−−− → X ⏐ ⏐ ⏐ ⏐h ⏐ ⏐f σ′ h 1 σ3′ Y ′ −−−1−→ Y ′ −−− −→ Y u

Commutativity of the remaining subdiagrams is clear. So we can indeed deﬁne βσ′ as indicated above. Condition (i) in (4.8.10) is then obvious. As for (ii), referring to a decomposition (4.8.10.1) of σ ∈ (where wi = v and hi = g), note that by (4.8.9)(4) the square σ2 σ1 is in , so by (4.8.2)(3) the diagram v ∗f ! ⏐ ⏐ ≃

βσ

−−−−−−−−−−−−−−−−→

g ! u∗

i∗ w∗f ! −−∗−−→ i∗ h! u∗ −−−−→ g ! 1∗ u∗ βσ2 σ1

i βσ3

commutes. Also, (4.8.8)(ii) applied to σ2 σ1 shows that βσ2 σ1 factors as −→ g ! u∗ . −→ i! h! u∗ − i∗ h! u∗ −−−−1 βi

Hence the composite map (4.8) is equal to βσ , proving (ii). Having thus deﬁned B ′ , we are left with proving (1)–(3) in (4.8.2). For (1), assume, with notation as in (4.8.2), that σ1 ∈ ′ . Consider a commutative decomposition of σ j−1

v

1 • −−− −→ • −−−1−→ ⏐ ⏐ ⏐ ⏐ kj−1 k 1

• ⏐ ⏐ i1 h

w

σ′

1 • −−−− → ⏐ ⏐ τ h

j

• −−−−→ ⏐ ⏐ 1 j

• −−−−→ ⏐ ⏐f 1 σ ′′

• ⏐ ⏐ 1

• ⏐ ⏐f

→ • −−−−→ • −−−−→ • • −−−− −1 i1

u1

i

in which the middle third of the diagram is a decomposition of σ1 with τ ∈ a ﬁber square and k special, and v1 := w1 k; and the right third exists by assumption, σ ′′ being a ﬁber square because i and j are isomorphisms. (Note: i1hkj1−1 = i1 g1 j1−1 = g.) The composed ﬁber square σ ′′ τ σ ′, being isomorphic to τ , is in ; and thus, since kj1−1 is special (see (4.8.6.1)(a)), therefore σ ∈ ′ , proving (1).

4.8 Two Fundamental Theorems

225

Conditions (2) and (2)′ for B ′ follow from the same for B, because of (4.8.10)(ii). As for (3), consider a composite diagram σ0 = σ2 σ1: v

v

• −−−1−→ • −−−2−→ ⏐ ⏐ ⏐ ⏐ h σ1 g σ2

• ⏐ ⏐f

• −−− −→ • −−− −→ • u1 u2

with σ2 , σ1 and σ0 in ′. Using all the assumptions in (4.8.9), we ﬁnd that this decomposes further as • ⏐ v1 ⏐ j • −−−p−→ • ⏐ ⏐ ⏐ ⏐ h2 σ ′′ k

v2

(4.8.10.3)

−→ • −−−q−→ • −−− w ⏐ ⏐ ⏐ h1 τ σ ′ g1⏐

• ⏐ ⏐f

• −−− −→ • −−− −→ • u1 u2

where σ ′′, σ ′ and τ are ﬁber squares in ; the maps g1 , w, h1 , q, h2 , p are the natural projections; the maps j and k are special—whence so are h2 and h2 j (see (4.8.9)(5)); the triangles commute; g1 k = g and h1 h2 j = h. What (3) asserts is, ﬁrst, that the following natural diagram commutes: βσ0

(v2 v1 )∗f ! −−−−−−−−−−−−−−−−−−−−→ h! (u2 u1 )∗ ⏐ ⏐ ⏐ ⏐ v1∗ v2∗ f ! ⏐ ⏐ ≃

v1∗ g ! u∗2 ⏐≃ ⏐

−−−−→ βσ1

h! u∗1 u∗2

(4.8.10.4)

j ∗ p∗ v2∗ f ! −−−−→ j ∗ p∗ (g1 k)! u∗2 via βσ2

Expanding βσ2 , βσ1 , and βσ2 σ1 , as in (4.8), one sees that for this it is enough to show commutativity of the outer border of the natural diagram on the following page, or just to show that each of its twelve undecomposed subdiagrams commutes. But for the eight unlabeled subdiagrams, commutativity holds by 1 , one can elementary (pseudo)functorial considerations; for subdiagram use (4.8.7); for 2 and 4 , (4.8.2)(3); and for 3 , (4.8.8)(i). This completes the proof of the “horizontal” part of (3).

226

4 Abstract Grothendieck Duality for Schemes

The proof of the “vertical” part of (3) is similar. Alternatively, one can just dualize everything in sight, as indicated in (4.8.2.1)(c). The conditions in (4.8.6) deﬁning a special subcategory are self-dual, so that if E is special for a setup B, then E is also special for the dual setup B op . Likewise, conditions (4)–(6) in (4.8.9) hold for B iﬀ they hold for B op . Then, one checks, vertical transitivity for (B op )′ (constructed as above) is identical with the just-proved horizontal transitivity for B ′. This completes the proof of Proposition (4.8.10). Q.E.D. Corollary 4.8.10.5 With notation and assumptions as in (4.8.10), let E′ be a subcategory of S such that for every map i : X → Y ∈ E′ the diagonal map δi : X → X ×Y X is in E. Assume further that for any ﬁber square σv,f, g,u in S, if u (resp. f ) is in E′ then so is v (resp. g). Then : (i) E′ is B ′ -special; and conditions (4)-(6) in (4.8.9) hold for (B ′, E′ ). Thus it is meaningful to set B ′′ := (B ′ )′E′ . (ii) If a ﬁber square σ = σv,f, g,u with u ∈ E′ is in , then any commutative σv′,f, g′,u with v ′ ∈ E′ and g ′ ∈ P is B ′′ -distinguished. Proof. (i) The second diagram in (4.8.6)—call it σ—expands as v

1

• −−−−→ • −−−−→ ⏐ ⏐ ⏐ ⏐ 1 1

• ⏐ ⏐ i

• −−−v−→ • −−−−→ • i

which when i ∈ E′ can be further expanded in the form (4.8.10.3), with j = 1 and k ∈ E, whence (since σ ′′ ∈ ) h2 ∈ E, whence by (4.8.10)(i), σ ∈ ′. In a similar way, or by dualizing (see (4.8.6.1(b)), one ﬁnds that the ﬁrst diagram in (4.8.6) is in ′. For (4.8.9)(4), we can decompose, say, the horizontal σ2 σ1 of that condition as • ⏐ v1 j⏐ q

v

• −−−−→ • −−−2−→ ⏐ ⏐ ⏐ h1 σ ′ g⏐ σ2

• ⏐ ⏐f

• −−− −→ • −−− −→ • u1 u2

with j ∈ E, qj = v1 , h1 j = h, and σ2 , σ ′ ﬁber squares such that the ﬁber square σ2 σ ′ is in .

v1∗ v2∗ f ! ⏐ ⏐

j ∗ p∗ v2∗ f ! ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

∗ ∗ ∗

∗ !

−−−−→ (h2 j)∗ (wq)∗f ! −−−−→ (h2 j)∗ h!1 (u2 u1 )∗ −−−−→ (h2 j)! h!1 (u2 u1 )∗ −−−−→ (h1 h2 j)! (u2 u1 )∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1

j ∗ h∗2 (wq)∗f ! −−−−→ j ∗ h∗2 h!1 (u2 u1 )∗ −−−−→ j ! h!2 h!1 (u2 u1 )∗ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 2 −−−−→ j ∗ h∗2 q ∗ w∗f ! j ∗ h∗2 h!1 u∗1 u∗2 −−−−→ j ! h!2 h!1 u∗1 u∗2 ⏐ ⏐ ⏐ ⏐

j p k w f −−−−→

j ∗ h∗2 q ∗g1! u∗2 ⏐ ⏐ j ∗ p∗ k ∗g1! u∗2

−−−−→ 3

−−−−→

j ∗ h!2 q ∗g1! u∗2 ⏐ ⏐

−−−−→

j ∗ p∗ k !g1! u∗2

−−−−→

j ∗ p∗ (g1 k)! u∗2

−−−−→

⏐ ⏐

j ! h!2 q ∗g1! u∗2 ⏐ ⏐ j ! p∗ k !g1! u∗2 ⏐ ⏐

j ! p∗ (g1 k)! u∗2

(h1 h2 j)! u∗1 u∗2 ⏐ ⏐

4

4.8 Two Fundamental Theorems

(v2 v1 )∗f ! ⏐ ⏐

j ! h!2 h!1 u∗1 u∗2 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐

−−−−→ j ! (h1 h2 )! u∗1 u∗2

227

228

4 Abstract Grothendieck Duality for Schemes

It follows from (4.8.9)(4) for B that σ ′ ∈ , whence σ1 ∈ ′ , proving the horizontal part of (4.8.9)(4) for B ′. The vertical part is similar (or dual). Since any ﬁber square in ′ is in , (4.8.9)(5) is essentially the “further” assumption on E′ . Finally, (4.8.9)(6) for B ′ follows from (4.8.10)(i), (4.8.9)(6) for B, and (4.8.9.3). (ii) Consider a decomposition of σv′,f, g′,u Z ⏐ j⏐

v′ v

X ′ −−−−→ ⏐ ⏐ σ g

X ⏐ ⏐f

Y ′ −−− u−→ Y

with v ′ = vj. We need only show that j ∈ E′. With Γj the graph map of j and π2 : Z ×X X ′ → X ′ the projection, the map j factors as π

Γj

Z −−−−→ Z ×X X ′ −−−2−→ X ′ . The ﬁber square

π

Z ×X X ′ −−−2−→ ⏐ ⏐ π1 Z

X′ ⏐ ⏐v

−−−− → X ′ v

shows that π2 ∈ E′; and the ﬁber square Γj

Z −−−−→ Z ×X X ′ ⏐ ⏐ ⏐j×1 ′ ⏐ j X X ′ −−−−→ X ′ ×X X ′ δv

shows that Γj ∈ E′, whence the conclusion.

Q.E.D.

(4.8.11). Let us now complete the proof of (4.8.1) and (4.8.3) by doing what was indicated just before Deﬁnition (4.8.6). For either the setup B of (4.8.2.4) or the larger setup B of (4.8.5.2), the category E of open-and-closed immersions is special, see (4.8.6.2). The diagonal of a separated ´etale map is an open-and-closed immersion [EGA IV, (17.4.2)(b)]; and maps which are ´etale (resp. separated, resp. proper) remain so after arbitrary base change [EGA IV, (17.3.3)(iii)]. Therefore the category E′ of separated ´etale maps (resp. proper ´etale maps) satisﬁes the hypotheses of (4.8.10.5) with respect to (B, E) (resp. (B, E)). Keeping in mind the uniqueness part of (4.8.10), one see that the resulting

4.8 Two Fundamental Theorems

229

:= B′′ is the sought-after unique enlargement of B, and base-change setup B that B ′′ is the unique enlargement (4.8.2.4)′ of B. It remains to show that conditions (4.8.3)(ii) and (iii) hold for B. Using the deﬁnition (4.8) of βσ , one readily reduces the question to where σ is a ﬁber square. In that case, (ii) follows from the description of B ′ in (4.8.5.2). As for (iii), let f = f¯i be a compactiﬁcation, and apply vertical transitivity (4.8.2)(3), to reduce to where either f = i is an open immersion, a case covered by (ii), or f = f¯ is proper, a case covered by (4.8.1)(iii). Q.E.D. Exercises 4.8.12 (a) Let B S, P, F, !, *, (βσ )σ∈ be a base-change setup, and let ∼ × * ∼ # there be given pseudofunctorial isomorphisms ! −→ , −→ . For any σv,f, g,u ∈ ¯ let βσ be the natural composite isomorphism ∼ ! ∗ ∼ × # v #f × −→ v ∗f ! −−− σ−→ g u −→ g u . β Show that B S, P, F, × , #, (β¯σ )σ∈ is a base-change setup. (b) (generalizing (4.1.9)(c)). Notation is as in (4.8.2.4). For a ﬁnite ´etale scheme∼ map f : X → Y , the natural map is an isomorphism f∗ −→ Rf∗ of functors from Dqc (X) to Dqc (Y ), see proof of (3.10.2.2). Deﬁne the functorial “trace” map E ∈ Dqc (Y ) f∗ f ∗E ∼ =E = f∗ OX ⊗ E → OY ⊗ E ∼

(3.9.4)

to be trf ⊗ 1 where trf is the natural composition

f∗ OX −→ Hom• (f∗ OX , f∗ OX ) ∼ = Hom• (f∗ OX , OY ) ⊗ f∗ OX −→ OY , given locally by the usual linear-algebra trace map. (Note that, f being ﬂat and ﬁnitely presented, f∗ f ∗ OY is a locally free OY -module.) There corresponds a functorial map tf : f ∗ → f × . (i) Show that on ﬁnite ´etale maps, the map t(−) : (−)∗ → (−)× is pseudofunctorial, see (3.6.6). (Reduction to the aﬃne case may help.) Also, tidentity = identity. (ii) (Compatibility of trace with base change.) Given a ﬁber square σ = σv,f, g,u with f and g ﬁnite ´etale, u and v ﬂat, show that the following diagram commutes: u∗ trf

u∗f∗ f ∗ −−−−→ ⏐ ⏐ (3.7.2)≃

u∗ ⏐tr ⏐ g

g∗ v ∗f ∗ −natural −− −−→ g∗ g ∗ u∗

(iii) For σ as in (ii), show that the following diagram commutes: natural

v ∗f ∗ −−−−−→ g ∗ u∗ ⏐ ⏐ ⏐ ⏐t v∗tf g

v ∗f × −−− −→ g×u∗ β σ

(Commutativity of the adjoint diagram is a consequence of (ii).)

230

4 Abstract Grothendieck Duality for Schemes

(iv) For any ﬁnite ´etale f show, using, e.g., (i), (iii), and (4.8), that with ∼ βf : f × −→ f ∗ (see (4.8.7.0)) as in the base-change setup (4.8.2.4)′, βf tf is the identity (whence tf is an isomorphism—which can also be proved more directly). (v) Deduce from (iv) that when ! is constructed as in the proof of (4.8.1), via application of (4.8.4) to (4.8.2.4)′, then the canonical map f∗ f ∗ = f∗ f ! → 1 (arising from right-adjointness of f ! to f∗ ) is just the trace map. (vi) For any ﬁnite ´etale f : X → Y , and E, F ∈ Dqc (X), show, via (v) or otherwise, ∼ f ∗ (E ⊗ F ) that the map χE,F of (4.7.3.4) is just the isomorphism f ∗ E ⊗ f ∗ F −→ of (3.2.4). (vii) Suppose that on the category E of ﬁnite ´ etale maps of noetherian schemes there is associated to each f : X → Y a functorial map τf : f∗ f ∗ → 1 in such a way that the pairs (f ∗, τf ) (f ∈ E) form a pseudofunctorial right adjoint to the Dqc -valued direct image pseudofunctor, and such that furthermore, the diagram in (ii) above still commutes when trf is replaced by τf . Prove that τf = trf for all f. Deduce that (v) holds for any f ! satisfying (4.8.1). Hint. Show that τf = trf ◦ θf for some automorphism θf of the functor f ∗, i.e., θf = multiplication by ef for some unit ef ∈ H0 (X, OX ). Then check that f g pseudofunctoriality implies, for any composition X − →Y − → Z, that egf = eg (g ∗ ef ); ∗ and check that for any σ as in (ii), eg = v ef . Then deduce from (iii), mutatis mutandis, that for any open-and closed immersion δ, eδ = 1; and ﬁnally, from the diagram π

δ

2 X −−−−→ X ×Y X −−−−→ ⏐ ⏐ π1

X

X ⏐ ⏐f

−−− −→ Y f

(δ := diagonal), that ef = 1 for all f. (c) Show that a horizontal or vertical composite of admissible squares is admissible. (d) Adapt the arguments in §4.11 to extend [Nk, p. 268, Thm. 7.3.2]—which avoids noetherian hypotheses—to where s can be any admissible square σv,f, g,u with f and g composites of ﬁnitely-presentable proper ﬂat maps and ´etale maps. (Recall that ﬁnitely-presentable ﬂat maps are pseudo-coherent (4.3.1).)

4.9 Perfect Maps of Noetherian Schemes In this section all schemes are assumed noetherian and all scheme-maps ﬁnite-type and separated. The abbreviations introduced at the beginning of §4.4 will be used throughout. We will associate to any such scheme-map f : X → Y a canonical bifunctorial map, with f ! as in (4.8.1), and both E and E ⊗ F in D+ qc (Y ), f χE,F : f !E ⊗ f ∗F → f ! (E ⊗ F ),

agreeing with the map χE,F in (4.7.3.4) when f is proper,. and with the inverse of the isomorphism in (3.2.4) when f is ´etale.

4.9 Perfect Maps of Noetherian Schemes

231

Any functorial relation involving (−)! ought to be examined with regard to pseudofunctoriality and base change (cf., e.g., (4.2.3)(h)–(j)). For χ, this is done in Corollary (4.9.5) and Exercise (4.9.3)(c). The main result, Theorem (4.9.4), inspired by [V′, p. 396, Lemma 1 and Corollary 2], gives several criteria for f to be perfect (i.e., since f is pseudocoherent, to have ﬁnite tor-dimension). Included there is the implication f f perfect =⇒ χE,F an isomorphism. ′ In [Nk , Theorem 5.9] Nayak extends these results to separated maps that are only essentially of ﬁnite type. f

u

→ Y , u an open immersion, f proper, (4.9.1). For scheme-maps X − →X− we deﬁne the bifunctorial map !

∗

!

f : f E ⊗ f F −→ f (E ⊗ F ) χE,F

E, F ∈ Dqc (Y )

to be the map adjoint to the natural composite map !

∗

!

∼ f∗ (f E ⊗ f F ) −→ f∗ f E ⊗ F −→ E ⊗ F , (3.9.4)

and we deﬁne the bifunctorial map !

!

f,u : u∗f E ⊗ f ∗F −→ u∗f (E ⊗ F ) χE,F

to be the natural composite map !

!

∗

!

E, F ∈ Dqc (Y )

∗

f u∗χE,F

!

∼ ∼ u∗f E ⊗ f ∗F −→ u∗f E ⊗ u∗f F −→ u∗(f E ⊗ f F ) −−−−→ u∗f (E ⊗ F ). ! ∗ ! When E and E ⊗ F are in D+ qc (Y ), setting f := f u we can write f for u f . f,u depends only on f , not In that case, we’ll see below, in (4.9.2.2), that χE,F f,u by on the factorization f = f u, so we can denote the map χE,F f χE,F : f !E ⊗ f ∗F → f ! (E ⊗ F ).

(4.9.1.1)

In this connection, recall that by Nagata’s compactiﬁcation theorem, any (ﬁnite-type separated) scheme-map f factors as f = f u. Lemma 4.9.2 Let there be given a commutative diagram f

g

X −−−−−−−−−−−−→ Y −−−−−−−−−−−−−→ Z u

v

f

g ¯

X

Y ¯ h

w

X ¯ proper. with u, v and w open immersions, f , g¯ and h

232

4 Abstract Grothendieck Duality for Schemes

Then for all E, F ∈ D(Z) such that E and E ⊗ F are in D+ qc (Z), the following natural diagram commutes. ¯

(gf )! E ⊗ (gf )∗F ⏐ ⏐ ≃ f ! g ! E ⊗ f ∗g ∗F

g ¯h,wu χE,F

−−−−−−−−−−−−−−−−−−−−−−−−−→ (gf )! (E ⊗ F ) ⏐ ⏐≃ !

−−f,u −−−→ u∗f (g ! E ⊗ g ∗F ) −−−−−−→ f ! g ! (E ⊗ F ) !

g ¯,v u∗f χE,F

χ

g !E,g ∗F

¯ := g¯!E, F¯ := g¯∗F (so that v ∗E ¯ ∼ Proof (Sketch). Set E = g !E and ∗ ∼ v F = g F ). Let β be the natural composite functorial isomorphism ∗¯

!

∼ ∼ ¯ ! = (vf )! −→ ¯ ! −→ f v ∗. (hw) w∗ h

(4.9.2.1)

Straightforward—if a bit tedious—considerations, using the deﬁnitions of the maps involved (see, e.g., (4.8.4)), translate Lemma (4.9.2) into commutativity of the natural diagram ¯ g ¯h (wu)∗ χE,F ¯ ! (E ⊗ F ) ¯ ! E ⊗ (¯ ¯ ∗F −−−−−−−−−−−−− −−−−−−−→ (wu)∗ (¯ g h) (wu)∗ (¯ g h) g h) ⏐ ⏐ ⏐≃ ⏐ 1 ≃ ¯ g ¯

¯ ∗ F¯ ¯ ⊗ u∗ w ∗ h u w hE ⏐ ⏐ via β ≃ ∗

∗¯!

!

∗

¯ ⊗ u∗f v ∗F¯ u∗f v ∗E

h u∗w∗χE, ¯ F ¯

¯ !χ u∗w∗h E,F

∗¯!

¯ ⊗ F¯ ) −−→ −−→ u w h (E ⏐ ⏐ via β ≃ 3 2 ∗

! ¯ ⊗ F¯ ) −−→ −−−→ u∗f v ∗(E

! g ¯ u∗f v∗χE,F

u∗ χvf∗E,v ¯ ¯ ∗F

¯ ! g¯! (E ⊗ F ) u∗ w ∗ h ⏐ ⏐ ≃via β !

u∗f v ∗ g¯! (E ⊗ F ),

in which, commutativity of subdiagram 3 is obvious. Commutativity of subdiagram 1 follows from “transitivity” of χ with respect to proper maps (Exercise (4.7.3.4)(d)). As for the remaining subdiagram 2 , decomposing σw, h, ¯ f,v ¯ as w

i

1 X −−−−→ Y × Y X −−−− → ⏐ ⏐ f1 σ

Y

X ⏐ ⏐¯ h

−−−v−→ Y

with w1 i = w, f1 i = f¯, and σ an independent ﬁber square (since v is ﬂat), we see from (4.8) that β factors naturally as !

∼ ∼ ¯ ! −−−−→ i∗ f ! v ∗ −−−−→ i! f ! v ∗ −→ ¯ ! −→ i∗ w1∗ h w∗ h f v ∗. 1 1 −1 βσ

βi

4.9 Perfect Maps of Noetherian Schemes

233

Here i is an open and closed immersion, so that by (4.8.4), i! = i∗ and the map βi (see (4.8.7.0)) is the identity. Indeed, since if1 and f1 are both proper, therefore so is i [EGA, II, (5.4.3)(i)]; and since iw1 and w1 are both open immersions, therefore so is i (cf. (4.8.3.1)(a)). It is left now to the reader to expand β as above and then to verify, with the aid of (4.7.3.4)(c) and (d), and of Exercise (4.8.12)(b)(vi) for open-and-closed immersions, that 2 does commute. Q.E.D. Corollary 4.9.2.2 If a map f : X → Z factors in two ways as u

f

X −−−−→ Y −−−−→ Z,

g ¯

v

X −−−−→ Y −−−−→ Z

(f and g¯ proper, u and v open immersions) then for all E, F as in (4.9.2), it f,u g ¯,v holds that χE,F = χE,F . Proof. The given data determine uniquely a map w ¯ : X → Y ×Z Y , whose schematic image we denote by X, see [GD, p. 324, (6.10.1) and p. 325, (6.10.5)]. The map w ¯ factors as X → X ×Z X → Y ×Z Y , where the ﬁrst map is the diagonal, a closed immersion, and the second is an open immersion. So w ¯ is an immersion, and hence induces an open immersion w : X → X. Furthermore, the projections to Y and Y induce proper maps h : X → Y and ¯ : X → Y . It suﬃces then for (i) to prove the Corollary for each of the pairs h ¯ of factorizations f = g¯v = (¯ g h)w and f = f u = (f h)w. For the ﬁrst pair, one need only look at the case u = f = f = 1 of Lemma (4.9.2). The second pair, being of the same form as the ﬁrst, is handled similarly. Q.E.D. Corollary 4.9.2.3 For any ´etale g : Y → Z and E, F as in (4.9.2), the ∼ map χgE,F (4.9.1.1) is the isomorphism f ∗ E ⊗ f ∗ F −→ f ∗ (E ⊗ F ) coming from (3.2.4). Proof (Sketch). The idea is to redo everything in this section 4.9, up to this point, with “´etale” in place of “open immersion.” The ﬁrst diﬃculty which arises is that in the last paragraph of the proof of Lemma (4.9.2), the map i is now ﬁnite ´etale, making it necessary to know (4.9.2.3) for ﬁnite ´etale f, a fact given by Exercise (4.8.12)(b)(vi). The only other nontrivial modiﬁcation is in the proof of (4.9.2.2), where the map X ×Z X → Y ×Z Y should now be factored as X ×Z X ֒→ W → Y ×Z Y with the ﬁrst map an open immersion and the second proper, and then X should be deﬁned to be the schematic Q.E.D. image of X → X ×Z X ֒→ W . . . !

Exercises 4.9.3 (a) In Ex. (4.7.3.4)(e) replace f × by f and apply the functor u∗ ! ! to get a natural map u∗f E → HX (f ∗F , u∗f (E ⊗ F )). Then show that this map f,u corresponds via (2.6.1)′ to χE,F .

234

4 Abstract Grothendieck Duality for Schemes

(b) Let f = f¯u be as in (4.9.1). Show, for E, F ∈ Dqc (Y ), that the composite map !

∗

χ f,u

natural

!

!

u∗f HY (E, F ) ⊗ u∗f E −−−−→ u∗f (HY (E, F ) ⊗ E) −−−−−→ u∗f F depends only on f, not on its factorization. + (Y ), of a canonical Deduce the existence, for any E ∈ D−c (Y ) and F ∈ Dqc isomorphism ! ∼ f HY (E, F ) −→ HX (f ∗E, f !F ), inverse to u∗ ζ where ζ comes from (4.2.3)(e) applied to f¯. (This can also be done without recourse to χ.) (c) (Compatibility of χ with base change.) After replacing (−)× by (−)! , do exercise (4.7.3.4)(c), assuming that the square is an admissible square, and interpreting β as in (4.8.3). Do something similar with the map φ of (3.10.4) in place of β. (d) Proceeding as in (a), work out exercises (4.7.3.4)(a), (d), and (f), with (−)× replaced by (−)! . This will likely involve veriﬁcations of compatibility with restriction to open subschemes for a number of functorial maps. Do similarly for (4.2.3)(h)–(j). (e) Show that if f : X → Y is ` etale then the map in (b) is the same as the map coming from (3.5.4.5). (f) Explain the formal tensor-hom symmetry in the pair of natural isomorphisms ∼ f ∗E ⊗ f !F −→ f ! (E ⊗ F ) ∼ HX (f ∗E, f !F ) −→ f ! HY (E, F )

E, F ∈ Dqc(Y ) ,

+ (Y ) . E ∈ Dc− (Y ), F ∈ Dqc

Another such pair, coming from (3.9.4) and (3.2.3.2), is ∼ f∗ (f ∗ E ⊗ F ) E ⊗ f∗ F −→ ∼ HY (E, f∗ F ) −→ f∗ HX (f ∗E, F )

(I don’t have an answer.)

E, F ∈ Dqc (Y ) , E, F ∈ D(Y ) .

With respect to a scheme-map f : X → Y , an OX -complex E is f-perfect if E has coherent homology and ﬁnite ﬂat f -amplitude. As noted in (2.7.6), f is perfect (i.e., of ﬁnite tor-dimension) ⇐⇒ OX is f -perfect. When f is perfect, the natural map, taking 1 ∈ H0 (X, OX ) to the identity map of the relative dualizing complex f ! OY is an isomorphism ∼ ξ : OX −→ HX (f ! OY , f ! OY ).

In fact, the functor HX (−, f ! OY ) induces an antiequivalence of the full subcategory of f -perfect complexes in D(X) to itself [I, p. 259, 4.9.2]. Theorem 4.9.4 For any ﬁnite-type separated map f : X → Y of noetherian schemes, the following conditions are equivalent. (i) The map f is perfect, i.e., the complex OX is f-perfect. (ii) The complex f ! OY is f-perfect. + (iii) f ! OY ∈ D− qc (X), and for every F ∈ Dqc (Y ), the Dqc (X)-map f χO : f ! OY ⊗ f ∗F −→ f !F Y ,F

is an isomorphism.

4.9 Perfect Maps of Noetherian Schemes

235

(iii)′ For every perfect OY -complex E, f ! E is f-perfect; and for all E, F ∈ D(Y ) such that E and E ⊗ F are in D+ qc (Y ), the Dqc (X)-map f χE,F : f !E ⊗ f ∗F −→ f ! (E ⊗ F ).

is an isomorphism. + (iv) The functor f ! : D+ qc (Y ) → Dqc (X) is bounded. Proof. (i)⇔(ii). The question is local on X, and so we may assume that p i f factors as X − → Z − → Y where Z is an aﬃne open subscheme of Y ⊗Z Z[T1 , . . . , Tn ] (with independent indeterminates Ti ), i is a closed immersion, and p is the obvious map. By (4.4.2) (with F = OX ), we have a functorial isomorphism ∼ i∗ i! G −→ HZ (i∗ OX , G) G ∈ D+ (4.9.4.1) qc (Z) .

Also, with Ωnp the invertible OZ -module of relative K¨ ahler n-forms, there is a natural isomorphism p!E ∼ = Ωnp [n] ⊗ p∗E

(E ∈ D+ qc (Y )),

(4.9.4.2)

see [V′, p. 397, Thm. 3].16 Now, by [I, p. 250, 4.1, and p. 252, 4.4], (i) holds if and only if the OZ -complex i∗ OX is perfect; and (ii) holds if and only if the OZ -complex i∗ f ! OY ∼ = HZ (i∗ OX , Ωnp [n]) = HZ (i∗ OX , p! OY ) ∼ = i∗ i! p! OY ∼ is perfect. Hence the equivalence of (i) and (ii) results from the following fact, in the case F = i∗ OX . Lemma 4.9.4.3 On any noetherian scheme W , an OW -complex F is perfect ⇐⇒ F ∈ Dbc (W ) and HW (F, OW ) is perfect. Proof. The implication ⇒ results from [I, p. 148, 7.1]. For the converse, the question being local, we may assume that W is aﬃne, say W = Spec(R), that F is a bounded-above complex of ﬁnite-rank locally free OW -modules (see 4.3.2), and that HW (F , OW ) is D(W )-isomorphic to a strictly perfect OW -complex. 16

The proof in loc. cit. can be imitated, without the assumption of ﬁnite Krull dimension, and with E in place of OY ; but instead of Corollary 2 one should use [H, p. 180, Cor. 7.3], noting that the graph map denoted by Δ is a local complete intersection map of codimension n [EGA, IV, (17.12.3)]. It might appear simpler to use [V′, p. 396, Lemma 1], whose proof, however, seems to need an isomorphism of the form (4.9.4.2) when Z is P1Y . For this, see [H, p. 161, 5.1] (duality for Pn Y ), − (Y ). That suﬃces, neverexcept that the proof given there applies only to F ∈ Dqc theless, by (4.3.7) applied to the map φ : Ω1p → p! OY corresponding by duality to the ∼ canonical isomorphism R1p∗(Ω1p ) −→ OY [H, p. 155, 4.3].

236

4 Abstract Grothendieck Duality for Schemes

Then N := Γ(W , F ) is a bounded-above complex of ﬁnite-rank projective R-modules, and with ∼ the usual sheaﬁﬁcation functor, F ∼ = N ∼. Let R → I • be an R-injective resolution of R. By [H, p. 130, 7.14], the resulting map OW = R∼ → I •∼ is an injective resolution of OW . So HomW (N ∼, I •∼ ) ∼ = HW (F , OW ) is D(W )-isomorphic—and hence, )-isomorphic—to a strictly perfect OW -complex. Since by (3.9.6)(a), D(Aqc W Γ(W , −) is exact on Aqc , it follows that W RHomR (N , R) ∼ = Γ W , HomW (N ∼, I •∼ ) = HomR (N , I • ) ∼

is a perfect R-complex. So by [AIL, Prop. 4.1(ii)], N is perfect, whence so is F ∼ Q.E.D. = N ∼. p i (i)⇒(iii). One may assume f factors as above: X − →Z− →Y. By (4.9.4.2), for f ! OY = i! p! OY to be in D− qc (X) it suﬃces that the functor i! be bounded on D+ qc (Z), which it is, by (4.9.4.1), because i∗ OX is perfect. (For this boundedness, as in the proof of [I, p. 148, 7.1], after replacing i∗ OX by an arbitrary perfect OX -complex E and localizing, one may assume that E is a bounded complex of ﬁnite-rank free OZ -modules, and proceed by “d´evissage,” i.e., induction on the number of nonzero components of E, to reduce to noting that HZ (E, G) is a bounded functor of G ∈ D+ qc (Z) when E is a ﬁnite-rank free OZ -module.) Next, by (4.9.2), with (f , g, u, f¯) replaced by (i, p, 1, i), it suﬃces to show p that χpi !OY , p∗F and χO are isomorphisms. Y ,F By (4.9.3)(c), the question of whether χpi !OY , p∗F is an isomorphism is local on Y , so we may assume Y aﬃne, in which case every quasi-coherent OY -module is a homomorphic image of a free one. Since p is ﬂat and, by (4.9.4.2), the complex p! OY is perfect, therefore ! p OY ⊗p∗F is a bounded functor of F ; and again by (4.9.4.2), so is p!F. Hence, by (1.11.3.1), one need only note that by (4.7.5) applied to a compactiﬁcation of p, χpi !OY , p∗F is an isomorphism whenever F is a free OY -module. That χpi !OY , G is an isomorphism for any G ∈ Dqc (Z) can be checked after application of the functor i∗. The source and target of i∗ χpi !OY , G are i∗ (i! p! OY ⊗ i∗ G)

∼ =

(3.9.4)

i∗ i! p! OY ⊗ G

i∗ i! (p! OY ⊗ G)

∼ =

(4.9.4.1)

∼ =

(4.9.4.1)

HZ (i∗ OX , p! OY ) ⊗ G,

HZ (i∗ OX , p! OY ⊗ G).

Since i∗ OX is perfect, and, by (4.9.4.2), so is p! OY , therefore both the source and target are bounded functors of G, commuting with direct sums (3.8.2). As before, one reduces to where Z is aﬃne and G is a free OZ -module, in which case commutativity with direct sums gives a reduction to the trivial case G = OZ . (Alternatively, it is a nontrivial exercise to show that (4.9.4.2) with p! OY p . One also shows, with E := i∗ OX , F := p! OY , in place of Ωnp [n] is in fact χO Y ,E

4.9 Perfect Maps of Noetherian Schemes

237

i that i∗ χF, G is isomorphic to the map

ζ(E) : HZ (E, F ) ⊗ G → HZ (E, F ⊗ G) associated by (2.6.1)∗ to the natural map HZ (E, F ) ⊗ G ⊗ E → F ⊗ G, and then sees via d´evissage to the trivial case E = OZ that ζ(E) is an isomorphism for all perfect E. What is involved here is a concrete local interpretation of χ f .) (iii)⇔(iii)′ ⇒(ii). The implications (iii)′ ⇒ (ii) and (iii)′ ⇒ (iii) are trivial. Assume, conversely, that (iii) holds. To be shown ﬁrst is that for a perfect OY -complex E, f ! E is f-perfect. Since f ! commutes with open base change (4.8.3), one can replace Y by any open subset. Thus one may assume that E is a bounded complex of ﬁniterank free OY -modules, and then proceed by d´evissage to reduce to the case E = OY , treated as follows. Let μ : V ֒→ Y be the inclusion of an open subscheme, ν : f −1 V ֒→ X the inclusion, g : f −1 V → V the map induced by f, and M an OV -module. We have then the obvious isomorphisms ν ∗f ! OY ⊗ g ∗M ∼ = ν ∗f ! μ∗ M . = ν ∗(f ! OY ⊗ f ∗μ∗ M ) ∼ (iii)

Since μ∗ M is a bounded complex (3.9.2), and since f ! is bounded below and, by (iii), bounded above, therefore there is an interval [m, n] not depending on M such that H i (ν ∗f ! OY ⊗ g ∗M ) = 0 for all i ∈ / [m, n]. So by [I, p. 242, 3.3(iv)], f ! OY has ﬁnite ﬂat f-amplitude. Also, (4.9.4.1) and (4.9.4.2) imply that f ! OY ∈ Dc (X). Thus f ! OY is f-perfect. For the isomorphism in (iii)′ , apply (4.7.3.4)(a) with E = OY to a compactiﬁcation of f . (i)⇒(iv). Theorem (4.1) gives that f ! is bounded below. If (i) holds then by deﬁnition, the (derived) functor f ∗ is bounded above; and as shown above, (iii) holds, whence f ! is bounded above. Thus f ! is bounded. (iv)⇒(i). With notation as in the proof of (i) ⇔ (ii), we will show that if f ! is bounded then so is i! . By [LN, Thm. 1.2] (or (4.9.6(e) below), this implies that i is perfect, whence so is f = pi. γ g → X ×Y Z − → Z where γ is the graph of i and g is the Factor i as X − projection. The map γ, a local complete intersection [EGA, IV, (17.12.3)], is perfect, and so, as we’ve just seen, γ ! is bounded. Also, g arises from f by ﬂat base change, so, as in (4.7.3.1)(ii) with × replaced by ! , g ! is bounded: to imitate the proof of (4.7.3.1)(ii) one just ∼ needs to associate a functorial isomorphism v∗ g ! −→ f ! u∗ to each composite ﬁber square

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4 Abstract Grothendieck Duality for Schemes v

• −−−−→ ⏐ ⏐ s v ¯

• −−−−→ ⏐ ⏐ g ¯

• ⏐ ⏐ t

• ⏐ ⏐¯ f

• −−− −→ • u

with u, v¯ and v ﬂat, f¯ and g¯ proper, t and s open immersions, f = f¯t and g = g¯s. One such isomorphism is the natural composition !

∼ ∼ ∼ ∼ f ! u∗ . t∗ f u∗ −→ v∗ g ! −→ v∗ s∗ g¯! −→ t∗ v¯∗ g¯! −→ (3.10.4)

Thus i! ∼ = γ ! g ! is bounded.

Q.E.D.

Corollary 4.9.5 On the category of perfect maps there is a pseudofunctor (−)# which associates to each such map f : X → Y the functor + f # : D+ qc (Y ) → Dqc (X) given objectwise by f # F := f ! OY ⊗ F f

g

F ∈ D+ qc (Y ) .

For a composition X − →Y − → Z of perfect maps, the resulting functorial ∼ isomorphism f # g # G −→ (gf )# G (G ∈ D+ qc (Z)) is the left column of the following diagram of natural isomorphisms, whose commutativity results from (4.7.3.4)(a) and (d), as treated in (4.9.3)(d), or from (4.9.2) with E := OX and F := G. f # g # G = f ! OY ⊗ f ∗ (g ! OZ ⊗ g ∗ G) ⏐ ⏐

χ

f !χ

−−→ f ! (g ! OZ ⊗ g ∗ G) −−→ f ! g ! G ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (f ! OY ⊗ f ∗ g ! OZ ) ⊗ f ∗g ∗ G ⏐ ⏐ ⏐ ⏐ ⏐ χ⊗1 ⏐ ⏐ ⏐ ! ! ∗ ∗ f g OZ ⊗ f g G ⏐ ⏐ ⏐ ⏐ ⏐

(gf )# G = (gf )! OZ ⊗ (gf )∗ G

−−−−−−−−−−χ−−−−−−−−−→ (gf )!G

f Exercises 4.9.6 (a) Show that χE,F is an isomorphism whenever F ∈ Dqc (X) has ﬁnite tor-dimension. (Cf. (4.7.5).)

4.10 Appendix: Dualizing Complexes

239

(b) Noting Ex. (3.5.3)(g), establish a natural commutative diagram f !F ⊗ HX (f ∗E, f ∗G) −−−→ HX (f ∗E, f !F ⊗ f ∗G) ←−−− HX (f ∗E, f !F ) ⊗ f ∗G ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ f !F ⊗ f ∗ HY (E, G) ⏐ ⏐ f ! (F ⊗ HY (E, G))

−−−→

HX (f ∗E, f ! (F ⊗ G)) ⏐ ⏐ f ! HY (E, F ⊗ G)

f ! HY (E, F ) ⊗ f ∗G ⏐ ⏐

←−−− f ! (HY (E, F ) ⊗ G)

+ (Y ), (c) (Neeman, van den Bergh). Show, for any perfect f : X → Y and E ∈ Dqc f ∗ ! ! ′ that the map f E → HX (f OY , f E) induced via (2.6.1) by χOY ,E is an isomorphism. Hint. Factor f locally as pi—see proof of (4.9.4), and apply i∗ . (d) Let X be a noetherian scheme, E ∈ Dcb (X). Show that the functor HX (E, −) + (X) to itself is bounded if and only if E is perfect. from Dqc Hint. Reduce to where X = Spec(A), and where E is the sheaﬁﬁcaton E∼ of a bounded A-complex E of ﬁnitely generated A-modules. Use the fact that the sheaﬁﬁcation of an A-injective module is OX -injective [RD, p. 130, 7.14], to show that for any F ∈ D+ (A), HX (E, F∼ ) = RHomA (E,F)∼ , and hence to reduce further to the corresponding statement for A-modules. (e) Using (d) and (4.4.2) with F = OX , show that a ﬁnite map f : X → Y of + (Y ) → D+ (X) is bounded. noetherian schemes is perfect iﬀ the functor f ! : Dqc qc

4.10 Appendix: Dualizing Complexes Grothendieck’s original strategy for proving duality—at least the version in Corollary (4.2.2)—for proper not-necessarily-projective maps, is based on pseudofunctorial properties of dualizing complexes. In this section, we sketch the idea. The principal result, Thm. (4.10.4), makes clear how the basic problem—not treated here—in this approach is the construction of a “coherent” family of dualizing complexes (in other words, a “Dualizing Complex,” see below). What emerges is less than Thm. (4.8.1). But for formal schemes, this kind of approach yields results not otherwise obtainable (as of early 2009), see the remarks following Thm. (4.10.4). Throughout this section, without further mention we restrict to schemes which are noetherian and to scheme-maps that are separated, of ﬁnite type. Also, we continue to use the notations introduced at the beginning of §4.4. Let Ac (X) ⊂ A(X) be the full subcategory whose objects are the coherent OX -modules; it is a plump subcategory [GD, 113, 5.3.5]. Additional notation will be as in §(1.9.1), with # = c. For example, D+ c (X) is the Δ-subcategory of D(X) whose objects are the complexes whose homology modules vanish in all suﬃciently negative degrees, and are coherent in all degrees.

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4 Abstract Grothendieck Duality for Schemes

A dualizing complex R on a noetherian scheme X is a complex in Dc (X) which is D-isomorphic to a bounded injective complex, and has the following equivalent properties [H, p. 258, 2.1]: (i) For every F ∈ Dc (X), the map corresponding via (2.6.1)′ to the natural composition ∼ RHom(F , R) ⊗ F → R F ⊗ RHom(F , R) −→

is an isomorphism (called by some other authors the Grothendieck Duality isomorphism): ∼ F −→ RHom(RHom(F , R), R). (ii) Condition (i) holds for F = OX , i.e., the map OX → RHom(R, R) which takes 1 ∈ Γ(X, OX ) to the identity map of R is an isomorphism. For connected X, dualizing OX -complexes, if they exist, are unique up to tensoring with a complex of the form L[n] where L is an invertible OX -module and n ∈ Z [H, p. 266, 3.1]. The associated dualizing functor DR := RHomX (−, R) satisﬁes DR ◦DR ∼ = 1, and it induces antiequivalences from Dc (X) to itself, and between D+ (X) and Dc− (X) (in either direction). c The existence of a dualizing complex places restrictions on X—for instance, X must then be universally catenary and of ﬁnite Krull dimension [H, p. 300]. Suﬃcient conditions for the existence are given in [H, p. 299]. For example, any scheme of ﬁnite type over a regular (or even Gorenstein) scheme of ﬁnite Krull dimension has a dualizing complex.17 Henceforth we restrict schemes to those which, in addition to being noetherian, have dualizing complexes. The relation between dualizing complexes and the pseudofunctor ! of Thm. (4.8.1) is rooted in the following Proposition, see [H, Chapter V, §8], [V′, p. 396, Corollary 3], or [N′′, Theorems 3.12 and 3.14]. Proposition 4.10.1 Let f : X → Y be a scheme-map, and let R be a dualizing OY -complex. Then with Rf := f !R, (i) Rf is a dualizing OX -complex. (ii) There is a functorial isomorphism ∼ f ! D F −→ D Lf ∗F F ∈ D− (Y ) R

Rf

c

or equivalently,

∼ f !E −→ DRf Lf ∗DR E

17

E ∈ D+ c (Y ) .

In [N′′ ], Neeman studies a notion of dualizing complex which applies to inﬁnitedimensional schemes. Suresh Nayak observed, via [C, p. 121, Lemma 3.1.5], that Neeman’s dualizing complexes are the same as pointwise dualizing complexes with bounded cohomology, cf. [C, p. 127, Lemma 3.2.1].

4.10 Appendix: Dualizing Complexes

241

Proof. First, it follows from the construction of the functor f × (see just before (4.1.8)) that it preserves ﬁnite injective dimension. So when f is proper, f ! = f × preserves ﬁnite injective dimension. The same is clearly true for f ! = f ∗ when f is an open immersion, and hence—via compactiﬁcation— for any f . The question of whether f !R ∈ Dc (X) is local; hence an aﬃrmative answer is provided by (4.9.4.1) and (4.9.4.2). It remains to show that the natural map ψf : OX → DRf DRf OX is an isomorphism. Again, the question is local, so we reduce to the two cases (a) f is smooth, (b) f is a closed immersion. (a) For smooth f , (4.9.4.2) and (4.6.7) provide natural isomorphisms ∼ ∼ RHomX (p∗R, p∗R) −→ p∗ RHomY (R, R). RHomX (Rf , Rf ) −→

One veriﬁes then that ψf is isomorphic, via the preceding isomorphisms, to p∗ ∼ applied to the isomorphism OY −→ DR DR OY . (b) It suﬃces that f∗ ψf be an isomorphism, which it is, by (4.9.4.1) (with i = f ), since f∗ OX ∈ Dbc (Y ) and therefore the canonical map f∗ OX → DR DR f∗ OX is an isomorphism. Assertion (ii) follows immediately from Ex. (4.2.3)(e), as DR and DRf are antiequivalences. Q.E.D. Deﬁnition 4.10.2 A Dualizing Complex on a scheme Y is a map which associates to each f : X → Y a dualizing complex Rf on X, to each open ∼ immersion u : U → X a D(X)-isomorphism γf,u : u∗Rf −→ Rf u , and to each ′ proper map g : X → X a D(X)-map τf,g : g∗ Rfg → Rf , subject to the following conditions on each such f , u and g: (a) If v : V → U is an open immersion, then the following diagram commutes: v ∗ u∗Rf −−− −−→ (uv)∗Rf (3.6.4)∗ ⏐ ⏐ ⏐γ ⏐ v∗γf,u f,uv v ∗Rf u −− −−→ γ fu,v

Rf uv

(b) The pair (Rfg , τf,g ) represents the functor ′ + HomD(X) (g∗ E, Rf ) : D+ c (X ) → Dc (X),

that is, the natural composite map HomD(X ′ ) (E, Rfg ) −→ HomD(X) (g∗ E, g∗ Rfg ) −−−→ HomD(X) (g∗ E, Rf ) via τ

is an isomorphism.

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4 Abstract Grothendieck Duality for Schemes

Further, if h : X ′′ → X ′ is proper then the following diagram commutes: g∗ h∗⏐Rfgh −−− −−→ (gh)⏐ ∗ Rfgh (3.6.4)∗ ⏐τ f,gh

g∗ τfg,h ⏐

g∗ Rfg

(c) For any ﬁber square

Rf

−−τ−−→ f,g

v

V −−−−→ ⏐ ⏐ h

Z ⏐ ⏐g

U −−− −→ X u

with g (hence h) proper and u (hence v) an open immersion, the following natural diagram commutes: u∗g∗ Rfg −−−−−−− −−−−−−−→ ⏐ ⏐ ∗ u τf,g u∗Rf

h∗ v ∗Rfg ⏐ ⏐ ≃h∗ γfg,v

−−γ −−→ Rf u ←τ−−−− h∗ Rf uh = h∗ Rfgv f,u

fu,h

Remarks. In (4.10.2)(a) take U = V = X and let u and v be identity maps, to get γf,u ◦ γf,u = γf,u , whence the isomorphism γf,u is the identity map 1 of Rf . Similarly, when g is the identity map of X, one deduces from (b) that τf,g ◦ τf,g = τf,g ; but (Rf , τf,g ) and (Rf , 1) both represent the same functor, whence τf,g is an isomorphism, so τf,g = 1. Also, when Z = U = V and g = u is an open and closed immersion, (c) shows that γf,g ◦ g ∗ τf,g is the canonical ∼ isomorphism g ∗ g∗ Rfg −→ Rfg . Example 4.10.2.1 (A) If R is a dualizing OY -complex and ! is as in (4.8.1), one can associate to each map f : X → Y the dualizing OX -complex Rf := f !R, to each open immersion u : U → X the natural composition ∼ γf,u : u∗Rf = u!f ! Rf −→ (f u)! R = Rf u ,

and to each proper map g : X ′ → X the map τ = τf,g : g∗ (f g)! R → f ! R resulting from (4.1.1). Condition (a) is then clear, (b) follows from (4.1.2), and (c) from (4.4.4)(d). (B) Let R = (R, γ, τ ) be a Dualizing Complex on Y . Then for any map e : Y ′ → Y we have a Dualizing Complex R ×Y Y ′ := (R′ , γ ′ , τ ′ ) on Y ′, where ′ ′ ′ := γef,u and τf,g := τef,g . for all f : X → Y ′ we set Rf′ := Ref , γf,u ′ That R ×Y Y satisﬁes conditions (a), (b) and (c) is simple to check. (C) Let R = (R, γ, τ ) be a Dualizing Complex on Y . Then for any invertible OY -module L and any locally constant function n : Y → Z,

4.10 Appendix: Dualizing Complexes

243

we have a Dualizing Complex R ⊗ L[n] = (R ⊗ L[n], γ ⊗ L[n], τ ⊗ L[n]) on Y , where for all f : X → Y , • (R ⊗ L[n])f := Rf ⊗ f ∗L[n] (easily seen to be a dualizing OX -complex), • (γ ⊗ L[n])f,u is the natural composition ∼ ∼ u∗ Rf ⊗ f ∗L[n] −→ u∗Rf ⊗ u∗f ∗L[n] −→ Rf u ⊗ (f u)∗ L[n], • τ ⊗ L[n] f,g is the natural composition ∼ ∼ g∗ Rfg ⊗ f ∗L[n] g∗ Rfg ⊗ (f g)∗ L[n] −→ g∗ Rfg ⊗ g ∗f ∗L[n] −→ (3.9.4)

−→ Rf ⊗ f ∗L[n].

Here, condition (a) is given by the (readily veriﬁed) commutativity of the natural diagram v ∗ u∗ Rf ⊗ f ∗L[n] −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ (uv)∗ Rf ⊗ f ∗L[n] ⏐ ⏐ ⏐ ⏐ v ∗ u∗Rf ⊗ u∗f ∗L[n] −−−→ v ∗ u∗Rf ⊗ v ∗ u∗f ∗L[n] −−−→ (uv)∗Rf ⊗ (uv)∗f ∗L[n] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ v ∗ Rf u ⊗ (f u)∗L[n] −−−→ v ∗Rf u ⊗ v ∗ (f u)∗L[n] −−−→ Rf uv ⊗ (f uv)∗L[n]

∼ Fix a D(X)-isomorphism α : L[n] ⊗ L−1 [−n] −→ OY . The ﬁrst part of condition (b) results from commutativity of the natural diagram

HomD(X) E, Rfg ⊗ (f g)∗L[n] ⏐ ⏐≃ HomD(X) E, Rfg ⊗ g ∗f ∗L[n] ⏐ ⏐ HomD(Y ) g∗ E, g∗ Rfg ⊗g ∗f ∗L[n] ⏐ ⏐ via (3.9.4)≃ HomD(Y ) g∗ E, g∗ Rfg ⊗ f ∗L[n] ⏐ ⏐ HomD(Y ) g∗ E, Rf ⊗ f ∗L[n]

HomD(X) E ⊗ (f g)∗L−1 [−n], Rfg ⏐ ⏐ ≃ −−−→ HomD(X) E ⊗g ∗f ∗L−1 [−n],Rfg ⏐ ⏐ HomD(Y ) g∗ E ⊗g ∗f ∗L−1 [−n] , g∗ Rfg ⏐ ⏐ ≃via (3.9.4) −− HomD(Y ) g∗ E ⊗ f ∗L−1 [−n], g∗ Rfg ←− ⏐ ⏐ −− ←− HomD(Y ) g∗ E ⊗ f ∗L−1 [−n], Rf

−− −→

−1 where, with Ln := L[n] and L−1 [−n], the ﬁrst row takes a map −n := L ∗ η : E → Rfg ⊗ (f g) Ln to the natural composition

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4 Abstract Grothendieck Duality for Schemes

via η E ⊗ (f g)∗L−1 −−→ Rfg ⊗ (f g)∗ Ln ⊗ (f g)∗ L−1 −n − −n via α −1 ∼ ∗ ∼ −→ Rfg ⊗ (f g) Ln ⊗ L−n −−−→ Rfg ⊗ (f g)∗ OY −→ Rfg

and the second row takes η ′ : E → Rfg ⊗ g ∗f ∗ Ln to the natural composition via η ′ ∗ ∗ ∗ ∗ −1 E ⊗ g ∗f ∗L−1 −n −−−→ Rfg ⊗ g f Ln ⊗ g f L−n via α ∗ ∗ ∼ ∼ −→ Rfg ⊗ g ∗f ∗ Ln ⊗ L−1 −n −−−→ Rfg ⊗ g f OY −→ Rfg .

The arrows in the last two rows are deﬁned in a similar manner. Commutativity of the bottom subrectangle is obvious. Checking commutativity of the other two subdiagrams is left as an exercise. (For the middle one, a variant of diagram (3.4.7)(iv) may prove useful.) The second part of condition (b) follows from (3.7.1). (Details left as an exercise.) Condition (c) is given by commutativity of the following natural diagram, where L[n] has been abbreviated to L: u∗ (Rf ⊗ f ∗L) ←− u∗ (g∗ Rfg ⊗ f ∗L) ←− u∗g∗ (Rfg ⊗ g ∗f ∗L) −→ h∗ v ∗ (Rfg ⊗ g ∗f ∗L) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 ⏐ u∗Rf ⊗ u∗f ∗L ←− u∗g∗ Rfg ⊗ u∗f ∗L ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⏐ h∗ v Rfg ⊗ u f L ←− h∗ (v Rfg ⊗ h u f L) ←− h∗ (v Rfg ⊗ v ∗g ∗f ∗L) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ h∗ (Rfgv ⊗ (f gv)∗L) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Rfu ⊗ (f u)∗L ←− h∗ Rfuh ⊗ (f u)∗L ←− h∗ (Rfuh ⊗ h∗ (f u)∗L) ←− h∗ (Rfuh ⊗ (f uh)∗L)

Commutativity of subdiagram 1 is given by (3.7.3). Commutativity of the other subdiagrams is easy to check. ∼ A morphism of Dualizing Complexes on Y , ψ : (R, γ, τ ) −→ (R′, γ ′, τ ′ ) is a ∼ map associating to each scheme-map f : X → Y a D(X)-map ψf : Rf −→ Rf′ , ′ such that for each open immersion u : U → X (resp. each proper map g : X → X) the following diagrams commute: γf,u

u∗Rf −−−−→ Rf u ⏐ ⏐ ⏐ψ ⏐ u∗ψf fu u∗Rf′ −−− −→ Rf′ u ′ γf,u

τf,g

g∗ Rf g −−−−→ ⏐ ⏐ g∗ ψfg

Rf ⏐ ⏐ψ f

g∗ Rf′ g −−−′ −→ Rf′ τf,g

In the next Proposition, 1 denotes the identity map of Y .

(4.10.2.2)

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245

Proposition 4.10.3 Let (R, γ, τ ) and (R′, γ ′, τ ′ ) be Dualizing Complexes on Y , and let ψ0 : R1 → R1′ be a D(Y )-map. Then there exists a unique ∼ (R′, γ ′, τ ′ ) with ψ1 = ψ0 . morphism ψ : (R, γ, τ ) −→ Corollary 4.10.3.1 (Uniqueness of Dualizing Complexes). If R and R′ are Dualizing Complexes on Y then there exists an invertible OY -module L, unique up to isomorphism, and a unique locally constant function n : Y → Z such that R′ ∼ = R ⊗ L[n]. Moreover, if ψ and χ are two isomorphisms from R′ to R ⊗ L[n] then ψ −1 χ is multiplication by a unit in H0 (Y , OY ). Proof of (4.10.3.1). One reduces easily to where Y is connected. In view of (4.10.3), the ﬁrst assertion follows then from the corresponding assertion for dualizing OY -complexes [H, p. 266, Thm. 3.1]. The second assertion results from the sequence of natural ring isomorphisms and anti-isomorphisms—with R a dualizing OY -complex and DR (−) := RHomX (−, R): HomD(Y ) (R, R) ∼ = HomD(Y ) (OY , OY ) = HomD(Y ) DR (R), DR (R) ∼ 0 ∼ H0 RΓ(OY ) = ∼ H0 (Y , OY ). ∼ H RΓRHom(OY , OY ) = = ′ Proof of (4.10.3). For any proper map g : X → Y , since (Rg′ , τ1,g ) ′ represents the functor HomD(Y ) (g∗ E, R1 ) (see (4.10.2)(b)), there exists a unique D(X)-map ψg : Rg → Rg′ making the following diagram commute: g∗ ψg

g∗ Rg −−−−→ g∗ Rg′ ⏐ ⏐ ⏐τ ′ τ1,g ⏐ 1,g R1

−−−−→ R1′ ψ0

g u A general map f : X → Y factors as X −−− −→ Z −−−−→ Y with g proper ′ and u an open immersion. Let ψg,u : Rf → Rf be the unique D(X)-map making the following diagram commute: ψg,u

Rf −−−−→ Rf′ ⏐ ′ γg,u ⏐ ≃⏐γg,u ⏐≃ u∗Rg −−− −→ u∗Rg′ ∗ u ψg

Let us show that ψg,u depends only on f , allowing us to write ψf instead u ˜ u an open of ψg,u . So let X −−− −−−g˜−→ Y also be a factorization of f (˜ −→ Z immersion, g˜ proper). There results a natural diagram

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4 Abstract Grothendieck Duality for Schemes

Z u

X

g

u ¯

Z

u ˜

g ¯

Y

g ˜

Z

with Z the scheme-theoretic image [GD, p. 324, §6.10] of the composite (u,˜ u) diag and u ¯ : X → Z¯ the resulting immersion X −−→ X ×Y X −−−→ Z ×Y Z, open immersion; and where the vertical maps, induced by the canonical projections, are proper. We need only show that ψu,g = ψu¯,¯g = ψu˜,˜g ; so it’s enough to treat the case →Z (˜ u, g˜) = (¯ u, g¯), that is, we may assume that there is a proper map p : Z such that gp = g˜ and p˜ u = u, and that furthermore u ˜(X) is a dense open subset of Z: X ⏐ ⏐ u ˜

1

X ⏐ ⏐ u

X ⏐ ⏐g˜

−−−−→ Z −−−−→ Y Z p g

Here subdiagram 1 is a ﬁber square, since the map u ˜0 : X → p−1 (uX) induced by u ˜ is both an open immersion (clearly) and a closed immersion ˜X is open, (because u ˜0 has a left inverse, essentially p|p−1 (uX) ), so that u closed and dense in p−1 (uX), hence equal to p−1 (uX). Consequently, there ∼ u ˜∗. is a natural functorial isomorphism θ : u∗ p∗ −→ It will be enough to show that the following diagram—whose top and bottom rows compose to ψg,u and ψg˜,˜u respectively—commutes: u∗ψg

′ −′ −→ Rgu − −→ u∗Rg −−−−→ u∗Rg′ −− Rgu −− −1 γg,u γg,u ⏐u∗τ ′ ⏐ u∗τg,p ⏐ 3 ⏐ g,p ∗ ′ ∗ − − − − → u p R u p R 2 5 ∗ gp ∗ gp ∗ u p∗ ψgp ⏐ ⏐ ⏐≃ ⏐ ≃θ θ 4

Rg˜u˜ = Rgp˜u −− − −→ −1 γgp,˜ u

u ˜∗Rgp

−−∗−−→ u ˜ ψgp

′ u ˜∗Rgp

′ ′ −− − −→ Rgp˜ u = Rg ˜u ˜ ′ γgp,˜ u

Commutativity of subdiagram 4 is clear. Subdiagrams 2 and 5 commute by condition (c) in (4.10.2), applied to the above ﬁber square 1 . Finally,

4.10 Appendix: Dualizing Complexes

247

′ the ﬁrst part of (4.10.2)(b) guarantees the existence of a map ψˆgp : Rgp → Rgp such that the following diagram commutes: ψg

Rg τg,p ⏐ ⏐

−−−−→

Rg′ ⏐τ ′ ⏐ g,p

′ p∗ Rgp −−−ˆ−→ p∗ Rgp ; p∗ ψgp

and in view of the of the commutative diagram in (4.10.2)(b), and of the deﬁnition of ψf for proper f, application of the functor g∗ to the preceding diagram shows that ψˆgp = ψgp , whence 3 commutes. We have now deﬁned ψf for all f. The commutativity in (4.10.2.2) shows that no other family (ψf ) can satisfy (4.10.3). It remains to be proved that with the present (ψf ), commutativity does hold for the two diagrams in (4.10.2.2). For the ﬁrst of those diagrams, the problem is to show, given a sequence g u v →X − →Z − → Y with u and v open immersions and g proper, that the U − following natural diagram commutes: ′ u∗Rgv −− −−→ u∗Rgv −−→ u∗ v ∗Rg −−−−→ u∗ v ∗Rg′ −− ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ′ Rgvu −−−−→ (vu)∗Rg −−−−→ (vu)∗Rg′ −−−−→ Rgvu ;

but this is an immediate consequence of (4.10.2)(a). For the second diagram in (4.10.2.2), suppose there is given a sequence g

v

h

X′ − → X − → Z − → Y with u an open immersion and g, h both proper. g ¯ w →W − → Z such that w maps X ′ isomorphically As above, there are maps X ′ − onto a dense open subscheme of W, g¯ is proper, and g¯w = vg: w

X ′ −−−−→ ⏐ ⏐ g

W ⏐ ⏐g¯

X −−−v−→ Z −−− −→ Y h

The proper map g factors naturally as X ′ → g¯−1 (vX) → X, whence w(X ′ ) is open, closed and dense in—hence equal to—¯ g −1 (vX), and so there is a ∗ ∼ ∗ natural isomorphism θ : v g¯∗ −→ g∗ w .

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4 Abstract Grothendieck Duality for Schemes

The problem is to show commutativity of the natural diagram g∗ Rhvg −−−−−−−−−−−−−−−−−−→ Rhv ⏐ ⏐ ⏐ ⏐ ⏐ ⏐≃ ⏐ g∗ Rh¯gw 6 ⏐ ⏐ ⏐ ⏐ ≃ θ −1

g∗ w∗Rh¯g −−−−→ v ∗ g¯∗ Rh¯g −−−−→ v ∗Rh ⏐ ⏐ ⏐ ⏐ ⏐ ⏐v∗ψ g∗w∗ψh¯ v∗g ¯∗ψh¯ 7 g g h ′ ′ ∗ ′ g∗ w∗Rh¯ −−→ v ∗ g¯∗ Rh¯ g −− g −−−−→ v Rh θ −1 ⏐ ⏐ ⏐ ⏐ ⏐ ≃ ⏐ ⏐ ⏐≃ ′ ⏐ g∗ Rh¯ 8 gw ⏐ ⏐ ′ g∗ Rhvg −−−−−−−−−−−−−−−−−−→ Rh′

Commutativity of subdiagrams 6 and 8 is given by (4.10.2)(c). The argument that subdiagram 7 commutes is similar to that used above for 3 . Commutativity of the remaining subdiagram is obvious. Q.E.D. Here is the main result of this section.18 Theorem 4.10.4 Let S be a category of noetherian schemes such that if Y ∈ S and f : X → Y is a separated ﬁnite-type map then X ∈ S. Suppose every scheme in S has a Dualizing Complex. ! which is uniquely Then there exists on S a D+ c -valued pseudofunctor determined up to isomorphism by the properties that it restricts to the inverseimage pseudofunctor * on the subcategory of open immersions, that for a + proper f ∈ S, the functor f ! is right-adjoint to f∗ : D+ c (X) → Dc (Y ) (see (3.9.2.6)(c)), and that for any ﬁber square σ in S j′

X ′ −−−−→ ⏐ ⏐ p′

X ⏐ ⏐p

Y ′ −−−−→ Y j

with j an open immersion and p proper, the base-change map βσ of (4.4.3) is the natural composite isomorphism ∼ ∼ (pj ′ )! = (jp′ )! −→ p′ ! j ! = p′ ! j ∗ . j ′∗p! = j ′ ! p ! −→ 18

Cf. [H, p. 383, Cor. 3.4], and its proof.

4.10 Appendix: Dualizing Complexes

249

¯ γ¯ , τ¯) on Y is isomorphic to the With this ! , each Dualizing Complex (R, ¯ (identity of Y ) . one in (4.10.2.1)(A) with R := R Remarks. This says less than Theorem (4.8.1): the restriction to S of the pseudofunctor in that Theorem satisﬁes this one. The point is, however, that Theorem (4.10.4) captures Grothendieck’s strategy for constructing a duality pseudofunctor by means of Dualizing Complexes. Indeed, showing the existence of Dualizing Complexes is a major theme of the second half of [H]. (See also the discussion and clariﬁcation of this material in [C, §§3.1–3.4].)19 Let us add a few words, in passing, about noetherian formal schemes. Applying his results about pasting pseudofunctors to the duality theory in [AJL′ ], Nayak gets the existence of a duality pseudofunctor for composites of any number of pseudo-proper maps and open immersions [Nk, §7.1]. (As of 2009, one doesn’t know whether or not any pseudo-ﬁnite separated map of formal schemes is such a composite.) On the other hand, using an analog of Theorem (4.10.4), Sastry constructs a duality pseudofunctor on the category of all formal schemes admitting a dualizing complex (suitably deﬁned for formal schemes), with “essentially pseudo-ﬁnite type” maps; and he shows that this pseudofunctor agrees with Nayak’s whenever both are deﬁned [S′, §9]. Sastry’s approach has some resemblance to the one in [H], but there are a number of new techniques involved in the construction of Dualizing Complexes. In short, Chapter 6 of [H] is localized, generalized, and extended to the context of formal schemes in [LNS]; and then, among other things, the main results of Chapter 7 of [H], are extended to this context in [S′]. Thus, at the present time (2008), the theory of Dualizing Complexes for formal schemes gives rise in certain situations to the only way to construct dualizing pseudofunctors. Proof of (4.10.4) (Outline only). For each Y ∈ S choose a Dualizing Complex RY = (RY , γ Y , τ Y ). For any S-map f : X → Y let DfY be the functor from Dc (X) to Dc (X) given by DfY (E) := HX (E, RfY )

(HX := RHom•X ).

We set RY := R1YY and DY := D1YY where 1Y is the identity map of Y . For any S-map f : X → Y , the functor f ! : Dc (Y ) → Dc (X) is deﬁned to be f ! := DfY f ∗ DY . This functor has the following properties. 19 Recently, Yekutieli and Zhang have exploited the notion of “rigid dualizing complex,” introduced by Van den Bergh in the context of noncommutative algebra, to give an elegant new approach to the existence question, at least for ﬁnite tordimension maps of schemes of ﬁnite type over a regular scheme. See [YZ] for a preliminary account.

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4 Abstract Grothendieck Duality for Schemes

(1) If f is an open immersion, there is a natural functorial isomorphism ∼ f ∗, namely, the natural composition, with E ∈ Dc (Y ), f ! −→ −−−→ f ! E = HX f ∗ DY E, RfY −−− HX HX (f ∗E, f ∗RY ), RfY via (4.6.7) ∼ −−− −− −→ HX HX (f ∗E, RfY ), RfY −→ f ∗E −1 via γ1

Y ,u

(the last isomorphism resulting from RfY being a dualizing OX -complex). (2) If f is proper then f ! is right-adjoint to f∗ : Dc (X) → Dc (Y ). Indeed, for E ∈ Dc (X), F ∈ Dc (Y )we have, in view of (4.10.2)(b), natural functorial isomorphisms HomD(X) E, f ! F −− −−→′ HomD(X) E ⊗ f ∗ HY (F , RY ), RfY (2.6.1) −− −−→ HomD(Y ) f∗ (E ⊗ f ∗ HY (F , RY )), RY −− −−→ HomD(Y ) f∗ E ⊗ HY (F , RY ), RY (3.9.4) −− −−→′ HomD(Y ) f∗ E, HY (HY (F , RY ), RY ) (2.6.1) −− −−→ HomD(Y ) f∗ E, RY . ∼ (3) There is a natural isomorphism f !RY −→ RfY . This follows easily from ∼ OY . the natural isomorphism DY RY −→

(4) The functor ! extends to a pseudofunctor. For the proof, we need: h

g

f

Lemma 4.10.4.1 For any sequence V − → W − → X − → Y in S there is a natural isomorphism X ∗ X ∼ Y Y φf,g,h : Dgh h Dg −→ Dfgh h∗ Dfg ,

such that ∼ Y Y φf,g,h ◦ φg,1W ,h = φfg,1W ,h : DhW h∗ DW −→ Dfgh h∗ Dfg .

Proof. By (4.10.3.1) there is an invertible OX -module Lf , a locally constant, integer-valued function nf , and an isomorphism of Dualizing Complexes ∼ (RY ×Y X) ⊗ Lf [nf ], α : RX −→ see (4.10.2.1), (B) and (C). Set I := Lf [nf ] and I−1 := HX (I, OX ), so that ∼ OX . For any map e : Z → X there is a canonical isomorphism I ⊗ I−1 −→ and F , G ∈ D(Z), the map coming from (3.5.3)(g) is an isomorphism ∼ HZ (F , G) ⊗ e∗ I−1 −→ HZ (F ⊗ e∗ I, G). (The question being local, the proof reduces easily to the simple case I = OX .)

4.10 Appendix: Dualizing Complexes

251

There results, for any E ∈ Dc (W ), a composite isomorphism X ∗ X ∼ Y ϕα,L : Dgh h Dg E −→ Dfgh (h∗ DgXE) ⊗ (gh)∗ I ∼ Y Y Dfgh (h∗ Dfg E ⊗ (gh)∗ I) ⊗ (gh)∗ I −→ ∼ Y Y Dfgh h∗ Dfg E ⊗ (gh)∗ I−1 ⊗ (gh)∗ I −→ ∼ Y Y Dfgh h∗ Dfg E ⊗ (gh)∗ (I−1 ⊗ I) −→ ∼ Y Y Dfgh h∗ Dfg E. −→

It is easily checked that ϕα,L is independent of the choice of α and of L, i.e., if μ is a unit in H0 (X, OX ), and if L′ ∼ = L, then ϕα,L = ϕμα,L′ . So we can set φf,g,h = ϕα,L . The ﬁnal assertion is left to the very patient reader. (A direct approach seems to involve a formidable diagram—although the analogous statement (3.3.13) in [C, p. 135] is said there to be “easy to check.”) Q.E.D. Next, with f , g, h as in (4.10.4.1), we deﬁne the functorial isomorphism ∼ (f g)! to be the natural composition dg,f : g !f ! −→ Y ∗ Y − −−→ Dfg g Df DfY f ∗ DY g !f ! = DgX g ∗ DX DfY f ∗ DY − φf,1 ,g V

−− −−→

Y ∗ ∗ Y Dfg gf D

Y −− −−→ Dfg (f g)∗ DY = (f g)!

Pseudofunctoriality requires the following diagram to commute:20 dh,f g

(f gh)! ←−−−− h! (f g)! ⏐ ! ⏐ dgh,f ⏐ ⏐h df ,g (gh)! f ! ←−−−− h! g ! f ! dg,h

Expanding this diagram according to the deﬁnition of dg,f , one ﬁnds quickly that the problem is to show commutativity of the following diagram of natural isomorphisms: Y (gh)∗ DfY −−−→ Dfgh ⏐ φf ,1X,gh ⏐

X Dgh (gh)∗ DX ←−−−

Y Dfgh h∗g ∗ DfY

X ∗ ∗ X Dgh h g D ⏐ ⏐

Y Y Y ∗ Y ←−−− Dfgh h∗ Dfg Dfg g Df ⏐φ ⏐ fg,1W ,h Y ∗ Y DhW h∗ DW Dfg g Df ⏐ DhW h∗ DW⏐(φf ,1X,g )

X ∗ X X ∗ X Dgh h Dg Dg g D ← −−−− DhW h∗ DW DgXg ∗ DX φ 20

g,1W ,h

Strictly speaking, we need also to “normalize” ! , i.e., to replace (1Y )! by the identity functor of Dc (Y ) for every Y ∈ S.

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4 Abstract Grothendieck Duality for Schemes

Using the equality in (4.10.4.1), one transforms the question to commutativity of Y Dfgh (gh)∗ DfY −−−→ ⏐ φf ,1X,gh ⏐

X Dgh (gh)∗ DX ←−−−

Y Dfgh h∗g ∗ DfY

X ∗ ∗ X Dgh h g D ⏐ ⏐

Y Y Y ∗ Y ←−−− Dfgh h∗ Dfg Dfg g Df ⏐ ⏐ ⏐ ⏐ Y Y⏐ Dfgh h∗ Dfg ⏐(φf ,1X,g ) ⏐ ⏐ ⏐

Y Y X ∗ X X ∗ X Dgh h∗ Dfg DgX g ∗ DX −−− Dfgh h Dg Dg g D ← φ f,g,h

Checking this commutativity is left to the few (if any) extremely patient readers who might be willing to do it. Again, the complete expansion according to deﬁnitions is intimidating—but the analogous associativity statement is said in [C, p. 136] to be “straightforward to check.” Pseudofunctoriality being thus established, one must now verify that the isomorphism in (1) above is pseudofunctorial; that on proper maps, * and ! are adjoint as pseudofunctors (see (2) and (3.6.7(d)); that the isomorphism in (3) extends to an isomorphism of Dualizing Complexes; and that βσ is as described in Theorem (4.10.4). And ﬁnally, the uniqueness (up to isomorphism) of the pseudofunctor ! can be veriﬁed as at the beginning of the proof of (4.8.4). Each of these veriﬁcations amounts, upon expansion according to deﬁnitions, to checking commutativity of a rather unpleasant diagram. For the purposes of these Notes, Thm. (4.10.4) is not one of the “main results” referred to in Section (0.3) of the Introduction; so I leave it at that.

Introduction

Let S be a scheme, G a ﬂat S-group scheme of ﬁnite type, X and Y noetherian S-schemes with G-actions, and f : X → Y a ﬁnite-type separated G-morphism. The purpose of these notes is to construct an equivariant version of the twisted inverse functor f ! and study its basic properties. One of the main motivations of the work is applications to invariant theory. As an example, we give a short proof of a generalized version of Watanabe’s theorem on the Gorenstein property of invariant subrings [45]. Also, there might be some meaning in formulating the equivariant duality theorem, of which Serre duality for representations of reductive groups (see [21, (II.4.2)]) is a special case, in a reasonably general form. As a byproduct, we give some foundations for G-equivariant sheaf theory. More generally, we discuss sheaves over diagrams of schemes. In the case where G is trivial, f ! is deﬁned as follows. For a scheme Z, we denote the category of OZ -modules by Mod(Z). By deﬁnition, a plump subcategory of an abelian category is a non-empty full subcategory which is closed under kernels, cokernels, and extensions [26, (1.9.1)]. We denote the plump subcategory of Mod(Z) consisting of quasi-coherent OZ -modules by Qch(Z). By Nagata’s compactiﬁcation theorem [34], [27], there exists some factorization i ¯ p →X − →Y X− such that p is proper and i an open immersion. We call such a factorization + + a compactiﬁcation. We deﬁne f ! : DQch(Y ) (Mod(Y )) → DQch(X) (Mod(X)) to + + ¯ be the composite i∗ p× , where p× : DQch(Y ¯ (Mod(X)) ) (Mod(Y )) → DQch(X) is the right adjoint of Rp∗ , and i∗ is the restriction. This deﬁnition of f ! is independent of the choice of compactiﬁcation. In order to consider a non-trivial G, we need to replace Qch(X) and Mod(X) by some appropriate categories which respect G-actions. The category Qch(G, X) which corresponds to Qch(X) is fairly well-known. It is J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

267

268

Introduction

the category of G-linearized quasi-coherent OX -modules deﬁned by Mumford [32]. The category Qch(G, X) is equivalent to the category of quasi-coherent sheaves over the diagram of schemes ⎞ ⎛ 1G ×a −−→ a −−→ ⎟ ⎜ M μ×1X (X) := ⎝G ×S G ×S X −−→ BG G ×S X p2 X ⎠ , −−→ p23 −−→ where a : G × X → X is the action, μ : G × G → G the product, and p23 and p2 are appropriate projections. Thus it is natural to embed the cateM gory Qch(G, X) into the category of all OBGM (X) -modules Mod(BG (X)), and M Mod(BG (X)) is a good substitute of Mod(X). As G is ﬂat, Qch(G, X) is a M (X)), and we may consider the triangulated plump subcategory of Mod(BG M subcategory DQch(G,X) (Mod(BG (X))). However, our construction utilizes an intermediate category Lqc(G, X) (the category of locally quasi-coherent sheaves), and is not an obvious interpretation of the non-equivariant case. M (X)) → Mod(X), Note that there is a natural restriction functor Mod(BG which sends Qch(G, X) to Qch(X). This functor is regarded as the forgetful functor, forgetting the G-action. The equivariant duality theorem which we are going to establish must be compatible with this restriction functor, otherwise the theory would be something diﬀerent from the usual scheme theory and probably useless. Most of the discussion in these notes treats more general diagrams of schemes. This makes the discussion easier, as some of the important properties are proved by induction on the number of objects in the diagram. Our main construction and theorems are only for the class of ﬁnite diagrams of M (X). schemes of certain type, which contains the diagrams of the form BG In Chapters 1–3, we review some general facts on homological algebra. In Chapter 1, we give some basic facts on commutativity of various diagrams of functors derived from an adjoint pair of almost-pseudofunctors over closed symmetric monoidal categories. In Chapter 2, we give basics about sheaves on ringed sites. In Chapter 3, we give some basics about unbounded derived categories. The construction of f ! is divided into ﬁve steps. The ﬁrst is to study the functoriality of sheaves over diagrams of schemes. Chapters 4–7 are devoted to this step. The second is the derived version of the ﬁrst step. This will be done in Chapters 8, 13, and 14. Note that not only the categories of all module sheaves Mod(X• ) and the category of quasi-coherent sheaves Qch(X• ), but also the category of locally quasi-coherent sheaves Lqc(X• ) plays an important role in our construction. The third is to prove the existence of the right adjoint p× • of R(p• )∗ for (componentwise) proper morphism p• of diagrams of schemes. This is not so diﬃcult, and is done in Chapter 17. We use Neeman’s existence theorem on the right adjoint of triangulated functors. Not only to utilize Neeman’s theorem, but to calculate composites of various left and right derived functors,

Introduction

269

it is convenient to utilize unbounded derived functors. A short survey on unbounded derived functors is given in Chapter 3. The fourth step is to prove various commutativities related to the welldeﬁnedness of the twisted inverse pseudofunctors, Chapters 16, 18, and 19. Among them, the compatibility with restrictions (Proposition 18.14) is the key to our construction. Given a separated G-morphism of ﬁnite type f : M (f ) : X → Y between noetherian G-schemes, the associated morphism BG M M BG (X) → BG (Y ) is cartesian, see (4.2) for the deﬁnition. If we could ﬁnd a compactiﬁcation p• i• M M (X)− →Z• −→BG (Y ) BG such that p• is proper and cartesian, and i• an image-dense open immersion, then the construction of f ! and the proof of commutativity of various diagrams would be very easy. However, it seems that this is almost the same as the problem of equivariant compactiﬁcations. Equivariant compactiﬁcations are known to exist only in very restricted cases, see [40]. We avoid this diﬃcult open problem, and prove the commutativity of various diagrams without assuming that p• is cartesian. The ﬁfth part is the existence of a factorization f• = p• i• , where p• is proper and i• an image-dense open immersion. This is easily done utilizing Nagata’s compactiﬁcation theorem, and is done in Chapter 20. This completes the basic construction of the equivariant twisted inverse pseudofunctor. Theorem 20.4 is our main theorem. In Chapters 21–28, we prove equivariant versions of most of the known results on twisted inverses including equivariant Grothendieck duality and the ﬂat base change, except that equivariant dualizing complexes are treated later. We also prove that the twisted inverse functor preserves quasi-coherence of cohomology groups. As we already know the corresponding results on single schemes and the commutativity with restrictions, this consists in straightforward (but not easy) checking of commutativity of various diagrams of functors. Almost all results above are valid for any diagram of noetherian schemes with ﬂat arrows over a ﬁnite ordered category. Although our construction can M (X), some readers might ask why this were be done using the diagram BG not done over the simplicial scheme BG (X) associated with the action of G on X. We explain the simplicial method and the related descent theory in Chapters 9 and 10. In the literature, it seems that equivariant sheaves with respect to the action of G on X has been regarded as equivariant sheaves on the diagram BG (X), see for example, [6, Appendix B]. The relation beM (X) and Y• := BG (X) is subtle. The category Qch(X• ) and tween X• := BG Qch(Y• ) are equivalent, and the category of equivariant modules EM(X• ) and EM(Y• ) are equivalent. However, I do not know anything about the reM (X) only because it lationship between Mod(X• ) and Mod(Y• ). We use BG is a diagram over a ﬁnite ordered category.

270

Introduction

In Chapter 11, we prove that if X• is a simplicial groupoid of schemes, d0 (1) and d1 (1) are concentrated, and X0 is concentrated, then Qch(X• ) is Grothendieck. If, moreover, X0 is noetherian, then Qch(X• ) is locally noetherian, and M ∈ Qch(X• ) is a noetherian object if and only if M0 is coherent. In Chapter 12, we study groupoids of schemes and their relations with simplicial groupoids of schemes. In Chapter 15, we compare the two derived functors − (Qch(X• )) and N ∈ D+ (Qch(X• )). One is of Hom•OX• (M, N ) for M ∈ DCoh the derived functor taken in D(Mod(X• )) and the other is the derived functor taken in D(Qch(X• )). They coincide under mild noetherian hypothesis. Finally, we consider the group actions on schemes. In Chapter 29, we give a groupoid of schemes associated with a group action. The equivariant duality theorem for group actions is established. In Chapter 30, we prove that the equivariant twisted inverse is compatible with the derived G-invariance. In Chapter 31, we give a deﬁnition of the equivariant dualizing complexes. As an application, we give a short proof of a generalized version of Watanabe’s theorem on the Gorenstein property of invariant subrings in Chapter 32. In Chapter 33, we give some other examples of diagrams of schemes. Acknowledgement The author is grateful to Professor Luchezar Avramov, Professor Ryoshi Hotta, Professor Joseph Lipman, and Professor Jun-ichi Miyachi for valuable advice. Special thanks are due to Professor Joseph Lipman for correcting English of this introduction.

Chapter 1

Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors

(1.1) Let S be a category. A (covariant) almost-pseudofunctor # on S assigns to each object X ∈ S a category X# , to each morphism f : X → Y in S a functor f# : X# → Y# , and for each X ∈ S, a natural isomorphism eX : IdX# → (idX )# is assigned, and for each composable pair of morphisms f

g

X− →Y − → Z, a natural isomorphism ∼ =

c = cf,g : (gf )# − → g# f# is given, and the following conditions are satisﬁed. cf,id

1. For any f : X → Y , the map f# IdX# = f# = (f idX )# −−−→ f# (idX )# agrees with f# eX . cid,f 2. For any f : X → Y , the map IdY# f# = f# = (idY f )# −−−→ (idY )# f# agrees with eY f# . f

g

h

3. For any composable triple of morphisms X − →Y − →Z− → W , the diagram cf,hg

(hgf )# −−−→ (hg)# f# ↓ cgf,h ↓ cg,h cf,g h# (gf )# −−→ h# g# f# commutes. op op If (?)# is an almost-pseudofunctor on S, then (?)op # given by (X)# = X# op op op for X ∈ S and (f )op # = f# : X# → Y# for a morphism f : X → Y of S −1 −1 together with e and c is again an almost-pseudofunctor on S. Letting eX = id for each X, a pseudofunctor [26, (3.6.5)] is an almost-pseudofunctor.

(1.2) Let ∗ be an almost-pseudofunctor on S. Let gf = f ′ g ′ be a commutative diagram in S. The composite isomorphism c−1

c

→f∗′ g∗′ g∗ f∗ −−→(gf )∗ = (f ′ g ′ )∗ − is also denoted by c = c(gf = f ′ g ′ ), by abuse of notation. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

271

272

1 Commutativity of Diagrams

Lemma 1.3. Let f g ′ = gf ′ and hg ′′ = g ′ h′ be commutative squares in S. Then the diagram c

c

(f h)∗ g∗′′ − → f∗ h∗ g∗′′ − → f∗ g∗′ h′∗ ↓c ↓c c

g∗ (f ′ h′ )∗ −−−−− −−−−−→ g∗ f∗′ h′∗ is commutative. ⊓ ⊔

Proof. Easy.

Lemma 1.4. Let f g ′ = gf ′ and f ′ h′ = hf ′′ be commutative squares in S. Then the diagram c

c

f∗ (g ′ h′ )∗ − → g∗ f∗′ h′∗ → f∗ g∗′ h′∗ − ↓c ↓c c

(gh)∗ f∗′′ −−−−− −−−−−→ g∗ h∗ f∗′′ is commutative. ⊓ ⊔

Proof. Follows from Lemma 1.3.

(1.5) A contravariant almost-pseudofunctor (?)# is deﬁned similarly. For X ∈ S, a category X # is assigned, and for a morphism f : X → Y in S, a functor f # : Y # → X # is assigned, and for X ∈ S, a natural isomorphism f = fX : id# X → IdX # is assigned, and for a composable pair of morphisms f

g

→Y − → Z, a natural isomorphism X− df,g : f # g # → (gf )# is given, and ((?)# )op together with (fX )X∈S and (dg,f ) is a covariant almostpseudofunctor on S op . If (?)# is a contravariant almost-pseudofunctor on S, then ((?)# )op together with f−1 and (d−1 f,g ) is again a contravariant almostpseudofunctor on S. (1.6) Let (?)∗ be a contravariant almost-pseudofunctor on S. For a commutative diagram gf = f ′ g ′ in S, the composite map d−1

d

(g ′ )∗ (f ′ )∗ − →(f ′ g ′ )∗ = (gf )∗ −−→f ∗ g ∗ is also denoted by d = d(gf = f ′ g ′ ), by abuse of notation. (1.7) Let ∗ and # be almost-pseudofunctors on S such that X∗ = X# . A morphism of almost-pseudofunctors υ : ∗ → # is a family of natural maps υf : f∗ → f# (one for each f ∈ Mor(S)) such that for any X ∈ S the diagram id

IdX∗ −→ IdX# ↓e ↓e υ (idX )∗ − → (idX )#

1 Commutativity of Diagrams

273 f

g

commutes, and for any composable pair of morphisms X − → Y − → Z, the diagram υ / (gf )# (gf )∗ c

c

g∗ f∗

g∗ υ

υ

/ g∗ f#

/ g# f#

commutes. If υ is a morphism of almost-pseudofunctors, and υf is a natural isomorphism for each f , then we say that υ is an isomorphism of almostpseudofunctors. (1.8) Let # be an almost-pseudofunctor on S. We deﬁne ∗ by X∗ = X# for X ∈ S, (idX )∗ = IdX# for X ∈ S, and f∗ = f# if f = idX for any X. For a f

g

composable pair of morphisms X − →Y − → Z, we deﬁne cf,g : (gf )∗ → g∗ f∗ to be the identity map if f = idX or g = idY . If Z = X, g = f −1 and f = idX , then cf,f −1 : (idX )∗ → f∗−1 f∗ is deﬁned to be the composite cf,f −1

e

−1 → (idX )# −−−−→ f# f# . IdX# −−X

Otherwise, we deﬁne cf,g to be the original cf,g of #. It is easy to see that ∗ is a pseudofunctor on S. We deﬁne υ : f∗ → f# to be eX if f = idX , and the identity of f# otherwise. Then υ is an isomorphism of almost-pseudofunctors. Thus any almostpseudofunctor is isomorphic to a pseudofunctor. We call ∗ the associated pseudofunctor of the almost-pseudofunctor #. Similarly, any contravariant almost-pseudofunctor is isomorphic to a contravariant pseudofunctor, and the associated contravariant pseudofunctor of a contravariant almost-pseudofunctor is deﬁned. (1.9) In this paper, various (diﬀerent) adjoint pairs appears almost everywhere. By abuse of notation, the unit (resp. the counit) of adjunction is usually simply denoted by the same symbol u (resp. ε). When we mention an adjunction of functors, we implicitly (or occasionally explicitly) ﬁx the unit u and the counit ε. (1.10) We need the notion of the conjugation from [29, (IV.7)] and [26, (3.3.5)]. Let X and Y be categories, and f∗ and g∗ functors X → Y with respective left adjoints f ∗ and g ∗ . By Hom, we denote the set of natural transformations. Then Φ : Hom(f∗ , g∗ ) → Hom(g ∗ , f ∗ ) given by u

α

ε

→ g ∗ f∗ f ∗ − → g ∗ g∗ f ∗ − → f∗ Φ(α) : g ∗ − and Ψ : Hom(g ∗ , f ∗ ) → Hom(f∗ , g∗ ) given by u

β

ε

Ψ(β) : f∗ − → g∗ → g∗ g ∗ f∗ − → g∗ f ∗ f∗ −

274

1 Commutativity of Diagrams

are inverse each other. We say that Φ(α) is left conjugate to α, Ψ(β) is right conjugate to β, and α and Φ(α) are conjugate. The identity map in Hom(f∗ , f∗ ) is conjugate to the identity in Hom(f ∗ , f ∗ ). Let h∗ : X → Y be a functor with the left adjoint h∗ . If α ∈ Hom(f∗ , g∗ ) and α′ ∈ Hom(g∗ , h∗ ), and β ∈ Hom(g ∗ , f ∗ ) and β ′ ∈ Hom(h∗ , g ∗ ) are their respective conjugates, then α′ ◦ α and β ◦ β ′ are conjugate. In particular, α ∈ Hom(f∗ , g∗ ) is an isomorphism if and only if its conjugate β ∈ Hom(g ∗ , f ∗ ) is an isomorphism, and if this is the case, α−1 and β −1 are conjugate. Lemma 1.11. Let X, Y and f∗ , g∗ , f ∗ , g ∗ be as above. Let α ∈ Hom(f∗ , g∗ ), and β ∈ Hom(g ∗ , f ∗ ). Then the following are equivalent. 1 α and β are conjugate. 2 One of the following diagrams commutes. g ∗ f∗

α

/ g ∗ g∗

ε

/1

ε

β

f ∗ f∗

1

u

/ f∗ f ∗

β

/ g∗ f ∗

u

α

g∗ g ∗

⊓ ⊔

Proof. See [29, (IV.7), Theorem 2].

(1.12) Let S be a category, (?)∗ be an almost-pseudofunctor on S. Let (?)∗ be a left adjoint of (?)∗ . Namely, for each morphism f of S, we have a left adjoint f ∗ of f∗ (and the explicitly given unit u : 1 → f∗ f ∗ and counit ε : f ∗ f∗ → 1). For X ∈ S, X ∗ is deﬁned to be X∗ . For composable two morphisms f and g in S, we denote the map f ∗ g ∗ → (gf )∗ conjugate to c : (gf )∗ → g∗ f∗ by d = df,g . Thus df,g is the composite u

c

ε

ε

f ∗ g∗ − → f ∗ g ∗ (gf )∗ (gf )∗ − → f ∗ g ∗ g∗ f∗ (gf )∗ − → f ∗ f∗ (gf )∗ − → (gf )∗ . Being the conjugate of an isomorphism, d is an isomorphism. As d−1 is conjugate to c−1 , it is the composite u

c−1

u

ε

(gf )∗ − → (gf )∗ g∗ g ∗ − → (gf )∗ g∗ f∗ f ∗ g ∗ −−→ (gf )∗ (gf )∗ f ∗ g ∗ − → f ∗ g∗ . Lemma 1.13. Let f and g be morphisms in S, and assume that gf is deﬁned. Then the composite u

u

c−1

d

1− → g∗ g ∗ − → g∗ f∗ f ∗ g ∗ −−→ (gf )∗ f ∗ g ∗ − → (gf )∗ (gf )∗ is u. Proof. Follows immediately from Lemma 1.11.

⊓ ⊔

1 Commutativity of Diagrams

275

Lemma 1.14. Let f and g be morphisms in S, and assume that gf is deﬁned. Then the composite d−1

c

ε

ε

→ (gf )∗ g∗ f∗ −−→ f ∗ g ∗ g∗ f∗ − → f ∗ f∗ − →1 (gf )∗ (gf )∗ − is ε. ⊓ ⊔

Proof. Follows immediately from Lemma 1.11.

(1.15) For X ∈ S, IdX ∗ is left adjoint to IdX∗ (with u = id and ε = id). The morphism left conjugate to eX : IdX∗ → (idX )∗ is denoted by fX : (idX )∗ → IdX ∗ . Namely, fX is the composite id

e

ε

(idX )∗ −→ (idX )∗ IdX∗ IdX ∗ − → (idX )∗ (idX )∗ IdX ∗ − → IdX ∗ . (1.16) Let S, (?)∗ , and (?)∗ be as above. Then it is easy to see that (?)∗ together with d and f deﬁned above forms a contravariant almostpseudofunctor. We say that ((?)∗ , (?)∗ ) is an adjoint pair of almostpseudofunctors on S, with this situation. For a commutative diagram gf = f ′ g ′ in S, the composite maps c(gf = f ′ g ′ ) and d(gf = f ′ g ′ ) are conjugate. ∗ op ∗ The opposite ((?)op ∗ , ((?) ) ) of ((?) , (?)∗ ) is an adjoint pair of almostop pseudofunctors on S . cf,g , df,g , u : 1 → f∗ f ∗ , and ε : f ∗ f∗ → 1 of ∗ op ((?)∗ , (?)∗ ) correspond to dg,f , cg,f , ε, and u of ((?)op ∗ , ((?) ) ), respectively. (1.17) Let S be as above, and (?)∗ a given contravariant almostpseudofunctor, and (?)∗ its right adjoint. Then ((?)∗ )op is a covariant ∗ op almost-pseudofunctor on S op as in (1.5). Then ((?)op ∗ , ((?) ) ) is an adjoint pair of almost-pseudofunctors. So ((?)∗ , (?)∗ ) = (((?)∗ )opop , (?)∗opop ) is also an adjoint pair of almost-pseudofunctors. (1.18) Let S be as above, and (?)∗ a given covariant almost-pseudofunctor, and (?)! its right adjoint. For composable morphisms f and g, deﬁne df,g : f ! g ! → (gf )! to be the map right conjugate to cf,g : (gf )∗ → g∗ f∗ . For X ∈ S, deﬁne fX : (idX )! → IdX ! to be the map right conjugate to eX : IdX∗ → (idX )∗ . Then it is straightforward to check that (?)! is a contravariant almost-pseudofunctor on S. We say that ((?)∗ , (?)! ) is an opposite adjoint pair of almost-pseudofunctors on S. Opposite adjoint pair is also obtained from a given contravariant almost-pseudofunctor (?)! and its left adjoint (?)∗ . Note that ((?)# , (?)# ) is an adjoint pair of almost-pseudofunctors on S if and only # op if ((?)op # , ((?) ) ) is an opposite adjoint pair of almost-pseudofunctors on S. (1.19) Let ((?)∗ , (?)∗ ) be an adjoint pair of pseudofunctors on S. Let σ = (f g ′ = gf ′ ) be a commutative diagram in S. Lemma 1.20. The following composite maps agree: u

c

ε

u

d

ε

1 g ∗ f∗ − →f∗′ (g ′ )∗ ; →g ∗ g∗ f∗′ (g ′ )∗ − →g ∗ f∗ g∗′ (g ′ )∗ − 2 g ∗ f∗ − →f∗′ (f ′ )∗ g ∗ f∗ − →f∗′ (g ′ )∗ f ∗ f∗ − →f∗′ (g ′ )∗ .

276

1 Commutativity of Diagrams

For the proof and more information, see [26, (3.7.2)]. (1.21) We denote the composite map in the lemma by θ(σ) or θ, and call it Lipman’s theta. Note that θ(f g ′ = gf ′ ) of ((?)∗ , (?)∗ ) is θ(g ′ f = f ′ g) in the ∗ op opposite ((?)op ∗ , ((?) ) ). Lemma 1.22. Let f g ′ = gf ′ and hg ′′ = g ′ h′ be commutative squares in S. Then the diagram g ∗ (f h)∗ ↓θ

c

θ

− → g ∗ f∗ h∗ − → f∗′ (g ′ )∗ h∗ ↓θ c

(f ′ h′ )∗ (g ′′ )∗ −−−−− −−−−−→ f∗′ h′∗ (g ′′ )∗ is commutative. ⊓ ⊔

Proof. See [26, (3.7.2)].

Lemma 1.23. Let σ = (f g ′ = gf ′ ) and τ = (f ′ h′ = hf ′′ ) be commutative diagrams in S. Then the composite d−1

θ

θ

d

(gh)∗ f∗ −−→ h∗ g ∗ f∗ − → h∗ f∗′ (g ′ )∗ − → f∗′′ (h′ )∗ (g ′ )∗ − → f∗′′ (g ′ h′ )∗ agrees with θ for f (g ′ h′ ) = (gh)f ′′ . Proof. This is the ‘opposite assertion’ of Lemma 1.22. Namely, Lemma 1.22 ∗ op ⊓ ⊔ applied to the opposite pair ((?)op ∗ , ((?) ) ) is this lemma. Lemma 1.24. Let f g ′ = gf ′ be a commutative diagram in S. Then the composite u θ c f∗ − → g∗ g ∗ f∗ − → g∗ f∗′ (g ′ )∗ − → f∗ g∗′ (g ′ )∗ is u. Proof. Obvious by the commutativity of the diagram u

c

/ g∗ g ∗ g∗ f∗′ (g ′ )∗ . / g∗ g ∗ f∗ g∗′ (g ′ )∗ g; ∗ g ∗ f∗ w nn6 w unnnn u www ε n w nn ww nnn ww c id / g f ′ (g ′ )∗ / g∗ f∗′ (g ′ )∗ f∗ u / f∗ g∗′ (g ′ )∗ WW WWWWW ∗ ∗ WWWWW WWWWW c WWWWW id WWWW+ f∗ g∗′ (g ′ )∗

⊓ ⊔

Lemma 1.25. Let f g ′ = gf ′ be a commutative diagram in S. Then the composite c θ ε g ∗ g∗ f∗′ − → g ∗ f∗ g∗′ − → f∗′ (g ′ )∗ g∗′ − → f∗′ is ε.

1 Commutativity of Diagrams

277

Proof. Obvious by the commutativity of the diagram u

c

/ g ∗ f∗ g∗′ (g ′ )∗ g∗′ / g ∗ f∗ g∗′ g ∗ g∗ f∗′ LL NNN LL NNN LL ε N LL id NNN LL & LL LL g ∗ f∗ g∗′ LL id LL LL LL c LL L% g ∗ g∗ f∗′

c

/ g ∗ g∗ f∗′ (g ′ )∗ g∗′ .

⊓ ⊔

ε

f∗′ (g ′ )∗ g∗′ ε

/ f∗′

ε

Lemma 1.26. Let f g ′ = gf ′ be a commutative diagram in S. Then the composite u θ d g∗ − → g ∗ f∗ f ∗ − → f∗′ (g ′ )∗ f ∗ − → f∗′ (f ′ )∗ g ∗ is u. ⊓ ⊔

Proof. This is the opposite version of Lemma 1.25.

Lemma 1.27. Let f g ′ = gf ′ be a commutative diagram in S. Then the composite d

θ

ε

(g ′ )∗ f ∗ f∗ − → (f ′ )∗ g ∗ f∗ − → (f ′ )∗ f∗′ (g ′ )∗ − → (g ′ )∗ is ε. Proof. This is the opposite assertion of Lemma 1.24.

⊓ ⊔

(1.28) We say that (?)∗ is a covariant symmetric monoidal almostpseudofunctor on a category S, if (?)∗ is an almost-pseudofunctor on S, and the following conditions are satisﬁed. For each X ∈ S, X∗ = (X∗ , ⊗, OX , α, λ, γ, [?, ∗], π) is a (symmetric monoidal) closed category (see e.g., [26, (3.5.1)]), where X∗ on the right-hand side is the underlying category, ⊗ : X∗ × X∗ → X∗ the product structure, OX ∈ X∗ the unit object, α : (a ⊗ b) ⊗ c ∼ = a ⊗ (b ⊗ c) the associativity isomorphism, λ : OX ⊗ a ∼ = a the left unit isomorphism, op γ : a⊗b ∼ = b⊗a the twisting (symmetry) isomorphism, [?, −] : X∗ ×X∗ → X∗ the internal hom, and π : X∗ (a ⊗ b, c) ∼ = X∗ (a, [b, c])

(1.29)

the associative adjunction isomorphism of X, respectively. For a morphism f : X → Y in S, f∗ : X∗ → Y∗ is a symmetric monoidal functor [26, (3.4.2)], and eX : IdX∗ → (idX )∗ and cf,g are morphisms of symmetric monoidal functors, see [26, (3.6.7)].

278

1 Commutativity of Diagrams

(1.30) The unit map and the counit map arising from the adjunction (1.29) are denoted (by less worse abuse of notation, not using u or ε) by (the same symbol) tr : a → [b, a ⊗ b] (the trace map) and ev : [b, c] ⊗ b → c

(the evaluation map),

respectively. (1.31) By the deﬁnition of closed categories, the associative adjunction isomorphism (1.29) is natural on a, b and c. So not only that tr and ev are natural transformations, we have the following. Lemma 1.32. For a, b, b′ , c ∈ X∗ and a morphism ϕ : b → b′ , the diagrams a

tr

/ [b, a ⊗ b]

[ϕ,1a ⊗1b ]

[b, a ⊗ b′ ]

[1b ,1a ⊗ϕ]

tr

[b′ , a ⊗ b′ ]

[b′ , c] ⊗ b

/

[ϕ,1c ]⊗1b

/

[b, c] ⊗ b ev

[1b′ ,1c ]⊗ϕ

[b′ , c] ⊗ b′

ev

/c

are commutative. Proof. We only prove the commutativity of the ﬁrst diagram. By the naturality of π, the diagram (1a ⊗ϕ)∗

(1a ⊗ϕ)∗

[1b ,1a ⊗ϕ]∗

[ϕ,1a ⊗1 ′ ]∗

X∗ (a ⊗ b, a ⊗ b) −−−−−→ X∗ (a ⊗ b, a ⊗ b′ ) ←−−−−− X∗ (a ⊗ b′ , a ⊗ b′ ) ↓π ↓π ↓π X∗ (a, [b, a ⊗ b]) −−−−−−−→ X∗ (a, [b, a ⊗ b′ ]) ←−−−−−b−− X∗ (a, [b′ , a ⊗ b′ ]) is commutative. Considering the image of 1a⊗b ∈ X∗ (a ⊗ b, a ⊗ b) and 1a⊗b′ ∈ X∗ (a ⊗ b′ , a ⊗ b′ ) in X∗ (a, [b, a ⊗ b′ ]), we have [1b , 1a ⊗ ϕ] ◦ tr = π(1a ⊗ ϕ) = [ϕ, 1a ⊗ 1b ] ◦ tr . This is what we wanted to prove.

⊓ ⊔

(1.33) Let f : X → Y be a morphism. Then f∗ is a symmetric monoidal functor. The natural map f∗ a ⊗ f∗ b → f∗ (a ⊗ b) is denoted by m = m(f ), and the map OY → f∗ OX is denoted by η = η(f ).

1 Commutativity of Diagrams

279

A covariant symmetric monoidal almost-pseudofunctor which is a pseudofunctor is called a covariant symmetric monoidal pseudofunctor. Let ⋆ be the associated pseudofunctor of the symmetric monoidal almost-pseudofunctor ∗. Then letting the closed structure of X⋆ be the same as that of X∗ , and letting m : (idX )⋆ a ⊗ (idX )⋆ b → (idX )⋆ (a ⊗ b) and η : OX → (idX )⋆ OX to be the identity morphisms, ⋆ is a symmetric monoidal pseudofunctor which is isomorphic to ∗ as a symmetric monoidal almost-pseudofunctor. (1.34)

Let f : X → Y be a morphism in S. The composite natural map tr

via m

via ev

f∗ [a, b]− →[f∗ a, f∗ [a, b] ⊗ f∗ a]−−−→[f∗ a, f∗ ([a, b] ⊗ a)]−−−−→[f∗ a, f∗ b] is denoted by H. (1.35) G. Lewis proved a theorem which guarantee that some diagrams involving two symmetric monoidal closed categories and one symmetric monoidal functor commute [25]. By Lewis’s result, we have that the following diagrams are commutative for any morphism f : X → Y (also checked by a direct computation). H⊗1

f∗ [a, b] ⊗ f∗ a −−−→ [f∗ a, f∗ b] ⊗ f∗ a ↓m ↓ ev f∗ ev

f∗ ([a, b] ⊗ a) −−−→

(1.36)

f∗ b

f∗ tr

f∗ a −−−→ f∗ [b, a ⊗ b] ↓ tr ↓H m [f∗ b, f∗ a ⊗ f∗ b] − → [f∗ b, f∗ (a ⊗ b)] λ−1

η

f∗ a −−→ OY ⊗ f∗ a − → ↓1

f∗ OX ⊗ f∗ a

ev

(1.37)

via tr

−−−−→ f∗ [a, OX ⊗ a] ⊗ f∗ a ↓ via λ H⊗1

f∗ a ←−−−−−−−−−−−−− [f∗ a, f∗ a] ⊗ f∗ a ←−−−

(1.38)

f∗ [a, a] ⊗ f∗ a

Lemma 1.39. Let f : X → Y and g : Y → Z be morphisms in S. Then the diagram H H → g∗ [f∗ a, f∗ b] − → [g∗ f∗ a, g∗ f∗ b] g∗ f∗ [a, b] − ↑c ↑ [c−1 , c] H

(gf )∗ [a, b] −−−−−−− −−−−−−−→ [(gf )∗ a, (gf )∗ b] is commutative. Proof. Consider the diagram

280

H / [(gf )∗ a, (gf )∗ b] ⊗ g∗ f∗ a c / [(gf )∗ a, g∗ f∗ b] ⊗ g∗ f∗ a TTTT k k k TTTTc−1 c kkk TTTT k −1 −1 −1 (b) (c) k c c c k k TTTT (a) k ukkk c⊗c * c H o o / / [(gf )∗ a, g∗ f∗ b] ⊗ (gf )∗ a [(gf )∗ a, (gf )∗ b] ⊗ (gf )∗ a (gf )∗ [a, b] ⊗ (gf )∗ a g∗ f∗ [a, b] ⊗ g∗ f∗ a TTTT (d) [g∗ f∗ a, g∗ f∗ b] ⊗ g∗ f∗ a UUUU TTTT SSS SSS m U T T U TTTev TTTmT UUev SSS UUUU TTTT ev (f) (e) TTTT SSS UUUU (g) TTTT SS) TT) U* T* c ev / (gf )∗ b (gf )∗ ([a, b] ⊗ a) g∗ (f∗ [a, b] ⊗ f∗ a) H dd42/ g∗ f∗ b IITTTT ddddjdjdjdjjj d d II TTT m d d d d II TTTT c (i) (h) ddevddddd jjjjjjj II ddddddd d d jj d II H TTTT) d d d d jjjj II j g∗ f∗ ([a, b] ⊗ a) g∗ [f∗ a, f∗ b] ⊗ g∗ f∗ a j j II YYYYYY jjj ev II YYYYYY m jjjj II j YYYYYY j j (j) YYYYYY I jjj YYYY, II$ jjjj (gf )∗ [a, b] ⊗ g∗ f∗ a

ED

H

@A

(k)

BC

1 Commutativity of Diagrams

g∗ ([f∗ a, f∗ b] ⊗ f∗ a)

1 Commutativity of Diagrams

281

The commutativity of (a), (b), (c), (g), (h) and (i) is trivial. The commutativity of (d) is Lemma 1.32. (e) is commutative, since c is assumed to be a morphism of symmetric monoidal functors, see [26, (3.6.7.2)]. The commutativity of (f), (j), and (k) is the commutativity of (1.36). Thus the whole diagram is commutative. Taking the adjoint, we get the commutativity of the diagram in the lemma. ⊓ ⊔ (1.40) Let (?)∗ be a covariant symmetric monoidal almost-pseudofunctor on S. Let (?)∗ be its left adjoint. Namely, for each morphism f of S, a left adjoint f ∗ of f∗ (and the unit map 1 → f∗ f ∗ and the counit map f ∗ f∗ → 1) is given. For a morphism f : X → Y , the map f ∗ OY → OX adjoint to η : OY → f∗ OX is denoted by C. The composite map u⊗u

m

ε

→f ∗ f∗ (f ∗ a ⊗ f ∗ b)− →f ∗ a ⊗ f ∗ b f ∗ (a ⊗ b)−−−→f ∗ (f∗ f ∗ a ⊗ f∗ f ∗ b)−

(1.41)

is denoted by Δ. Almost by deﬁnition, the diagrams a⊗b ↓u ∗

u⊗u

−−−→ f∗ f ∗ a ⊗ f∗ f ∗ b ↓m ∆

∗

(1.42)

∗

f∗ f (a ⊗ b) − → f∗ (f a ⊗ f b) and m

f ∗ (f∗ a ⊗ f∗ b) − → f ∗ f∗ (a ⊗ b) ↓∆ ↓ε ε⊗ε ∗ ∗ f f∗ a ⊗ f f∗ b −−→ a⊗b

(1.43)

are commutative. If (?)⋆ is the associated pseudofunctor of (?)∗ and (?)⋆ is the associated contravariant pseudofunctor of (?)∗ , then ((?)⋆ , (?)⋆ ) is a monoidal adjoint pair. Note that u : Id → f⋆ f ⋆ is the identity if f = idX for some X, and u agrees with the unit map for the original adjoint pair ((?)∗ , (?)∗ ) otherwise. Similarly for the counit map ε. Lemma 1.44. Let ((?)∗ , (?)∗ ) be a monoidal adjoint pair on S. Let σ = (f g ′ = gf ′ ) be a commutative diagram in S. Then the diagram m

f ∗ (g∗ a ⊗ g∗ b) − → ↓∆ θ⊗θ

f ∗ g∗ (a ⊗ b)

θ

− →

g∗′ (f ′ )∗ (a ⊗ b) ↓∆

m

→ g∗′ ((f ′ )∗ a ⊗ (f ′ )∗ b) f ∗ g∗ a ⊗ f ∗ g∗ b −−→ g∗′ (f ′ )∗ a ⊗ g∗′ (f ′ )∗ b − is commutative. Proof. Utilize the commutativity of (1.42) and (1.43).

⊓ ⊔

282

(1.45)

1 Commutativity of Diagrams

For a symmetric monoidal category X∗ , the composite γ

λ

a ⊗ OX − → OX ⊗ a − →a is called the right unit isomorphism, and is denoted by ρ. Let f∗ : X∗ → Y∗ be a symmetric monoidal functor. Then it is easy to see that the diagram f∗ a ⊗ OY

ρ

/ f∗ a O ρ

1⊗η

f∗ a ⊗ f∗ OX

(1.46)

m

/ f∗ (a ⊗ OX )

is commutative for a ∈ X∗ . Lemma 1.47. Let f∗ : X∗ → Y∗ be a symmetric monoidal functor between closed categories. Then the diagram f∗ [OX , c ⊗ OX ] O

ρ

H

/ f∗ [OX , c]

/ [f∗ OX , f∗ c] η

tr tr

f∗ c

/ [OY , f∗ c ⊗ OY ]

ρ

/ [OY , f∗ c]

is commutative. Proof. Consider the diagram tr

GF

f∗ c tr

[f∗ OX , f∗ c ⊗ f∗ OX ] η

(c)

tr

(a) m

(d)

/ f∗ [OX , c ⊗ OX ] H

ρ

(b)

/ [f∗ OX , f∗ (c ⊗ OX )] η

/ f∗ [OX , c] . H ρ

(e)

/ [f∗ OX , f∗ c] η

@A

ρ m / [OY , f∗ c] / [OY , f∗ (c ⊗ OX )] [OY , f∗ c ⊗ f∗ OX ] 2 e e O eeeeee ρ eeeeeeee e e η e (f) e eeeeee eeeeee / [OY , f∗ c ⊗ OY ]

By the commutativity of (1.37), (a) commutes. Clearly, (b), (d), and (e) commute. By Lemma 1.32, (c) commutes. (f) is nothing but (1.46), and commutes. So the whole diagram commutes, and this is what we want to prove. ⊓ ⊔

1 Commutativity of Diagrams

283

Deﬁnition 1.48. A monoidal adjoint pair ((?)∗ , (?)∗ ) is said to be Lipman if Δ : f ∗ (a ⊗ b) → f ∗ a ⊗ f ∗ b and C : f ∗ OY → OX are isomorphisms for any morphism f : X → Y in S and any a, b ∈ Y∗ . (1.49) Note that Δ : f ∗ (a ⊗ b) → f ∗ a ⊗ f ∗ b is a natural isomorphism if and only if its right conjugate (see (1.10)) is an isomorphism. The right conjugate is the composite H

u

→[f∗ f ∗ b, f∗ c]− →[b, f∗ c]. f∗ [f ∗ b, c]− Let ((?)∗ , (?)∗ ) be a Lipman adjoint pair of monoidal almost-pseudofunctors over S. Then (?)∗ together with Δ−1 and C −1 form a covariant symmetric monoidal almost-pseudofunctor on S op . (1.50)

For a morphism f : X → Y in S, the composite map via ∆−1

via tr

via ev

f ∗ [a, b]−−−−→[f ∗ a, f ∗ [a, b] ⊗ f ∗ a]−−−−−→[f ∗ a, f ∗ ([a, b] ⊗ a)]−−−−→[f ∗ a, f ∗ b] is denoted by P . We can apply Lewis’s theorem to f ∗ . In particular, the following diagrams are commutative by (1.35) for a morphism f : X → Y . P ⊗1

f ∗ [a, b] ⊗ f ∗ a −−−→ [f ∗ a, f ∗ b] ⊗ f ∗ a ↓ ∆−1 ↓ ev f ∗ ev

∗

f ([a, b] ⊗ a) −−−→

∗

∗

f∗ b

f ∗ tr

f ∗a ↓ tr

f ∗ [b, a ⊗ b] ↓P

−−−→ ∆−1

∗

(1.51)

∗

(1.52)

∗

[f b, f a ⊗ f b] −−−→ [f b, f (a ⊗ b)] λ−1

C −1

f ∗ a −−→ OX ⊗ f ∗ a −−−→ ↓1

f ∗ OY ⊗ f ∗ a

ev

via tr

−−−−→ f ∗ [a, OY ⊗ a] ⊗ f ∗ a ↓ via λ P ⊗1

f ∗ a ←−−−−−−−−−−−−−−− [f ∗ a, f ∗ a] ⊗ f ∗ a ←−−−

f ∗ [a, a] ⊗ f ∗ a (1.53)

Lemma 1.54. Let f : X → Y and g : Y → Z be morphisms in S. Then the diagram P

P

f ∗ g ∗ [a, b] − → f ∗ [g ∗ a, g ∗ b] − → [f ∗ g ∗ a, f ∗ g ∗ b] −1 ↑d ↑ [d, d−1 ] P

(gf )∗ [a, b] −−−−−−− −−−−−−−→ [(gf )∗ a, (gf )∗ b] is commutative.

284

1 Commutativity of Diagrams

⊓ ⊔

Proof. Follows instantly by Lemma 1.39. Lemma 1.55. The diagram ev

[a, b] ⊗ a ↓ u⊗u

−−−−−−−−−−−−− −−−−−−−−−−−−−→ HP ⊗1

b ↓u

ev

f∗ f ∗ [a, b] ⊗ f∗ f ∗ a −−−−→ [f∗ f ∗ a, f∗ f ∗ b] ⊗ f∗ f ∗ a −→ f∗ f ∗ b is commutative. Proof. Consider the diagram [a, b] ⊗ a YYYY YYY

YYYYYY YYYYYYu YYYYYY YYYYYY YYY, m ∗ ∗ ∗ ∗ / f∗ (f [a, b] ⊗ f a) o f∗ f ∗ ([a, b] ⊗ a) f∗ f [a, b] ⊗ f∗ f a ∆ x xx (b) (d) P P xx x xx m / f∗ ([f ∗ a, f ∗ b] ⊗ f ∗ a) xxxx f∗ [f ∗ a, f ∗ b] ⊗ f∗ f ∗ a x ev xx x ev (c) (e) H xx xx { x ev u / f∗ f ∗ b o [f∗ f ∗ a, f∗ f ∗ b] ⊗ f∗ f ∗ a u⊗u

(a)

ED

ev

b.

Then (a), (c), and (d) are commutative by the commutativity of (1.42), (1.36), and (1.51), respectively. The commutativity of (b) and (e) is trivial. Thus the whole diagram is commutative, and the lemma follows. ⊓ ⊔ Lemma 1.56. The following diagrams are commutative. [a, b] ↓u

u

− →

[a, f∗ f ∗ b] ↑u

HP

f∗ f ∗ [a, b] −−→ [f∗ f ∗ a, f∗ f ∗ b] PH

f ∗ f∗ [a, b] −−→ [f ∗ f∗ a, f ∗ f∗ b] ↓ε ↓ε ε ∗ [a, b] − → [f f∗ a, b] Proof. We prove the commutativity of the ﬁrst diagram. Consider the diagram

GF tr

[a, b] u

/ [a, [a, b] ⊗ a]

tr

(d)

HP

ev

/ [a, b]

u

(b)

f∗ f ∗ [a, b]

ED

(a)

/ [a, f∗ f ∗ [a, b] ⊗ a] (c) WWWW WWWW u WWWW HP WWWW WWW+

tr / [a, [f∗ f ∗ a, f∗ f ∗ b] ⊗ a] (e) [a, f∗ f ∗ [a, b] ⊗ f∗ f ∗ a] UUUU WWWW WWWW u UUUUtr WWWW UUUU (f) HP WWWW UUUU WWW+ * u / [a, [f∗ f ∗ a, f∗ f ∗ b] ⊗ f∗ f ∗ a] (g) [f∗ f ∗ a, [f∗ f ∗ a, f∗ f ∗ b] ⊗ f∗ f ∗ a]

[f∗ f ∗ a, f∗ f ∗ b]

@A

ev id

/ [f∗ f ∗ a, f∗ f ∗ b]

ev

(h) u

/ [a, f∗ f ∗ b] o

ED

1 Commutativity of Diagrams

id

u

BC

285

286

1 Commutativity of Diagrams

The commutativity of (a), (b), (d), (e), (g), and (h) is trivial. The commutativity of (c) is Lemma 1.55. The commutativity of (f) is Lemma 1.32. Thus the whole diagram is commutative, and the ﬁrst part of the lemma follows. The commutativity of the second diagram is proved by a similar diagram drawing. The details are left to the reader. ⊓ ⊔ (1.57)

Let X be an object of S. We denote the composite isomorphism via λ

π

→X∗ (OX , [a, b]) X∗ (a, b)−−−→X∗ (OX ⊗ a, b)− by hX . Lemma 1.58. Let f : X → Y be a morphism in S. Then the composite map h

C

X X∗ (a, b)−−→X →X∗ (f ∗ OY , [a, b]) ∼ = Y∗ (OY , f∗ [a, b]) ∗ (OX , [a, b])−

h−1

H

Y − →Y∗ (OY , [f∗ a, f∗ b])−− →Y∗ (f∗ a, f∗ b)

agrees with the map given by ϕ → f∗ ϕ. The composite map f∗

h

Y ′ ′ ∗ ∗ ′ ′ Y∗ (a′ , b′ )−−→Y ∗ (OY , [a , b ])−→X∗ (f OY , f [a , b ])

h−1

X∗ (C −1 ,P )

X −−−−−−−→X∗ (OX , [f ∗ a′ , f ∗ b′ ])−− →X∗ (f ∗ a′ , f ∗ b′ )

agrees with f ∗ . Proof. We prove the ﬁrst assertion. The all maps are natural on a. By Yoneda’s lemma, we may assume that a = b and it suﬃces to show that the identity map 1b is mapped to 1f∗ b by the map. It is straightforward to check that 1b goes to the composite map λ−1

u

C

f∗ b−−→OY ⊗ f∗ b− →f∗ f ∗ OY ⊗ f∗ b− →f∗ OX ⊗ f∗ b via tr

λ

ev

−−−−→f∗ [b, OX ⊗ b] ⊗ f∗ b− →f∗ [b, b] ⊗ f∗ b−→f∗ b. By the commutativity of (1.38), we are done. The second assertion is proved similarly, utilizing the commutativity of (1.53). ⊓ ⊔ Lemma 1.59. Let σ = (f g ′ = gf ′ ) be a commutative diagram in S. Then the diagram g ∗ f∗ [a, b] ↓ PH

θ

−−−−−−−−−−−−−−−−−→ θ

f∗′ (g ′ )∗ [a, b] ↓ HP

θ

− [f∗′ (g ′ )∗ a, f∗′ (g ′ )∗ b] [g ∗ f∗ a, g ∗ f∗ b] − → [g ∗ f∗ a, f∗′ (g ′ )∗ b] ← is commutative. Proof. Follows from Lemma 1.56.

⊓ ⊔

Chapter 2

Sheaves on Ringed Sites

(2.1) We ﬁx a universe [16] U and a universe V such that U ∈ V or U = V. A set is said to be small or U-small if it is an element of V, and is bijective to an element of U. A category is said to be small if both the object set and the set of morphisms are small. Ringed spaces (including schemes) are required to be small, unless otherwise speciﬁed. The categories of small sets and small abelian groups are respectively denoted by Set and Ab. For example, M is an object of Ab if and only if M is a group whose underlying set is in V, and M is bijective to some set in U. A category C is said to be a U-category if for any objects x, y of C, the set C(x, y) is small. Note that Set and Ab are U-categories. (2.2) For categories I and C, we denote the functor category Func(I op , C) by P(I, C). An object of P(I, C) is sometimes referred as a presheaf over I with values in C. If C is a U-category and I is small, then P(I, C) is a U-category. For a small category X, we denote P(X, Ab) by PA(X). In these notes, a site (i.e., a category with a Grothendieck topology, in the sense of [42]) is required to be a small category whose topology is deﬁned by a pretopology (see [42]). Let C be a category with small products. If X is a site, then the category of sheaves on X with values in C is denoted by S(X, C). The inclusion q : S(X, C) → P(X, C) is fully faithful. If C is a U-category, then S(X, C) is a U-category. The category S(X, Ab) is denoted by AB(X). Let X be a site. The inclusion AB(X) → PA(X) is denoted by q(X, AB). We denote the sheaﬁﬁcation functor PA(X) → AB(X) by a(X, AB). Namely, a = a(X, AB) is the left adjoint of q(X, AB). Note that a is exact. We review the construction of the sheaﬁﬁcation described in [2, (II.1)]. For M ∈ PA(X), x ∈ X, and a covering U = (xi → x)i∈I of x, we denote the kernel of the map ϕ Γ(xi , M) − → Γ(xi ×x xj , M) i∈I

(i,j)∈I×I

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

287

288

2 Sheaves on Ringed Sites

ˇ 0 (U, M), where ϕ((mi )i∈I ) = (resx × x ,x mj −resx × x ,x mi )(i,j)∈I×I . by H i x j j i x j i 0 ˇ Note that H (U, ?) is a functor from PA(X) to Ab, which is compatible with arbitrary limits. The set of all coverings of x is a directed set. Let U = (φi : xi → x)i∈I and V = (ψj : yj → x)j∈J be coverings of x. We say that V is a reﬁnement of U if there are a map τ : J → I and a collection of morphisms (ηj : yj → xτ j )j∈J such that φτ j ◦ ηj = ψj for j ∈ J. If (τ, (ηj )) makes V a reﬁnement of U, then we deﬁne ˇ 0 (U, M) → H ˇ 0 (V, M) L = L(V, U; (τ, (ηj )))(M) : H by L((mi )i∈I ) = (resηj (mτ j ))j∈J . It is easy to see that L is independent of the choice of τ or ηj , and depends only on U and V, see [31, Lemma III.2.1]. If W is a reﬁnement of V, then L(W, U) = L(W, V) ◦ L(V, U). Thus we get ˇ 0 (U, M))U , where U runs through the all coverings an inductive system (H ˇ 0 (U, M) by H ˇ 0 (x, M). This is a small abelian group, of x. We denote lim H − → ˇ 0 (x, ?) is a left exact functor from PA(X) to Ab. and H Let x′ → x be a morphism. Then a covering U = (xi → x)i∈I gives a covering x′ ×x U = (x′ ×x xi → x′ )i∈I in a natural way. This correspondence ˇ 0 (x′ ×x U, M). So we have a canonical map ˇ 0 (U, M) → H induces a map H 0 0 ′ ˇ (x, M) → H ˇ (x , M). So we have a presheaf of abelian groups H ˇ 0 (M) H ˇ 0 (M)) = H ˇ 0 (x, M). Note that H ˇ 0 is an endofunctor of such that Γ(x, H PA(X). Note that there is a natural map 0

ˇ (M). Y = Y (M) : M → H The map Y at the object x ˇ 0 (x, M) ˇ 0 (M)) = H Y (x) : Γ(x, M) → Γ(x, H ˇ 0 (idx , M) → H ˇ 0 (x, M) for m ∈ Γ(x, M), is given by Y (x)(m) = m ∈ H where idx is the covering (x → x) consisting of the one morphism idx . Y = Y (M) is an isomorphism if and only if M is a sheaf. ˇ 0 (H ˇ 0 (M)) is a sheaf, and it is the sheaﬁﬁcation aM. It is known that H The composite map ˇ0

Y (H (M)) Y (M) ˇ 0 (H ˇ 0 (M)) = aM = qaM ˇ 0 (M) − −−−−−−→ H u : M −−−−→ H

is the unit of adjunction. By the naturality of Y , u also agrees with the composite map Y (M)

0

ˇ 0 (Y (M)) H

0

0

ˇ (M) −−−−−−−→ H ˇ (M)). ˇ (H M −−−−→ H

2 Sheaves on Ringed Sites

289

Note that the counit of adjunction ε : aq → Id is given as the unique natural map such that qε : qaq → q is the inverse of uq. (2.3) Let X = (X, OX ) be a ringed site. Namely, let X be a site and OX a sheaf of commutative rings on X. We denote the category of presheaves (resp. sheaves) of OX -modules by PM(X) (resp. Mod(X)). The inclusion Mod(X) → PM(X) is denoted by q(X, Mod). The sheaﬁﬁcation PM(X) → Mod(X) is denoted by a(X, Mod). Note that a(X, Mod) is constructed in the same way ˇ 0 (M) is in PM(X) in a natural way for M ∈ PM(X). as in (2.2), since H Since q is fully faithful, ε : aq → Id is an isomorphism. The forgetful functor Mod(X) → AB(X) is denoted by F (X). The forgetful functor PM(X) → PA(X) is denoted by F ′ (X). Thus F ′ (X) ◦ q(X, Mod) = q(X, AB) ◦ F (X) and a(X, AB) ◦ F ′ (X) = F (X) ◦ a(X, Mod). We say that a category A is Grothendieck if it is an abelian U-category with a generator which satisﬁes the (AB5) condition in [13] (the existence of arbitrary small coproducts, and the exactness of small ﬁltered inductive limits), see [37]. The categories AB(X) and Mod(X) are Grothendieck. In general, a Grothendieck category satisﬁes (AB3*), see [37, Corollary 7.10]. A ringed category (X, OX ) is a pair such that X is a small category, and OX is a presheaf of commutative rings on X. If X is a ringed category, then PA(X) and PM(X) are Grothendieck with (AB4*). (2.4) Let f : Y → X be a functor between small categories. Then the pull# # back PA(X) → PA(Y) is denoted by fPA . Note that fPA (F) := F ◦ f op . In general, the pull-back P(X, C) → P(Y, C) is deﬁned in a similar way, and # is denoted by f # . If f is a continuous functor (i.e., fSet carries sheaves to # sheaves) between sites, then fAB : AB(X) → AB(Y) is deﬁned to be the # restriction of fPA . Throughout these notes, we require that a continuous functor f : Y → X between sites satisﬁes the following condition. For y ∈ Y, a covering (yi → y)i∈I , and any i, j ∈ I, the morphisms f (yi ×y yj ) → f (yi ) and f (yi ×y yj ) → f (yj ) make f (yi ×y yj ) the ﬁber product f (yi ) ×f (y) f (yj ). The identity functor is continuous. A composite of continuous functors is again continuous. Thanks to the re-deﬁnition of sites and continuous functors, we have the following.

Lemma 2.5. Let f : Y → X be a functor between sites. Then f is continuous if and only if the following holds. If (ϕi : yi → y)i∈I is a covering, then (f ϕi : f yi → f y)i∈I is a covering, and for any i, j ∈ I, the morphisms f (yi ×y yj ) → f (yi ) and f (yi ×y yj ) → f (yj ) make f (yi ×y yj ) the ﬁber product f (yi ) ×f (y) f (yj ). For the proof, see [43, (1.6)]. (2.6) Let f : Y → X be a functor between small categories. The left adjoint # of fPA , which exists by Kan’s lemma (see e.g., [2, Theorem I.2.1]), is denoted PA by f# .

290

2 Sheaves on Ringed Sites

For x ∈ X, we deﬁne the small category Ixf as follows. An object of Ixf is a pair (y, φ) with y ∈ Y and φ ∈ X(x, f (y)). A morphism h : (y, φ) → (y ′ , φ′ ) is PA (F)) = a morphism h ∈ Y(y, y ′ ) such that f (h) ◦ φ = φ′ . Note that Γ(x, f# f op lim Γ(y, F), where the colimit is taken over (Ix ) . −→ The left adjoint of fC# : P(X, C) → P(Y, C) is constructed similarly, proC or vided C has arbitrary small colimits. The left adjoint is denoted by f# simply by f# . AB For a continuous functor f : Y → X between sites, the left adjoint f# of # AB PA fAB is given by f# = a(X, AB) ◦ f# ◦ q(Y, AB). Lemma 2.7. If (Ixf )op is pseudoﬁltered (see e.g., [16, 31]) for each x ∈ X, PA is exact. then f# ⊓ ⊔

Proof. This is a consequence of [16, Corollaire 2.10].

(2.8) We say that f : Y → X is admissible if f is continuous and the functor PA is exact. f# AB (2.9) Let f : Y → X be an admissible functor. Then f# is exact. Indeed, # AB f# is right exact, since it is a left adjoint of fAB . On the other hand, being AB PA a composite of left exact functors, f# = af# q is left exact. PA (2.10) If Y has ﬁnite limits and f preserves ﬁnite limits, then f# is exact by Lemma 2.7. It follows that a continuous map between topological spaces induces an admissible continuous functor between the corresponding sites. # , which we denote by f♭PA also exists, (2.11) The right adjoint functor of fPA op as Ab has arbitrary small colimits (i.e., Ab has small limits). The functor f♭PA is the composite op

(f op )#

op

Func(Yop , Ab)−→ Func(Y, Abop ) −−−−→ Func(X, Abop )−→ Func(Xop , Ab), where (f op )# is the left adjoint of (f op )# : Func(X, Abop ) → Func(Y, Abop ), where f op = f is the opposite of f , namely, f viewed as a functor Yop → Xop . (2.12) For M, N ∈ PM(X), the presheaf tensor product is denoted by ⊗pOX . It is deﬁned by Γ(x, M ⊗pOX N ) := Γ(x, M) ⊗Γ(x,OX ) Γ(x, N ) for x ∈ X. The sheaf tensor product a(qM ⊗pOX qN ) of M, N ∈ Mod(X) is denoted by M ⊗OX N .

2 Sheaves on Ringed Sites

291

Let M, N ∈ PM(X), x ∈ X, and U = (xi → x)i∈I and V = (x′j → x)j∈J be coverings of x. We deﬁne a map ˇ 0 (V, N ) → H ˇ 0 (U, M) ⊗Γ(x,O ) H ˇ 0 (U × V, M ⊗p N ) Z = Z(U, V; M, N ) : H X by Z((mi )i∈I ⊗ (nj )j∈J ) = (mi ⊗ nj )(i,j)∈I×J , where U × V denotes the covering (xi ×x x′j → x)(i,j)∈I×J of x. Note that Z induces ˇ 0 (N ) → H ˇ 0 (M ⊗p N ). ˇ 0 (M) ⊗p H Z = Z(M, N ) : H Lemma 2.13. The composite Y (M)⊗Y (N )

0

0

0

Z

ˇ N − ˇ M ⊗p H ˇ (M ⊗p N ) M ⊗p N −−−−−−−−−→ H →H agrees with Y (M ⊗p N ). ⊓ ⊔

Proof. This is straightforward, and we omit it. Lemma 2.14. The composite ˇ0

H (Y ⊗Y ) Z ˇ0 ˇ0N − ˇ 0 N) ˇ 0 M ⊗p H ˇ 0 M ⊗p H ˇ 0 (H →H (M ⊗p N ) −−−−−−→ H H 0

0

ˇ M ⊗p H ˇ N ). agrees with Y = Y (H ˇ 0 N ) at x is represented by data ˇ 0 M ⊗p H ˇ 0 (H Proof. Note that a section of H as follows. A covering V = (xi → x)i∈I , a collection of coverings Vi = (yji → i,l xi )j∈Ji (i ∈ I), and a collection of elements ( l (mi,l j )j∈Ji ⊗(nj )j∈Ji )i∈I subi,l i i ject to the patching conditions, where mi,l j ∈ Γ(yj , M) and nj ∈ Γ(yj , N ). ′ ′ Let U = (zl → x)l∈L and U = (zl′ → x)l′ ∈L′ be coverings of x, and ˇ 0 (U, M) and H ˇ 0 (U ′ , N ), respectively. (ml )l∈L and (nl′ )l′ ∈L′ elements of H Then Y ((ml ) ⊗ (nl′ )) is represented by the collection I = {idx }, V = (idx ), Jidx = L × L′ , Vidx = (zl ×x zl′′ → x)(l,l′ )∈L×L′ , and ((reszl ×x zl′′ ,zl ml ) ⊗ ˇ 0 N ). As an element of H ˇ 0 (V, H ˇ 0 M⊗p H ˇ 0 (x, H ˇ 0 M⊗p (resz × z′ ,z′ nl′ )) ∈ H l

x l′

l′

ˇ 0 N ), this element is the same as the element represented by the collecH tion I = L × L′ , V = U × U ′ , Jl1 ,l1′ = L × L′ for any (l1 , l1′ ) ∈ L × L′ , Vl1 ,l1′ = zl1 ×x U ×x zl′′ ×x U ′ for (l1 , l1′ ) ∈ L × L′ , and ((reszl ,l,l′ ,l′ ,zl ml ) ⊗ 1 1 1 (reszl ,l,l′ ,l′ ,zl′′ nl′ ))(l1 ,l1′ )∈L×L′ , where zl1 ,l,l1′ ,l′ := zl1 ×x zl ×x zl′′ ×x zl′′ . Since 1 1 1 reszl1 ×x zl ,zl ml = reszl1 ×x zl ,zl1 ml1 and reszl′′ ×x zl′′ ,zl′′ nl′ = reszl′′ ×x zl′′ ,zl′′ nl1′ , 1

1

1

this element agrees with the element represented by the collection I = L×L′ , V = U × U ′ , Jl1 ,l1′ = L × L′ for (l1 , l1′ ) ∈ L × L′ , Vl1 ,l1′ = zl1 ×x U ×x zl′′ ×x U ′ 1 for (l1 , l1′ ) ∈ L×L′ , and ((reszl ,l,l′ ,l′ ,zl1 ml1 )⊗(reszl ,l,l′ ,l′ ,zl′′ nl1′ ))(l1 ,l1′ )∈L×L′ . 1

1

1

1

1

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2 Sheaves on Ringed Sites

It also agrees with the element represented by the collection I = L × L′ , V = U × U ′ , Jl1 ,l1′ is the singleton {idzl1 ×x zl′′ } for (l1 , l1′ ) ∈ L × 1

L′ , Vl1 ,l1′ = (idzl1 ×x zl′′ ) for (l1 , l1′ ) ∈ L × L′ , and ((reszl1 ×x zl′′ ,zl1 ml1 ) ⊗ 1

1

(reszl1 ×x zl′′ ,zl′′ nl1′ ))(l1 ,l1′ )∈L×L′ , which agrees with the image of (ml ) ⊗ (nl′ ) 1

1

ˇ 0 (Y ⊗ Y ) ◦ Z. ˇ 0 (Y ⊗ Y ) ◦ Z. This shows that Y = H by H (2.15)

⊓ ⊔

We deﬁne a natural map m′ : qaM ⊗p qaN → qa(M ⊗p N )

as the composite Z ˇ0 ˇ0 ˇ0H ˇ 0 M ⊗p H ˇ0H ˇ0N − ˇ 0 N) qaM ⊗p qaN = H →H (H M ⊗p H ˇ0

H Z ˇ0H ˇ 0 (M ⊗p N ) = qa(M ⊗p N ). −−−→ H

Lemma 2.16. The composite u⊗u

m′

M ⊗p N −−−→ qaM ⊗p qaN −−→ qa(M ⊗p N ) agrees with the unit map u. Proof. Consider the diagram Y ⊗Y / hM ⊗p hN Y ⊗Y / hhM ⊗p hhN M ⊗p NP SSS PPP SSS Y PPPY SSS PPP Z Z SSS PP' SS) h(Y ⊗Y ) / h(hM ⊗p hN ) h(M ⊗p N ) RRR RRR hY RRR hZ RRR RR) hh(M ⊗p N ), 0

ˇ . Then the four triangles in the diagram commutes by where h = H Lemma 2.13 and Lemma 2.14. So the whole diagram commutes, and the lemma follows. ⊓ ⊔ Lemma 2.17. The composite m′

qa(u⊗p u)

qaM ⊗p qaN −−→ qa(M ⊗p N ) −−−−−−→ qa(qaM ⊗p qaN ) agrees with u.

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Proof. Consider the diagram hZ Z / hh(M ⊗p N ) / h(hM ⊗p hN ) hhM ⊗p hhN RRR S S SSSS RRR Y ShY RRR SSSS hh(Y ⊗p Y ) h(Y ⊗p Y ) RRR SSSS RR) ) hZ / hh(hM ⊗p hN ) h(hhM ⊗p hhN ) SSSS SSShY SSSS hh(Y ⊗p Y ) SSSS ) hh(hhM ⊗p hhN ),

0

ˇ . The four triangles in the diagram are commutative by where h = H Lemma 2.13 and Lemma 2.14. So the whole diagram is commutative, and the lemma follows. ⊓ ⊔ Lemma 2.18. For M, N ∈ PM(X), the natural map ¯ := a(u ⊗p u) : a(M ⊗p N ) → a(qaM ⊗p qaN ) Δ is an isomorphism. Proof. Consider the diagram ϕ / qaM ⊗p qaN M ⊗p NO RRR OOO RRR β OOτO RRR ′ OOO m RRR RR) O' ¯ q(∆) / p qa(M ⊗ N ) o qa(qaM ⊗p qaN ), q(ψ)

¯ = a(u ⊗p u), and ψ : a(qaM ⊗p qaN ) → where ϕ = u ⊗p u, τ = u, β = u, Δ p a(M ⊗ N ) is the unique map of sheaves such that m′ = q(ψ)β (this map exists by the universality of the sheaﬁﬁcation). By Lemma 2.16, τ = m′ ϕ. ¯ ′. By Lemma 2.17, β = q(Δ)m So ¯ = q(ψ)q(Δ)m ¯ ′ ϕ = q(ψ)βϕ = m′ ϕ = τ = q(id)τ. q(ψ Δ)τ ¯ = id. Moreover, By the universality of the sheaﬁﬁcation τ , we have that ψ Δ ¯ ¯ ¯ ′ = β = q(id)β. q(Δψ)β = q(Δ)q(ψ)β = q(Δ)m ¯ = id. This shows By the universality of the sheaﬁﬁcation β, we have that Δψ ¯ that Δ is an isomorphism. ⊓ ⊔

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(2.19) Let (Y, OY ) and (X, OX ) be ringed categories. We say that f : (Y, OY ) → (X, OX ) is a ringed functor if f : Y → X is a functor, and a morphism of presheaves of rings η : OY → f # OX is given. If, moreover, both (Y, OY ) and (X, OX ) are ringed sites and f is continuous, then we call f a ringed continuous functor. # , and its left adjoint is The pull-back PM(X) → PM(Y) is denoted by fPM PM PM denoted by f# . The left adjoint f# is deﬁned by PM Γ(x, f# M) := lim Γ(x, OX ) ⊗Γ(y,OY ) Γ(y, M) −→

for x ∈ X and M ∈ PM(Y), where the colimit is taken over the category # Mod (Ixf )op . Similarly, fMod : Mod(X) → Mod(Y) and its left adjoint f# = # # # # # afPM q is deﬁned. Note that qfMod = fPM q and qfAB = fPA q. We sometimes # # # # denote the identity map qfMod = fPM q and qfAB = fPA q and their inverses PA f op OY has a structure by c = c(f ). If (Ix ) is ﬁltered for any x ∈ X, then f# of a presheaf of rings in a natural way, and there is a canonical isomorphism p # PM PA M ∼ M. The right adjoint of fPM , which exists as in f# = OX ⊗f PA OY f# #

(2.11), is denoted by f♭PM . (2.20) Let f : (Y, OY ) → (X, OX ) be a ringed continuous functor. For later # ♥ use, we need the explicit description of the unit u : Id → f♥ f# and the ♥ # counit ε : f# f♥ → Id, where ♥ denotes either PM or Mod. The unit u for the case ♥ = PM is induced by the map

Γ(y, M) → Γ(f y, OX ) ⊗Γ(y,OY ) Γ(y, M) → lim Γ(f y, OX ) ⊗Γ(y′ ,OY ) Γ(y ′ , M) = Γ(f y, f# M) = Γ(y, f # f# M), −→ where the ﬁrst map sends m to 1 ⊗ m, and the second map is the obvious map. The counit ε for the case ♥ = PM is induced by the map Γ(x, f# f # M) = lim Γ(x, OX ) ⊗Γ(y,OY ) Γ(f y, M) → Γ(x, M), −→ where the colimit is taken over (Ixf )op , and the last map is given by a ⊗ m → a resx,f y (m). It is easy to verify that the composite u

ε

→ f# f # f# − → f# f# − is the identity, and the composite u

ε

→ f # f# f # − → f# f# − is the identity, and thus certainly (f# , f # ) is an adjoint pair. # Mod f# is the composite The unit u : Id → fMod ε−1

u

θ

# # # PM PM Mod Id −−→ aq − → afPM f# q− → fMod af# q = fMod f# ,

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where θ is the composite u

c

ε

af # − → af # qa − → aqf # a − → f # a, see Lemma 2.32 below. Mod # fMod → Id is the composite The counit ε : f# c

ε

ε

# Mod # PM PM # f# fMod = af# qfMod − → af# fPM q − → aq − → Id .

(2.21) If there is no confusion, the ♥ attached to the functors of sheaves # # deﬁned above are omitted. For example, f # stands for f♥ . Note that f♥ (F) viewed as a presheaf of abelian groups is independent of ♥. (2.22) Let X be a ringed site, and x ∈ X. The category X/x is a site with the same topology as that of X. The canonical functor Rx : X/x → X is # a ringed continuous functor, and yields the pull-backs (Rx )# AB and (Rx )PA , PA AB which we denote by (?)|x and (?)|x , respectively. Their left adjoints are and LPA denoted by LAB x , respectively. Note that Rx is admissible, see [31, x p.78]. (2.23) Note that X/x is a ringed site with the structure sheaf OX |x . Thus, are deﬁned in an obvious way, and their left adjoints LMod and (?)|PM (?)|Mod x x x PM and Lx are also deﬁned. Note that LMod and LPM are faithful and exact. x x (2.24) For a morphism φ : x → y, we have an obvious admissible ringed continuous functor Rφ : X/x → X/y. The corresponding pull-back is denoted by φ⋆♥ , and its left adjoint is denoted by φ♥ ⋆ , where ♥ is AB, PA, Mod or PM. For M, N ∈ ♥(X), we deﬁne Hom♥(X) (M, N ) to be the object of ♥(X) given by ♥ Γ(x, Hom♥(X) (M, N )) := Hom♥(X/x) (M|♥ x , N |x ), where ♥ = PA, AB, PM, or Mod. For φ : x → y, the restriction map ♥ ♥ ♥ Hom♥(X/y) (M|♥ y , N |y ) → Hom♥(X/x) (M|x , N |x )

is given by φ⋆♥ . It is easy to see that if N is a sheaf, then Hom♥(X) (M, N ) is a sheaf. Note that Hom♥(X) (M, N ) is a functor from ♥(X)op × ♥(X) to ♥(X). (2.25) Let f : Y → X be a ringed continuous functor, y ∈ Y, and U = (yi → y)i∈I a covering of y. Then f U = (f yi → f y)i∈I is a covering of f y, since f is continuous, see Lemma 2.5. Let M ∈ PM(X). Then we have a canonical isomorphism ˇ 0 (f U, M), ˇ 0 (U, f # M) ∼ ν: H =H

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2 Sheaves on Ringed Sites

since the canonical map f (yi ×y yj ) → f yi ×f y f yj is an isomorphism. This induces a natural map ˇ 0 M. ˇ 0 f #M → f # H ν: H Lemma 2.26. Let f and M be as above. Then the composite Y ˇ0 # ν ˇ0M f # M −→ H f M− → f# H

is f # Y . Proof. This is straightforward, and left to the reader.

⊓ ⊔

Lemma 2.27. Let f and M be as above. Then the composite ν Y ˇ0 # ˇ0 ˇ0M − ˇ 0 f #M − H → f# H →H f H M

ˇ 0 f #Y . agrees with H Proof. This is proved quite similarly to Lemma 2.14, and we omit the proof. ⊓ ⊔ (2.28) Let f : Y → X be a ringed continuous functor, and M ∈ PM(X). Then we deﬁne the natural map θ¯: af # M → f # aM to be the unique map such that q θ¯ is the composite ˇ0

H ν ˇ 0 f# H ˇ0H ˇ 0 f #M − ˇ0M q θ¯: qaf # M = H −−→ H c ν ˇ 0 M = f # qaM − ˇ0H → qf # aM. − → f# H

Lemma 2.29. Let f and M be as above. Then the composite u

q θ¯

c

f #M − → qaf # M −→ qf # aM − → f # qaM is f # u. Proof. Consider the diagram Y / hY / hhf # M hf # MK f # MI KKK # II f # Y II Khf KKK Y II ν II ν KK $ % Y / hf # hM f # hMK KKK # KfKK Y ν KKK % f # hhM,

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297

0

ˇ . The four triangles in the diagram commutes by Lemma 2.26 where h = H and Lemma 2.27. So the whole diagram commutes, and the lemma follows. ⊓ ⊔ (2.30) Let f : Y → X be a ringed continuous functor between ringed sites. The following is a restricted version of the results on cocontinuous functors in [43]. We give a proof for convenience of readers. Lemma 2.31. Assume that for any y ∈ Y and any covering (xλ → f y)λ∈Λ of f y, there is a covering (yμ → y)μ∈M of y such that there is a map φ : M → Λ such that f yμ → f y factors through xφ(μ) → f y for each μ. Then the pullback f # is compatible with the sheaﬁﬁcation in the sense that the canonical natural transformation # # → fMod a(X, Mod) θ¯ : a(Y, Mod)fPM # is a natural isomorphism. If this is the case, fMod has the right adjoint f♭Mod , and in particular, it preserves arbitrary limits and arbitrary colimits.

Proof. Let M ∈ PM(X) and y ∈ Y. Recall that ˇ 0 ((f yi → f y)i∈I , M) ˇ 0 ((yi → y)i∈I , f # M) → H ν: H is an isomorphism, and induces ν ˇ 0 ((yi → y)i∈I , f # M) − ˇ 0 (y, f # M) = lim H ν: H → −→ ˇ 0 ((f yi → f y)i∈I , M) → lim H ˇ 0 ((xj → f y)j∈J , M) = H ˇ 0 (f y, M). lim H −→ −→

This is also an isomorphism, since coverings of the form (f yi → f y) is ﬁnal in ¯ θ¯ is an isomorphism. the category of all coverings of f y. By the deﬁnition of θ, # Next we show that fMod has a right adjoint. To prove this, it suﬃces to show that f♭PM (M) is a sheaf if so is M for M ∈ PM(X). Let u : IdPM(X) → # # f♭PM fPM be the unit of adjunction, ε : fPM f♭PM → IdPM(Y) the counit of adjunction, v(X) : IdPM(X) → q(X, Mod)a(X, Mod) the unit of adjunction, and v(Y) : IdPM(Y) → q(Y, Mod)a(Y, Mod) the unit of adjunction. Then the diagram of functors uf PM

id

f PM ε

∼ =

f PM qaε

# # ♭ −→ f♭PM − → f♭PM fPM f♭PM −− f♭PM fPM # PM # PM PM ↓ f♭ fPM v(X)f♭ ↓ f♭ v(Y)fPM f♭PM

♭ f♭PM −−− → PM ↓ v(X)f♭

uqaf PM

f♭PM ↓ f♭PM v(Y)

# # ♭ → f♭PM qafPM f♭PM −− −−−→ f♭PM qa qaf♭PM −−−−♭−→ f♭PM fPM qaf♭PM −

is commutative, where ∼ = is the inverse of the canonical map caused by q θ¯

c

# # # # −→ qfMod a− → fPM qa, which exists by the ﬁrst part. As (fPM , f♭PM ) is qafPM an adjoint pair, the composite of the ﬁrst row of the diagram is the identity. As v(Y)(M) is an isomorphism, the right-most vertical arrow evaluated at M is an isomorphism. Hence, v(X)f♭PM (M), which is the left-most vertical

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arrow evaluated at M, is a split monomorphism. As it is a direct summand of a sheaf, f♭PM (M) is a sheaf, as desired. # Mod As it is a right adjoint of f# , the functor fMod preserves arbitrary limits. # Mod As it is a left adjoint of f♭ , the functor fMod preserves arbitrary colimits. ⊓ ⊔ Lemma 2.32. Let f : Y → X be a ringed continuous functor. Then # # θ¯: afPM → fMod a agrees with the composite u

c

ε

→ af # qa − → aqf # a − → f # a. θ : af # − Proof. Consider the diagram f# u

qaf #

u

/ f # qa

c

/ qf # a JJ JJ id J u (a) (b) u (c) JJ JJ $ c / ε u / # # / qf # a . qaf qa qaqf a

(a) and (b) are commutative by the naturality of u. The commutativity of (c) is basics on adjunction. So the adjoint qθ ◦ u of θ agrees with the composite u

c

→ f # qa − → qf # a. f# − ¯ Since the adjoint By Lemma 2.29, this agrees with the adjoint q θ¯ ◦ u of θ. ¯ maps agree, we have θ = θ. ⊓ ⊔ Lemma 2.33. Let f : Y → X be a ringed continuous functor. Then the composite u θ c # # # # fPM − → qafPM − → qfMod a− → fPM qa agrees with u. Proof. Follows from Lemma 2.29 and Lemma 2.32.

⊓ ⊔

Lemma 2.34. Let f : Y → X be a ringed continuous functor. Then the # # conjugate of c : qfMod q agrees with → fPM u

PM PM Mod − → af# qa = f# a. af#

(2.35)

In particular, being a conjugate of an isomorphism, (2.35) is an isomorphism. Proof. Straightforward.

⊓ ⊔

(2.36) Let X be a ringed site, and x ∈ X. It is easy to see that Rx : X/x → X satisﬁes the condition in Lemma 2.31. So (?)|Mod preserves arbitrary limits x and colimits. In particular, (?)|Mod is exact. Similarly, for a morphism φ : x x → y in X, φ⋆Mod preserves arbitrary limits and colimits.

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299

(2.37) Let X be a ringed site, M ∈ PM(X), and N ∈ Mod(X). We deﬁne an isomorphism V : q HomMod(X) (aM, N ) → HomPM(X) (M, qN ) as follows. For x ∈ X, the map V at x is the composite θ¯

V(x) : HomMod(X/x) ((aM)|x , N |x ) − → HomMod(X/x) (a(M|x ), N |x ) c ∼ → HomPM(X/x) (M|x , (qN )|x ). = HomPM(X/x) (M|x , q(N |x )) − The ∼ = is an isomorphism coming from the adjunction. Note that θ¯: a(M|x ) → (aM)|x is an isomorphism by Lemma 2.31. A morphism ϕ : (aM)|x → N |x is mapped to the composite u|x

qϕ

c

c

M|x −−→ (qaM)|x − → q((aM)|x ) −→ q(N |x ) − → (qN )|x . Note that V is a map of presheaves, that is, V is compatible with the restriction. Indeed, for ϕ ∈ HomMod(X/x) ((aM)|x , N |x ) and φ : y → x, φ⋆ Vϕ is the composite φ⋆ (u|x )

c

→ φ⋆ (M|x ) −−−−−→ φ⋆ ((qaM)|x ) − φ⋆ (qϕ)

c

φ⋆ (q((aM)|x )) −−−−→ φ⋆ (q(N |x )) − → φ⋆ ((qN )|x ), which can be identiﬁed with the composite u|y

q(φ⋆ ϕ)

c

c

M|y −−→ (qaM)|y − → q((aM)|y ) −−−−→ q(N |y ) − → (qN )|y . This map is Vφ⋆ ϕ, and V is compatible with the restriction maps. Lemma 2.38. Let X be a ringed site, and M, N ∈ Mod(X). Then the composite ε V ¯ : q Hom → q HomMod(X) (aqM, N ) − → HomPM(X) (qM, qN ) H Mod(X) (M, N ) −

¯ ∈ is given as follows. For x ∈ X and ϕ ∈ HomMod(X/x) (M|x , N |x ), H(x)(ϕ) HomPM(X/x) ((qM)|x , (qN )|x ) is the composite c

qϕ

c

(qM)|x − → q(M|x ) −→ q(N |x ) − → (qN )|x . Proof. By the deﬁnition of V, the map in question is the composite u

c

ε

qϕ

c

(qM)|x − → (qaqM)|x − → q((aqM)|x ) − → q(M|x ) −→ q(N |x ) − → (qN )|x .

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It agrees with the composite u

ε

c

qϕ

c

→ (qaqM)|x − → (qM)|x − → q(M|x ) −→ q(N |x ) − → (qN )|x . (qM)|x − ⊓ ⊔

Since εu = id, the assertion follows. (2.39)

Let M, N ∈ PM(X). The composite V−1

u

→ a HomPM(X) (M, qaN ) −−→ a HomPM(X) (M, N ) − ε

→ HomMod(X) (aM, aN ) aq HomMod(X) (aM, aN ) − is denoted by P¯ . Lemma 2.40. Let (X, OX ) be a ringed site. The category PM(X) is a closed symmetric monoidal category (see [29, (VII.7)]) with ⊗pOX the multiplication, qOX the unit object, HomPM(X) (?, ?) the internal hom, etc., etc. The proof of the lemma (including the precise statement) is straightforward, but we give some remarks on non-trivial natural maps. (2.41)

The evaluation map ev : HomPM(X) (M, N ) ⊗p M → N

at the section Γ(x, ?), Γ(x, HomPM(X) (M, N ) ⊗p M) = HomMod(X/x) (M|x , N |x ) ⊗Γ(x,OX ) Γ(x, M) → Γ(x, N ), is given by ϕ ⊗ a → ϕ(idx )(a) for ϕ ∈ HomMod(X/x) (M|x , N |x ) and a ∈ Γ(x, M). (2.42)

The trace map tr : M → HomPM(X) (N , M ⊗pOX N )

at the section Γ(x, ?), Γ(x, M) → HomPM(X/x) (N |x , (M ⊗pOX N )|x ), maps α ∈ Γ(x, M) to the map tr(x)(α) ∈ HomPM(X/x) (N |x , (M ⊗pOX N )|x ) as follows. For φ : x′ → x, β ∈ Γ(φ, N |x ) = Γ(x′ , N ) is mapped to resx′ ,x (α) ⊗ β ∈ Γ(x′ , M) ⊗Γ(x′ ,OX ) Γ(x′ , N ) = Γ(φ, (M ⊗pOX N )|x ) by tr(x)(α).

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301

Lemma 2.43. The category Mod(X) is a closed symmetric monoidal category with ⊗OX the multiplication, OX the unit object, HomMod(X) (?, ?) the internal hom, etc., etc. The precise statement and the proof is left to the reader. We only remark the following. (2.44)

Let M, N , P ∈ Mod(X). Then the associativity morphism α : (M ⊗ N ) ⊗ P → M ⊗ (N ⊗ P)

is the composite u

(M ⊗ N ) ⊗ P = a(qa(qM ⊗p qN ) ⊗p qP) − → a(qa(qM ⊗p qN ) ⊗p qaqP) ¯ −1 ∆

aα′

−−−→ a((qM ⊗p qN ) ⊗p qP) −−→ a(qM ⊗p (qN ⊗p qP)) u

− → a(qM ⊗p qa(qN ⊗p qP)) = M ⊗ (N ⊗ P), ¯ −1 is the inverse of the map Δ, ¯ see Lemma 2.18, and α′ is the where Δ associativity morphism for presheaves. (2.45) The left unit isomorphism λ : OX ⊗ M → M is deﬁned to be the composite aλ′

ε

OX ⊗ M = a(qOX ⊗p qM) −−→ aqM − → M, where λ′ is the left unit isomorphism for presheaves. (2.46)

The twisting isomorphism γ: M ⊗ N → N ⊗ M

is nothing but aγ ′

M ⊗ N = a(qM ⊗p qN ) −−→ a(qN ⊗p qM) = N ⊗ M, where γ ′ is the twisting map for presheaves. (2.47)

The natural map ev is the composite ¯ H

HomMod(X) (M, N ) ⊗OX M = a(q HomMod(X) (M, N ) ⊗p qM) −→ a ev′

ε

a(HomPM(X) (qM, qN ) ⊗p qM) −−−→ aqM − → M, where ev′ is the evaluation map for presheaves.

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(2.48)

The natural map tr is the composite

ε−1

tr′

M −−→ aqM −→ a HomPM(X) (qN , qM ⊗p qN ) P¯

− → HomMod(X) (aqN , a(qM ⊗p qN )) ε−1

= HomMod(X) (aqN , M ⊗ N ) −−→ HomMod(X) (N , M ⊗ N ), where tr′ is the trace map for presheaves. Lemma 2.49. The inclusion q : Mod(X) → PM(X) and the natural transformations u

m : qM ⊗pOX qN − →qa(qM ⊗pOX qN ) = q(M ⊗OX N ) and

id

η : qOX − →qOX form a symmetric monoidal functor, see [26, (3.4.2)]. Letting S be the connected category with two objects and one non-trivial morphism, (a, q) forms a Lipman monoidal adjoint pair. The map H (see ¯ (see Lemma 2.38). The map P (see for the deﬁnition, (1.34)) agrees with H ¯ for the deﬁnition, (1.50)) agrees with P (see (2.39)). The map Δ (see (1.40)) ¯ (see Lemma 2.18). agrees with Δ Proof. We prove the ﬁrst assertion. First we prove that the diagram m

qOX ⊗p qM −→ q(OX ⊗ M) ↑ η ⊗p 1 ↓ qλ λ′

qOX ⊗p qM −→

qM

is commutative for M ∈ Mod(X). This is trivial from the deﬁnition of the sheaf tensor product (2.12), the deﬁnition of λ (2.45), and the commutativity of the diagram u / qOX ⊗p qM qa(qOX ⊗ qM) qaλ′

λ′

u / qaqM qM QQ QQQ QQQid QQQ ε QQQ ( qM . Next we prove that the diagram m

qM ⊗p qN −→ q(M ⊗ N ) ↓ γ′ ↓ qγ m qN ⊗p qM −→ q(N ⊗ M)

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303

is commutative for M, N ∈ Mod(X). By the deﬁnition of m and γ (2.46), the diagram is nothing but u

qM ⊗p qN − → qa(qM ⊗p qN ) ′ ↓γ ↓ qaγ ′ , u p qN ⊗ qM − → qa(qN ⊗p qM) which is commutative by the naturality of u. To prove that q is a symmetric monoidal functor, it remains to prove that (qM ⊗p qN ) ⊗p qP

α′

m

m

qM ⊗p q(N ⊗ P)

q(M ⊗ N ) ⊗p qP m

q((M ⊗ N ) ⊗ P)

/ qM ⊗p (qN ⊗p qP)

m qα

/ q(M ⊗ (N ⊗ P))

is commutative, where α′ is the associativity map for presheaves. By the deﬁnition of m, the diagram equals (qM ⊗p qN ) ⊗p qP

α′

u

u

qM ⊗p qa(qN ⊗p qP)

qa(qM ⊗p qN ) ⊗p qP u

qa(qa(qM ⊗p qN ) ⊗p qP)

/ qM ⊗p (qN ⊗p qP)

u qα

/ qa(qM ⊗p qa(qN ⊗p qP)) .

By the naturality of u, the commutativity of this diagram is reduced to the commutativity of (qM ⊗p qN ) ⊗p qP u

qa((qM ⊗p qN ) ⊗p qP) u

qa(qa(qM ⊗p qN ) ⊗p qP)

α′

(a) qaα′

(b) qα

/ qM ⊗p (qN ⊗p qP) u

/ qa(qM ⊗p (qN ⊗p qP)) u

/ qa(qM ⊗p qa(qN ⊗p qP)) .

304

2 Sheaves on Ringed Sites

The commutativity of (a) is obvious by the naturality of u. The commutativity of (b) follows from the deﬁnition of α (2.44). We have proved that q is a symmetric monoidal functor. ¯ By deﬁnition (1.40), Δ is the comNext we prove that Δ agrees with Δ. posite ¯ ∆

u

ε

→ aqa(qaM ⊗p qaN ) − → a(qaM ⊗p qaN ), a(M ⊗p N ) −→ a(qaM ⊗p qaN ) − ¯ which agrees with Δ. ¯ For b, c ∈ Mod(X), the diagram Next we prove that H = H.

q[b, c]

tr′

(a)

¯ H

[qb, qc]

/ [qb, q[b, c] ⊗p qb]

tr

′

¯ H

/ [qb, [qb, qc] ⊗p qb]

OOO OOOid OOO OO'

u

/ [qb, qa(q[b, c] ⊗p qb)]

(b) u

¯ H

/ [qb, qa([qb, qc] ⊗p qb)]

(c) ev′ ev′ u / [qb, qaqc] [qb, qc] T TTTT TTTTid TTTT ε TTTT ) [qb, qc]

is commutative, where [d, e] stands for Hom(d, e). Indeed, (a) is commutative by the naturality of tr′ , and (b) and (c) are commutative by the naturality of u. The commutativity of the two triangles are obvious. ¯ tr′ By the deﬁnition of H (1.34) and ev (2.47), the composite ε ev′ Hu ¯ agrees with H. By the commutativity of the diagram, we have that H = H. Now we prove that (a, q) forms a Lipman adjoint pair. That is, Δ and C ¯ is an isomorphism by Lemma 2.18 and Δ = Δ, ¯ Δ are isomorphisms. Since Δ is an isomorphism. Note that C : aqOX → OX is nothing but ε by deﬁnition. Since q is fully faithful, ε is an isomorphism (apply [19, Lemma I.1.2.6, 4] for the adjoint pair (q op , aop )), and we are done. So the deﬁnition of P makes sense. We prove that P = P¯ . For b, c ∈ PM(X), the diagram

ED

2 Sheaves on Ringed Sites

id / a[b, c] 4 a[b, c] j j j j j j id jjj (a) u=ε−1 jjjj j j j ′ jjj tr / a[qab, [b, c] ⊗p qab] a[b, c] aqa[b, c] UUUU n n n U U n UUUuU id nn n u UUUU tr′ tr′ (c) (d) nnn (b) UUU* n n vn ev′ u / a[b, [b, c] ⊗p qab] a[b, [b, c] ⊗p b] a[qab, qa[b, c] ⊗p qab] a[b, c] o (g) UUUU p iii i UUu⊗ UUUuU (f) u iiii i ¯ ¯ ¯ u P P P (e) UUUU iii UU* iiii t i ′ ev a[b, qa[b, c] ⊗p qab] [ab, ac] o [ab, a([b, c] ⊗p b)] (i) [aqab, a(qa[b, c] ⊗p qab)] UUUU (h) hPPP p iiii PPP UUUu⊗ UUUUu iiii PPP i ¯ i P (j) i UUUU −1 PPP ii ev′ U* tiiii u=ε (u⊗p u)−1 p p o [ab, a(qa[b, c] ⊗ qab)] [ab, a([b, c] ⊗ b)] O id id (k) tr [ab, a([b, c] ⊗p b)] o [ab, a[b, c] ⊗ ab] o

BC

∆−1

305

306

2 Sheaves on Ringed Sites

is commutative. Note that ε : aqa → a has an inverse, which must agree with u. The commutativity of (a) is the naturality of tr′ . The commutativity of (b) is the basics on adjunction. The commutativity of (c) is Lemma 1.32. The commutativity of (d) is the deﬁnition of tr, see (2.48). The commutativity of (e), (h) and (i) is the naturality of P¯ . The commutativity of (f), (g) and (j) ¯ Thus the diagram is trivial. The commutativity of (k) is the deﬁnition of Δ. is commutative, and we have P¯ = P , by the deﬁnition of P . ⊓ ⊔ (2.50) Let f : Y → X be a ringed continuous functor. For M, N ∈ PM(X), we deﬁne # # # (M ⊗pOX N ) M ⊗pOY fPM N → fPM m = mPM (f ) : fPM

by # # Γ(y, fPM N ) = Γ(f y, M) ⊗Γ(y,OY ) Γ(f y, N ) M ⊗pOY fPM # → Γ(f y, M) ⊗Γ(f y,OX ) Γ(f y, N ) = Γ(y, fPM (M ⊗pOX N ))

(m ⊗ n → m ⊗ n) for each y ∈ Y. We also deﬁne η = ηPM (f ) to be the canonical map c

# # → fPM qOX . OX − qOY → qfMod # together with m and η above forms a symLemma 2.51. The functor fPM metric monoidal functor.

Proof. Consider the diagram m

f # qOX ⊗p f # M −→ f # (qOX ⊗p M) ↑ η ⊗p 1 ↓ f #λ , qOY ⊗p f # M

λ

− →

f #M

whose commutativity we need to prove. For y ∈ Y, applying Γ(y, ?) to this diagram, we get Γ(f y, OX ) ⊗Γ(y,OY ) Γ(f y, M) → Γ(f y, OX ) ⊗Γ(f y,OX ) Γ(f y, M) ↑ ↓ . Γ(f y, M) Γ(y, OY ) ⊗Γ(y,OY ) Γ(f y, M) → By the bottom horizontal arrow, a ⊗ m ∈ Γ(y, OY ) ⊗Γ(y,OY ) Γ(f y, M) goes to η(a)m. We get the same result when we keep track the other path in the diagram. So the diagram in question is commutative. The rest of the proof is similar, and we leave it to the reader. ⊓ ⊔

2 Sheaves on Ringed Sites

307

(2.52) Let f : Y → X be a ringed continuous functor as above. We deﬁne m = mMod (f ) to be the composite map c

# # # # # # qN ) N) − → a(fPM qM ⊗pOY fPM fMod M ⊗OY fMod N = a(qfMod M ⊗pOY qfMod via m

via θ

# # # PM −−−−− →afPM (qM ⊗pOX qN )−−−→fMod a(qM ⊗pOX qN ) = fMod (M ⊗OX N ), # # where θ : afPM → fMod a is the composite map via u

c

via ε

# # # # afPM −−−→afPM qa − → aqfMod a−−−→fMod a. # We deﬁne η = ηMod (f ) to be the given map of sheaves of rings OY → fMod OX . # :Mod(X)→ Lemma 2.53. Let the notation be as above. Then the functor fMod Mod(Y), together with m and η, is a symmetric monoidal functor.

Proof. The diagram u

c

ε

→ af # qaq − → aqf # aq − → f # aq af # q − ց id ↓ ε ↓ε ↓ε c ε af # q − → aqf # − → f#

(2.54)

is commutative. For M ∈ Mod(X), consider the diagram m

θ

−→ af # (qOX ⊗p qM) − → f # a(qOX ⊗p qM) a(f # qOX ⊗p f # qM) −−PM ↑ ηPM (a) ↓λ (b) ↓λ a(qOY ⊗p f # qM) ↓c

− → (c)

λ

af # qM ↓c

− → (d)

θ

f # aqM ↓ε

a(qOY ⊗p qf # M)

− →

λ

aqf # M

− →

ε

f #M

.

(a) is commutative by Lemma 2.51. (b) is commutative by the naturality of θ. (c) is commutative by the naturality of λ. (d) is commutative by the commutativity of (2.54). So the whole diagram is commutative. This shows that the diagram m

# # # Mod fMod OX ⊗ fMod M −−− −→ fMod (OX ⊗ M) ↑η ↓λ # OY ⊗ fMod M

λ

− →

# fMod M

is commutative. The other axioms are checked similarly. The details are left to the reader. ⊓ ⊔ Now the following is easy to prove.

308

2 Sheaves on Ringed Sites

Lemma 2.55. Let S ′ denote the category of ringed categories and ringed # functors. Then ((?)PM # , (?)PM ) is an adjoint pair of monoidal almostpseudofunctors on (S ′ )op . Let S denote the category of ringed sites and # ringed continuous functors. Then ((?)Mod # , (?)Mod ) is an adjoint pair of op monoidal almost-pseudofunctors on S , see (1.40). (2.56) We give some comments on Lemma 2.55. Let g : Z → Y and f : # # Y → X be morphisms in S. Then c : (f g)# ♥ → g♥ f♥ is the identity map for ♥ = PM, Mod. For z ∈ Z, Γ(z, (f g)# M) = Γ(f gz, M) = Γ(gz, f # M) = Γ(z, g # f # M). In particular, the diagram c

q(f g)# −−−−− −−−−−→ (f g)# q ↓c ↓c c c qf # g # − → f # qg # − → f # g# q is commutative. Note also that (idX )# : ♥(X) → ♥(X) is the identity functor, and eX : Id → (idX )# is the identity. PM PM A straightforward computation shows that d : f# g# → (f g)PM is # given by Γ(x, f# g# M) = lim Γ(x, OX ) ⊗Γ(y,OY ) lim Γ(y, OY ) ⊗Γ(z,OZ ) Γ(z, M) → −→ −→ lim Γ(x, OX ) ⊗Γ(z,OZ ) Γ(z, M) = Γ(x, (f g)# M), −→ where → is given by a ⊗ b ⊗ m → ab ⊗ m. A straightforward diagram chasing (using Lemma 2.33) shows that d : Mod Mod f# g# → (f g)Mod agrees with the composite # u−1

d

Mod Mod PM PM PM PM g# = af# qag# q −−→ af# g# q − → a(f g)PM f# # q = (f g)# ,

where u−1 is the inverse of the isomorphism u : af# → af# qa, see Lemma 2.34. (2.57)

Let g′

X −→ X′ ↑f ↑ f′ g Y − → Y′

(2.58)

be a commutative diagram of ringed sites and ringed (continuous) functors. ′ Then the natural map θ : g# f # → (f ′ )# g# is deﬁned, see (1.21). Using ′ the explicit description of the unit map u : 1 → (g ′ )# g# and the counit

2 Sheaves on Ringed Sites

309

map ε : g# g # → 1 in (2.20) and c : f # (g ′ )# ∼ = g # (f ′ )# in (2.56), it is straightforward to check the following. For M ∈ PM(X) and y ′ ∈ Y, Γ(y ′ , θ) : Γ(y ′ , g# f # M) = lim Γ(y ′ , OY′ ) ⊗Γ(y,OY ) Γ(f y, M) −→ ′ y →gy

→ Γ(y

′

′ , (f ′ )# g# M)

=

lim −→

Γ(f ′ y ′ , OX′ ) ⊗Γ(x,OX ) Γ(x, M)

f ′ y ′ →g ′ x

is induced by the map Γ(y ′ , OY′ ) ⊗Γ(y,OY ) Γ(f y, M) → Γ(f ′ y ′ , OX′ ) ⊗Γ(f y,OX ) Γ(f y, M) (a ⊗ m → a ⊗ m) for (y ′ → gy) ∈ (Iyg′ )op . θMod is described by θPM as follows. Lemma 2.59. Let (2.58) be a commutative diagram of ringed sites and ringed ′ continuous functors. Then θMod : g# f # → (f ′ )# g# is the composite c

θ

θ

′ ′ ′ g# f # = ag# qf # − → qg# f # q −−PM − → a(f ′ )# g# q− → (f ′ )# ag# q = (f ′ )# g# .

Proof. Left to the reader as an exercise (utilize Lemma 2.60).

⊓ ⊔

Lemma 2.60. Let f : Y → X be a ringed continuous functor. Then the diagram of functors PM(X) → PM(Y) u

f# − → f # qa ↓u ↓ c(f ) θ

qaf # − → qf # a is commutative. Proof. Left to the reader as an exercise.

⊓ ⊔

(2.61) Let S ′ and S be as in Lemma 2.55. Then the monoidal adjoint pairs # # Mod ((?)PM # , (?)PM ) and ((?)# , (?)Mod ) are not Lipman, see (6.9). Lemma 2.62. Let A be an abelian category which satisﬁes the (AB3) condition, I a small category, and ((aλ )λ∈I , (ϕf )f ∈Mor(I) ) a direct system in A. Assume that I has an initial object λ0 , and ϕf is an isomorphism for any f ∈ Mor(I). Then aλ → lim aλ is an isomorphism for any λ. −→ Proof. It suﬃces to show that aλ0 → lim aλ has an inverse. For each λ, −→ consider ϕ−1 f (λ) : aλ → aλ0 , where f (λ) is the unique map λ0 → λ. Then the collection (ϕ−1 lim aλ → aλ0 . This gives the desired f (λ) ) induces a morphism − → inverse. ⊓ ⊔

310

2 Sheaves on Ringed Sites

Lemma 2.63. Let Y and X be ringed categories, and f : Y → X be a ringed functor. Assume that for each x ∈ X, Ixf has a terminal object. Then C : f# OY → OX is an isomorphism. Proof. Let x ∈ X. Then Γ(x, f# OY ) = lim Γ(x, OX ) ⊗Γ(y,OY ) Γ(y, OY ), where −→ the colimit is taken over (Ixf )op , which has an initial object (y0 , (h : x → f y0 )) by assumption. By Lemma 2.62, the canonical maps Γ(x, OX ) ∼ = Γ(x, OX ) ⊗Γ(y0 ,OY ) Γ(y0 , OY ) → Γ(x, f# OY ) ⊓ ⊔

are isomorphisms. So C is an isomorphism, as can be seen easily.

Corollary 2.64. Let f : Y → X be a morphism of ringed sites. If has a Mod OY → OX is an isomorphism. terminal object for x ∈ X, then C : f# Ixf

Proof. Note that C is the composite C′

ε

Mod OY = af# qOY −→ aqOX − → OX , f#

where C ′ : f# qOY → qOX is the C associated with the ringed functor (Y, qOY ) → (X, qOX ). Note that C ′ is an isomorphism by the lemma. As ε : aq → 1 is also an isomorphism, C is also an isomorphism, as desired. ⊓ ⊔ Corollary 2.65. Let f : X → Y be a morphism of ringed spaces. Then C : f ∗ OY → OX is an isomorphism. Proof. For each open subset U of X, IUf (U ֒→ X)).

−1

has the terminal object (Y, ⊓ ⊔

Corollary 2.66. Let f : Y → X be a morphism of ringed sites, and y ∈ Y. Then we may consider the induced morphism of ringed sites f /y : Y/y → X/f y. The canonical map C : (f /y)# (OY |y ) → OX |x is an isomorphism. f /y

Proof. For ϕ : x → f y in X/f y, Iϕ

has a terminal object (idy , ϕ).

⊓ ⊔

Chapter 3

Derived Categories and Derived Functors of Sheaves on Ringed Sites

We utilize the notation and terminology on triangulated categories in [44]. However, we usually write the suspension (translation) functor of a triangulated category by Σ or (?)[1]. Let T be a triangulated category. Lemma 3.1. Let

fλ

gλ

h

λ (aλ −→bλ −→cλ −→Σa λ)

be

a small

family of

distinguished triangles in T . Assume that the coproducts aλ , bλ , and cλ exist. Then the triangle

fλ

aλ −−−→

gλ

bλ −−−→

H◦ hλ cλ −−−−−→Σ( aλ )

is distinguished, where H : Σaλ → Σ( aλ ) is the canonical isomorphism. Similarly, a product of distinguished triangles is a distinguished triangle. We refer the reader to [36, Proposition 1.2.1] for the proof of the second assertion. The proof for the ﬁrst assertion is similar [36, Remark 1.2.2]. (3.2) Let A be an abelian category. The category of unbounded (resp. bounded below, bounded above, bounded) complexes in A is denoted by C(A) (resp. C + (A), C − (A), C b (A)). The corresponding homotopy category and the derived category are denoted by K ? (A) and D? (A), where ? is either ∅ (i.e., nothing), +, − or b. The localization K ? (A) → D? (A) is denoted by Q. We denote the homotopy category of complexes in A with unbounded (resp. ¯ ? (A). bounded below, bounded above, bounded) cohomology groups by K ¯ ? (A). The corresponding derived category is denoted by D ? ¯? For a plump subcategory A′ of A, we denote by KA ′ (A) (resp. KA′ (A)) ? ? ¯ (A)) consisting of complexes whose the full subcategory of K (A) (resp. K ? cohomology groups are objects of A′ . The localization of KA ′ (A) by the ´epaisse subcategory (see for the deﬁnition, [44, Chapitre 1, §2, (1.1)]) of ? ¯? exact complexes is denoted by DA ′ (A). The category DA′ (A) is deﬁned sim? ilarly. Note that the canonical functor DA′ (A) → D(A) is fully faithful, J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

311

312

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

? and hence DA ′ (A) is identiﬁed with the full subcategory of D(A) consisting of unbounded (resp. bounded below, bounded above, bounded) complexes whose cohomology groups are in A′ . Note also that the canonical functor ? ¯? DA ′ (A) → DA′ (A) is an equivalence.

(3.3) Let A and B be abelian categories, and F : K ? (A) → K ∗ (B) a triangulated functor. Let C be a triangulated subcategory of K ? (A) such that 1 If c ∈ C is exact, then F c is exact. 2 For any a ∈ K ? (A), there exists some quasi-isomorphism a → c. The condition 2 implies that the canonical functor i(C) : C/(E ∩ C) → K ? (A)/E = D? (A) is an equivalence, where E denotes the ´epaisse subcategory of exact complexes in K ? (A), see [44, Chapitre 2, §1, (2.3)]. We ﬁx a quasi-inverse p(C) : D? (A) → C/(E ∩ C). On the other hand, the composite Q

F

C ֒→ K ? (A)− →K ∗ (B) − → D∗ (B)

(3.4)

is factorized as QC

F

C −−→C/(E ∩ C)− →D∗ (B)

(3.5)

up to a unique natural isomorphism, by the universality of localization and the condition 1 above. Under the setting above, we have the following [17, (I.5.1)]. Lemma 3.6. The composite functor p(C)

F

RF : D? (A)−−→C/(E ∩ C)− →D∗ (B) is a right derived functor of F . For more about the existence of a derived functor, see [26, (2.2)]. We denote the map QF → (RF )Q in the deﬁnition of RF (see [17, p.51]) by Ξ or Ξ(F ). (3.7) Here we are going to review Spaltenstein’s work on unbounded derived categories [39]. A chain complex I of A is called K-injective if for any exact sequence E of A, the complex of abelian groups Hom•A (E, I) is also exact. A morphism f : C → I in K(A) is called a K-injective resolution of C, if I is K-injective and f is a quasi-isomorphism. The following is pointed out in [9]. Lemma 3.8. Let A be an abelian category, and I ∈ C(A). Then the following are equivalent.

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

313

1 I is K-injective, and In is an injective object of A for each n ∈ Z. 2 For an exact sequence 0 → A → B → C → 0 in C(A) with C exact, any chain map A → I lifts to B. Proof. 1⇒2. The sequence of complexes of abelian groups 0 → Hom•A (C, I) → Hom•A (B, I) → Hom•A (A, I) → 0 is exact, since each term of I is injective. So H 0 (Hom•A (B, I)) → H 0 (Hom•A (A, I)) → H 1 (Hom•A (C, I)) is exact. But H 1 (Hom•A (C, I)) = 0, since C is exact and I is K-injective. So H 0 (Hom•A (B, I)) → H 0 (Hom•A (A, I)) is surjective. Since Hom−1 A (B, I) → Hom−1 (A, I) is surjective, the commutative diagram with exact rows A Hom−1 A (B, I) Hom−1 A (A, I)

∂

/ Z 0 (Hom• (B, I))

/ H 0 (Hom• (B, I)) A

/0

/ Z 0 (Hom• (A, I)) A

/ H 0 (Hom• (A, I))

/0

A

∂

A

shows that Z 0 (Hom•A (B, I)) → Z 0 (Hom•A (A, I)) is surjective. This is what we wanted to prove. 2⇒1 First we prove that I is K-injective. It suﬃces to show that for any exact complex F, any chain map ϕ : F → I is null-homotopic. Let C = Cone(ϕ), where Cone denotes the mapping cone. Consider the exact sequence 0 → I → C → F[1] → 0. So the identity map I → I lifts to ψ : C → I. Let s be the restriction of ψ to F[1] ⊂ C. It is easy to see that ϕ = sd + ds. So ϕ is null-homotopic, as desired. Next we show that In is injective for any n. To prove this, let f : A → B be a monomorphism in A, and ϕ : A → In a morphism. Let C be the cokernel of f . Deﬁne a complex A by An = An+1 = A, dnA = id, and Ai = 0 (i = n, n + 1). Replacing A by B and C, we deﬁne the complexes B and C, respectively. Deﬁne f • : A → B by f n = f n+1 = f and f i = 0 for i = n, n+1. Obviously, Coker f • ∼ = C is exact. Deﬁne a chain map Φ : A → I by Φn = ϕ and Φn+1 = dnI ◦ ϕ. By assumption, there is a chain map Ψ : B → I such that Φ = Ψf • . So ϕ = ⊓ ⊔ Φn = Ψn ◦ f n = Ψn ◦ f , and Ψn lifts ϕ. For I ∈ C(A), we say that I is strictly injective if I satisﬁes the equivalent conditions in the lemma. A strictly injective resolution is a quasi-isomorphism F → I with I strictly injective. The following is proved in [9]. See also [39] and [1].

314

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

Lemma 3.9. If A is Grothendieck, then for any chain complex F ∈ C(A) admits a strictly injective resolution F → I which is a monomorphism. A chain complex I is K-injective if and only if K(A)(E, I) = 0 for any exact sequence E. It is easy to see that the K-injective complexes form an ´epaisse subcategory I(A) of K(A). (3.10) Let F : K(A) → K(B) be a triangulated functor, and assume that A is Grothendieck. Let I be the full subcategory of K-injective complexes of K(A). It is easy to see that I is triangulated, and I ∩ E = 0. By Lemma 3.6, the composite p(I)

F

→D(B) D(A)−−→I − is a right derived functor RF of F . Note that to ﬁx p(I) and the isomorphism IdD(A) → i(I)p(I) is nothing but to ﬁx a functorial K-injective resolution F → pQF = IF in K(A). (3.11) Let F : K(A) → K(B) be a triangulated functor. Assume that A is Grothendieck. For F ∈ K(A), F is (right) F -acyclic (more precisely, Q ◦ F acyclic, where Q : K(B) → D(B) is the localization. See for the deﬁnition, [26, (2.2.5)]) if and only if for some K-injective resolution F → I, F (F) → F (I) is a quasi-isomorphism, if and only if for any K-injective resolution F → I, F (F) → F (I) is a quasi-isomorphism. Note that the set of F -acyclic objects in K(A) forms a localizing subcategory of K(A), see [26, (2.2.5.1)]. Lemma 3.12. Let A and B be abelian categories, and F : A → B an exact functor with the right adjoint G. Assume that B is Grothendieck. Then KG : K(B) → K(A) preserves K-injective complexes. Moreover, RG : D(B) → ˙ = F. D(A) is the right adjoint of LF Proof. Let M ∈ K(A), and I a K-injective complex of K(B). Then HomK(A) (M, (KG)I) ∼ = H 0 (Hom•A (M, GI)) ∼ = H 0 (Hom• (F M, I)) = HomK(B) (F M, I). B

If M is exact, then the last group is zero. This shows (KG)I is K-injective. Now let M ∈ D(A) and N ∈ D(B) be arbitrary. Then by the ﬁrst part, we have a functorial isomorphism HomD(A) (M, (RG)N) ∼ = HomK(A) (M, (KG)IN ) ∼ = HomK(B) (F M, IN ) ∼ = HomD(B) (F M, N). This proves the last assertion. ⊓ ⊔ Remark 3.13. Note that for an abelian category A, we have ob(C(A)) = ob(K(A)) = ob(D(A)). Thus, an object of one of the three categories is sometimes viewed as an object of another.

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

315

(3.14) Let A be a closed symmetric monoidal abelian category which satisﬁes the (AB3) and (AB3*) conditions. Let ⊗ be the multiplication and [?, ?] be the internal hom. For a ﬁxed b ∈ A, (? ⊗ b, [b, ?]) is an adjoint pair. In particular, ? ⊗ b preserves colimits, and [b, ?] preserves limits. By symmetry, a⊗? also preserves colimits. As we have an isomorphism A(a, [b, c]) ∼ = A(a ⊗ b, c) ∼ = A(b ⊗ a, c) ∼ = A(b, [a, c]) ∼ = Aop ([a, c], b), we have [?, c] : Aop → A is right adjoint to [?, c] : A → Aop . This shows that [?, c] changes colimits to limits. As in [17], we deﬁne the tensor product F ⊗• G of F, G ∈ C(A) by Fp ⊗ Gq . (F ⊗• G)n := p+q=n

The diﬀerential dn on Fp ⊗ Gq is deﬁned to be dn = dF ⊗ 1 + (−1)p 1 ⊗ dG (the sign convention is slightly diﬀerent from [17], but this is not essential). We have F ⊗• G ∈ C(A). Similarly, [F, G]• ∈ C(A) is deﬁned by [Fp , Gn+p ] [F, G]n := p∈Z

and dn := [dF , 1] + (−1)n+1 [1, dG ]. It is straightforward to prove the following. Lemma 3.15. Let A be as above. Then the category of chain complexes C(A) is closed symmetric monoidal with ⊗• the multiplication and [?, ?]• the internal hom. The bi-triangulated functors ⊗• : K(A) × K(A) → K(A) and

[?, ?]• : K(A)op × K(A) → K(A),

are induced, and K(A) is a closed symmetric monoidal triangulated category (see [26, (3.5), (3.6)]). (3.16) Let A be an abelian category, and P a full subcategory of C(A). An inverse system (Fi )i∈I in C(A) is said to be P-special if the following conditions are satisﬁed. i I is well-ordered. ii If i ∈ I has no predecessor, then the canonical map Ii → limj
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3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

iii If i ∈ I has a predecessor i − 1, then the natural chain map Ii → Ii−1 is an epimorphism, the kernel Ci is isomorphic to some object of P, and the exact sequence 0 → Ci → Ii → Ii−1 → 0 is semi-split. Similarly, P-special direct systems are also deﬁned, see [39]. The full subcategory of C(A) consisting of inverse (resp. direct) limits of P-special inverse (resp. direct) systems is denoted by P (resp. P). ← − − → (3.17) Let (X, OX ) be a ringed site. Various deﬁnitions and results on unbounded complexes of sheaves over a ringed space by Spaltenstein [39] is generalized to those for ringed sites. However, note that we can not utilize the notion related to closed subsets, points, or stalks of sheaves. (3.18) We say that a complex F ∈ C(Mod(X)) is K-ﬂat if G ⊗• F is exact whenever G is an exact complex in Mod(X). We say that A ∈ C(Mod(X)) is weakly K-injective if A is Hom•Mod(X) (F, ?)-acyclic for any K-ﬂat complex F. (3.19) Let (X, OX ) be a ringed site. For x ∈ X, we deﬁne Oxp to be Mod LPM (OX |x ) ∼ = aOxp . We denote by P0 = x ((qOX )|x ), and Ox := Lx P0 (X, OX ) the full subcategory of C(Mod(X)) consisting of complexes of the form Ox [n] with x ∈ X. We deﬁne P = P(X, OX ) to be P0 . We call an object − → of P a strongly K-ﬂat complex. We also deﬁne Q to be the full subcategory of C(Mod(X)) consisting of bounded above complexes whose terms are direct sums of copies of Ox . We say that A ∈ C(Mod(X)) is K-limp if A is Hom•Mod(X) (F, ?)-acyclic for any strongly K-ﬂat complex F. Lemma 3.20. Let f : (Y, OY ) → (X, OX ) be a ringed continuous functor. Then we have an isomorphism Mod (Oy ) ∼ f# = Of y

for y ∈ Y. In particular, if F ∈ P(Y), then f# F ∈ P(X). Proof. For y ∈ Y, we denote the canonical continuous ringed functor (Y/y, OY |y ) → (X/f y, OX |f y ) by f /y. We have Rf y ◦ (f /y) = f ◦ Ry . Hence by Corollary 2.66, f# Oy = f# Ly (OY |y ) ∼ = Lf y (f /y)# (OY |y ) ∼ = Lf y (OX |f y ) = Of y . ⊓ ⊔ Lemma 3.21. Let (X, OX ) be a ringed site, and F, G ∈ C(Mod(X)). Then the following hold:

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

317

1 F is K-ﬂat if and only if Hom•Mod(X) (F, I) is K-injective for any K-injective complex I. 2 If F is K-ﬂat exact, then G ⊗•OX F is exact. 3 The inductive limit of a pseudo-ﬁltered inductive system of K-ﬂat complexes is again K-ﬂat. 4 The tensor product of two K-ﬂat complexes is again K-ﬂat. See [39] for the proof. For 2, utilize 3 of Lemma 3.25 and Corollary 3.23 below. Proposition 3.22. Let (X, OX ) be a ringed site, and x ∈ X. Then Ox is K-ﬂat. Proof. It suﬃces to show that for any exact complex E, E ⊗ Ox is exact. To verify this, it suﬃces to show that for any K-injective complex I, the complex Hom•OX (E ⊗ Ox , I) is exact. Indeed, then if we consider the K-injective resolution E ⊗ Ox → I, it must be null-homotopic and thus E ⊗ Ox must be exact. Note that we have Hom•OX (E ⊗ Ox , I) ∼ = Hom•OX (Ox , Hom•OX (E, I)) ∼ = ∼ Hom• (Lx (E|x ), I). Hom•Mod(X/x) (E|x , I|x ) = OX As (?)|x and Lx are exact (2.36), (2.23), the last complex is exact, and we are done. ⊓ ⊔ Corollary 3.23. A strongly K-ﬂat complex is K-ﬂat. Proof. Follows immediately from the proposition. (3.24)

Let A be an abelian category. For an object dn

dn+1

F : · · · → F n −→F n+1 −−−→ → · · · in C(A), we denote the truncated complex dn

dn+1

0 → F n / Im dn−1 −→F n+1 −−−→ → · · · by τ≥n F. Similarly, the truncated complex dn−2

dn−1

· · · → F n−2 −−−→F n−1 −−−→ Ker dn → 0 is denoted by τ≤n F. Lemma 3.25. Let (X, OX ) be a ringed site and F ∈ C(Mod(X)). 1 We have Q ⊂ P.

⊓ ⊔

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3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

2 A K-injective complex is weakly K-injective, and a weakly K-injective complex is K-limp. 3 For any H ∈ C(Mod(X)), there is a Q-special direct system (Fn ) and a direct system of chain maps (fn : Fn → τ≤n H) such that fn is a quasiisomorphism for each n ∈ N, and (Fn )l = 0 for l ≥ n + 1. We have lim Fn → H is a quasi-isomorphism, and lim Fn ∈ P. Moreover, QH is −→ −→ the homotopy colimit of the inductive system (τ≤n QH) in the category D(Mod(X)). 4 The following are equivalent. i ii iii iv v

F is K-limp. F is K-limp as a complex of sheaves of abelian groups. F is Hom•Mod(X) (Ox , ?)-acyclic for x ∈ X. F is Γ(x, ?)-acyclic for x ∈ X. If G ∈ P and G is exact, then Hom•Mod(X) (G, F) is exact.

5 The following are equivalent. i F is weakly K-injective. ii If G is K-ﬂat exact, then Hom•Mod(X) (G, F) is exact. iii For any K-ﬂat complex G, Hom•Mod(X) (G, F) is weakly K-injective. Proof. 1 is trivial. 2 follows from the deﬁnition and Corollary 3.23. The proof of 3 and 4 are left to the reader, see [39, (3.2), (3.3), (5.16), (5.17), (5.21)]. 5 is similar. ⊓ ⊔ Lemma 3.26. Let (X, OX ) be a ringed site and F, G ∈ C(Mod(X)). If F is weakly K-injective and G is K-ﬂat, then F is Hom•Mod(X) (G, ?)-acyclic. Proof. Let F → I be the K-injective resolution, and J the mapping cone. Let ϕ : H → Hom•Mod(X) (G, J) be a P-resolution. As H ⊗• G is K-ﬂat and J is weakly K-injective exact, Hom•Mod(X) (H, Hom•Mod(X) (G, J)) ∼ = Hom•Mod(X) (H ⊗• G, J) is exact. So ϕ must be null-homotopic, and hence Hom•Mod(X) (G, J) is exact. This is what we wanted to prove. ⊓ ⊔ (3.27) Let (X, OX ) be a ringed site. For G ∈ C(Mod(X)), it is easy to see that G⊗•OX ? induces a functor from K(Mod(X)) to itself. By Lemma 3.25, 3 and the dual assertion of [17, Theorem I.5.1], the derived functor L(G⊗•OX ?) is induced, and it is calculated using any K-ﬂat resolution of ?. If we ﬁx ?, then L(G⊗•OX ?) is a functor on G, and it induces a bifunctor ∗⊗•,L OX ? : D(Mod(X)) × D(Mod(X)) → D(Mod(X)). G⊗•,L OX F is calculated using any K-ﬂat resolution of F or any K-ﬂat resolution of G. Note that ⊗•,L OX is a △-functor as in [26, (2.5.7)].

3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

319

We deﬁne the hyperTor functor as follows: •,L −i X TorO i (F, G) := H (F ⊗OX G).

(3.28) Let (X, OX ) be a ringed site. For F ∈ C(Mod(X)), the functor Hom•OX (F, ?) induces a functor from K(Mod(X)) to itself. As Mod(X) is Grothendieck, we can take K-injective resolutions, and hence the right derived functor R Hom•OX (F, ?) is induced. Thus a bifunctor R HomOX (∗, ?) : D(Mod(X))op × D(Mod(X)) → D(Mod(X)) is induced. For F, G ∈ D(Mod(X)), we deﬁne the hyperExt sheaf of F and G by ExtiOX (F, G) := H i (R Hom•OX (F, G)). Similarly, the functor Hom•OX (∗, ?) induces R Hom•OX (∗, ?) : D(Mod(X))op × D(Mod(X)) → D(Ab). Almost by deﬁnition, we have H i (R Hom•OX (F, G)) ∼ = HomD(Mod(X)) (F, G[i]). Sometimes we denote these groups by ExtiOX (F, G). Lemma 3.29. Let (X, OX ) be a ringed site. Then D(Mod(X)) is a closed symmetric monoidal triangulated category with ⊗•,L OX its product and R Hom•OX (∗, ?) its internal hom. Proof. This is straightforward.

⊓ ⊔

Lemma 3.30. Let f : Y → X be a continuous functor between sites. If I ∈ C # (AB(X)) is K-limp and exact, then fAB I is exact. # # I be a P(AB)-resolution of fAB I. It suﬃces to show Proof. Let ξ : F → fAB # • HomAB(Y) (F, fAB I) is exact (if so, then ξ must be null-homotopic). Since AB f# F ∈ P(AB) and I is K-limp exact, this is obvious. ⊓ ⊔

By the lemma, a K-limp complex is f # -acyclic. Lemma 3.31. Let f : (Y, OY ) → (X, OX ) be an admissible ringed continuous functor. Then the following hold: 1 If I ∈ C(AB(X)) is a K-injective (resp. K-limp) complex of sheaves of # abelian groups, then so is fAB I. Mod F is strongly 2 If F ∈ C(Mod(Y)) is strongly K-ﬂat and exact, then f# K-ﬂat and exact. # AB has an exact left adjoint f# , the assertion for K-injectivity Proof. As fAB in 1 is obvious.

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3 Derived Categories and Derived Functors of Sheaves on Ringed Sites

We prove the assertion for the K-limp property in 1. Let P ∈ P(Y) be an AB AB is exact, f# P is exact and a complex in P(X) by exact complex. As f# Lemma 3.20. Hence, # AB Hom•AB(Y) (P, fAB I) ∼ P, I) = Hom•AB(X) (f# # is exact. This shows fAB I is K-limp. We prove 2. We already know that f# F is strongly K-ﬂat by Lemma 3.20. Mod Mod F, I) is exact, where η : f# F → I is a We prove that Hom•Mod(X) (f# Mod K-injective resolution. Then η must be null-homotopic, and we have f# F is # exact and the proof is complete. Clearly, I is K-limp, and hence so is fMod I by 1 and Lemma 3.25, 4. The assertion follows immediately by adjunction. ⊓ ⊔ Mod (3.32) By the lemma, there is a derived functor Lf# : D(Mod(Y)) → Mod D(Mod(X)) of f# for an admissible ringed continuous functor f . It is calculated via strongly K-ﬂat resolutions. Now as in [39, section 6] and [26], the following is proved.

Lemma 3.33. Let S be the category of ringed sites and admissible ringed # continuous functors. Then (L(?)Mod # , R(?)Mod ) is a monoidal adjoint pair of Δ-almost-pseudofunctors (deﬁned appropriately as in [26, (3.6.7)]) on S op . The proof is basically the same as that in [26, Chapter 1–3], and left to the reader. Remark 3.34. Later we will treat ringed continuous functor f which may not be admissible. In this case, we may use Rf # and related functorialities, but not Lf# .

Chapter 4

Sheaves over a Diagram of S-Schemes

(4.1) Let S be a (small) scheme, and I a small category. We call an object of P(I op , Sch/S) an I-diagram of S-schemes, where Sch/S denotes the category of (small) S-schemes. We denote Sch/ Spec Z simply by Sch. So an object of P(I, Sch/S) is referred as an I op -diagram of S-schemes. Let X• ∈ P(I, Sch/S). We denote X• (i) by Xi for i ∈ I, and X• (φ) by Xφ for φ ∈ Mor(I). Let P be a property of schemes (e.g., quasi-compact, locally noetherian, regular). We say that X• satisﬁes P if Xi satisﬁes P for any i ∈ I. Let Q be a property of morphisms of schemes (e.g., quasi-compact, locally of ﬁnite type, smooth). We say that X• is Q over S if the structure map Xi → S satisﬁes Q for any i ∈ I. We say that X• has Q arrows if Xφ satisﬁes Q for any φ ∈ Mor(I). (4.2) Let f• : X• → Y• be a morphism in P(I, Sch/S). For i ∈ I, we denote f• (i) : Xi → Yi by fi . For a property Q of morphisms of schemes, we say that f• satisﬁes Q if so does fi for any i ∈ I. We say that f• is cartesian if the canonical map (fj , Xφ ) : Xj → Yj ×Yi Xi is an isomorphism for any morphism φ : i → j of I. (4.3) Let S, I and X• be as above. We deﬁne the Zariski site of X• , denoted by Zar(X• ), as follows. An object of Zar(X• ) is a pair (i, U ) such that i ∈ I and U is an open subset of Xi . A morphism (φ, h) : (j, V ) → (i, U ) is a pair (φ, h) such that φ ∈ I(i, j) and h : V → U is the restriction of Xφ . For a given morphism φ : i → j, U , and V , such an h exists if and only if V ⊂ Xφ−1 (U ), and it is unique. We denote this h by h(φ; U, V ). The composition of morphisms is deﬁned in an obvious way. Thus Zar(X• ) is a small category. For (i, U ) ∈ Zar(X• ), a covering of (i, U ) is a family of morphisms of the form ((idi , h(idi ; U, Uλ )) : (i, Uλ ) → (i, U ))λ∈Λ

such that λ∈Λ Uλ = U . This deﬁnes a pretopology of Zar(X• ), and Zar(X• ) is a site. As we will consider only the Zariski topology, a presheaf or sheaf on Zar(X• ) will be sometimes referred as a presheaf or sheaf on X• , if there J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

321

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4 Sheaves over a Diagram of S-Schemes

is no danger of confusion. Thus P(X• , C) and S(X• , C) mean P(Zar(X• ), C) and S(Zar(X• ), C), respectively. (4.4) Let S and I be as above, and let σ : J ֒→ I be a subcategory of I. Then we have an obvious restriction functor σ # : P(I, Sch/S) → P(J, Sch/S), which we denote by (?)|J . If ob(J) is ﬁnite and I(j, i) is ﬁnite for each j ∈ J and i ∈ I, then (?)|J has a right adjoint functor coskIJ = op (?)op σ# (?)op , because Sch/S has ﬁnite limits. Note that (coskIJ X• )i = op lim Xj , where the limit is taken over Iiσ , where σ op : J op → I op is the ←− opposite of σ. See [10, pp. 9–12]. (4.5) Let X• ∈ P(I, Sch/S). Then we have an obvious continuous functor Q(X• , J) : Zar((X• )|J ) ֒→ Zar(X• ). Note that Q(X• , J) may not be admis# sible. The restriction functors Q(X• , J)# AB and Q(X• , J)PA are denoted by PA (?)AB J and (?)J , respectively. For i ∈ I, we consider that i is the subcategory of I whose object set is {i} with Homi (i, i) = {id}. The restrictions (?)♥ i for ♥ = AB, PA are deﬁned. (4.6) Let F ∈ PA(X• ) and i ∈ I. Then Fi ∈ PA(Xi ), and thus we have a family of sheaves (Fi )i∈I . Moreover, for (i, U ) ∈ Zar(X• ) and φ : i → j, we have the restriction map res

Γ(U, Fi ) = Γ((i, U ), F)−→Γ((j, Xφ−1 (U )), F) = Γ(Xφ−1 (U ), Fj ) = Γ(U, (Xφ )∗ Fj ), which induces βφ (F) ∈ HomPA(Xi ) (Fi , (Xφ )∗ Fj ).

(4.7)

The corresponding map in HomPA(Xj ) ((Xφ )∗PA (Fi ), Fj ) is denoted by αφPA (F). If F is a sheaf, then (4.7) yields αφAB (F) ∈ HomAB(Xj ) ((Xφ )∗AB (Fi ), Fj ). It is straightforward to check the following. Lemma 4.8 ([10]). Let ♥ be either AB or PA. The following hold: ♥ 1 For any i ∈ I, we have αid : (Xidi )∗♥ (Fi ) → Fi is the canonical identiﬁi cation fXi . 2 If φ ∈ I(i, j) and ψ ∈ I(j, k), then the composite map d−1

♥ (Xψ )∗ ♥ αφ

α♥ ψ

(Xψφ )∗♥ (Fi ) −−→ (Xψ )∗♥ (Xφ )∗♥ (Fi )−−−−−−→(Xψ )∗♥ (Fj )−−→Fk

(4.9)

♥ . agrees with αψφ 3 Conversely, a family ((Gi )i∈I , (αφ )φ∈Mor(I) ) such that Gi ∈ ♥(Xi ), αφ ∈ Hom♥(Xj ) ((Xφ )∗♥ (Gi ), Gj ) for φ ∈ I(i, j), and that the conditions corresponding to 1,2 are satisﬁed yields G ∈ ♥(X• ), and this correspondence gives an equivalence.

4 Sheaves over a Diagram of S-Schemes

(4.10)

323

Similarly, a family ((Gi )i∈ob(I) , (βφ )φ∈Mor(I) ) with Gi ∈ ♥(Xi ) and βφ ∈ Hom♥(Xi ) (Gi , (Xφ )♥ ∗ Gj )

satisfying the conditions 1’ For i ∈ ob(I), βidi : Gi → (Xidi )∗ Gi is the canonical identiﬁcation eXi ; 2’ For φ ∈ I(i, j) and ψ ∈ I(j, k), the composite βφ

(Xφ )∗ βψ

c−1

Fi −→(Xφ )∗ (Fj )−−−−−→(Xφ )∗ (Xψ )∗ (Fk ) −−→ (Xψφ )∗ (Fk ) agrees with βψφ is in one to one correspondence with G ∈ ♥(X• ). (4.11) Let F ∈ AB(X• ). We say that F is an equivariant abelian sheaf if αφAB are isomorphisms for all φ ∈ Mor(I). For F ∈ PA(X• ), we say that F is an equivariant abelian presheaf if αφPA are isomorphisms for all φ ∈ Mor(I). An equivariant sheaf may not be an equivariant presheaf. However, an equivariant presheaf which is a sheaf is an equivariant sheaf. We denote the category of equivariant sheaves and presheaves by EqAB(X• ) and EqPA(X• ), respectively. As (Xφ )∗♥ is exact for ♥ = AB, PA and any φ, we have that EqAB(X• ) is plump in AB(X• ), and EqPA(X• ) is plump in PA(X• ). (4.12)

Let X• ∈ P(I, Sch/S). The data ((OXi )i∈I , (βφ = η : OXi → (Xφ )∗ OXj )φ∈Mor(I) )

gives a sheaf of commutative rings on X• , which we denote by OX• , and thus Zar(X• ) is a ringed site. The categories PM(Zar(X• )) and Mod(Zar(X• )) are denoted by PM(X• ) and Mod(X• ), respectively. Let ♥ = PM, Mod. Note that for M ∈ ♥(X• ) and φ : i → j, βφ : Mi → (Xφ )∗ Mj is a morphism in ♥(Xi ), which we denote by βφ♥ . The adjoint morphism Xφ∗ Mi → Mj is denoted by αφ♥ . α is not compatible with the forgetful functors in general. (4.13) For J ⊂ I, we have OX• |J = (OX• )J by deﬁnition. The continuous functor Q(X• , J) : (Zar(X• |J ), OX• |J ) → (Zar(X• ), OX• ) is actually a ringed continuous functor. ♥ The corresponding restriction Q(X• , J)# ♥ is denoted by (?)J for ♥ = PM, Mod. For subcategories J1 ⊂ J ⊂ I of I, we denote the restriction ♥ (?)♥ J1 : ♥(X• |J ) → ♥(X• |J1 ) by (?)J1 ,J , to emphasize J. (4.14)

Let ♥ be PM or Mod. Note that M ∈ ♥(X• ) is nothing but a family Dat(M) := ((Mi )i∈I , (αφ♥ )φ∈Mor(I) )

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4 Sheaves over a Diagram of S-Schemes

such that Mi ∈ ♥(Xi ), αφ♥ : (Xφ )∗♥ (Mi ) → Mj is a morphism of ♥(X• ) for any φ : i → j, and the conditions corresponding to 1,2 in Lemma 4.8 are satisﬁed. We say that M ∈ ♥(X• ) is equivariant if αφ♥ is an isomorphism for any φ ∈ Mor(I). Note that equivariance depends on ♥, and is not preserved by the forgetful functors in general. We denote the full subcategory of Mod(X• ) consisting of equivariant objects by EM(X• ). (4.15) Let ♥ be Mod or AB, and M ∈ ♥(X• ). For a morphism φ : i → j, the diagram β

−−−−−−−→ q(Xφ )∗ Mj qMi −−−−−−− ↓c ↓c β

(4.16)

c

(qM)i − → (Xφ )∗ (qM)j − → (Xφ )∗ qMj is commutative. This is checked at the section level directly. Utilizing this fact, we have the following. Lemma 4.17. Let M ∈ PM(X• ). For φ : i → j, the diagram β

θ

aMi − → a(Xφ )∗ Mj − → (Xφ )∗ aMj ↓θ ↓θ β

(aM)i −−−−−−−−−−−−→ (Xφ )∗ (aM)j is commutative. ⊓ ⊔

Proof. Straightforward diagram drawing.

(4.18) Let X• ∈ P(I, Sch/S), and φ : i → j be a morphism of I. Let M ∈ PM(X• ). Then αφ : Xφ∗ Mi → Mj is the composite β

ε

→ Xφ∗ (Xφ )∗ Mj − → Mj . Xφ∗ Mi −

(4.19)

Thus for U ∈ Zar(Xj ), αφ is given by Γ(U, Xφ∗ Mi ) = lim Γ(U, OXj ) ⊗Γ(V,OXi ) Γ((i, V ), M) −→ → Γ((j, U ), M) = Γ(U, Mj ), where the colimit is taken over the open subsets V of Xi such that U ⊂ Xφ−1 (V ), and the arrow is given by a ⊗ m → a res(j,U ),(i,V ) m, see (2.20). (4.20) Let X• and φ : i → j be as in (4.18). Let M ∈ Mod(X• ). Then αφ : Xφ∗ Mi → Mj is also given by the composite (4.19). By (2.20), it is given by the composite β

c

ε

ε

→ aq(?)j − → (?)j . Xφ∗ (?)i = aXφ∗ q(?)i − → aXφ∗ q(Xφ )∗ (?)j − → aXφ∗ (Xφ )∗ q(?)j −

4 Sheaves over a Diagram of S-Schemes

325

By the commutativity of (4.16), it is easy to see that it agrees with c

αφ

c

ε

→ aq(?)j − → (?)j . Xφ∗ (?)i = aXφ∗ q(?)i − → aXφ∗ (?)i q −−→ a(?)j q − Thus if M ∈ Mod(X• ) is equivariant as an object of PM(X• ), then it is equivariant as an object of Mod(X• ).

Chapter 5

The Left and Right Inductions and the Direct and Inverse Images

Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). ♥ (5.1) Let J be a subcategory of I. The left adjoint Q(X• , J)♥ # of (?)J (see (4.13)) is denoted by L♥ J for ♥ = PA, AB, PM, Mod. The right adjoint ♥ Q(X• , J)♥ of (?) , which exists by Lemma 2.31, is denoted by RJ♥ for ♥ = J ♭ ♥ PA, AB, PM, Mod. We call L♥ J and RJ the left and right induction functor, respectively. Let J1 ⊂ J ⊂ I be subcategories of I. The left and right adjoints of (?)♥ J1 ,J ♥ ♥ are denoted by L♥ and R , respectively. As (?) has both a left adjoint J,J1 J,J1 J and a right adjoint, we have

Lemma 5.2. The functor (?)♥ J preserves arbitrary limits and colimits (hence is exact) for ♥ = PA, AB, PM, Mod.

The functor ♥(X• ) → i∈I ♥(Xi ) given by F → (Fi )i∈I is faithful for ♥ = PA, AB, PM, Mod. (5.3) Let f• : X• → Y• be a morphism in P(I, Sch/S). This induces an obvious ringed continuous functor f•−1 : (Zar(Y• ), OY• ) → (Zar(X• ), OX• ). We have id−1 = id, and (g• ◦ f• )−1 = f•−1 ◦ g•−1 for g• : Y• → Z• . −1 # We deﬁne the direct image (f• )♥ ∗ to be (f• )♥ , and the inverse image (f• )∗♥ to be (f•−1 )♥ # for ♥ = Mod, PM, AB, PA. Lemma 5.4. Let f• : X• → Y• be a morphism in P(I, Sch/S), and K ⊂ J ⊂ I. Then we have 1 Q(X• , J) ◦ Q(X• |J , K) = Q(X• , K) 2 f•−1 ◦ Q(Y• , J) = Q(X• , J) ◦ (f• |J )−1 .

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5 The Left and Right Inductions and the Direct and Inverse Images

(5.5) Let us ﬁx I and S. By Lemma 2.43 and Lemma 2.55, we have various natural maps between functors on sheaves arising from the closed structures and the monoidal pairs, involving various J-diagrams of schemes, where J varies subcategories of I. In the sequel, many of the natural maps are referred as ‘the canonical maps’ or ‘the canonical isomorphisms’ without any explicit deﬁnitions. Many of them are deﬁned in [26] and Chapter 1, and various commutativity theorems are proved there. Example 5.6. Let I be a small category, S a scheme, and f• : X• → Y• and g• : Y• → Z• are morphisms in P(I, Sch/S). Let K ⊂ J ⊂ I be subcategories, and ♥ denote PM, Mod, PA, or AB. 1 There is a natural isomorphism ♥ ∼ ♥ ♥ c♥ I,J,K : (?)K,I = (?)K,J ◦ (?)J,I .

Taking the conjugate, ♥ ♥ ∼ ♥ d♥ I,J,K : LI,J ◦ LJ,K = LI,K

is induced. 2 There is a natural isomorphism ♥ ♥ ♥ ∼ ♥ c♥ J,f• : (?)J ◦ (f• )∗ = (f• |J )∗ ◦ (?)J

and its conjugate ♥ ♥ ∗ ∼ ∗ d♥ J,f• : LJ ◦ (f• |J )♥ = (f• )♥ ◦ LJ .

3 We have ♥ ♥ ♥ ♥ ♥ ♥ ♥ −1 (c♥ (f• )♥ ∗ ). K,f• |J (?)J ) ◦ ((?)K,J cJ,f• ) = ((f• |K )∗ cI,J,K ) ◦ cK,f• ◦ ((cI,J,K )

4 We have ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ −1 ((g• |J )♥ ), ∗ cJ,f• ) ◦ (cJ,g• (f• )∗ ) = (cf• |J ,g• |J (?)J ) ◦ cJ,g• ◦f• ◦ ((?)J (cf• ,g• ) ♥ ∼ ♥ ♥ where c♥ f• ,g• : (g• ◦ f• )∗ = (g• )∗ ◦ (f• )∗ is the canonical isomorphism, and ♥ similarly for cf• |J ,g• |J . 5 The canonical map

mJ : MJ ⊗OX• |J NJ → (M ⊗OX• N )J is an isomorphism, as can be seen easily (the corresponding assertion for PM is obvious. Utilize Lemma 5.7 below to show the case of Mod). The canonical map Δ : LJ (M ⊗OX• |J N ) ∼ = (LJ M) ⊗OX• (LJ N ). is deﬁned, which may not be an isomorphism.

5 The Left and Right Inductions and the Direct and Inverse Images

329

Lemma 5.7. Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Let J be a subcategory of I. Then the natural map θ = θ¯ : a(?)PM → (?)Mod a J J is an isomorphism. Proof. Obvious by Lemma 2.31.

⊓ ⊔

Chapter 6

Operations on Sheaves Via the Structure Data

Let I be a small category, S a scheme, and P := P(I, Sch/S). To study sheaves on objects of P, it is convenient to utilize the structure data of them, and then utilize the usual sheaf theory on schemes. (6.1) Let X• ∈ P. Let ♥ be any of PA, AB, PM, Mod, and M, N ∈ Hom♥(Xi ) (Mi , Ni ) is given by some ϕ ∈ ♥(X• ). An element (ϕi ) in

♥(Xi )), Hom♥(X• ) (M, N ) (by the canonical faithful functor ♥(X• ) → if and only if ϕj ◦ αφ (M) = αφ (N ) ◦ (Xφ )∗♥ (ϕi )

(6.2)

holds (or equivalently, βφ (N ) ◦ ϕi = (Xφ )∗ ϕj ◦ βφ (M) holds) for any (φ : i → j) ∈ Mor(I). We say that a family of morphisms (ϕi )i∈I between structure data ϕi : Mi → Ni is a morphism of structure data if ϕi is a morphism in ♥(Xi ) for each i, and (6.2) is satisﬁed for any φ. Thus the categories of structure data of sheaves, presheaves, modules, and premodules on X• , denoted by D♥ (X• ) are deﬁned, and the equivalence Dat♥ : ♥(X• ) ∼ = D♥ (X• ) are given. This is the precise meaning of Lemma 4.8. (6.3) Let X• ∈ P and M, N ∈ Mod(X• ). As in Example 5.6, 5, we have an isomorphism mi : Mi ⊗OXi Ni ∼ = (M ⊗OX• N )i . This is trivial for presheaves, and utilize the fact the sheaﬁﬁcation is compatible with (?)i for sheaves. At the section level, for M, N ∈ PM(X• ), i ∈ I, and U ∈ Zar(Xi ), mpi : Γ(U, Mi ⊗pOX Ni ) → Γ(U, (M ⊗pOX• N )i ) i

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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6 Operations on Sheaves Via the Structure Data

is nothing but the identiﬁcation Γ(U, Mi ) ⊗Γ(U,OXi ) Γ(U, Ni ) = Γ((i, U ), M) ⊗Γ((i,U ),OX• ) Γ((i, U ), N ) = Γ((i, U ), M ⊗pOX• N ). For M, N ∈ Mod(X• ) and i ∈ I, mi is given as the composite mp

c

i → a((qM)i ⊗pOX (qN )i ) −−→ Mi ⊗OXi Ni = a(qMi ⊗pOX qNi ) − i

i

θ

a(qM ⊗pOX• qN )i − → (a(qM ⊗pOX• qN ))i = (M ⊗OX• N )i , see (2.52). Utilizing this identiﬁcation, the structure map αφ of M ⊗ N can be completely described via those of M and N . Namely, Lemma 6.4. Let X• ∈ P(I, Sch/S), and M, N ∈ ♥(X• ), where ♥ is PM or Mod. For φ ∈ I(i, j), αφ (M ⊗ N ) agrees with the composite map m−1

αφ ⊗αφ

∆

mj

i →Xφ∗ (Mi ⊗Ni )− →Xφ∗ Mi ⊗Xφ∗ Ni −−−−→Mj ⊗Nj −−→(M⊗N )j , Xφ∗(M⊗N )i −−−

where ⊗ should be replaced by ⊗p when ♥ = PM. Proof (sketch). It is not so diﬃcult to show that it suﬃces to show that βφ (M ⊗ N ) agrees with the composite m−1

β⊗β

m

i (M ⊗ N )i −−− → Mi ⊗ Ni −−−→ (Xφ )∗ Mj ⊗ (Xφ )∗ Nj −→

mj

(Xφ )∗ (Mj ⊗ Nj ) −−→ (Xφ )∗ (M ⊗ N )j . (6.5) First we prove this for the case that ♥ = PM. For an open subset U of Xi , this composite map evaluated at U is res ⊗ res

Γ((i, U ), (M ⊗ N )) = Γ((i, U ), M) ⊗Γ((i,U ),OX• ) Γ((i, U ), N ) −−−−−→ p

→ Γ((j, Xφ−1 (U )), M) ⊗Γ((i,U ),OX• ) Γ((j, Xφ−1 (U )), N ) − Γ((j, Xφ−1 (U )), M) ⊗Γ((j,X −1 (U )),OX φ

•)

Γ((j, Xφ−1 (U )), N )

= Γ((j, Xφ−1 (U )), M ⊗ N ), where p(m ⊗ n) = m ⊗ n. This composite map is nothing but the restriction map of M ⊗ N . So by deﬁnition, it agrees with βφ : Γ(U, (M ⊗ N )i ) → Γ(U, (Xφ )∗ (M ⊗ N )j ). Next we consider the case ♥ = Mod. First note that the diagram

6 Operations on Sheaves Via the Structure Data

333

β

(a(qM ⊗p qN ))i −−−−−−−−−−−−−−−−−−−−→ (Xφ )∗ (a(qM ⊗p qN ))j ↑θ ↑θ β

θ

a(qM ⊗p qN )i − → a(Xφ )∗ (qM ⊗p qN )j − → (Xφ )∗ a(qM ⊗p qN )j is commutative by Lemma 4.17. By the presheaf version of the lemma, which has been proved in the last paragraph, the diagram mi

a(qM ⊗p qN )i o

a((qM)i ⊗p (qN )i ) β⊗β

a((Xφ )∗ (qM)j ⊗p (Xφ )∗ (qN )j )

β

m

a(Xφ )∗ (qM ⊗p qN )j o

mj

a(Xφ )∗ ((qM)j ⊗p (qN )j )

is commutative. By the commutativity of the diagram (4.16), the diagram c⊗c

a((qM)i ⊗p (qN )i )

/ a(qMi ⊗p qNi ) β⊗β

a(q(Xφ )∗ Mj ⊗p q(Xφ )∗ Nj )

β⊗β

c⊗c

a((Xφ )∗ (qM)j ⊗p (Xφ )∗ (qN )j )

c⊗c

/ a((Xφ )∗ qMj ⊗p (Xφ )∗ qNj )

is commutative. Combining the commutativity of these three diagrams (and some other easy commutativity), it is not so diﬃcult to show that the map β : (M⊗N )i = (a(qM⊗p qN ))i → (Xφ )∗ (a(qM⊗p qN ))j = (Xφ )∗ (M⊗N )j agrees with the composite m−1

θ −1

i (M ⊗ N )i = (a(qM ⊗p qN ))i −−→ a(qM ⊗p qN )i −−− → a((qM)i ⊗p (qN )i )

β⊗β

c⊗c

c⊗c

−−→ a(qMi ⊗p qNi ) −−−→ a(q(Xφ )∗ Mj ⊗p q(Xφ )∗ Nj ) −−→ m

θ

a((Xφ )∗ qMj ⊗p (Xφ )∗ qNj ) −→ a(Xφ )∗ (qMj ⊗p qNj ) − → (Xφ )∗ a(qMj ⊗p qNj ) c⊗c

mj

θ

−−→ (Xφ )∗ a((qM)j ⊗p (qN )j ) −−→ (Xφ )∗ a((qM ⊗p qN )j ) − → p (Xφ )∗ (a(qM ⊗ qN ))j = (Xφ )∗ (M ⊗ N )j . This composite map agrees with the composite map (6.5). This proves the lemma. ⊓ ⊔

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6 Operations on Sheaves Via the Structure Data

(6.6) Let X• ∈ P, and J a subcategory of I. The left adjoint functor ♥ ♥ L♥ J = Q(X• , J)# of (?)J is given by the structure data as follows explicitly. For M ∈ ♥(X• |J ) and i ∈ I, we have Lemma 6.7. There is an isomorphism ♥ ∼ λJ,i : (L♥ (X )∗ (Mj ), J (M))i = lim −→ φ ♥ (J op →I op )

where the colimit is taken over the subcategory (Ii )op of I/i whose objects are (φ : j → i) ∈ I/i with j ∈ ob(J) and morphisms are morphisms ϕ of I/i such that ϕ ∈ Mor(J). The translation map of the direct system is given as follows. For morphisms φ : j → i and ψ : j ′ → j, the translation ∗ map Xφψ Mj ′ → Xφ∗ Mj is the composite d

αψ

∗ Xφψ Mj ′ − → Xφ∗ Xψ∗ Mj ′ −−→ Xφ∗ Mj .

Proof. We prove the lemma for the case that ♥ = PM, Mod. The case that ♥ = PA, AB is similar and easier. Consider the case ♥ = PM ﬁrst. For any object (φ, h) : (i, U ) → (j, V ) of Zar(X• |J )֒→Zar(X• ) I(i,U ) , consider the obvious map Γ((i, U ), OX• ) ⊗Γ((j,V ),OX• |J ) Γ((j, V ), M) = Γ(U, OXi ) ⊗Γ(V,OXj ) Γ(V, Mj ) →

lim −→ −1

Γ(U, OXi ) ⊗Γ(V ′ ,OXj ) Γ(V ′ , Mj )

Xφ (V ′ )⊃U

= Γ(U, Xφ∗ Mj ) → lim Γ(U, Xφ∗′ Mj ′ ), −→ (J op →I op ) op

where the last lim is taken over (φ′ : j ′ → i) ∈ (Ii −→ induces a unique map

) . This map

Γ(U, (LJ M)i ) = Γ((i, U ), LJ M) = lim Γ((i, U ), OX• ) ⊗Γ((j,V ),OX• |J ) Γ((j, V ), M) → lim Γ(U, Xφ∗′ Mj ′ ). −→ −→ It is easy to see that this deﬁnes λJ,i . (J op →I op ) op We deﬁne the inverse of λJ,i explicitly. Let (φ : j → i) ∈ (Ii ) . Let U ∈ Zar(Xi ) and V ∈ Zar(Xj ) such that U ⊂ Xφ−1 (V ). We have an obvious map Γ(U, OXi ) ⊗Γ(V,OXj ) Γ(V, Mj ) = Γ((i, U ), OX• ) ⊗Γ((j,V ),OX• |J ) Γ((j, V ), M) → lim Γ((i, U ), OX• ) ⊗Γ((j,V ),OX• |J ) Γ((j, V ), M) −→ = Γ((i, U ), LJ M) = Γ(U, (LJ M)i ), which induces Γ(U, Xφ∗ M) = lim Γ(U, OXi ) ⊗Γ(V,OXj ) Γ(V, Mj ) → Γ(U, (LJ M)i ). −→

6 Operations on Sheaves Via the Structure Data

335

This gives a morphism Xφ∗ M → (LJ M)i . It is easy to see that this deﬁnes lim Xφ∗ M → (LJ M)i , which is the inverse of λJ,i . This completes the proof −→ for the case that ♥ = PM. Now consider the case ♥ = Mod. Deﬁne λMod J,i to be the composite λPM J,i

θ −1

PM (LMod M)i = (?)i aLPM −−→ a lim Xφ∗ (qM)j J J qM −−→ a(?)i LJ qM − −→ c ∼ aXφ∗ (qM)j − → lim aXφ∗ qMj = lim Xφ∗ Mj . = lim −→ −→ −→

As the morphisms appearing in the composition are all isomorphisms, λMod J,i is an isomorphism. ⊓ ⊔ In particular, we have an isomorphism ♥ ∼ (Xφ )∗♥ (M). (6.8) λj,i : (L♥ j (M))i = φ∈I(j,i)

(6.9) As announced in (2.61), we show that the monoidal adjoint pair # ((?)Mod # , (?)Mod ) in Lemma 2.55 is not Lipman. We deﬁne a ﬁnite category K by ob(K) = {s, t}, and K(s, t) = {u, v}, o u K(s, s) = {ids }, and K(t, t) = {idt }. Pictorially, K looks like t o v s . Let k be a ﬁeld, and deﬁne X• ∈ P(K, Sch) by Xs = Xt = Spec k, and Xu = Xv = id. Then Γ(Xt , (Ls OXs )t ) is two-dimensional by (6.8). So Ls OXs and OX• are not isomorphic by the dimension reason. Similarly, Ls (OXs ⊗OXs OXs ) cannot be isomorphic to Ls OXs ⊗OX• Ls OXs . # Similarly, ((?)PM # , (?)PM ) in Lemma 2.55 is not Lipman. (6.10)

Let ψ : i → i′ be a morphism. The structure map ♥ ♥ ♥ αψ : (Xψ )∗♥ ((L♥ J (M))i ) → (LJ (M))i′

is induced by (Xψ )∗♥ ((Xφ )∗♥ (Mj )) ∼ = (Xψφ )∗♥ (Mj ). More precisely, for ψ : i → i′ , the diagram λJ,i Xψ∗ ((LJ M)i ) −−→ Xψ∗ lim Xφ∗ Mj ∼ X ∗ X ∗M = lim −→ −→ ψ φ j ↓ αψ ↓h

(LJ M)i′

λJ,i′

−−−−−−−−−−−−−−−−→ lim Xφ∗′ Mj ′ −→

is commutative, where φ : i → j runs through (Iif )op , and φ′ : i′ → j ′ runs through (Iif′ )op , where f : J op → I op is the inclusion. The map h is induced ∗ by d : Xψ∗ Xφ∗ → (Xφ Xψ )∗ = Xψφ . This is checked at the section level directly when ♥ = PM.

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6 Operations on Sheaves Via the Structure Data

We consider the case that ♥ = Mod. Then the composite λJ,i h Xψ∗ (?)i LJ −−→ Xψ∗ lim Xφ∗ (?)j ∼ X ∗ X ∗ (?) − → lim Xφ∗′ (?)j ′ = lim −→ −→ ψ φ j −→

agrees with the composite λPM J,i

θ −1

Xψ∗ (?)i LJ = aXψ∗ q(?)i aLJ q −−→ aXψ∗ qa(?)i LJ q −−−→ aXψ∗ qa lim Xφ∗ (?)j q −→ ∼ =

∼ =

c

u−1

− → aXψ∗ q lim aXφ∗ (?)j q − → aXψ∗ q lim aXφ∗ q(?)j − → lim aXψ∗ qaXφ∗ q(?)j −−→ −→ −→ −→ d ∗ q(?)j → lim aXφ∗′ q(?)j ′ = lim Xφ∗′ (?)j ′ . → lim aXψφ lim aXψ∗ Xφ∗ q(?)j − −→ −→ −→ −→ Using Lemma 2.60, it is straightforward to show that this map agrees with αψ

c

c

Xψ∗ (?)i LJ = aXψ∗ q(?)i aLJ q − → aXψ∗ (?)i qaLJ q −−→ a(?)i′ qaLJ q − → λJ,i′

θ −1

ε

∼ =

aq(?)i′ aLJ q − → (?)i′ aLJ q −−→ a(?)i′ LJ q −−−→ a lim Xφ∗′ (?)j ′ q − → −→ c ∗ ∗ ∗ lim aXφ′ (?)j ′ q − → lim aXφ′ q(?)j ′ = lim Xφ′ (?)j ′ . −→ −→ −→ This composite map agrees with αψ

λJ,i′

Xψ∗ (?)i LJ −−→ (?)i′ LJ −−−→ lim Xφ∗′ (?)j ′ −→ by (4.20) and the deﬁnition of λJ,i′ for sheaves (see the proof of Lemma 6.7). This is what we wanted to prove. The case that ♥ = PA, AB is proved similarly. (6.11) In the remainder of this chapter, we do not give detailed proofs, since the strategy is similar to the above (just check the commutativity at the section level for presheaves, and sheaﬁfy it). (6.12) The counit map ε : LJ (?)J → Id is given as a morphism of structure data as follows. εi : (?)i LJ (?)J → (?)i agrees with λJ,i

c

α

→ lim Xφ∗ (?)j − →(?)i , (?)i LJ (?)J −−→ lim Xφ∗ (?)j (?)J − −→ −→ where α is induced by αφ : Xφ∗ (?)j → (?)i . (6.13)

The unit map u : Id → (?)J LJ is also described, as follows. uj : (?)j → (?)j (?)J LJ

6 Operations on Sheaves Via the Structure Data

337

agrees with λ−1

−1

f J,j ∗ (?)j −−→ Xid (?)j → lim Xφ∗ (?)k −−→(?)j LJ ∼ = (?)j (?)J LJ , j −→ (J op ⊂I op ) op

where the colimit is taken over (φ : k → j) ∈ (Ij

) .

(6.14) Let X• ∈ P, and J a subcategory of I. The right adjoint functor RJ♥ of (?)♥ J is given as follows explicitly. For M ∈ ♥(X• |J ) and i ∈ I, we have ∼ (X )♥ (Mj ), ρJ,i : (RJ♥ (M))♥ i = lim ←− φ ∗ (J→I)

where the limit is taken over Ii , see (2.6) for the notation. The descriptions of α, u, and ε for the right induction are left to the reader. Lemma 6.15. Let X• ∈ P, and J a full subcategory of I. Then we have the following. ♥ 1 The counit of adjunction ε : (?)♥ J ◦ RJ → Id is an isomorphism. In ♥ particular, RJ is full and faithful. ♥ 2 The unit of adjunction u : Id → (?)♥ J ◦ LJ is an isomorphism. In particular, L♥ J is full and faithful.

Proof. 1 For i ∈ J, the restriction ♥ ♥ εi : (?)♥ lim(Xφ )♥ ∗ (Mj ) → (Xidi )∗ Mi = Mi = (?)i M i (?)J RJ M = ← −

is nothing but the canonical map from the projective limit, where the limit (J→I) (J→I) is taken over (φ : i → j) ∈ Ii . As J is a full subcategory, we have Ii equals i/J, and hence idi is its initial object. So the limit is equal to Mi , and εi is the identity map. Since εi is an isomorphism for each i ∈ J, we have that ε is an isomorphism. The proof of 2 is similar, and we omit it. ⊓ ⊔ Let C be a small category. A connected component of C is a full subcategory of C whose object set is one of the equivalence classes of ob(C) with respect to the transitive symmetric closure of the relation ∼ given by c ∼ c′ ⇐⇒ C(c, c′ ) = ∅. Deﬁnition 6.16. We say that a subcategory J of I is admissible if 1 2

(J op ⊂I op )

For i ∈ I, the category (Ii )op is pseudoﬁltered. For j ∈ J, we have idj is the initial object of one of the connected (J op ⊂I op ) components of Ij (i.e., idj is the terminal object of one of the (J op ⊂I op ) op

connected components of (Ij

) ).

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6 Operations on Sheaves Via the Structure Data

Note that for j ∈ I, the subcategory j = ({j}, {idj }) of I is admissible. In Lemma 6.7, the colimit in the right hand side is pseudo-ﬁltered and hence it preserves exactness, if 1 is satisﬁed. In particular, if 1 is satisﬁed, then Q(X• , J) : Zar(X• |J ) → Zar(X• ) is an admissible functor. As in the proof of Lemma 6.15, (?)j is a direct summand of (?)j ◦ LJ for j ∈ J so that LJ is faithful, if 2 is satisﬁed. We have the following. Lemma 6.17. Let X• ∈ P(I, Sch/S), and K ⊂ J ⊂ I be admissible subcategories of I. Then LPA J,K is faithful and exact. The morphism of sites Q(X• |J , K) is admissible. If, moreover, Xφ is ﬂat for any φ ∈ I(k, j) with j ∈ J and k ∈ K, then L♥ J,K is faithful and exact for ♥ = Mod. 0, and LJ,K M = 0. There exists Proof. Assume that M ∈ ♥(X• |K ), M = some k ∈ K such that Mk = 0. Since LJ,K M = 0, we have that 0 ∼ = (?)k LI,J LJ,K M ∼ = (?)k LI,K M. This contradicts the fact that Mk is a direct summand of (LI,K M)k . Hence LJ,K is faithful. We prove that L♥ J,K is exact. It suﬃces to show that for any j ∈ J, (?)j LJ,K is exact. As J is admissible, (?)j is a direct summand of (?)j LI,J . Hence it suﬃces to show that (?)j LI,K ∼ = (?)j LI,J LJ,K is exact. By Lemma 6.7, (Xφ )∗♥ (?)k , where the colimit is taken over (φ : k → j) ∈ (?)j LI,K ∼ = lim −→ op op (IjK ⊂I )op . By assumption, (Xφ )∗♥ is exact for any φ in the colimit. As op op (IjK ⊂I )op is pseudo-ﬁltered by assumption, (?)j LI,K is exact, as desired. ⊓ ⊔ (6.18)

As in Example 5.6, 2, we have an isomorphism ci,f• : (?)i ◦ (f• )∗ ∼ = (fi )∗ ◦ (?)i .

(6.19)

The translation αφ is described as follows. Lemma 6.20. Let f• : X• → Y• be a morphism in P(I, Sch/S). For φ ∈ I(i, j), αφ (f• )∗ : Yφ∗ (?)i (f• )∗ → (?)j (f• )∗ agrees with ci,f

via θ

• ∗ Yφ∗ (?)i (f• )∗ −−−→Y −−→(fj )∗ Xφ∗ (?)i φ (fi )∗ (?)i −

(fj )∗ αφ

c−1 j,f

• −−−−−→(fj )∗ (?)j −−−→(?) j (f• )∗ ,

(6.21)

where θ is Lipman’s theta [26, (3.7.2)]. One of the deﬁnitions of θ is the composite via u

c

via ε

θ : Yφ∗ (fi )∗ −−−→Yφ∗ (fi )∗ (Xφ )∗ Xφ∗ − → Yφ∗ (Yφ )∗ (fj )∗ Xφ∗ −−−→(fj )∗ Xφ∗ .

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339

Proof. Note that the diagram β

c

(?)i (f• )∗ − → (Yφ )∗ (?)j (f• )∗ − → (Yφ )∗ (fj )∗ (?)j ↓c ↓c

(6.22)

(fi )∗ β

(fi )∗ (?)i −−−−−−−− −−−−−−−−→ (fi )∗ (Xφ )∗ (?)j is commutative. Indeed, when we apply the functor Γ(U, ?) for an open subset U of Yi , then we get an obvious commutative diagram res

id

Γ((i, fi−1 (U )), ?) −−→ Γ((j, fj−1 (Yφ−1 (U ))), ?) −→ Γ((j, fj−1 (Yφ−1 (U ))), ?) ↓ id ↓ id res

−−−−−−−−−−−−→ Γ((j, Xφ−1 (fi−1 (U ))), ?). Γ((i, fi−1 (U )), ?) −−−−−−−−−−−− Now the assertion of the lemma follows from the commutativity of the diagram α

Yφ∗ (?)i (f• )∗ − → ↓ id (a)

(?)j (f• )∗ ↑ε

β

c

− → (b)

(fj )∗ (?)j ↑ε

id

−→ (fj )∗ (?)j

c

Yφ∗ (?)i (f• )∗ − → Yφ∗ (Yφ )∗ (?)j (f• )∗ − → Yφ∗ (Yφ )∗ (fj )∗ (?)j ↓c (c) ↓c β

Yφ∗ (fi )∗ (?)i −−−−−−−−−−−−−−−−−−→ Yφ∗ (fi )∗ (Xφ )∗ (?)j (f) ↓θ (d) ↓θ

↓ id

β

(fj )∗ Xφ∗ (?)i −−−−−−−−−−−−−−−−−−→ (fj )∗ Xφ∗ (Xφ )∗ (?)j ↓ id (e) ↓ε α

(fj )∗ Xφ∗ (?)i −−−−−−−−−−−−−−−−−−→

(fj )∗ (?)j

id

←− (fj )∗ (?)j .

Indeed, the commutativity of (a) and (e) is the deﬁnition of α. The commutativity of (b) follows from the naturality of ε. The commutativity of (c) follows from the commutativity of (6.22). The commutativity of (d) is the naturality of θ. The commutativity of (f) follows from the deﬁnition of θ and the fact that the composite u

ε

→ (Xφ )∗ Xφ∗ (Xφ )∗ − → (Xφ )∗ (Xφ )∗ − ⊓ ⊔

is the identity.

Proposition 6.23. Let f• : X• → Y• be a morphism in P, J a subcategory of I, and i ∈ I. Then the composite map via θ

via ci,f

• (?)i LJ (f• |J )∗ −−−→(?)i (f• )∗ LJ −−−−−→(f i )∗ (?)i LJ

agrees with the composite map

340

6 Operations on Sheaves Via the Structure Data via cj,f• |

via λJ,i

J (?)i LJ (f• |J )∗ −−−−−→ lim Yφ∗ (?)j (f• |J )∗ −−−−−−−→ lim Yφ∗ (fj )∗ (?)j −→ −→

via λ−1 J,i

via θ

−−−→ lim(fi )∗ Xφ∗ (?)j → (fi )∗ lim Xφ∗ (?)j −−−−−→(fi )∗ (?)i LJ . −→ −→ Proof. Note that θ in the ﬁrst composite map is the composite via u

c

ε

θ = θ(J, f• ) : LJ (f• |J )∗ −−−→LJ (f• |J )∗ (?)J LJ − →LJ (?)J (f• )∗ LJ − →(f• )∗ LJ . The description of u and ε are already given, and the proof is reduced to the iterative use of (6.10), (6.12), (6.13), and Lemma 6.20. The detailed argument is left to a patient reader. The reason why the second map involves θ is Lemma 6.20. ⊓ ⊔ Similarly, we have the following. Proposition 6.24. Let f• : X• → Y• be a morphism in P, J a subcategory of I, and i ∈ I. Then the composite map via θ(f• ,i)

via df

,J

(fi )∗ (?)i LJ −−−−−−→(?)i (f• )∗ LJ −−−−−•−→(?)i LJ (f• |J )∗ agrees with the composite map via λJ,i (fi )∗ (?)i LJ −−−−−→(fi )∗ lim Yφ∗ (?)j ∼ (f )∗ Y ∗ (?) = lim −→ −→ i φ j d

via λ−1 J,i

via θ(f• |J ,j)

− → lim Xφ∗ (fj )∗ (?)j −−−−−−−−→ lim Xφ∗ (?)j (f• |J )∗ −−−−−→(?)i LJ (f• |J )∗ . −→ −→ The proof is left to the reader. The proof of Proposition 6.23 and Proposition 6.24 are formal, and the propositions are valid for ♥ = PM, Mod, PA, and AB. Let f• : X• → Y• be a morphism in P, and J ⊂ I a subcategory. The inverse image (f• )∗♥ is compatible with the restriction (?)J . Lemma 6.25. The natural map θ♥ = θ♥ (f• , J) : ((f• )|J )∗♥ ◦ (?)J → (?)J ◦ (f• )∗♥ is an isomorphism for ♥ = PA, AB, PM, Mod. In particular, f•−1 : Zar(Y• ) → Zar(X• ) is an admissible continuous functor. Proof. We consider the case where ♥ = PM. Let M ∈ PM(Y• ), and (j, U ) ∈ Zar(X• |J ). We have Γ((j, U ), (f• |J )∗ MJ ) = lim Γ((j, U ), OX• ) ⊗Γ((j ′ ,V ),OY• ) Γ((j ′ , V ), M), −→ (f | )−1 op

• J where the colimit is taken over (j ′ , V ) ∈ (I(j,U ) have

) . On the other hand, we

Γ((j, U ), (?)J f•∗ M) = lim Γ((j, U ), OX• ) ⊗Γ((i,V ),OY• ) Γ((i, V ), M), −→

6 Operations on Sheaves Via the Structure Data

341

f −1

op • where the colimit is taken over (i, V ) ∈ (I(j,U ) ) . There is an obvious map from the ﬁrst to the second. This obvious map is θ, see (2.57). To verify that this is an isomorphism, it suﬃces to show that the category f•−1 op (f• |J )−1 op (I(j,U ) ) is ﬁnal in the category (I(j,U ) ) . In fact, any (φ, h) : (j, U ) → −1 (i, fi (V )) with (i, V ) ∈ Zar(Y• ) factors through

(idj , h) : (j, U ) → (j, fj−1 Yφ−1 (V )). Hence, θ♥ is an isomorphism for ♥ = PM. The construction for the case where ♥ = PA is similar. As (?)J is compatible with the sheaﬁﬁcation by Lemma 2.31, we have that θ is an isomorphism for ♥ = Mod, AB by Lemma 2.59. ⊓ ⊔ Corollary 6.26. The conjugate ♥ ξ♥ = ξ♥ (f• , J) : (f• )♥ ∗ RJ → RJ (f• |J )∗

of θ♥ (f• , J) is an isomorphism for ♥ = PA, AB, PM, Mod. ⊓ ⊔

Proof. Obvious by Lemma 6.25. (6.27)

By Corollary 6.26, we may deﬁne the composite u

→ f•∗ RJ (f• |J )∗ (f• |J )∗ μ♥ = μ♥ (f• , J) : f•∗ RJ − ξ −1

ε

−−→ f•∗ (f• )∗ RJ (f• |J )∗ − → RJ (f• |J )∗ . Observe that the diagram (?)i f•∗ RJ

θ −1

ρ

/ fi∗ (?)i RJ

/ fi∗ lim(Yφ )∗ (?)j ←−

/ lim fi∗ (Yφ )∗ (?)j ←− mmm θ mmm m mmm mv mm

μ

(?)i RJ f• |∗ J

ρ

/ lim(Xφ )∗ (?)j f• |∗J ←−

θ −1

/ lim(Xφ )∗ fj∗ (?)j ←−

is commutative. Lemma 6.28. Let the notation be as above, and M, N ∈ ♥(Y• ). Then the diagram (f• |J )∗♥ (MJ ⊗ NJ ) ↓∆

m

− → θ⊗θ

(f• |J )∗♥ ((M ⊗ N )J )

θ

− →

((f• )∗♥ (M ⊗ N ))J ↓ (?)J ∆

m

→ ((f• )∗♥ M ⊗ (f• )∗♥ N )J (f• |J )∗♥ MJ ⊗ (f• |J )∗♥ NJ −−→ ((f• )∗♥ M)J ⊗ ((f• )∗♥ N )J −

(6.29) is commutative. Proof. This is an immediate consequence of Lemma 1.44.

⊓ ⊔

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6 Operations on Sheaves Via the Structure Data

Corollary 6.30. The adjoint pair ((?)∗Mod , (?)Mod ) over the category ∗ P(I, Sch/S) is Lipman. Proof. Let f• : X• → Y• be a morphism of P(I, Sch/S). It is easy to see that the diagram (?)i OY•

=

NNN NNN η NNN NNN ' = / (fi )∗ (?)i OX / (fi )∗ OX • i

(?)i η

(?)i (f• )∗ OX•

/ OYi

c

is commutative. So utilizing Lemma 1.25, it is easy to see that (?)i f•∗ OY•

θ −1

i

(?)i C

(?)i OX•

=

/ OXi

=

/ fi∗ OYi s ss sss s s C yss

/ f ∗ (?)i OY•

is also commutative. Since C : fi∗ OYi → OXi is an isomorphism by Corollary 2.65, (?)i C is an isomorphism for any i ∈ I. Hence C : f•∗ OY• → OX• is also an isomorphism. Let us consider M, N ∈ ♥(Y• ). To verify that Δ is an isomorphism, it suﬃces to show that (?)i Δ : (f•∗ (M ⊗ N ))i → (f•∗ M ⊗ f•∗ N )i is an isomorphism for any i ∈ ob(I). Now consider the diagram (6.29) for J = i. Horizontal maps in the diagram are isomorphisms by (6.3) and Lemma 6.25. The left Δ is an isomorphism, since fi is a morphism of single schemes. By Lemma 6.28, (?)i Δ is also an isomorphism. ⊓ ⊔ (6.31) The description of the translation map αφ for f•∗ is as follows. For φ ∈ I(i, j), αφ : Xφ∗ (?)i f•∗ → (?)j f•∗ is the composite ∗ −1 Xφ θ

d

fj∗ αφ

θ

→(?)j f•∗ . → fj∗ Yφ∗ (?)i −−−→fj∗ (?)j − Xφ∗ (?)i f•∗ −−−−→Xφ∗ fi∗ (?)i − (6.32)

Let X• ∈ P, and M, N ∈ ♥(X• ). Although there is a canonical map Hi : Hom♥(X• ) (M, N )i → Hom♥(Xi ) (Mi , Ni )

arising from the closed structure for i ∈ I, this may not be an isomorphism. However, we have the following.

6 Operations on Sheaves Via the Structure Data

343

Lemma 6.33. Let i ∈ I. If M is equivariant, then the canonical map Hi : Hom♥(X• ) (M, N )i → Hom♥(Xi ) (Mi , Ni ) is an isomorphism of presheaves. In particular, it is an isomorphism in ♥(Xi ). Proof. It suﬃces to prove that Hi : Hom♥(Zar(X• )/(i,U )) (M|(i,U ) , N |(i,U ) ) → Hom♥(U ) (Mi |U , Ni |U ) is an isomorphism for any Zariski open set U in Xi . To give an element of ϕ ∈ Hom♥(Zar(X• )/(i,U )) (M|(i,U ) , N |(i,U ) ) is the same as to give a family (ϕφ )φ:i→j with ϕφ ∈ Hom♥(X −1 (U )) (Mj |X −1 (U ) , Nj |X −1 (U ) ) φ

φ

φ

such that for any φ : i → j and ψ : j → j ′ , ϕψφ ◦ (αψ (M))|X −1 (U ) = (αψ (N ))|X −1 (U ) ◦ ((Xψ )|X −1 (U ) )∗♥ (ϕφ ). ψφ

ψφ

ψφ

(6.34)

As αφ (M) is an isomorphism for any φ : i → j, we have that such a (ϕφ ) is uniquely determined by ϕidi by the formula . ϕφ = (αφ (N ))|X −1 (U ) ◦ ((Xφ )|X −1 (U ) )∗♥ (ϕidi ) ◦ (αφ (M))|−1 X −1 (U ) φ

φ

(6.35)

φ

Conversely, ﬁx ϕidi , and deﬁne ϕφ by (6.35). Consider the diagram ∗ Xψφ Mi

d−1

ϕidi

(a)

∗ Xψϕ Ni

d−1

/ Xψ∗ Xφ∗ Mi ϕidi

/ Xψ∗ Xφ∗ Ni

αφ

/ Xψ∗ Mj

αψ

ϕφ

(c)

(b) αφ

/ Xψ∗ Nj

αψ

/ Mj ′ ϕψϕ

/ Nj ′ .

The diagram (a) is commutative by the naturality of d−1 . The diagram (b) and (a)+(b)+(c) are commutative, by the deﬁnition of ϕφ and ϕψφ (6.35), respectively. Since d−1 and αφ (M) are isomorphisms, the diagram (c) is ⊓ ⊔ commutative, and hence (6.34) holds. Hence Hi is bijective, as desired. Lemma 6.36. Let J be a subcategory of I. If M is equivariant, then the canonical map HJ : Hom♥(X• ) (M, N )J → Hom♥(X• |J ) (MJ , NJ ) is an isomorphism of presheaves. In particular, it is an isomorphism in ♥(X• |J ).

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6 Operations on Sheaves Via the Structure Data

Proof. It suﬃces to show that (HJ )i : (Hom♥(X• ) (M, N )J )i → Hom♥(XJ ) (MJ , NJ )i is an isomorphism for each i ∈ J. By Lemma 1.39, the composite map Hom♥(X• ) (M, N )i ∼ = (Hom♥(X• ) (M, N )J )i (HJ )i

H

i −−−→ Hom♥(XJ ) (MJ , NJ )i −→ Hom♥(Xi ) (Mi , Ni )

agrees with Hi . As MJ is also equivariant, we have that the two Hi are isomorphisms by Lemma 6.33, and hence (HJ )i is an isomorphism for any i ∈ J. ⊓ ⊔ (6.37)

By the lemma, the sheaf Hom♥(X• ) (M, N ) is given by the collection (Hom♥(Xi ) (Mi , Ni ))i∈I

provided M is equivariant. The structure map is the canonical composite map P

αφ : (Xφ )∗♥ Hom♥(Xi ) (Mi , Ni )− → Hom♥(Xj ) ((Xφ )∗♥ Mi , (Xφ )∗♥ Ni ) Hom♥(X ) (α−1 φ ,αφ ) j

−−−−−−−−−−−−→ Hom♥(Xj ) (Mj , Nj ). Similarly, the following is also easy to prove. Lemma 6.38. Let i ∈ I be an initial object of I. Then the following hold: 1 If M ∈ ♥(X• ) is equivariant, then (?)i : Hom♥(X• ) (M, N ) → Hom♥(Xi ) (Mi , Ni ) is an isomorphism. 2 (?)i : EM(X• ) → Mod(Xi ) is an equivalence, whose quasi-inverse is Li . The fact that Li (M) is equivariant for M ∈ Mod(Xi ) is checked directly from the deﬁnition.

Chapter 7

Quasi-Coherent Sheaves Over a Diagram of Schemes

Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). (7.1) Let M ∈ Mod(X• ). We say that M is locally quasi-coherent (resp. locally coherent) if Mi is quasi-coherent (resp. coherent) for any i ∈ I. We say that M is quasi-coherent if for any (i, U ) ∈ Zar(X• ) with U = Spec A being aﬃne, there exists an exact sequence in Mod(Zar(X• )/(i, U )) of the form (OX• |(i,U ) )(T ) → (OX• |(i,U ) )(Σ) → M|(i,U ) → 0, (7.2) where T and Σ are arbitrary small sets. Lemma 7.3. Let M ∈ Mod(X• ). Then the following are equivalent. 1 M is quasi-coherent. 2 M is locally quasi-coherent and equivariant. 3 For any morphism (φ, h) : (j, V ) → (i, U ) in Zar(X• ) such that V = Spec B and U = Spec A are affine, the canonical map B⊗A Γ((i, U ), M) → Γ((j, V ), M) is an isomorphism. Proof. 1⇒2 Let i ∈ I and U an aﬃne open subset of Xi . Then there is an exact sequence of the form (7.2). Applying the restriction functor Mod(Zar(X• )/(i, U )) → Mod(U ), we get an exact sequence (T )

OU

(Σ)

→ OU

→ (Mi )|U → 0,

which shows that Mi is quasi-coherent for any i ∈ I. We prove that αφ (M) is an isomorphism for any φ : i → j, to show that M is equivariant. Take an aﬃne open covering (Uλ ) of Xi , and we prove that αφ (M) is an isomorphism over Xφ−1 (Uλ ) for each λ. But this is obvious by the existence of an exact sequence of the form (7.2) and the ﬁve lemma. 2⇒3 Set W := Xφ−1 (U ), and let ι : V ֒→ W be the inclusion map. Obviously, we have h = (Xφ )|W ◦ ι. As M is equivariant, we have that the canonical map J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

345

346

7 Quasi-Coherent Sheaves Over a Diagram of Schemes ∗

αφ |W (M) : (Xφ )|W Mod (Mi )|U → (Mj )|W is an isomorphism. Applying ι∗Mod to the isomorphism, we have that h∗Mod ((Mi )|U ) ∼ = (Mj )|V . The assertion follows from the assumption that Mi is quasi-coherent. 3⇒1 Let (i, U ) ∈ Zar(X• ) with U = Spec A aﬃne. There is a presentation of the form A(T ) → A(Σ) → Γ((i, U ), M) → 0. It suﬃces to prove that the induced sequence (7.2) is exact. To verify this, it suﬃces to prove that the sequence is exact after taking the section at ((φ, h) : (j, V ) → (i, U )) ∈ Zar(X• )/(i, U ) with V = Spec B being aﬃne. We have a commutative diagram B ⊗A A(T ) → B ⊗A A(Σ) → B ⊗A Γ((i, U ), M) → 0 ↓∼ ↓∼ ↓∼ = = = (T ) (Σ) B → B → Γ((j, V ), M) →0 whose ﬁrst row is exact and vertical arrows are isomorphisms. Hence, the second row is also exact, and (7.2) is exact. ⊓ ⊔ Deﬁnition 7.4. We say that M ∈ Mod(X• ) is coherent if it is equivariant and locally coherent. We denote the full subcategory of Mod(X• ) consisting of coherent objects by Coh(X• ). (7.5) Let J ⊂ I be a subcategory. We say that J is big in I if for any (ψ : j → k) ∈ Mor(I), there exists some (φ : i → j) ∈ Mor(J) such that ψ ◦ φ ∈ Mor(J). Note that ob(J) = ob(I) if J is big in I. Let Q be a property of morphisms of schemes. We say that X• has Q J-arrows if (X• )|J has Q-arrows. Lemma 7.6. Let J ⊂ I be a subcategory, and M ∈ Mod(X• ). 1 The full subcategory Lqc(X• ) of Mod(X• ) consisting of locally quasicoherent objects is a plump subcategory. 2 If M is equivariant (resp. locally quasi-coherent, quasi-coherent), then so . is MMod J 3 If J is big in I and MJ is equivariant (resp. locally quasi-coherent, quasicoherent), then so is M. 4 If J is big in I and X• has ﬂat J-arrows, then the full subcategory EM(X• ) (resp. Qch(X• )) of Mod(X• ) consisting of equivariant (resp. quasi-coherent) objects is a plump subcategory. 5 If J is big in I, then (?)J is faithful and exact. Proof. 1 and 2 are trivial. We prove 3. The assertion for the local quasi-coherence is obvious, because we have ob(J) = ob(I). By Lemma 7.3, it remains to show the assertion for the equivariance. Let us assume that MJ is equivariant and ψ : j → k is

7 Quasi-Coherent Sheaves Over a Diagram of Schemes

347

a morphism in I, and take φ : i → j such that φ, ψφ ∈ Mor(J). Then the composite map (Xψ )∗

αMod

αMod

Mod φ ψ ∗ ∗ ∗ (Xψφ )∗Mod (Mi )∼ =(Xψ )Mod (Xφ )Mod (Mi )−−−−−−−−−→(Xψ )Mod (Mj )−−−→Mk ,

Mod which agrees with αψφ , is an isomorphism by assumption. As we have αφMod Mod is also an isomorphism, we have that αψ is an isomorphism. Thus M is equivariant. We prove 4. By 1 and Lemma 7.3, it suﬃces to prove the assertion only for the equivariance. Let

M1 → M2 → M3 → M4 → M5 be an exact sequence in Mod(X• ), and assume that Mi is equivariant for i = 1, 2, 4, 5. We prove that M3 is equivariant. The sequence remains exact after applying the functor (?)Mod . By 3, replacing I by J and X• by (X• )|J , J we may assume that X• has ﬂat arrows. Now the assertion follows easily from the ﬁve lemma. The assertion 5 is obvious, because ob(J) = ob(I). ⊓ ⊔ Lemma 7.7. Let (Mλ ) be a diagram in Mod(X• ). If each Mλ is locally quasi-coherent (resp. equivariant, quasi-coherent), then so is lim Mλ . −→ Proof. As (?)i preserves colimits, the assertion for local quasi-coherence is trivial. Assume that each Mλ is equivariant. For (φ : i → j) ∈ Mor(I), αφ (Mλ ) is an isomorphism. As αφ (lim Mλ ) is nothing but the composite −→ lim α (M )

φ λ − → −−−−−−→ lim(Mλ )j ∼ M ) , (X )∗ (Mλ )i − (Xφ )∗Mod ((lim Mλ )i )∼ = (lim = lim −→ −→ λ j −→ −→ φ Mod

it is an isomorphism. The rest of the assertions follow.

⊓ ⊔

By Lemma 6.7, we have the following. Lemma 7.8. Let J ⊂ I be a subcategory, and M ∈ Lqc(X• |J ). Then we (M) ∈ Lqc(X• ). have LMod J Similarly, we have the next lemma. We say that a morphism f : X → Y of schemes is quasi-separated if the diagonal map X → X×Y X is quasi-compact. A quasi-compact quasi-separated morphism is said to be concentrated. If f : X → Y is concentrated, and M ∈ Qch(X), then f∗ M ∈ Qch(Y ) [14, (9.2.1)], where Qch(X) and Qch(Y ) denote the category of quasi-coherent sheaves on X and Y , respectively. Lemma 7.9. Let j ∈ I. Assume that X• has concentrated arrows, and that I(i, j) is ﬁnite for any i ∈ I. If M ∈ Qch(Xj ), then we have Rj M ∈ Lqc(X• ). The following is also proved easily, using (6.3) and Lemma 6.4.

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7 Quasi-Coherent Sheaves Over a Diagram of Schemes

Lemma 7.10. Let M and N be locally quasi-coherent (resp. equivariant, quasi-coherent) OX• -modules. Then M ⊗OX• N is also locally quasi-coherent (resp. equivariant, quasi-coherent). The following is a consequence of the observation in (6.37). Lemma 7.11. Let M be a coherent OX• -module, and N a locally quasicoherent OX• -module. Then HomMod(X• ) (M, N ) is locally quasi-coherent. If, moreover, there is a big subcategory J of I such that X• has ﬂat J-arrows and N is quasi-coherent, then HomMod(X• ) (M, N ) is quasi-coherent. Lemma 7.12. Let f : X → Y be a concentrated morphism of schemes, and gY : Y ′ → Y a ﬂat morphism of schemes. Set X ′ := X ×Y Y ′ , gX : X ′ → X the ﬁrst projection, and f ′ : X ′ → Y ′ the second projection. Then for M ∈ Qch(X), the canonical morphism ∗ θ : gY∗ f∗ M → f∗′ gX M

is an isomorphism. Proof. First note that the assertion is true if gY is an open immersion. Indeed, it is easy to check that θPM and θ in the composition in Lemma 2.59 are isomorphisms in this case. Using Lemma 1.23, we may assume that both Y and Y ′ are aﬃne. Thus X is quasi-compact. Let (Ui ) be a ﬁnite aﬃne open covering of X, which ˜ → X be the obvious map. Since ˜ = Ui , and let p : X exists. Set X i f is quasi-separated and Y is aﬃne, Ui ∩ Uj is quasi-compact for any i, j. ˜ is aﬃne. Let Thus p is quasi-compact. Note also that p is separated, since X ˜ ˜ ˜ pi : X ×X X → X be the ith projection for i = 1, 2, and set q = pp1 = pp2 . Note that p1 , p2 and q are quasi-compact separated. Almost by the deﬁnition of a sheaf, there is an exact sequence of the form u

0→M− → p∗ p∗ M → q∗ q ∗ M. Since q∗ q ∗ M ∼ = p∗ ((p1 )∗ q ∗ M), and p∗ M and (p1 )∗ q ∗ M are quasi-coherent, ˜ by the ﬁve lemma. By we may assume that M = p∗ N for some N ∈ Qch(X) Lemma 1.22, replacing f by p and f p, we may assume that f is quasi-compact separated. Then repeating the same argument as above, we may assume that p is aﬃne now. Replacing f by p and f p again, we may assume that f is aﬃne. That is, X is aﬃne. But this case is trivial. ⊓ ⊔ (7.13) Let f• : X• → Y• be a morphism in P = P(I, Sch/S). As θ in (6.21) is not an isomorphism in general, (f• )♥ ∗ (M) need not be equivariant even if M is equivariant. However, we have Lemma 7.14. Let f• : X• → Y• be a morphism in P, and J a big subcategory of I. Then we have the following:

7 Quasi-Coherent Sheaves Over a Diagram of Schemes

349

1 f• is cartesian if and only if (f• )|J is cartesian. 2 If f• is concentrated and M ∈ Lqc(X• ), then (f• )∗ (M) ∈ Lqc(Y• ). 3 If f• is cartesian concentrated, Y• has ﬂat J-arrows, and M ∈ Qch(X• ), then we have (f• )∗ (M) ∈ Qch(Y• ). Proof. 1 Assume that f• |J is cartesian, and let ψ : j → k be a morphism in I. Take φ : i → j such that φ, ψφ ∈ Mor(J). Consider the commutative diagram Xψ

Xφ

Yψ

Yφ

Xk −−→ Xj −−→ Xi ↓ fk(a) ↓ fj(b) ↓ fi Yk −−→ Yj −→ Yi . By assumption, the square (b) and the whole rectangle ((a)+(b)) are ﬁber squares. Hence (a) is also a ﬁber square. This shows that f• is cartesian. The converse is obvious. The assertion 2 is obvious by the isomorphism ((f• )∗ M)i ∼ = (fi )∗ (Mi ) for i ∈ I. We prove 3. By Lemma 7.6, we may assume that J = I. Then (f• )∗ (M) is locally quasi-coherent by 2. As M is equivariant and θ in (6.21) is an isomorphism by Lemma 7.12, we have that (f• )∗ (M) is equivariant. Hence ⊓ ⊔ by Lemma 7.3, (f• )∗ (M) is quasi-coherent. (7.15) Let the notation be as in Lemma 7.14. If f• is concentrated, then (f• )Lqc : Lqc(X• ) → Lqc(Y• ) is deﬁned as the restriction of (f• )Mod . If f• is ∗ ∗ concentrated cartesian and Y• has ﬂat J-arrows, then (f• )Qch : Qch(X• ) → ∗ Qch(Y• ) is induced. Lemma 7.16. Let f• : X• → Y• and g• : Y• → Z• be morphisms in P. Then the following hold. 0 1 2 3

An isomorphism is a cartesian morphism. If f• and g• are cartesian, then so is g• ◦ f• . If g• and g• ◦ f• are cartesian, then so is f• . If f• is faithfully ﬂat cartesian and g• ◦f• is cartesian, then g• is cartesian.

Proof. Trivial.

⊓ ⊔

Lemma 7.17. Let f• : X• → Y• and g• : Y•′ → Y• be morphisms in P. Let f•′ : X•′ → Y•′ be the base change of f• by g• . 1 If f• is cartesian, then so is f•′ . 2 If f•′ is cartesian and g• is faithfully ﬂat, then f• is cartesian. Proof. Obvious.

⊓ ⊔

(7.18) Let f : X → Y be a morphism of schemes. If f is concentrated, then f∗ is compatible with pseudo-ﬁltered inductive limits.

350

7 Quasi-Coherent Sheaves Over a Diagram of Schemes

Lemma 7.19 ([23, p. 641, Proposition 6]). Let f : X → Y be a concentrated morphism of schemes, and (Mi ) a pseudo-ﬁltered inductive system of OX -modules. Then the canonical map lim f∗ Mi → f∗ lim Mi −→ −→ is an isomorphism. By the lemma, the following follows immediately. Lemma 7.20. Let f• : X• → Y• be a morphism in P(I, Sch/S). If f• is conand (f• )Lqc preserve pseudo-ﬁltered inductive limits. centrated, then (f• )Mod ∗ ∗ If, moreover, f• is cartesian and Y• has ﬂat arrows, then (f• )Qch preserves ∗ pseudo-ﬁltered inductive limits. Lemma 7.21. Let f• : X• → Y• be a morphism in P. Let J be an admissible subcategory of I. If Y• has ﬂat arrows and f• is cartesian and concentrated, then the canonical map θ(J, f• ) : LJ ◦ (f• |J )∗ → (f• )∗ ◦ LJ is an isomorphism of functors from Lqc(X• |J ) to Lqc(Y• ). Proof. This is obvious by Proposition 6.23, Lemma 7.19, and Lemma 7.12. ⊓ ⊔ The following is obvious by Lemma 6.25 and (6.31). Lemma 7.22. Let f• : X• → Y• be a morphism in P. If M ∈ Mod(Y• ) is equivariant (resp. locally quasi-coherent, quasi-coherent), then so is (f• )∗Mod (M). If M ∈ Mod(Y• ), f• is faithfully ﬂat, and (f• )∗Mod (M) is equivariant, then we have M is equivariant. The restriction (f• )∗ : Qch(Y• ) → Qch(X• ) is sometimes denoted by (f• )∗Qch .

Chapter 8

Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes

(8.1) Let I be a small category, and S a scheme. Set P := P(I, Sch/S), and let X• ∈ P. In these notes, we use some abbreviated notation for derived categories of modules over diagrams of schemes. In the sequel, + D(Mod(X• )) may be denoted by D(X• ). DEM(X (Mod(X• )) may be denoted •) + b b by DEM (X• ). DCoh(X (Qch(X )) may be denoted by DCoh (Qch(X• )), and • •) so on. This notation will be also used for a single scheme. For a scheme X, + + (Mod(X)) will be denoted by DQch (X), where Mod(X) is the cateDQch(X) gory of OX -modules. Proposition 8.2. Let X• ∈ P, and I ∈ K(Mod(X• )). We have I is K-limp if and only if so is Ii for i ∈ I. Proof. The only if part follows from Lemma 3.31 and Lemma 3.25, 4. We prove the if part. Let I → J be a K-injective resolution, and let C be the mapping cone. Note that Ci is exact for each i. Let (U, i) ∈ Zar(X• ). We have an isomorphism Γ((U, i), C) ∼ = Γ(U, Ci ). As Ci is K-limp by the only if part, these are exact for each (U, i). It follows that I is K-limp. ⊓ ⊔ Corollary 8.3. Let J be a subcategory of I, and f• : X• → Y• a morphism in P(I, Sch/S). Then there is a canonical isomorphism c(J, f• ) : (?)J R(f• )∗ ∼ = R(f• |J )∗ (?)J . Lemma 8.4. Let J be an admissible subcategory of I. Assume that X• has ﬂat arrows. If I is a K-injective complex in Mod(X• ), then IJ is K-injective. Proof. This is simply because (?)J has an exact left adjoint LJ .

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

⊓ ⊔

351

352 8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes

Lemma 8.5. Let f• : X• → Y• be a concentrated morphism in P(I, Sch/S). Then R(f• )∗ takes DLqc (X• ) to DLqc (Y• ). R(f• )∗ : DLqc (X• ) → DLqc (Y• ) is way-out in both directions if Y• is quasi-compact and I is ﬁnite. ⊓ ⊔

Proof. Follows from [26, (3.9.2)] and Corollary 8.3 easily.

Lemma 8.6. Let X• ∈ P. Assume that X• has ﬂat arrows. For a complex F in Mod(X• ), F has equivariant cohomology groups if and only if αφ : Xφ∗ Fi → Fj is a quasi-isomorphism for any morphism φ : i → j in I. Proof. This is easy, since Xφ∗ is an exact functor.

⊓ ⊔

Lemma 8.7. Let f• : X• → Y• be a morphism in P. Assume that f• is concentrated and cartesian, and Y• has ﬂat arrows. If F ∈ DQch (X• ), then R(f• )∗ F ∈ DQch (Y• ). Proof. By the derived version of Lemma 6.20, αφ : Yφ∗ (?)i R(f• )∗ F → (?)j R(f• )∗ F

(8.8)

agrees with the composite c

θ

αφ

c

Yφ∗ (?)i R(f• )∗ F − → Yφ∗ R(fi )∗ Fi − → R(fj )∗ Xφ∗ Fi −−→ R(fj )∗ Fj − → (?)j R(f• )∗ F. The ﬁrst and the fourth map c’s are isomorphisms. The second map θ is an isomorphism by [26, (3.9.5)]. The third map αφ is an isomorphism by assumption and Lemma 8.6. Thus (8.8) is an isomorphism. Again by Lemma 8.6, we have the desired assertion. ⊓ ⊔ (8.9) Let X be a scheme, x ∈ X, and M an OX,x -module. We deﬁne ξx (M ) ∈ Mod(X) by Γ(U, ξx (M )) = M if x ∈ U , and zero otherwise. The restriction maps are deﬁned in an obvious way. For an exact complex H of OX,x -modules, ξx (H) is exact not only as a complex of sheaves, but also as a complex of presheaves. For a morphism of schemes f : X → Y , we have that f∗ ξx (M ) ∼ = ξf (x) (M ). Lemma 8.10. Let F ∈ C(Mod(X• )). The following are equivalent. 1 F is K-ﬂat. 2 Fi is K-ﬂat for i ∈ ob(I). 3 Fi,x is a K-ﬂat complex of OXi ,x -modules for any i ∈ ob(I) and x ∈ Xi . Proof. 3⇒1 Let G ∈ C(Mod(X• )) be exact. We are to prove that F ⊗OX• G is exact. For i ∈ ob(I) and x ∈ Xi , we have (F ⊗OX• G)i,x ∼ = (Fi ⊗OXi Gi )x ∼ = Fi,x ⊗OXi ,x Gi,x . Since Gi,x is exact, (F ⊗OXi ,x G)i,x is exact. So F ⊗OX• G is exact.

8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes 353

1⇒3 Let H ∈ C(Mod(OXi ,x )) be an exact complex, and we are to prove that Fi,x ⊗OXi ,x H is exact. For each j ∈ ob(I), (?)j Ri ξx (H) ∼ =

(Xφ )∗ ξx (H) ∼ =

φ∈I(j,i)

ξXφ (x) (H)

φ∈I(j,i)

is exact, since a direct product of exact complexes of presheaves is exact. So Ri ξx (H) is exact. It follows that F ⊗OX• Ri ξx (H) is exact. Hence (F ⊗OX• Ri ξx (H))i ∼ = Fi ⊗OXi

ξXφ (x) H

φ∈I(i,i)

is also exact. So Fi ⊗OXi ξidXi (x) (H) = Fi ⊗OXi ξx H is exact. So (Fi ⊗OXi ξx H)x ∼ = Fi,x ⊗OXi ,x H = Fi,x ⊗OXi ,x (ξx H)x ∼ is also exact. Applying 1⇔3, which has already been proved, to the complex Fi over ⊓ ⊔ the single scheme Xi , we get 2⇔3. Hence by [39], we have the following. Lemma 8.11. Let f• : X• → Y• be a morphism in P. Then we have the following. 1 If F ∈ C(Mod(Y• )) is K-ﬂat, then so is f•∗ F. 2 If F ∈ C(Mod(Y• )) is K-ﬂat exact, then so is f•∗ F. 3 If I ∈ C(Mod(X• )) is weakly K-injective, then so is (f• )∗ I. (8.12) By the lemma, the left derived functor Lf•∗ , which we already know its existence by Lemma 6.25, can also be calculated by K-ﬂat resolutions. Lemma 8.13. Let J be a subcategory of I, and f• : X• → Y• a morphism in P. Then we have the following. 1 The canonical map θ(f• , J) : L(f• |J )∗ (?)J → (?)J Lf•∗ is an isomorphism. 2 The diagram id

(?)J

/ (?)J

u

R(f• |J )∗ L(f• |J )∗ (?)J is commutative.

u θ

/ R(f• |J )∗ (?)J Lf•∗

c

−1

/ (?)J R(f• )∗ Lf•∗

354 8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes

3 The diagram id

(?)J O

/ (?)J O

ε

ε

L(f• |J )∗ R(f• |J )∗ (?)J

c

−1

/ L(f• |J )∗ (?)J R(f• )∗

θ

/ (?)J L(f• )∗ R(f• )∗

is commutative. Proof. Since (?)J preserves K-ﬂat complexes by Lemma 8.10, we have L(f• |J )∗ (?)J ∼ = L((f• |J ) ◦ (?)J ). On the other hand, it is obvious that we have (?)J Lf•∗ ∼ = L((?)J f•∗ ). By Lemma 6.25, we have a composite isomorphism Lθ θ : L(f• |J )∗ (?)J ∼ = L((f• |J )∗ ◦ (?)J )−→L((?)J ◦ f•∗ ) ∼ = (?)J L(f• )∗ ,

and 1 is proved. 2 and 3 follow from the proofs of Lemma 1.24 and Lemma 1.25, respectively. ⊓ ⊔ Lemma 8.14. Let X• ∈ P, and F, G ∈ D(X• ). Then we have the following. 1 FJ ⊗•,L OX

• |J

•,L GJ ∼ = (F ⊗OX• G)J for any subcategory J ⊂ I. O

2 If F and G have locally quasi-coherent cohomology groups, then Tori X• (F, G) is also locally quasi-coherent for any i ∈ Z. 3 Assume that there exists some big subcategory J of I such that X• has ﬂat J-arrows. If both F and G have equivariant (resp. quasi-coherent) cohomolO ogy groups, then Tori X• (F, G) is also equivariant (resp. quasi-coherent). Proof. The assertion 1 is an immediate consequence of Lemma 8.10 and Example 5.6, 5. 2 In view of 1, we may assume that X = X• is a single scheme. As the question is local, we may assume that X is even aﬃne. We may assume that F = lim Fn , where (Fn ) is the P(X• )-special direct −→ system such that each Fn is bounded above and has locally quasi-coherent cohomology groups as in Lemma 3.25, 3. Similarly, we may assume that G = lim Gn . As ﬁltered inductive limits are exact and compatible with tensor −→ products, and the colimit of locally quasi-coherent sheaves is locally quasicoherent, we may assume that both F and G are bounded above, ﬂat, and has locally quasi-coherent cohomology groups. By [17, Proposition I.7.3], we may assume that both F and G are single quasi-coherent sheaves. This case is trivial. 3 In view of 1, we may assume that J = I and X• has ﬂat arrows. By 2, it suﬃces to show the assertion for equivariance. Assuming that F and G

8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes 355

are K-ﬂat with equivariant cohomology groups, we prove that F ⊗ G has equivariant cohomology groups. This is enough. Let φ : i → j be a morphism of I. As Xφ is ﬂat and F and G have equivariant cohomology groups, αφ : Xφ∗ Fi → Fj and αφ : Xφ∗ Gi → Gj are quasi-isomorphisms. The composite αφ ⊗αφ Xφ∗ (F ⊗•OX• G)i ∼ = (F ⊗•OX• G)j = Xφ∗ Fi ⊗•OXj Xφ∗ Gi −−−−→Fj ⊗•OXj Gj ∼

is a quasi-isomorphism, since Xφ∗ Gi and Fj are K-ﬂat. By (6.3), αφ (F⊗•OX• G) is a quasi-isomorphism. As X• has ﬂat arrows, this shows that F ⊗•OX• G has equivariant cohomology groups. ⊓ ⊔ (8.15) Let X• ∈ P, and J an admissible subcategory of I. By Lemma 6.17, the left derived functor LLMod : D(X• |J ) → D(X• ) J of LMod is deﬁned, since Q(X• , J) is admissible. This is also calculated using J K-ﬂat resolutions. Namely, Lemma 8.16. Let X• and J be as above. If F ∈ K(Mod(X• |J )) is K-ﬂat, then so is LJ F. If F is K-ﬂat exact, then so is LJ F. Proof. This is trivial by Lemma 6.7.

⊓ ⊔

Corollary 8.17. Let X• ∈ P(I, Sch/S), J an admissible subcategory of I, and I ∈ K(Mod(X• )). If I is weakly K-injective, then IJ is weakly K-injective. Proof. Let F be a K-ﬂat exact complex in K(Mod(X• |J )). Then, Hom•Mod(X• |J ) (F, IJ ) ∼ = Hom•Mod(X• ) (LJ F, I) is exact by the lemma. By Lemma 3.25, 5, we are done.

⊓ ⊔

Lemma 8.18. Let f : X → Y be a morphism of schemes, and M ∈ Qch(Y ). Then for any i ≥ 0, Li f ∗ M ∈ Qch(X). Proof. Note that Lf ∗ is computed using a ﬂat resolution, and a ﬂat object is preserved by f ∗ . If g is a ﬂat morphism of schemes, then g ∗ is exact. Thus using a spectral sequence argument, it is easy to see that the question is local both on Y and X. So we may assume that X = Spec B and Y = Spec A are aﬃne. If Γ(Y, M) = M and F → M is an A-projective resolution, then ˜ = Hi ((B ⊗A F)∼ ) = TorA (B, M )∼ . Li f ∗ M = Hi (f ∗ F) i Thus Li f ∗ M is quasi-coherent for any i ≥ 0, as desired.

⊓ ⊔

356 8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes

Lemma 8.19. Let X• ∈ P and J an admissible subcategory of I. Let F ∈ DLqc (X• |J ). Then, LLJ F ∈ DLqc (X• ). Proof. First we consider the case that F = M is a single locally quasi-coherent sheaf. Then by the uniqueness of the derived functor, (?)i H −n (LLJ M) = Ln ((?)i LJ )M = lim Ln Xφ∗ Mj −→ for i ∈ I. Thus LLJ M ∈ DLqc (X• ) by Lemma 8.18. Now using the standard spectral sequence argument (or the way-out lemma [17, (I.7.3)]), the case that F is bounded above follows. The general case follows immediately by Lemma 3.25, 3. ⊓ ⊔ Lemma 8.20. Let f• : X• → Y• be a morphism in P. If F ∈ DLqc (Y• ), then Lf•∗ F ∈ DLqc (X• ). If Y• and X• have ﬂat arrows and F ∈ DEM (Y• ), then Lf•∗ F ∈ DEM (X• ). Proof. For the ﬁrst assertion, we may assume that f : X → Y is a morphism of single schemes by Lemma 8.13. If F is a single quasi-coherent sheaf, this is obvious by Lemma 8.18. So the case that F is bounded above follows from the way-out lemma. The general case follows from Lemma 3.25, 3. We prove the second assertion. If F is a K-ﬂat complex in Mod(Y• ) with equivariant cohomology groups, then αφ : Yφ∗ Fi → Fj is a quasiisomorphism for any morphism φ : i → j of I by Lemma 8.6. As the mapping cone Cone(αφ ) is K-ﬂat exact by Lemma 8.10, fj∗ Cone(αφ ) is also exact. Thus fj∗ αφ : fj∗ Yφ∗ Fi → fj∗ Fj is a quasi-isomorphism. This shows that αφ : Xφ∗ (f•∗ F)i → (f•∗ F)j is a quasi-isomorphism for any φ. So f•∗ F has equivariant cohomology groups by Lemma 8.6. This is what we wanted to prove. ⊓ ⊔ Lemma 8.21. Let f• : X• → Y• be a ﬂat morphism in P. If F ∈ DEM (Y• ), then Lf•∗ F ∈ DEM (X• ). Proof. Let F be a K-ﬂat complex in Mod(Y• ) with equivariant cohomology groups. Then H n (f•∗ F) ∼ = f•∗ (H n F) is equivariant by Lemma 7.22. This is what we wanted to prove. ⊓ ⊔ (8.22) Let I be a small category, and S a scheme. Set P := P(I, Sch/S). As we have seen, for a morphism f• : X• → Y• , f•−1 : Zar(Y• ) → Zar(X• ) is an admissible ringed continuous functor by Lemma 6.25. Moreover, if J and K are admissible subcategories of I such that J ⊂ K, then Q(X• |J , K) : Zar(X• |K ) → Zar(X• |J ) is also admissible. Utilizing Lemma 3.33 and Lemma 5.4, we have the following. Example 8.23. Let I be a small category, S a scheme, and f• : X• → Y• and g• : Y• → Z• are morphisms in P(I, Sch/S). Let K ⊂ J ⊂ I be admissible subcategories. Then we have the following.

8 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes 357

1 There is a natural isomorphism cI,J,K : (?)K,I ∼ = (?)K,J ◦ (?)J,I . Taking the conjugate, dI,J,K : LLI,J ◦ LLJ,K ∼ = LLI,K is induced. 2 There are natural isomorphism cJ,f• : (?)J ◦ R(f• )∗ ∼ = R(f• |J )∗ ◦ (?)J and its conjugate dJ,f• : LLJ ◦ L(f• |J )∗ ∼ = L(f• )∗ ◦ LLJ . 3 We have (cK,f• |J (?)J ) ◦ ((?)K,J cJ,f• ) = (R(f• |K )∗ cI,J,K ) ◦ cK,f• ◦ (c−1 I,J,K R(f• )∗ ). 4 We have (R(g• |J )∗ cJ,f• ) ◦ (cJ,g• R(f• )∗ ) = (cf• |J ,g• |J (?)J ) ◦ cJ,g• ◦f• ◦ ((?)J c−1 f• ,g• ), ∼ R(g• )∗ ◦ R(f• )∗ is the canonical isomorphism, where cf• ,g• : R(g• ◦ f• )∗ = and similarly for cf• |J ,g• |J . ) over the category P(I, Sch/S) is 5 The adjoint pair (L(?)∗Mod , R(?)Mod ∗ Lipman.

Chapter 9

Simplicial Objects

(9.1) For n ∈ Z with n ≥ −1, we deﬁne [n] to be the totally ordered ﬁnite set {0 < 1 < . . . < n}. Thus, [−1] = ∅, [0] = {0}, [1] = {0 < 1}, and so on. We deﬁne (Δ+ ) to be the small category given by ob(Δ+ ) := {[n] | n ∈ Z, n ≥ −1} and Mor(Δ+ ) := {monotone maps}. For a subset S of {−1, 0, 1, . . .}, we deﬁne (Δ+ )S to be the full subcategory of (Δ+ ) such that ob((Δ+ )S ) = {[n] | n ∈ S}. We deﬁne (Δ) := (Δ+ )[0,∞) . If −1 ∈ / S, then (Δ+ )S is also denoted by (Δ)S . We deﬁne (Δ+ )mon to be the subcategory of (Δ+ ) by ob((Δ+ )mon ) := ob(Δ+ ) and Mor((Δ+ )mon ) := {injective monotone maps}. For S ⊂ {−1, 0, 1, . . .}, the full subcategories (Δ+ )mon and (Δ)mon of (Δ+ )mon S S are deﬁned similarly. + + mon We denote (Δ)mon {0,1,2} and (Δ ){−1,0,1,2} by ΔM and ΔM , respectively. Let C be a category. We call an object of P((Δ+ ), C) (resp. P((Δ), C), an augmented simplicial object (resp. simplicial object) of C. For a subcategory D of (Δ+ ) and an object X• ∈ P(D, C), we denote X[n] by Xn . As [−1] is the initial object of (Δ+ ), an augmented simplicial object X• of C with X−1 = c is identiﬁed with a simplicial object of C/c. We deﬁne some particular morphisms in (Δ+ ). The unique map [−1] → [n] is denoted by ε(n). The unique injective monotone map [n − 1] → [n] such that i is not in the image is denoted by δi (n) for i ∈ [n]. The unique surjective monotone map [n + 1] → [n] such that i has two inverse images is denoted by σi (n) for i ∈ [n]. The unique map [0] → [n] such that i is in the image is denoted by ρi (n). The unique map [n] → [0] is denoted by λn . Let D be a subcategory of (Δ+ ). For X• ∈ P(D, C), we denote X• (ε(n)) (resp. X• (δi (n)), X• (σi (n)), X• (ρi (n)), and X• (λn )) by e(n, X• ) (resp. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

359

360

9 Simplicial Objects

di (n, X• ), si (n, X• ), ri (n, X• ), and ln (X• )), or simply by e(n) (resp. di (n), si (n), ri (n), ln ), if there is no danger of confusion. Note that (Δ) is generated by δi (n), σi (n) for various i and n. (9.2) Note that (Δ+ )([m], [n]) is a ﬁnite set for any m, n. Assume that C ˇ has ﬁnite limits and let f : X → Y be a morphism in C. Then the Cech (∆+ ) (∆+ ) nerve is deﬁned to be Nerve(f ) := cosk(∆+ ){−1,0} (f ), where cosk(∆+ ){−1,0} is the right adjoint of the restriction. It is described as follows. Nerve(f )n = X ×Y × · · · ×Y X ((n + 1)-fold ﬁber product) for n ≥ 0, and Nerve(f )−1 = Y . Note that di (n) is given by i

di (n)(xn , . . . , x1 , x0 ) = (xn , · · ˇ· · · , x1 , x0 ), and si (n) is given by si (n)(xn , . . . , x1 , x0 ) = (xn , . . . , xi+1 , xi , xi , xi−1 , . . . , x1 , x0 ) if C = Set. (9.3) Let S be a scheme. A simplicial object (resp. augmented simplicial object) in Sch/S, in other words, an object of P((Δ), Sch/S) (resp. P((Δ+ ), Sch/S)), is called a simplicial (resp. augmented simplicial) S-scheme. If I is a subcategory of (Δ+ ), X• ∈ P(I, Sch/Z), ♥ = Mod, PM, AB, PA, M ∈ ♥(X• ) and [n] ∈ I, then we sometimes denote M[n] by Mn . The following is well-known. Lemma 9.4. Let X• ∈ P((Δ), Sch/S). Then the restriction (?)∆M : EM(X• ) → EM(X• |∆M ) is an equivalence. With the equivalence, quasi-coherent sheaves correspond to quasi-coherent sheaves. Proof. We deﬁne a third category A as follows. An object of A is a pair (M0 , ϕ) such that, M0 ∈ Mod(X0 ), ϕ ∈ HomMod(X1 ) (d∗0 (M0 ), d∗1 (M0 )), ϕ an isomorphism, and that d∗1 (ϕ) = d∗2 (ϕ) ◦ d∗0 (ϕ) (more precisely, the composite map d−1

d∗ ϕ

d

d

d∗ ϕ

1 → d∗1 d∗1 M0 − → r0∗ M0 r2∗ M0 −−→ d∗1 d∗0 M0 −−

agrees with the composite map d−1

d∗ ϕ

d

0 2 r2∗ M0 −−→ d∗0 d∗0 M0 −− → d∗0 d∗1 M0 − → d∗2 d∗0 M0 −− → d∗2 d∗1 M0 − → r0∗ M0 .

We use such a simpliﬁed notation throughout the proof of this lemma). Note that applying l2∗ to the last equality, we get l1∗ (ϕ) = l1∗ (ϕ) ◦ l1∗ (ϕ). As ϕ is an isomorphism, we get l1∗ (ϕ) = id. A morphism γ0 : (M0 , ϕ) → (N0 , ψ) is an element γ0 ∈ HomMod(X0 ) (M0 , N0 )

9 Simplicial Objects

361

such that ψ ◦ d∗0 (γ0 ) = d∗1 (γ0 ) ◦ ϕ. We deﬁne a functor Φ : EM(X• |∆M ) → A by ◦ αd0 (1) ). Φ(M) := (M0 , αd−1 1 (1) It is easy to verify that this gives a well-deﬁned functor. Now we deﬁne a functor Ψ : A → EM(X• ). Note that an object M of EM(X• ) is identiﬁed with a family (Mn , αw )[n]∈(∆), w∈Mor((∆)) such that Mn ∈ Mod(Xn ), αw ∈ HomMod(Xn ) ((Xw )∗Mod (Mm ), Mn ) for w ∈ Δ(m, n), αw is an isomorphism, and αww′ = αw ◦ Xw∗ αw′ ◦ d−1

(9.5)

whenever ww′ is deﬁned, see (4.6). For (M′0 , ϕ) ∈ A, we deﬁne Mn,i := (ri (n))∗ (M′0 ), and Mn := Mn,0 for n ≥ 0 and 0 ≤ i ≤ n. We deﬁne ψi (n) : Mn,i+1 → Mn,i to be (Xq(i,n) )∗ (ϕ) for n ≥ 1 and 0 ≤ i < n, where q(i, n) : [1] → [n] is the unique injective monotone map with {i, i + 1} = Im q(i, n). We deﬁne ϕi (n) : Mn,i ∼ = Mn to be the composite map ϕi (n) := ψ0 (n) ◦ ψ1 (n) ◦ · · · ◦ ψi−1 (n) for n ≥ 0 and 0 ≤ i ≤ n. Now we deﬁne αw ∈ HomMod(Xn ) (Xw∗ (Mm ), Mn ) to be the map d

ϕw(0) (n)

Xw∗ Mm = Xw∗ r0 (m)∗ M′0 − → rw(0) (n)∗ M′0 = Mn,w(0) −−−−−→Mn for w ∈ Δ([m], [n]). Thus (M′0 , ϕ) yields a family (Mn , αw ), and this gives the deﬁnition of Ψ : A → EM(X• ). The details of the proof of the well-deﬁnedness is left to the reader. It is also straightforward to check that (?)∆M , Φ, and Ψ give the equivalence of these three categories. The proof is also left to the reader. The last assertion is obvious from the construction. ⊓ ⊔

Chapter 10

Descent Theory

Let S be a scheme. (10.1) Consider the functor shift : (Δ+ ) → (Δ) given by shift[n] := [n + 1], shift(δi (n)) := δi+1 (n + 1), shift(σi (n)) := σi+1 (n + 1), and shift(ε(0)) := δ1 (1). We have a natural transformation (δ0+ ) : Id(∆+ ) → ι ◦ shift given by (δ0+ )n := δ0 (n + 1) for n ≥ 0 and (δ0+ )−1 := ε(0), where ι : (Δ) ֒→ (Δ+ ) is the inclusion. We denote (δ0+ )ι by (δ0 ). Note that (δ0 ) can be viewed as a natural map (δ0 ) : Id(∆) → shift ι. Let X• ∈ P((Δ), Sch/S). We deﬁne X•′ to be the augmented simplicial scheme shift# (X• ) = X• shift. The natural map X• (δ0 ) : X•′ |(∆) = X• shift ι → X• is denoted by (d0 )(X• ) or (d0 ). Similarly, if Y• ∈ P((Δ+ ), Sch/S), then Y• (δ + )

0 ′ (d+ 0 )(Y• ) : (Y• |(∆) ) = Y• ι shift −−−−→Y•

is deﬁned as well. (10.2) We say that X• ∈ P((Δ), Sch/S) is a simplicial groupoid of S-schemes if there is a faithfully ﬂat morphism of S-schemes g : Z → Y such that there is a faithfully ﬂat cartesian morphism f• : Z• → X• of P((Δ), Sch/S), where Z• = Nerve(g)|(∆) . Lemma 10.3. Let X• ∈ P((Δ), Sch/S). 1 If X• ∼ = Nerve(g)|(∆) for some faithfully ﬂat morphism g of S-schemes, then X• is a simplicial groupoid. 2 If f• : Z• → X• is a faithfully ﬂat cartesian morphism of simplicial S-schemes and Z• is a simplicial groupoid, then we have X• is also a simplicial groupoid.

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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3 X• is a simplicial groupoid if and only if (d0 ) : X•′ |(∆) → X• is cartesian, the canonical unit map (∆+ )

X•′ → Nerve(d1 (1)) = cosk(∆+ )mon

{−1,0}

) (X•′ |(∆+ )mon {−1,0}

is an isomorphism, and d0 (1) and d1 (1) are ﬂat. 4 If f• : Z• → X• is a cartesian morphism of simplicial S-schemes and X• is a simplicial groupoid, then Z• is a simplicial groupoid. 5 A simplicial groupoid has faithfully ﬂat (Δ)mon -arrows. 6 If X• is a simplicial groupoid of S-schemes such that d0 (1) and d1 (1) are separated (resp. quasi-compact, quasi-separated, of ﬁnite type, smooth, ´etale), then X• has separated (resp. quasi-compact, quasi-separated, of ﬁnite type, smooth,´etale) (Δ)mon -arrows, and (d0 ) : X•′ |(∆) → X• is separated (resp. quasi-compact, quasi-separated, of ﬁnite type, smooth, ´etale). Proof. 1 and 2 are obvious by deﬁnition. We prove 3. We prove the ‘if’ part. As d0 (1)s0 (0) = id = d1 (1)s0 (0), we have that d0 (1) and d1 (1) are faithfully ﬂat by assumption. As d0 (1) = (d0 )0 is faithfully ﬂat and (d0 ) is cartesian, it is easy to see that (d0 ) is also faithfully ﬂat. So this direction is obvious. We prove the ‘only if’ part. As X• is a simplicial groupoid, there is a faithfully ﬂat S-morphism g : Z → Y and a faithfully ﬂat cartesian morphism f• : Z• → X• of simplicial S-schemes, where Z• = Nerve(g)|(∆) . It is easy to see that (d0 ) : Z•′ |(∆) → Z• is nothing but the base change by g, and it is faithfully ﬂat cartesian. It is also obvious that Z•′ ∼ = Nerve(d1 (1)(Z• )) and d0 (1)(Z• ) and d1 (1)(Z• ) are ﬂat. It is obvious that f•′ : Z•′ → X•′ is faithfully ﬂat cartesian. Now by Lemma 7.16, (d0 )(X• ) is cartesian. As f• is faithfully ﬂat cartesian and d0 (1)(Z• ) and d1 (1)(Z• ) are ﬂat, we have that d0 (1)(X• ) and d1 (1)(X• ) are ﬂat. When we base change X•′ → Nerve(d1 (1)(X• )) by f0 : Z0 → X0 , then we have the isomorphism Z•′ ∼ = Nerve(d1 (1)(Z• )). As f0 is faithfully ﬂat, we have that X•′ → Nerve(d1 (1)) is also an isomorphism. The assertions 4, 5 and 6 are proved easily. ⊓ ⊔ (10.4) Let X• ∈ P((Δ), Sch/S). Then we deﬁne F : Zar(X•′ ) → Zar(X• ) by F (([n], U )) = (shift[n], U ) and F ((w, h)) = (shift w, h). The corresponding # pull-back FMod is denoted by (?)′ . It is easy to see that (?)′ has a left and a right adjoint. It also preserves equivariant and locally quasi-coherent sheaves. Let M ∈ Mod(X• ). Then we deﬁne (α) : (d0 )∗ M → M′(∆) by θ −1

αδ

(n)

0 (α)n : ((d0 )∗ M)n −−→ d0 (n)∗ Mn −−− −→Mn+1 = M′n .

It is easy to see that (α) : (d0 )∗ → (?)(∆) ◦ (?)′ is a natural map. Similarly, for Y• ∈ P((Δ+ ), Sch/S), ∗ ′ (α+ ) : (d+ 0 ) → (?) ◦ (?)(∆)

is deﬁned.

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(10.5) Let X• ∈ P((Δ+ ), Sch/S), and M ∈ Mod(X• |(∆) ). Then, we have a cosimplicial object Cos(M) of Mod(X−1 ) (i.e., a simplicial object of Mod(X−1 )op ). We have Cos(M)n := e(n)∗ (Mn ), and βw

c−1

Cos(M)w : e(m)∗ (Mm ) −−→ e(m)∗ (w∗ (Mn )) −−→ e(n)∗ (Mn ) for a morphism w : [m] → [n] in Δ. Similarly, the augmented cosimplicial object Cos+ (N ) of Mod(X−1 ) is deﬁned for N ∈ Mod(X• ). Mod (M) is M on X• |(∆) , M+ By (6.14), it is easy to see that M+ := R(∆) −1 is lim Cos(M), and βε(n) (M+ ) is nothing but the canonical map ←− lim Cos(M) → Cos(M)n = e(n)∗ (Mn ) = e(n)∗ (M+ n ). ←− Note that Cos(M) can be viewed as a (co)chain complex such that Cos(M)n = e(n)∗ (Mn ) for n ≥ 0, and the boundary map ∂ n : Cos(M)n → Cos(M)n+1 is given by ∂ n = d0 − d1 + · · · + (−1)n+1 dn+1 , where di = di (n + 1) = Cos(M)δi (n+1) . Similarly, for N ∈ Mod(X• ), Cos+ (N ) can be viewed as an augmented cochain complex. Note also that for M ∈ Mod(X• |(∆) ), we have lim Cos(M) = Ker(d0 (1) − d1 (1)) = H 0 (Cos(M)), ←−

(10.6)

which is determined only by M(∆){0,1} . Lemma 10.7. Let f• : X• → Y• be a morphism of P((Δ+ ), Sch/S). If f• |(∆+ ){−1,0,1} is ﬂat cartesian, and Y• has concentrated (Δ+ )mon {−1,0,1} -arrows, then the canonical map μ : f•∗ ◦ R(∆) → R(∆) ◦ (f• |(∆) )∗ (see (6.27)) is an isomorphism of functors from Lqc(Y• |(∆) ) to Lqc(X• ). Proof. To prove that the map in question is an isomorphism, it suﬃces to show that the map is an isomorphism after applying the functor (?)n for n ≥ −1. This is trivial if n ≥ 0. On the other hand, if n = −1, the map restricted at −1 and evaluated at M ∈ Lqc(Y• |(∆) ) is nothing but ∗ ∗ (H 0 (Cos(M))) ∼ (Cos(M))) → H 0 (Cos((f• |(∆) )∗ (M))). f−1 = H 0 (f−1

The ﬁrst map is an isomorphism as f−1 is ﬂat. Although the map ∗ f−1 (Cos(M)) → Cos((f• |(∆) )∗ (M))

may not be a chain isomorphism, it is an isomorphism at the degrees −1, 0, 1, and it induces the isomorphism of H 0 . ⊓ ⊔

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Lemma 10.8. Let X• ∈ P((Δ), Sch/S), and M ∈ Mod(X• ). Then the (associated chain complex of the) augmented cosimplicial object Cos+ (M′ ) of M′ ∈ Mod(X•′ ) is split exact. In particular, the unit map u : M′ → R(∆) M′(∆) is an isomorphism. Proof. Deﬁne sn : Cos+ (M′ )n → Cos+ (M′ )n−1 to be (r0 )(n+1)∗ βσ

(n)

0 Cos+ (M′ )n = (r0 )(n + 1)∗ (Mn+1 )−−−−−−−−−− −→

c−1

(r0 )(n + 1)∗ s0 (n)∗ (Mn ) −−→ (r0 )(n)∗ (Mn ) = Cos+ (M′ )n−1 for n ≥ 0, and s−1 : Cos+ (M′ )−1 → 0 to be 0. It is easy to verify that s is a chain deformation of Cos+ (M′ ). ⊓ ⊔ Corollary 10.9. Let the notation be as in the lemma. Then there is a functorial isomorphism R(∆) (d0 )∗ (M) → M′ (10.10) for M ∈ EM(X• ). In particular, there is a functorial isomorphism (R(∆) (d0 )∗ (M))−1 → M0 .

(10.11)

Proof. The ﬁrst map (10.10) is deﬁned to be the composite R(∆) (α)

u−1

R(∆) (d0 )∗ −−−−−→R(∆) (?)(∆) (?)′ −−→(?)′ . As (α)(M) is an isomorphism if M is equivariant, this is an isomorphism. The second map (10.11) is obtained from (10.10), applying (?)−1 . ⊓ ⊔ The following well-known theorem in descent theory contained in [33] is now easy to prove. Proposition 10.12. Let f : X → Y be a morphism of S-schemes, and set X•+ := Nerve(f ), and X• := X•+ |(∆) . Let M ∈ Mod(X• ). Then we have the following. 0 The counit of adjunction ε : (R(∆) M)(∆) → M is an isomorphism. 1 If f is concentrated and M ∈ Lqc(X• ), then R(∆) M ∈ Lqc(X•+ ). 2 If f is faithfully ﬂat concentrated and M ∈ Qch(X• ), then we have R(∆) M ∈ Qch(X•+ ). 3 If f is faithfully ﬂat concentrated, N ∈ EM(X•+ ), and N(∆) ∈ Qch(X• ), then the unit of adjunction u : N → R(∆) (N(∆) ) is an isomorphism. In particular, N is quasi-coherent.

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4 If f is faithfully ﬂat concentrated, then the restriction functor (?)(∆) : Qch(X•+ ) → Qch(X• ) is an equivalence, with R(∆) its quasi-inverse. Proof. The assertion 0 follows from Lemma 6.15. We prove 1. By 0, it suﬃces to prove that (R(∆) M)−1 = Ker(e(0)∗ βδ0 (1) − e(0)∗ βδ1 (1) ) is quasi-coherent. This is obvious by [14, (9.2.2)]. Now we assume that f is faithfully ﬂat concentrated, to prove the assertions 2, 3, and 4. We prove 2. As we already know that R(∆) M is locally quasi-coherent, it ′ suﬃces to show that it is equivariant. As (d+ 0 ) : (X• |(∆) ) → X• is faithfully + ∗ ﬂat, it suﬃces to show that (d0 ) R(∆) M is equivariant, by Lemma 7.22. Now the assertion is obvious by Lemma 10.7 and Corollary 10.9, as M′ is quasi-coherent. We prove 3. Note that the composite map + ∗

(d0 ) u μ + ∗ ∗ ∗ (d+ → R(∆) (d0 )∗ N(∆) ∼ = R(∆) ((d+ 0 ) N −−−−→(d0 ) R(∆) (N(∆) ) − 0 ) N )(∆) (10.13) + ∗ ∗ + is nothing but the unit of adjunction u((d+ 0 ) N ). As (α ) : (d0 ) N → ′ ∗ (N(∆) ) is an isomorphism since N is equivariant, we have that u((d+ 0 ) N) is an isomorphism by Lemma 10.8. As μ in (10.13) is an isomorphism by + ∗ Lemma 10.7, we have that (d+ 0 ) u is an isomorphism. As (d0 ) is faithfully ﬂat, we have that u : N → R(∆) (N(∆) ) is an isomorphism, as desired. The last assertion is obvious by 2, and 3 is proved. The assertion 4 is a consequence of 0, 2 and 3. ⊓ ⊔

Corollary 10.14. Let f : X → Y be a faithfully ﬂat quasi-compact morphism of schemes, and M ∈ Mod(Y ). Then M is quasi-coherent if and only if f ∗ M is. Proof. The ‘only if’ part is trivial. We prove the ‘if’ part. We may assume that Y is aﬃne. So X is quasicompact, and has a ﬁnite aﬃne open covering (Ui ). Replacing X by i Ui , we may assume that X is also aﬃne. Thus f is faithfully ﬂat concentrated. If f ∗ M is quasi-coherent, then N := L−1 M satisﬁes the assumption of 3 of the proposition, as can be seen easily. So M ∼ ⊔ = (N )−1 is quasi-coherent. ⊓ Corollary 10.15. Let the notation be as in the proposition, and assume that f is faithfully ﬂat concentrated. The composite functor A := (?)(∆) ◦ L−1 : Qch(Y ) → Qch(X• )

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is an equivalence with D := (?)−1 ◦ R(∆) its quasi-inverse. Proof. Follows immediately by the proposition and Lemma 6.38, 2, since [−1] is the initial object of (Δ+ ). ⊓ ⊔ We call A in the corollary the ascent functor, and D the descent functor. Corollary 10.16. Let the notation be as in the proposition. Then the composite functor A ◦ D : Lqc(X• ) → Qch(X• ) is the right adjoint functor of the inclusion Qch(X• ) ֒→ Lqc(X• ). Proof. Note that D : Lqc(X• ) → Qch(Y ) is a well-deﬁned functor, and hence A ◦ D is a functor from Lqc(X• ) to Qch(X• ). For M ∈ Qch(X• ) and N ∈ Lqc(X• ), we have HomQch(X• ) (M, ADN ) ∼ = HomQch(Y ) (DM, DN ) ∼ + (R(∆) M, R(∆) N ) = Hom Lqc(X• )

∼ = HomLqc(X• ) (M, N ) = HomLqc(X• ) ((R(∆) M)(∆) , N ) ∼ by the proposition, Corollary 10.15, and Lemma 6.38, 1.

⊓ ⊔

Corollary 10.17. Let X• be a simplicial groupoid of S-schemes, and assume that d0 (1) and d1 (1) are concentrated. Then (d0 )Qch ◦ A : Qch(X0 ) → Qch(X• ) ∗ is a right adjoint of (?)0 : Qch(X• ) → Qch(X0 ), where A : Qch(X0 ) → Qch(X•′ |(∆) ) is the ascent functor deﬁned in Corollary 10.15. Proof. Note that (d0 )Qch is well-deﬁned, because (d0 ) is concentrated carte∗ sian, and the simplicial groupoid X• has ﬂat (Δ)mon -arrows, see (7.15) and Lemma 10.3. It is obvious that D ◦ (d0 )∗Qch is the left adjoint of (d0 )Qch ◦A ∗ by Corollary 10.15. On the other hand, we have (?)0 ∼ = D ◦ (d0 )∗Qch by ◦ A is a right adjoint of (?)0 , as desired. ⊓ ⊔ Corollary 10.9. Hence, (d0 )Qch ∗ (10.18) Let f : X → Y be a morphism of S-schemes, and set X•+ := Nerve(f ), and X• = X•+ |(∆) . It seems that even if f is concentrated and faithfully ﬂat, the canonical descent functor EM(X• ) → Mod(Y ) may not be an isomorphism. However, we have this kind of isomorphism for special morphisms. Let f : X → Y be a morphism of schemes. We say that f is a locally an open immersion if there exists some open covering (Ui ) of X such that f |Ui is an open immersion for any i. Assume that f is locally an open immersion.

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Lemma 10.19. Let f : X → Y be locally an open immersion. Let g : Y ′ → Y be any morphism, X ′ := Y ′ ×Y X, g ′ : X ′ → X the second projection, and f ′ : X ′ → Y ′ the ﬁrst projection. Then the canonical map θ : f ∗ g∗ → g∗′ (f ′ )∗ between the functors from Mod(Y ′ ) to Mod(X) is an isomorphism. ⊓ ⊔

Proof. Use Lemma 2.59.

Lemma 10.20. Let f : X → Y , X•+ , and X• be as in (10.18). Assume that f is faithfully ﬂat and locally an open immersion. Then the descent functor D = (?)−1 R(∆) : EM(X• ) → Mod(Y ) is an equivalence with A = (?)(∆) L−1 its quasi-inverse. Proof. Similar to the proof of Proposition 10.12.

⊓ ⊔

X•+ ,

Lemma 10.21. Let f : X → Y , and X• be as in (10.18). Assume that f is faithfully ﬂat and locally an open immersion. Then for M ∈ EM(X•′ ), the direct image (d0 )∗ M is equivariant. The restriction EM(X• ) → Mod(X0 ) = Mod(X) has the right adjoint (d0 )∗ A. Proof. Easy.

⊓ ⊔

Chapter 11

Local Noetherian Property

An abelian category A is called locally noetherian if it is a U-category, satisﬁes the (AB5) condition, and has a small set of noetherian generators [11]. For a locally noetherian category A, we denote the full subcategory of A consisting of its noetherian objects by Af . Lemma 11.1. Let A be an abelian U-category which satisﬁes the (AB3) condition, and B a locally noetherian category. Let F : A → B be a faithful exact functor, and G its right adjoint. If G preserves ﬁltered inductive limits, then the following hold. 1 A is locally noetherian. 2 a ∈ A is a noetherian object if and only if F a is. Proof. The ‘if’ part of 2 is obvious, as F is faithful and exact. Note that A satisﬁes the (AB5) condition, as F is faithful exact and colimit preserving, and B satisﬁes the (AB5) condition. Note also that, for a ∈ A, the set of subobjects of a is small, because the set of subobjects of F a is small [13] and F is faithful exact. Let S be a small set of noetherian generators of B. As any noetherian object is a quotient of a ﬁnite sum of objects in S, we may assume that any noetherian object in B is isomorphic to an element of S, replacing S by some larger small set, if necessary. For each s ∈ S, the set of subobjects of Gs is small by the last paragraph. Hence, there is a small subset T of ob(A) such that, any element t ∈ T admits a monomorphism t → Gs for some s ∈ S, F t is noetherian, and that if a ∈ A admits a monomorphism a → Gs for some s ∈ S and F a noetherian then a ∼ = t for some t ∈ T . We claim that any a ∈ A is a ﬁltered inductive limit lim aλ of subobjects −→ aλ of a, with each aλ is isomorphic to some element in T . If the claim is true, then 1 is obvious, as T is a small set of noetherian generators of A, and A satisﬁes the (AB5) condition, as we have already seen. The ‘only if’ part of 2 is also true if the claim is true, since if a ∈ A is noetherian, then it is a quotient of a ﬁnite sum of elements of T , and hence F a is noetherian. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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11 Local Noetherian Property

It suﬃces to prove the claim. As B is locally noetherian, we have F a = lim bλ , where (bλ ) is the ﬁltered inductive system of noetherian subobjects −→ of F a. Let u : Id → GF be the unit of adjunction, and ε : F G → Id be the counit of adjunction. It is well-known that we have (εF )◦(F u) = idF . As F u is a split monomorphism, u is also a monomorphism. We deﬁne aλ := u(a)−1 (Gbλ ). As G preserves ﬁltered inductive limits and A satisﬁes the (AB5) condition, we have lim aλ = u(a)−1 (G lim bλ ) = u(a)−1 (GF a) = a. −→ −→ Note that aλ → Gbλ is a monomorphism, with bλ being noetherian. It remains to show that F aλ is noetherian. Let iλ : aλ ֒→ a be the inclusion map, and jλ : bλ → F a the inclusion. Then the diagram F u(a)

εF (a)

F a −−−−→ F GF a −−−→ F a F iλ ↑ ↑ F Gjλ ↑ jλ ε(bλ )

F aλ −−−−→ F Gbλ −−−→ bλ is commutative. As the composite of the ﬁrst row is the identity map and F iλ is a monomorphism, we have that the composite of the second row F aλ → bλ is a monomorphism. As bλ is noetherian, we have that F aλ is also noetherian, as desired. ⊓ ⊔ Lemma 11.2. Let A be an abelian U-category which satisﬁes the (AB3) condition, and B a Grothendieck category. Let A → B be a faithful exact functor, and G its right adjoint. If G preserves ﬁltered inductive limits, then A is Grothendieck. Proof. Similar. (11.3)

⊓ ⊔

Let S be a scheme, and X• ∈ P((Δ), Sch/S).

Lemma 11.4. The restriction functor (?)0 : EM(X• ) → Mod(X0 ) is faithful exact. Proof. This is obvious, because for any [n] ∈ (Δ), there is a morphism [0] → [n]. Lemma 11.5. Let X• be a simplicial groupoid of S-schemes, and assume that d0 (1) and d1 (1) are concentrated. If Qch(X0 ) is Grothendieck, then Qch(X• ) is Grothendieck. Assume moreover that Qch(X0 ) is locally noetherian. Then we have 1 Qch(X• ) is locally noetherian. 2 M ∈ Qch(X• ) is a noetherian object if and only if M0 is a noetherian object. Proof. Let F := (?)0 : Qch(X• ) → Qch(X0 ) be the restriction. By Lemma 11.4, F is faithful exact. Let G := (d0 )Qch ◦ A be the right adjoint ∗ of F , see Corollary 10.17. As A is an equivalence and (d0 )Qch preserves ∗

11 Local Noetherian Property

373

ﬁltered inductive limits by Lemma 7.20, G preserves ﬁltered inductive limits. As Qch(X• ) satisﬁes (AB3) by Lemma 7.7, the assertion is obvious by Lemma 11.1 and Lemma 11.2. ⊓ ⊔ The following is well-known, see [18, pp. 126–127]. Corollary 11.6. Let Y be a noetherian scheme. Then Qch(Y ) is locally noetherian, and M ∈ Qch(Y ) is a noetherian object if and only if it is coherent. Proof. This is obvious if Y = Spec A is aﬃne. Now consider the general case. Let (Ui )1≤i≤r be an aﬃne open covering of Y , and set X := i Ui . Let p : X → Y be the canonical map, and set X• := Nerve(f )|(∆) . Note that p is faithfully ﬂat quasi-compact separated. By assumption and the lemma, we have that Qch(X• ) is locally noetherian, and M ∈ Qch(X• ) is a noetherian object if and only if M0 is noetherian, i.e., coherent. As A : Qch(Y ) → Qch(X• ) is an equivalence, we have that Qch(Y ) is locally noetherian, and M ∈ Qch(Y ) is a noetherian object if and only if (AM)0 = p∗ M is coherent if and only if M is coherent. ⊓ ⊔ A scheme X is said to be concentrated if the structure map X → Spec Z is concentrated. Corollary 11.7. Let Y be a concentrated scheme. Then Qch(Y ) is Grothendieck. Proof. Similar. Now the following is obvious.

⊓ ⊔

Corollary 11.8. Let X• be a simplicial groupoid of S-schemes, with d0 (1) and d1 (1) concentrated. If X0 is concentrated, then Qch(X• ) is Grothendieck. If, moreover, X0 is noetherian, then Qch(X• ) is locally noetherian, and M ∈ Qch(X• ) is a noetherian object if and only if M0 is coherent. Lemma 11.9. Let I be a ﬁnite category, S a scheme, and X• ∈ P(I, Sch/S). If X• is noetherian, then Mod(X• ) and Lqc(X• ) are locally noetherian. M ∈ Lqc(X• ) is a noetherian object if and only if M is locally coherent. Proof. Let J be the discrete subcategory of I such that ob(J) = ob(I). Obviously, the restriction (?)J is faithful and exact. For i ∈ ob(I), there is an

isomorphism of functors (?)i RJ ∼ = j∈ob(J) φ∈I(i,j) (Xφ )∗ (?)j . The product is a ﬁnite product, as I is ﬁnite. As each Xφ is concentrated, (Xφ )∗ (?)j preserves ﬁltered inductive limits by Lemma 7.19. Hence RJ preserves ﬁltered inductive limits. Note also that RJ preserves local quasi-coherence. Hence we may assume that I is a discrete ﬁnite category, which case is trivial by [17, Theorem II.7.8] and Corollary 11.6. ⊓ ⊔ Lemma 11.10. Let I be a ﬁnite category, S a scheme, and X• ∈ P(I, Sch/S). If X• is concentrated, then Lqc(X• ) is Grothendieck. Proof. Similar.

Chapter 12

Groupoid of Schemes

(12.1) Let C be a category with ﬁnite limits. A C-groupoid X∗ is a functor from C op to the category of groupoids ∈ U (i.e., category ∈ U all of whose morphisms are isomorphisms) such that the set valued functors X0 := ob ◦X∗ and X1 := Mor ◦X∗ are representable. Let X∗ be a C-groupoid. Let us denote the source (resp. target) X1 → X0 by d1 (resp. d0 ). Then X2 := X1 d1×d0 X1 represents the functor of pairs (f, g) of morphisms of X∗ such that f ◦ g is deﬁned. Let d′0 : X2 → X1 (resp. d′2 : X2 → X1 ) be the ﬁrst (resp. second) projection , and d′1 : X2 → X1 the composition. By Yoneda’s lemma, d0 , d1 , d′0 , d′1 , and d′2 are morphisms of C. d1 : X1 (T ) → X0 (T ) is surjective for any T ∈ ob(C). Note that the squares d′

0 X2 −→ X1 ′ ↓ d2 ↓ d1

d0

d′

d′

1 X1 X2 −→ ′ ↓ d0 ↓ d0

1 X1 X2 −→ ′ ↓ d2 ↓ d1

d1

X1 −→ X0

(12.2)

d0

X1 −→ X0

X1 −→ X0

are ﬁber squares. In particular, d′

0 −−→ ′

d1 X∗ := X2 −−→ X1 d′2

d

0 −−→

d

1 −−→

X0

(12.3)

−−→ forms an object of P(ΔM , C). Finally, by the associativity, ◦(◦ × 1) = ◦(1 × ◦),

(12.4)

where ◦ : X1 d1×d0 X1 → X1 denotes the composition, or ◦ is the composite ′

d1 X1 d1×d0 X1 ∼ = X2 −→X1 .

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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Conversely, a diagram X∗ ∈ P(ΔM , C) as in (12.3) such that the squares in (12.2) are ﬁber squares, d1 (T ) : X1 (T ) → X0 (T ) are surjective for all T ∈ ob(C), and the associativity (12.4) holds gives a C-groupoid [12]. In the sequel, we mainly consider that a C-groupoid is an object of P(ΔM , C). Let S be a scheme. We say that X• ∈ P(ΔM , Sch/S) is an S-groupoid, if X• is a (Sch/S)-groupoid with ﬂat arrows. (12.5) Let X• be a (Sch/S)-groupoid, and set Xn := X1 d1×d0 X1 d1×d0 · · · d1×d0 X1 (X1 appears n times) for n ≥ 2. For n ≥ 2, di : Xn → Xn−1 is deﬁned by d0 (xn−1 , . . . , x1 , x0 ) = (xn−1 , . . . , x1 ), dn (xn−1 , . . . , x1 , x0 ) = (xn−2 , . . . , x0 ), and di (xn−1 , . . . , x1 , x0 ) = (xn−1 , . . . , xi ◦ xi−1 , . . . , x0 ) for 0 < i < n. si : Xn → Xn+1 is deﬁned by si (xn−1 , . . . , x1 , x0 ) = (xn−1 , . . . , xi , id, xi−1 , . . . , x0 ). It is easy to see that this gives a simplicial S-scheme Σ(X• ) such that Σ(X• )|∆M = X• . For any simplicial S-scheme Z• and ψ• : Z• |∆M → X• , there exists some unique ϕ• : Z• → Σ(X• ) such that ϕ|∆M : Z• |∆M → Σ(X• )|∆M = X• equals ψ. Indeed, ϕ is given by ϕn (z) = (ψ1 (Qn−1 (z)), . . . , ψ1 (Q0 (z))), where qi : [1] → [n] is the injective monotone map such that Im qi = {i, i + 1} for 0 ≤ i < n, and Qi : Zn → Z1 is the associated morphism. This shows (∆) (∆) that Σ(X• ) ∼ = cosk∆M X• , and the counit map (cosk∆M X• )|∆M → X• is an isomorphism. Note that under the identiﬁcation Σ(X• )n+1 ∼ = Xn r0×d0 X1 , the morphism d0 : Σ(X• )n+1 → Σ(X• )n is nothing but the ﬁrst projection. So (d0 ) : Σ(X• )′ → Σ(X• ) is cartesian. If, moreover, d0 (1) is ﬂat, then (d0 ) is faithfully ﬂat. We construct an isomorphism h• : Σ(X• )′ → Nerve(d1 (1)). Deﬁne h−1 = id and h0 = id. Deﬁne h1 to be the composite (d0 ⊠d2 )−1

d ⊠d

2 X1 d1×d0 X1 −−−−−−−→X2 −−1−−→X 1 d1×d1 X1 .

Now deﬁne hn to be the composite 1×h

1 X1d1×d0 X1d1×d0 · · ·d1×d0 X1d1×d0 X1 −−−→X 1d1×d0 X1d1×d0 · · ·d1×d0 X1d1×d1 X1

via h

h ×1

1 1 −−−−→ · · · −− −→X1 d1×d1 X1 d1×d1 · · · d1×d1 X1 d1×d1 X1 .

It is straightforward to check that this gives a well-deﬁned isomorphism h• : Σ(X• )′ → Nerve(d1 (1)). In conclusion, we have (∆)

Lemma 12.6. If X• is an S-groupoid, then cosk∆M X• is a simplicial (∆)

S-groupoid, and the counit ε : (cosk∆M X• )|∆M → X• is an isomorphism.

12 Groupoid of Schemes

377

Conversely, the following holds. Lemma 12.7. If Y• is a simplicial S-groupoid, then Y• |∆M is an S-groupoid, (∆) and the unit map u : Y• → cosk∆M (Y• |∆M ) is an isomorphism. Proof. It is obvious that Y• |∆M has ﬂat arrows. So it suﬃces to show that Y• |∆M is a (Sch/S)-groupoid. Since Y•′ is isomorphic to Nerve d1 , the square d

2 Y2 −→ Y1 ↓ d1 ↓ d1

d

1 Y1 −→ Y0

is a ﬁber square. Since (d0 ) : Y•′ |(∆) → Y• is a cartesian morphism, the squares d

d

1 Y2 −→ Y1 ↓ d0 ↓ d0

2 Y2 −→ Y1 ↓ d0 ↓ d0

d

d

0 Y1 −→ Y0

1 Y1 −→ Y0

are ﬁber squares. As d1 s0 = id, d1 (T ) : Y1 (T ) → Y0 (T ) is surjective for any S-scheme T . Let us denote the composite d1 Y1 d1×d0 Y1 ∼ = Y2 −→Y1

by ◦. It remains to show the associativity. As the three squares in the diagram d

d

d

d

3 2 Y3 −→ Y2 −→ Y1 ↓ d0 ↓ d0 ↓ d0 2 1 Y2 −→ Y1 −→ Y0 ↓ d0 ↓ d0

d

1 Y1 −→ Y0

are all ﬁber squares, the canonical map Q := Q2 ⊠ Q1 ⊠ Q0 : Y3 → Y1 d1×d0 Y1 d1×d0 Y1 is an isomorphism. So it suﬃces to show that the maps Q

◦×1

◦

Q

1×◦

◦

Y3 − →Y1 d1×d0 Y1 d1×d0 Y1 −−→Y1 d1×d0 Y1 − →Y1 and

→Y1 Y3 − →Y1 d1×d0 Y1 d1×d0 Y1 −−→Y1 d1×d0 Y1 − agree. But it is not so diﬃcult to show that the ﬁrst map is d1 d2 , while the second one is d1 d1 . So Y• |∆M is an S-groupoid. (∆) Set Z• := cosk∆M (Y• |∆M ), and we are to show that the unit u• : Y• → Z• is an isomorphism. Since Y• |∆M is an S-groupoid, ε• : Z• |∆M → Y• |∆M is an

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12 Groupoid of Schemes

isomorphism. It follows that u• |∆M : Y• |∆M → Z• |∆M is also an isomorphism. Hence Nerve(d1 (1)(u• )) : Nerve(d1 (1)(Y• )) → Nerve(d1 (1)(Z• )) is also an isomorphism. As both Y• and Z• are simplicial S-groupoids by Lemma 12.6, u′• : Y•′ → Z•′ is an isomorphism. So un : Yn → Zn are all isomorphisms, and we are done. ⊓ ⊔ Lemma 12.8. Let S be a scheme, and X• an S-groupoid, with d0 (1) and d1 (1) concentrated. If X0 is concentrated, then Qch(X• ) is Grothendieck. If, moreover, X0 is noetherian, then Qch(X• ) is locally noetherian, and M ∈ Qch(X• ) is a noetherian object if and only if M0 is coherent. Proof. This is immediate by Corollary 11.8 and Lemma 9.4.

⊓ ⊔

(12.9) Let f : X → Y be a faithfully ﬂat concentrated S-morphism. Set X•+ := (Nerve(f ))|∆+ and X• := (X•+ )|∆M . We deﬁne the descent functor M

D : Lqc(X• ) → Qch(Y ) to be the composite (?)[−1] R∆M . The left adjoint (?)∆M L[−1] is denoted by A, and called the ascent functor. Lemma 12.10. Let the notation be as above. Then D : Qch(X• ) → Qch(Y ) is an equivalence, with A its quasi-inverse. The composite A ◦ D : Lqc(X• ) → Qch(X• ) is the right adjoint of the inclusion Qch(X• ) ֒→ Lqc(X• ). Proof. Follows easily from Lemma 9.4, Corollary 10.15, and Corollary 10.16. ⊓ ⊔ Lemma 12.11. Let X• be an S-groupoid, and assume that d0 (1) and d1 (1) (∆) are concentrated. Set Y•+ := ((cosk∆M X• )′ )|∆+ , and Y• := (Y•+ )|∆M . Let M (d0 ) : Y• → X• be the canonical map (d0 )|∆ (∆) (∆) Y• = ((cosk∆M X• )′ )|∆M −−−−−M →(cosk∆M X• )|∆M ∼ = X• .

Then (d0 ) is concentrated faithfully ﬂat cartesian, and (d0 )Qch ◦ A : Qch(X0 ) → Qch(X• ) ∗ is a right adjoint of (?)0 : Qch(X• ) → Qch(X0 ), where A : Qch(X0 ) → Qch(Y• ) is the ascent functor. Proof. Follows easily from Corollary 10.17.

⊓ ⊔

Utilizing Lemma 9.4, Lemma 10.20 and Lemma 10.21, we have the following easily.

12 Groupoid of Schemes

379

Lemma 12.12. Let f : X → Y be a faithfully ﬂat locally an open immersion of schemes. Set X• = Nerve(f )|∆M . Then the descent functor D = (?)−1 R(∆M ) : EM(X• ) → Mod(Y ) is an equivalence with the ascent functor A = (?)∆M L−1 its quasi-inverse. The restriction (?)0 : EM(X• ) → Mod(X0 ) = Mod(X) is faithfully exact with (d0 )∗ A its right adjoint. Proof. Easy.

⊓ ⊔

(12.13) We say that X• ∈ P(ΔM , Sch/S) is an almost-S-groupoid if the three squares in (12.2) are cartesian, (12.4) holds, and d0 (1) and d1 (1) are faithfully ﬂat. By deﬁnition, an S-groupoid is an almost-S-groupoid. If X• is an almost-S-groupoid, then there is a faithfully ﬂat cartesian morphism p• : Nerve(d1 (1))|∆M → X• such that p0 : X1 → X0 is d0 (1) (prove it). So Lemma 12.8 is true when we replace S-groupoid by almost-S-groupoid.

Chapter 13

B¨ okstedt–Neeman Resolutions and HyperExt Sheaves

(13.1) Let T be a triangulated category with small direct products. Note that a direct product of distinguished triangles is again a distinguished triangle (Lemma 3.1). Let s3 s2 →t1 (13.2) · · · → t3 −→t 2−

be a sequence of morphisms

in T . We deﬁne d : i≥1 ti → i≥1 ti by pi ◦ d = pi −si+1 ◦pi+1 , where pi : i ti → ti is the projection. Consider a distinguished triangle of the form d q m M− → ti − → ti − →ΣM, i≥1

i≥1

where Σ denotes the suspension. We call M , which is determined uniquely up to isomorphisms, the homotopy limit of (13.2) and denote it by holim ti . (13.3) Dually, homotopy colimit is deﬁned and denoted by hocolim, if T has small coproducts. (13.4) Let A be an abelian category which satisﬁes (AB3*). Let (Fλ )λ∈Λ be a small family of objects in K(A). Then for any G ∈ K(A), we have that HomK(A) (G,

Fλ ) = H 0 (Hom•A (G,

λ

λ

∼ =

λ

That is, the direct product

λ

H

0

Fλ )) ∼ = H 0 ( Hom•A (G, Fλ ))

(Hom•A (G, Fλ ))

λ

=

HomK(A) (G, Fλ ).

λ

Fλ in C(A) is also a direct product in K(A).

(13.5) Let A be a Grothendieck abelian category, and (tλ ) a small family of objects of D(A). Let (Fλ ) be a family of K-injective objects of K(A) such that Fλ represents tλ for each λ. Then Q( λ Fλ ) is a direct product of tλ in J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

381

382

13 B¨ okstedt–Neeman Resolutions and HyperExt Sheaves

D(A) (note that the direct product Hence D(A) has small products.

λ

Fλ exists, see [37, Corollary 7.10]).

Lemma 13.6. Let I be a small category, S be a scheme, and let X• ∈ P(I, Sch/S). Let F be an object of C(Mod(X• )). Assume that F has locally quasi-coherent cohomology groups. Then the following hold. 1 Let I denote the full subcategory of C(Mod(X• )) consisting of bounded below complexes of injective objects of Mod(X• ) with locally quasi-coherent cohomology groups. There is an I-special inverse system (In )n∈N with the index set N and an inverse system of chain maps (fn : τ≥−n F → In ) such that i fn is a quasi-isomorphism for any n ∈ N. ii Ini = 0 for i < −n. 2 If (In ) and (fn ) are as in 1, then the following hold. i For each i ∈ Z, the canonical map H i (lim In ) → H i (In ) is an isomor←− phism for n ≥ max(1, −i), where the projective limit is taken in the category C(Mod(X• )), and H i (?) denotes the ith cohomology sheaf of a complex of sheaves. ii lim fn : F → lim In is a quasi-isomorphism. ←− ←− iii The projective limit lim In , viewed as an object of K(Mod(X)), is the ←− homotopy limit of (In ). iv lim In is K-injective. ←− Proof. The assertion 1 is [39, (3.7)]. We prove 2, i. Let j ∈ ob(I) and U an aﬃne open subset of Xj . Then for any n ≥ 1, Ini and H i (In ) are Γ((j, U ), ?)-acyclic for each i ∈ Z. As In is bounded below, each Z i (In ) and B i (In ) are also Γ((j, U ), ?)-acyclic, and the sequence 0 → Γ((j, U ), Z i (In )) → Γ((j, U ), Ini ) → Γ((j, U ), B i+1 (In )) → 0

(13.7)

and 0 → Γ((j, U ), B i (In )) → Γ((j, U ), Z i (In )) → Γ((j, U ), H i (In )) → 0 (13.8) are exact for each i, as can be seen easily, where B i and Z i respectively denote the ith coboundary and the cocycle sheaves. In particular, the inverse system (Γ((j, U ), B i (In ))) is a Mittag-Leﬄer inverse system of abelian groups by (13.7), since (Γ((j, U ), Ini )) is. On the other hand, as we have H i (In ) ∼ = H i (F) for n ≥ max(1, −i), the inverse system i (Γ((j, U ), H (In ))) stabilizes, and hence we have (Γ((j, U ), Z i (In ))) is also Mittag-Leﬄer. Passing through the projective limit, 0 → Γ((j, U ), Z i (lim In )) → Γ((j, U ), lim In ) → Γ((j, U ), lim B i+1 (In )) → 0 ←− ←− ←−

13 B¨ okstedt–Neeman Resolutions and HyperExt Sheaves

383

is exact. Hence, the canonical map B i (lim In ) → lim B i (In ) is an isomor←− ←− phism, since (j, U ) with U an aﬃne open subset of Xj generates the topology of Zar(X• ). Taking the projective limit of (13.8), we have 0 → Γ((j, U ), B i (lim In )) → Γ((j, U ), Z i (lim In )) → Γ((j, U ), lim H i (In )) → 0 ←− ←− ←− is an exact sequence for any j and any aﬃne open subset U of Xj . Hence, the canonical maps Γ((j, U ), H i (In )) ∼ H i (In )) ← Γ((j, U ), H i (lim In )) = Γ((j, U ), lim ←− ←− are all isomorphisms for n ≥ max(1, −i), and we have H i (In ) ∼ I ) = H i (lim ←− n for n ≥ max(1, −i). The assertion ii is now trivial. The assertion iii is now a consequence of [7, Remark 2.3] (one can work at the presheaf level where we have the (AB4*) property). The assertion iv is now obvious. ⊓ ⊔ Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Lemma 13.9. Assume that X• has ﬂat arrows. Let J be a subcategory of I, and let F ∈ DEM (X• ) and G ∈ D(X• ). Assume one of the following. a G ∈ D+ (X• ). + (X• ). b F ∈ DEM c G ∈ DLqc (X• ). Then the canonical map HJ : (?)J R Hom•Mod(X• ) (F, G) → R Hom•Mod(X• |J ) (FJ , GJ ) is an isomorphism of functors to D(PM(X• |J )) (here Hom•Mod(X• ) (?, ∗) is viewed as a functor to PM(X• ), and similarly for HomMod(X• |J ) (?, ∗)). In particular, it is an isomorphism of functors to D(X• |J ). Proof. By Lemma 1.39, we may assume that J = i for an object i of I. So what we want to prove is for any complex in Mod(X• ) with equivariant cohomology groups F and any K-injective complex G in Mod(X• ), Hi : HomMod(X• ) (F, G)i → HomMod(Xi ) (Fi , Gi ) is a quasi-isomorphism of complexes in PM(Xi ) (in particular, it is a quasiisomorphism of complexes in Mod(Xi )), under the additional assumptions corresponding to a, b, or c. Indeed, if so, Gi is K-injective by Lemma 8.4. First consider the case that F is a single equivariant object. Then the assertion is true by Lemma 6.36. By the way-out lemma [17, Proposition I.7.1], the case that F is bounded holds. Under the assumption of a, the case that F is bounded above holds.

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13 B¨ okstedt–Neeman Resolutions and HyperExt Sheaves

Now consider the general case for a. As the functors in question on F changes coproducts to products, the map in question is a quasi-isomorphism if F is a direct sum of complexes bounded above with equivariant cohomology groups. Indeed, a direct product of quasi-isomorphisms of complexes of PM(Xi ) is again quasi-isomorphic. In particular, the lemma holds if F is a − (X• ). As any object F of DEM (X• ) is the homotopy colimit of objects of DEM homotopy colimit of (τ≤n F), we are done. The proof for the case b is similar. As F has bounded below cohomology groups, τ≤n F has bounded cohomology groups for each n. We prove the case c. By Lemma 13.6, we may assume that G is a homotopy limit of K-injective complexes with locally quasi-coherent bounded below cohomology groups. As the functors on G in consideration commute with homotopy limits, the problem is reduced to the case a. ⊓ ⊔ Lemma 13.10. Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Assume that X• has ﬂat arrows and is locally noetherian. − + + Let F ∈ DCoh (X• ) and G ∈ DLqc (X• ) (resp. DLch (X• )), where Lch denotes the plump subcategory of Mod consisting of locally coherent sheaves. Then ExtiOX• (F, G) is locally quasi-coherent (resp. locally coherent) for i ∈ Z. If, moreover, G has quasi-coherent (resp. coherent) cohomology groups, then ExtiOX• (F, G) is quasi-coherent (resp. coherent) for i ∈ Z. Proof. We prove the assertion for the local quasi-coherence and the local coherence. By Lemma 13.9, we may assume that X• is a single scheme. This case is [17, Proposition II.3.3]. We prove the assertion for the quasi-coherence (resp. coherence), assuming that G has quasi-coherent (resp. coherent) cohomology groups. By [17, Proposition I.7.3], we may assume that F is a single coherent sheaf, and G is an injective resolution of a single quasi-coherent (resp. coherent) sheaf. As X• has ﬂat arrows and the restrictions are exact, it suﬃces to show that αφ : Xφ∗ (?)i Hom•Mod(X• ) (F, G) → (?)j Hom•Mod(X• ) (F, G) is a quasi-isomorphism for any morphism φ : i → j in I. As Xφ is ﬂat, αφ : Xφ∗ Fi → Fj and αφ : Xφ∗ Gi → Gj are quasiisomorphisms. In particular, the latter is a K-injective resolution. By the derived version of (6.37), it suﬃces to show that P : Xφ∗ R Hom•OX (Fi , Gi ) → R Hom•OX (Xφ∗ Fi , Xφ∗ Gi ) i

j

is an isomorphism. This is [17, Proposition II.5.8].

⊓ ⊔

Chapter 14

The Right Adjoint of the Derived Direct Image Functor

(14.1) Let X be a scheme. A right adjoint of the inclusion FX : Qch(X) ֒→ Mod(X) is called the quasi-coherator of X, and is denoted by qch = qch(X). If f : Y → X is a concentrated morphism of schemes and qch(X) and qch(Y ) exist, then there is a canonical isomorphism f∗ qch(Y ) ∼ = qch(X)f∗ , which is the conjugate to f ∗ FX ∼ = FY f ∗ . Note also that if qch(X) exists, then the unit u : Id → qch(X)FX is an isomorphism, see [19, (I.1.2.6)]. (14.2) Let S be a scheme, I a small category, and X• ∈ P(I, Sch/S). Assume that for each i ∈ I, there exists some qch(Xi ) and that X• has concentrated arrows. Then we deﬁne lqc(X• ) : Mod(X• ) → Lqc(X• ) as follows. Let M ∈ Mod(X• ). Then M is expressed in terms of the data ((Mi )i∈I , (βφ )φ∈Mor(I) ). lqc(M) is then deﬁned in terms of the data as follows. (lqc(M))i = qch Mi for i ∈ I, and βφ is the composite qch βφ qch Mi −−−−→ qch(Xφ )∗ Mj ∼ = (Xφ )∗ qch Mj

for φ : i → j. It is easy to see that lqc(X• ) is the right adjoint of the inclusion FX : Lqc(X• ) → Mod(X• ). We call lqc the local quasi-coherator. Lemma 14.3. Let X be a concentrated scheme. 1 There is a right adjoint qch(X) : Mod(X) → Qch(X) of the canonical inclusion FX : Qch(X) ֒→ Mod(X). qch(X) preserves ﬁltered inductive limits. 1’ qch(X) preserves K-injective complexes. R qch(X) : D(Mod(X)) → D(Qch(X)) is right adjoint to FX : D(Qch(X)) → D(Mod(X)). 2 Assume that X is separated or noetherian. Then the functor FX : D(Qch(X)) → D(Mod(X)) is full and faithful, and induces an equivalence D(Qch(X)) → DQch(X) (Mod(X)). 3 Assume that X is separated or noetherian. Then the unit of adjunction u : Id → R qch(X)FX is an isomorphism, and ε : FX R qch(X) → Id is an isomorphism on DQch(X) (Mod(X)). J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

385

386

14 The Right Adjoint of the Derived Direct Image Functor

Proof. The existence assertion of 1 is proved in [20, Lemme 3.2]. If Spec A = X is aﬃne, then it is easy to see that the functor M → Γ(X, M)∼ is a desired qch(X). In fact, for a quasi-coherent N , a morphism N → M is uniquely determined by the A-linear map Γ(X, N ) → Γ(X, M). So qch(X) preserves ﬁltered inductive limits by [23, Proposition 6]. Next consider the general case. As X is quasi-compact, there is a ﬁnite aﬃne open covering (Ui ) of X. Set Y = i Ui , and let p : Y → X be the canonical map. Note that p is locally an open immersion and faithfully ﬂat. Let X• = Nerve(p)|∆M . Assume that each Xi admits qch(Xi ). This is the case if X is separated (and hence Xi is aﬃne for each i). Note that the inclusion Qch(X) ֒→ Mod(X) is equivalent to the composite A

D

Qch(X) − → Qch(X• ) ֒→ Lqc(X• ) ֒→ Mod(X• ) − → Mod(X). By Lemma 12.10, A is an equivalence. So it suﬃces to show that D : Qch(X• ) → Mod(X) has a right adjoint. By Lemma 12.12 and Lemma 12.10, for M ∈ Qch(X• ) and N ∈ Mod(X), we have HomMod(X) (DM, N ) ∼ = HomEM(X• ) (M, AN ) ∼ = HomMod(X• ) (M, AN ) ∼ ∼ = HomLqc(X• ) (M, lqc AN ) = HomQch(X• ) (M, AD lqc AN ). Thus D : Qch(X• ) → Mod(X) has a right adjoint AD lqc A, as desired (so qch(X) = D lqc A, as can be seen easily). So the case that X is quasi-compact separated is done. Now repeating the same argument for the general X (then Xi is quasi-compact separated for each i), the construction of qch is done. We prove that qch is compatible with ﬁltered inductive limits. Assume ﬁrst that X is separated. Then lqc : Mod(X• ) → Lqc(X• ) preserves ﬁltered inductive limits by the aﬃne case. As A, lqc, and D preserves ﬁltered inductive limits, so is qch(X) = D lqc A. Now repeating the same argument, the general case follows. The assertion 1’ follows from 1 and Lemma 3.12. Clearly, 2 and 3 are equivalent. 2 for the case that X is separated is proved in [7]. We remark that Verdier’s example [20, Appendice I] shows that the assertions are not true for a general concentrated scheme X which is not separated or noetherian. We give a proof for 3 for the case that X is noetherian, using the result for the case that X is separated. It suﬃces to show that if I is a K-injective complex in K(Mod(X)) with quasi-coherent cohomology groups, then qch(I) → I is a quasi-isomorphism. Since Mod(X) is Grothendieck, there is a strictly injective resolution I → J by Lemma 3.9. As Cone(I → J) is null-homotopic, replacing I by J, we may assume that I is strictly injective. Let U = (Ui )1≤i≤m be a ﬁnite aﬃne open covering of X. For a ﬁnite subset I of {1, . . . , m}, we denote i∈I Ui by UI . Note that each UI is noetherian

14 The Right Adjoint of the Derived Direct Image Functor

387

and separated. Let gI : UI ֒→ X be the inclusion. For M ∈ Mod(X), the ˇ ˇ ˇ Cech complex Cech(M ) = Cech U (M ) of M is deﬁned to be (gI )∗ gI∗ M → (gI )∗ gI∗ M → · · · → (gI )∗ gI∗ M → 0, 0→ #I=1

#I=2

#I=m

where (gI )∗ gI∗ M → (gJ )∗ gJ∗ M is the ± of the unit of adjunction if J ⊃ I, and ˇ zero if J ⊃ I. The augmented Cech complex 0 → M → Cech(M ) is denoted

+ + ˇ ˇ (gI )∗ g ∗ M is denoted by (M ). The lth term by Cech (M ) = Cech U

#I=l+1

I

l ˇ Cech (M ). + ˇ Note that if Ui = X for some i, then Cech U (M ) is split exact. In par+ + ∗ ∼ ˇ ˇ (g M ) is split exact,where Ui ∩ U = Cech (M )) ticular, gi∗ (Cech = Ui ∩U i U (Ui ∩ Uj )1≤j≤m is the open covering of Ui . Let g : Y = i Ui → X be the + ˇ canonical map. Since g is faithfully ﬂat and g ∗ (Cech U (M )) is split exact, + ˇ CechU (M ) is exact. Note also that if M = (gi )∗ N for some N ∈ Mod(Ui ), + + ∼ ˇ ˇ then Cech U (M ) = (gi )∗ (CechUi ∩U (N )) is split exact. In particular, if M is + ˇ (M ) is split a direct summand of g∗ N for some N ∈ Mod(Y ), then Cech exact. This is the case if M is injective, since M ֒→ g∗ g ∗ M splits. Now we want to prove that qch(I) → I is a quasi-isomorphism. Since + ˇ ˇ I is strictly injective, Cech (I) is split exact. So I → Cech(I) is a quasi+ ˇ ˇ isomorphism. As qch(Cech (I)) is split exact, qch(I) → qch(Cech(I)) ˇ is a quasi-isomorphism. So it suﬃces to show that qch(Cech(I)) → ˇ Cech(I) is a quasi-isomorphism. To verify this, it suﬃces to show that l l ˇ ˇ qch(Cech (I)) → Cech (I) is a quasi-isomorphism for l = 0, . . . , m − 1. To verify this, it suﬃces to show that for each non-empty subset I of {1, . . . , m}, ε : FX qch((gI )∗ gI∗ I) → (gI )∗ gI∗ I is a quasi-isomorphism. This map can be identiﬁed with (gI )∗ ε : (gI )∗ FUI qch gI∗ I → (gI )∗ gI∗ I. By the case that X is separated, ε : FUI qch gI∗ I → gI∗ I is a quasi-isomorphism, since gI∗ I is a K-injective complex and UI is noetherian separated. Note that gI∗ I is (gI )∗ acyclic simply because it is K-injective. On the other hand, since each term of qch gI∗ I is an injective object of Qch(UI ), it is also an injective object of Mod(UI ), see [17, (II.7)]. In particular, each term of qch gI∗ I is quasicoherent and (gI )∗ -acyclic. By [26, (3.9.3.5)], qch gI∗ I is (gI )∗ -acyclic. Hence ε : FX qch((gI )∗ gI∗ I) → (gI )∗ gI∗ I is a quasi-isomorphism, as desired. ⊓ ⊔

By the lemma, the following follows immediately. Corollary 14.4. Let X be a concentrated scheme. Then Qch(X) has arbitrary small direct products. This also follows from Corollary 11.7 and [37, Corollary 7.10]. (14.5) Let f : X → Y be a concentrated morphism of schemes. Then f∗Qch : Qch(X) → Qch(Y ) is deﬁned, and we have FY ◦ f∗Qch ∼ = f∗Mod ◦ FX , where FY and FX are the forgetful functors. Note that Rf∗ (DQch (X)) ⊂ DQch (Y ),

388

14 The Right Adjoint of the Derived Direct Image Functor

see [26, (3.9.2)]. If, moreover, X is concentrated, then there is a right derived functor Rf∗Qch of f∗Qch by Corollary 11.7. Lemma 14.6. Let f : X → Y be a morphism of schemes. Then, we have the following. 1 If X is noetherian or both Y and f are quasi-compact separated, then the canonical maps FY ◦ Rf∗Qch ∼ = R(FY ◦ f∗Qch ) ∼ = R(f∗Mod ◦ FX ) → Rf∗Mod ◦ FX are all isomorphisms. 2 Assume that both X and Y are either noetherian or quasi-compact separated. Then there are a left adjoint F of Rf∗Qch and an isomorphism ∗ FX F ∼ FY . = LfMod 3 Let X and Y be as in 2. Then there is an isomorphism Rf∗Qch R qch ∼ = R qch Rf∗Mod . Proof. We prove 1. It suﬃces to show that, if I ∈ K(Qch(X)) is K-injective, then FX I is f∗Mod -acyclic. By Corollary 11.7 and Lemma 3.9, we may assume that each term of I is injective. By [26, (3.9.3.5)] (applied to the plump subcategory A# = Qch(X) of Mod(X)), it suﬃces to show that an injective object I of Qch(X) is f∗Mod -acyclic. This is trivial if X is noetherian, since then I is injective in Mod(X), see [17, (II.7)]. Now assume that both Y and f are quasi-compact separated. Let g : Z → X be a faithfully ﬂat morphism such that Z is aﬃne. Such a morphism exists, as X is quasi-compact. Then it is easy to see that I is a direct summand of g∗ J for some injective object J in Qch(Z). So it suﬃces to show that FX g∗ J is f∗Mod -acyclic. Let FZ J → J be an injective resolution. As g is aﬃne (since X is separated and Z is aﬃne) and J is quasi-coherent, we have Ri g∗Mod FZ J = 0 for i > 0. Hence g∗ FZ J → g∗ J is a quasi-isomorphism, and hence is a K-limp resolution of FX g∗ J ∼ = g∗ FZ J . As f ◦ g is also aﬃne, f∗ g∗ FZ J → f∗ g∗ J is still a quasi-isomorphism, and this shows that FX g∗ J is f∗ -acyclic. ∗ 2 Deﬁne F : D(Qch(Y )) → D(Qch(X)) by F := R qch LfMod FY . Note ∗ that LfMod FY (D(Qch(Y ))) ⊂ DQch (X), see [26, (3.9.1)]. So we have via ε

∗ ∗ FY −−−→LfMod FY FX F = FX R qch LfMod

is an isomorphism by Lemma 14.3. Hence HomD(Qch(X)) (F F, G) ∼ = = HomD(X) (FX F F, FX G) ∼ ∗ ∼ HomD(Y ) (FY F, Rf Mod FX G) ∼ FY F, FX G) = HomD(X) (Lf = Mod

∗

HomD(Y ) (FY F, FY Rf∗Qch G) ∼ = HomD(Qch(Y )) (F, Rf∗Qch G). This shows that F is left adjoint to Rf∗Qch . 3 Take the conjugate of 2.

⊓ ⊔

14 The Right Adjoint of the Derived Direct Image Functor

389

Lemma 14.7. Let I be a small category, and f• : X• → Y• an affine morphism in P(I, Sch). Let F ∈ DLqc (X• ). Then R0 (f• )∗ F ∼ = (f• )∗ (H 0 (F)). Proof. Clearly, (f• )∗ (H 0 (F)) ∼ = R0 (f• )∗ (H 0 (F)). Since Ri (f• )∗ (τ>0 F) = 0 for i ≤ 0 is obvious, Ri (f• )∗ (τ≤0 F) ∼ = Ri (f• )∗ F for i ≤ 0. So it suﬃces to show that Ri (f• )∗ (τ<0 F) = 0 for i ≥ 0. To verify this, we may assume that f• = f : X → Y is a map of single schemes, and Y is aﬃne. By Lemma 14.3, we may assume that F = FX G for some D(Qch(X)). By Lemma 14.6, it suﬃces to show that Ri f∗Qch (τ<0 G) = 0 for i ≥ 0. But this is trivial, since f∗Qch is an exact functor. ⊓ ⊔ Corollary 14.8. Let I and f• be as in the lemma. If F ∈ DLqc (X• ), then Rn (f• )∗ F ∼ = (f• )∗ (H n (F)). (14.9) Let C be an additive category, and c ∈ C. We say that c is a compact object, if for any

small family of objects (tλ )λ∈Λ of C such that the coproduct (direct sum) λ∈Λ tλ exists, the canonical map

HomC (c, tλ ) → HomC (c,

tλ )

λ

λ

is an isomorphism. A triangulated category T is said to be compactly generated, if T has small coproducts, and there is a small set C of compact objects of T such that HomT (c, t) = 0 for all c ∈ C implies t = 0. The following was proved by A. Neeman [35]. Theorem 14.10. Let S be a compactly generated triangulated category, T any triangulated category, and F : S → T a triangulated functor. Suppose that F preserves coproducts, that is to say, family of objects (sλ )

for any small

of S, the canonical maps F (sλ ) → F ( λ sλ ) make F ( λ sλ ) the coproduct of F (sλ ). Then F has a right adjoint G : T → S. For the deﬁnition of triangulated category and triangulated functor, see [36]. Related to the theorem, we remark the following. Lemma 14.11 (Keller and Vossieck [22]). Let S and T be triangulated categories, and F : S → T a triangulated functor. If G is a right adjoint of F , then G is also a triangulated functor. Proof. By the opposite assertion of [29, (IV.1), Theorem 3], G is additive. Let φF : F Σ → ΣF be the canonical isomorphism. Then its inverse induces an isomorphism φ−1

F ψ : F Σ−1 = Σ−1 ΣF Σ−1 −− → Σ−1 F ΣΣ−1 = Σ−1 F.

Taking the conjugate of ψ, we get an isomorphism φG : GΣ ∼ = ΣG.

390

14 The Right Adjoint of the Derived Direct Image Functor a

b

c

Let A − →B− →C− → ΣA be a distinguished triangle in T . Then there exists some distinguished triangle of the form Ga

β

α

GA −−→ GB − →X− → Σ(GA). Then there exists some d : F X → C such that F GA

F Ga

/ F GB

a

/B

ε

Fα

/ FX

b

/C

ε

A

φ◦F β

/ Σ(F GA) Σε

d

/ ΣA

c

is a map of triangles. Then taking the adjoint, we get a commutative diagram GA

Ga

/ GB

Ga

/ GB

id

GA

α

/X

Gb

/ GC

id

β

/ ΣGA

(14.12)

δ

/ ΣGA,

φG ◦Gc

where δ is the adjoint of d, and the right most vertical arrow is the composite φF

u

φG

ε

ΣGA − → GF ΣGA −−→ GΣF GA − → GΣA −−→ ΣGA, which agrees with id, as can be seen easily. This induces a commutative diagram S(?, GA)

(Ga)∗

/ S(?, GB)

id

S(?, GA)

α∗

/ S(?, GB)

/ S(?, ΣGA) (ΣGa)∗ / S(?, ΣGB)

δ∗

id (Ga)∗

β∗

/ S(?, X)

(Gb)∗

id (φG ◦Gc)∗

/ S(?, GC)

id

/ S(?, ΣGA)

(ΣGa)∗

/ S(?, ΣGB).

The ﬁrst row is exact, since it comes from a distinguished triangle, see [17, Proposition I.1.1]. As the second row is isomorphic to the sequence a

b

c

(Σa)∗

∗ ∗ ∗ T (F ?, A) −→ T (F ?, B) −→ T (F ?, C) −→ T (F ?, ΣA) −−−−→ T (F ?, ΣB)

a

b

c

→B− →C − → ΣA is a distinguished triangle, it is also exact. By the and A − ﬁve lemma, δ∗ is an isomorphism. By Yoneda’s lemma, δ is an isomorphism. This shows that (14.12) is an isomorphism, and hence the second row of (14.12) is distinguished. This is what we wanted to show. ⊓ ⊔ Lemma 14.13. Let S and T be triangulated categories, and F : S → T a triangulated functor with a right adjoint G. If both S and T have t-structures

14 The Right Adjoint of the Derived Direct Image Functor

391

and F is way-out left (i.e., F (τ≤0 S) ⊂ τ≤d T for some d), then G is way-out right. Proof. For the deﬁnition of t-structures on triangulated categories, see [5]. Assuming that F (τ≤0 S) ⊂ τ≤d T , we show G(τ≥0 T ) ⊂ τ≥−d S. Let t ∈ τ≥0 T and s ∈ τ≤−d−1 S. Then since F s ∈ τ≤−1 T , we have ⊓ ⊔ S(s, Gt) ∼ = T (F s, t) = 0. By [5, (1.3.4)], Gt ∈ τ≥−d S. The following was proved by A. Neeman [35] for the quasi-compact separated case, and was proved generally by A. Bondal and M. van den Bergh [8]. Theorem 14.14. Let X be a concentrated scheme. Then c ∈ DQch (X) is a compact object if and only if c is isomorphic to a perfect complex, where we say that C ∈ C(Qch(X)) is perfect if C is bounded, and each term of C is locally free of ﬁnite rank. Moreover, DQch (X) is compactly generated. Lemma 14.15. Let f : X → Y be a concentrated morphism of schemes. Then Rf∗Mod : DQch (X) → D(Y ) preserves coproducts. Proof. See [35] or [26, (3.9.3.2), Remark (b)].

⊓ ⊔

(14.16) Let f : X → Y be a concentrated morphism of schemes such that X is concentrated. Then by Theorem 14.10, Theorem 14.14 and Lemma 14.15, there is a right adjoint f × : D(Y ) → DQch (X) of Rf∗Mod . By restriction, f × : DQch (Y ) → DQch (X) is a right adjoint of Rf∗Mod : DQch (X) → DQch (Y ). Note that (R(?)∗ , (?)× ) is an adjoint pair of Δ-pseudofunctors on the opposite of the category of concentrated schemes. In other words, (R(?)∗ , (?)× ) is an opposite adjoint pair (of Δ-pseudofunctors) on the category of concentrated schemes, see (1.18).

Chapter 15

Comparison of Local Ext Sheaves

(15.1) Let S be a scheme, and X• an almost-S-groupoid. Assume that d0 (1) and d1 (1) are aﬃne, and X0 is locally noetherian. − Lemma 15.2. Let F ∈ KCoh (Qch(X• )) and G ∈ K + (Qch(X• )). If G is a bounded below complex consisting of injective objects of Qch(X• ), then G is Hom•OX• (F, ?)-acyclic as a complex of Mod(X• ).

Proof. It is easy to see that we may assume that F is a single coherent sheaf, and G is a single injective object of Qch(X• ). To prove this case, it suﬃces to show that ExtiOX• (F, G) = 0 for i > 0. Set X•′ := Nerve(d1 (1))|∆M , and let p• : X•′ → X• be a cartesian morphism such that p0 : X1 → X0 is d0 (1), see (12.13). In particular, p• is aﬃne and faithfully ﬂat. Let A : Mod(X0 ) → Mod(X•′ ) be the ascent functor, and D : Mod(X•′ ) → Mod(X0 ) be the descent functor. As (p• )∗ : Qch(X•′ ) → Qch(X• ) has a faithful exact left adjoint p∗• , there exists some injective object I of Qch(X•′ ) such that G is a direct summand of (p• )∗ I. We may assume that G = (p• )∗ I. As R Hom•OX• (F, R(p• )∗ I) ∼ = R(p• )∗ R Hom•OX ′ (p∗• F, I), •

R Hom•OX ′

(p∗• F, I) has quasi-coherent cohop• is aﬃne by assumption, and • mology groups, we may assume that X• = Nerve(f )|∆M for some faithfully ﬂat aﬃne morphism f : X → Y between S-schemes with Y locally noetherian (but we may lose the assumption that X0 is locally noetherian). For each l, we have that (?)l R Hom•OX• (F, G) ∼ = R Hom•OXl ((?)l ADF, (?)l ADG) ∼ = • ∗ ∗ ∗ ∼ R HomOX (e(l) DF, e(l) DG) = e(l) R Hom•OY (DF, DG) l

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

393

394

15 Comparison of Local Ext Sheaves

by [17, (II.5.8)] (note that DF is coherent). As D : Qch(X• ) → Qch(Y ) is an equivalence, DG is an injective object of Qch(Y ). Hence it is also an injective object of Mod(Y ) by [17, (II.7)]. Hence ExtiOY (DF, DG) = 0 for i > 0, as desired. ⊓ ⊔ The following is a generalization of [19, Theorem II.1.1.12]. Corollary 15.3. Let the notation be as in the lemma. Then G0 is Hom•OX (F0 , ?)-acyclic as a complex of OX0 -modules. 0

Proof. Let G → I be a K-injective resolution in Mod(X• ) such that I is bounded below. Let C be the mapping cone of this. Since (?)0 has an exact left adjoint, G0 → I0 is a K-injective resolution in K(Mod(X0 )). So it suﬃces to show that Hom•OX (F0 , C0 ) is exact. As each term of F is equivariant, this 0 complex is isomorphic to Hom•OX• (F, C)0 , which is exact by the lemma. ⊓ ⊔

Chapter 16

The Composition of Two Almost-Pseudofunctors

Deﬁnition 16.1. We say that C = (A, F, P, I, D, D+ , (?)# , (?)♭ , ζ) is a composition data of contravariant almost-pseudofunctors if the following eighteen conditions are satisﬁed: 1 2 3 4 5 6 7

A is a category with ﬁber products. P and I are sets of morphisms of A. Any isomorphism in A is in P ∩ I. The composite of two morphisms in P is again a morphism in P. The composite of two morphisms in I is again a morphism in I. A base change of a morphism in P is again a morphism in P. Any f ∈ Mor(A) admits a factorization f = pi such that p ∈ P and i ∈ I.

Before stating the remaining conditions, we give some deﬁnitions for convenience. i Let C be a set of morphisms in A containing all identity maps and being closed under compositions. We deﬁne AC by ob(AC ) := ob(A) and Mor(AC ) := C. In particular, the subcategories AP and AI of A are deﬁned. ii We call a commutative diagram of the form p ◦ i = i′ ◦ p′ with p, p′ ∈ P and i, i′ ∈ I a pi-square. We denote the set of all pi-squares by Π. 8 D = (D(X))X∈ob(A) is a family of categories. 9 (?)# is a contravariant almost-pseudofunctor on AP , (?)♭ is a contravariant almost-pseudofunctor on AI , and we have X # = X ♭ = D(X) for each X ∈ ob(A). 10 ζ = (ζ(σ))σ=(pi=jq)∈Π is a family of natural transformations ζ(σ) : i♭ p# → q # j ♭ .

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

395

396

16 The Composition of Two Almost-Pseudofunctors

11 If j1

i

j

i

1 U1 −→ V1 − → X1 ′ ↓ pU σ ↓ pV σ ↓ pX

U − → V − → X is a commutative diagram in A such that σ, σ ′ ∈ Π, then the composite map d−1

ζ(σ ′ )

ζ(σ)

d

♭ ♭ # ♭ # ♭ ♭ ♭ ♭ (i1 j1 )♭ p# −−→p# → p# X −−→ j1 i1 pX −−→j1 pV i − Uj i − U (ij)

agrees with ζ(σ ′ σ), where σ ′ σ is the pi-square pX (i1 j1 ) = (ij)pU . 12 For any morphism p : X → Y in P, the composite f−1

ζ(p1X =1Y p)

f

→ p# p# −−→ 1♭X p# −−−−−−−−→ p# 1♭Y − is the identity. 13 If i

2 U2 − → X2 ↓ qU σ ′ ↓ qX

i

1 U1 − → X1 ↓ pU σ ↓ pX i U − → X

is a commutative diagram in A such that σ, σ ′ ∈ Π, then the composite ζ(σ ′ )

d−1

ζ(σ)

d

# # # ♭ # # # ♭ i♭2 (pX qX )# −−→ i♭2 qX pX −−−→qU i1 pX −−→qU pU i − → (pU qU )# i♭

agrees with ζ((pX qX )i2 = i(pU qU )). 14 For any morphism i : U → X in I, the composite f−1

ζ

f

♭ → 1# → i♭ i♭ −−→ i♭ 1# X − Ui −

is the identity. 15 F is a subcategory of A, and any isomorphism in A between objects of F is in Mor(F). 16 D+ = (D+ (X))X∈ob(F ) is a family of categories such that D+ (X) is a full subcategory of D(X) for each X ∈ ob(F). 17 If f : X → Y is a morphism in F, f = p ◦ i, p ∈ P and i ∈ I, then we have i♭ p! (D+ (Y )) ⊂ D+ (X). 18 If j i1 V − → U1 − → X1 ↓ pU σ ↓ pX q i U − → X − →Y

16 The Composition of Two Almost-Pseudofunctors

397

is a diagram in A such that σ ∈ Π, V, U, Y ∈ ob(F), pU j ∈ Mor(F) and qi ∈ Mor(F), then # ♭ # ♭ # j ♭ ζ(σ)q # : j ♭ i♭1 p# X q → j pU i q

is an isomorphism between functors from D+ (Y ) to D+ (V ). (16.2) Let C = (A, F, P, I, D, D+ , (?)# , (?)♭ , ζ) be a composition data of contravariant almost-pseudofunctors. We call a commutative diagram of the form f = pi with p ∈ P, i ∈ I and f ∈ Mor(A) a compactiﬁcation. We call a commutative diagram of the form pi = qj with p, q ∈ P, i, j ∈ I and pi = qj ∈ Mor(F) an independence diagram. Lemma 16.3. Let

i

1 U − → X1 ↓ i τ ↓ p1 p X− → Y

be an independence diagram. Then the following hold: 1 There is a diagram of the form j

q1

U − → Z −→ X1 ↓q ↓ p1 p X − → Y such that qj = i, q1 j = i1 , pq = p1 q1 , q, q1 ∈ P, and j ∈ I. 2 ζ(qj = i1U )p# : j ♭ q # p# → i♭ p# is an isomorphism between functors from D+ (Y ) to D+ (U ). + 3 ζ(q1 j1 = i1 1U )p# 1 is also an isomorphism between functors from D (Y ) + to D (U ). 4 The composite isomorphism f−1

(ζ(qj=i1U )p# )−1

♭ # Υ(τ ) : i♭ p# −−→ 1# −−−−−−−−−−→j ♭ q # p# Ui p − d

ζ(q1 j1 =i1 1U )p#

f

# ♭ # 1 − → j ♭ q1# p# → i♭1 p# 1 −−−−−−−−−−→1U i1 p1 − 1

(between functors deﬁned over D+ (Y ), not over D(Y )) depends only on τ . 5 If τ ′ = (p1 i1 = p2 i2 ) is an independence diagram, then we have Υ(τ ′ ) ◦ Υ(τ ) = Υ(pi = p2 i2 ). The proof is left to the reader. We call Υ(τ ) the independence isomorphism of τ . (16.4) Any f ∈ Mor(A) has a compactiﬁcation by assumption. We ﬁx a family of compactiﬁcations T := (τ (f ) : (f = p(f ) ◦ i(f )))f ∈Mor(A) .

398

16 The Composition of Two Almost-Pseudofunctors

For X ∈ ob(F), we deﬁne X ! := D+ (X). For a morphism f : X → Y in F, we deﬁne f ! := i(f )♭ p(f )# , which is a functor from Y ! to X ! by assumption. Let f : X → Y and g : Y → Z be morphisms in F. Let i(g)p(f ) = qj be a compactiﬁcation of i(g)p(f ). Then by 18 in Deﬁnition 16.1 and Lemma 16.3, the composite map Υ(p(gf )i(gf )=(p(g)q)(ji(f )))

(gf )! = i(gf )♭ p(gf )# −−−−−−−−−−−−−−−−−−→(j ◦ i(f ))♭ (p(g) ◦ q)# ♭

#

i(f ) ζ(qj=i(g)p(f ))p(g) ∼ = i(f )♭ j ♭ q # p(g)# −−−−−−−−−−−−−−−−→i(f )♭ p(f )# i(g)♭ p(g)# = f ! g !

is an isomorphism. We deﬁne df,g : f ! g ! → (gf )! to be the inverse of this composite. Lemma 16.5. The deﬁnition of df,g is independent of choice of q and j above. The proof of the lemma is left to the reader. For X ∈ ob(F), we deﬁne fX : id!X → IdX ! to be the composite ζ

f

f

♭ id!X = i(idX )♭ p(idX )# − → IdX ! . → id♭X − → id# X idX −

Proposition 16.6. Let the notation be as above. 1 (?)! together with (df,g ) and (fX ) form a contravariant almost-pseudofunctor on F. 2 For j ∈ I ∩ Mor(F), deﬁne ψ : j ! → j ♭ to be the composite Υ

f

j ! = i(j)♭ p(j)# − → j ♭ id# − → j♭. Then ψ : (?)! → (?)♭ is an isomorphism of contravariant almostpseudofunctors on AI ∩ F. 3 For q ∈ P ∩ Mor(F), deﬁne ψ : q ! → q # to be the composite Υ

f

→ q# . → id♭ q # − q ! = i(q)♭ p(q)# − Then ψ : (?)! → (?)# is an isomorphism of contravariant almostpseudofunctors on AP ∩ F. 4 Let us take another family of compactiﬁcations (f = p1 (f )i1 (f ))f ∈Mor(F ) , and let (?)⋆ be the resulting contravariant almost-pseudofunctor deﬁned by f ⋆ = i1 (f )♭ p1 (f )# . Then Υ : f ! → f ⋆ induces an isomorphism of contravariant almost-pseudofunctors (?)! ∼ = (?)⋆ . The proof is left to the reader. We call (?)! the composite of (?)# and (?)♭ . The composite is uniquely deﬁned up to isomorphisms of almostpseudofunctors on F. The discussion above has an obvious triangulated version. Composition data of contravariant triangulated almost-pseudofunctors

16 The Composition of Two Almost-Pseudofunctors

399

are deﬁned appropriately, and the composition of two contravariant triangulated almost-pseudofunctors is obtained as a contravariant triangulated almost-pseudofunctor. (16.7) Let S be a scheme. Let A be the category whose objects are noetherian S-schemes and morphisms are morphisms separated of ﬁnite type. Set F = A. Let I be the class of open immersions. Let P be the class of proper + (X) for X ∈ A. Let (?)♭ := (?)∗ , morphisms. Set D(X) = D+ (X) = DQch the (derived) inverse image almost-pseudofunctor for morphisms in I, where X ♭ := D(X). Let (?)# := (?)× , the twisted inverse almost-pseudofunctor (see [26, Chapter 4]) for morphisms in P, where X # := D(X) again. Note that the left adjoint R(?)∗ is way-out left for morphisms in P so that (?)× is way-out right by Lemma 14.13, and (?)# is well-deﬁned. The conditions 1–9 in Deﬁnition 16.1 hold. Note that 7 is nothing but Nagata’s compactiﬁcation theorem [34]. The conditions 15–17 are trivial. Let σ0 : pi = jq be a pi-diagram, which is also a ﬁber square. Then the canonical map θ : j ∗ (Rp∗ ) → (Rq∗ )i∗ is an isomorphism of triangulated functors, see [26, (3.9.5)]. Hence, taking the inverse of the conjugate, we have an isomorphism ξ = ξ(σ0 ) : (Ri∗ )q × ∼ = p× (Rj∗ ).

(16.8)

So we have a morphism of triangulated functors via u

via ξ −1

via ε

ζ0 (σ0 ) : i∗ p× −−−→i∗ p× (Rj∗ )j ∗ −−−−−→i∗ (Ri∗ )q × j ∗ −−−→q × j ∗ , which is an isomorphism, see [41]. The statements 10, 12 and 14, and corresponding statements to 11, 13 only for ﬁber square pi-diagrams, are readily proved. In particular, for a closed open immersion η : U → X, we have an isomorphism f−1

ζ0 (η1U =η1U )−1

f

∗ ∗ × v(η) : η ∗ −−→ 1× → η× . U η −−−−−−−−−−→ 1U η −

Let σ = (pi = qj) be an arbitrary pi-diagram. Let j1 be the base change of j by p, and let p1 be the base change of p by j. Let η be the unique morphism such that q = p1 η and i = j1 η. Note that η is a closed open immersion. Deﬁne ζ(σ) to be the composite isomorphism via ζ0 ∗ via v(η) × × ∗ ∼ × ∗ −−−−→η p1 j = q j . i∗ p× ∼ = η ∗ j1∗ p× −−−−→η ∗ p× 1j −

Now the proof of conditions 11, 13 consists in diagram chasing arguments, while 18 is trivial, since ζ(σ) is always an isomorphism. Thus the twisted inverse triangulated almost-pseudofunctor (?)! on A is deﬁned to be the composite of (?)× and (?)∗ .

Chapter 17

The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams

Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Lemma 17.1. Assume that X• is concentrated. That is, Xi is concentrated for each i ∈ I. Let Ci be a small set of compact generators of DQch (Xi ), which exists by Theorem 14.14. Then C := {LLi c | i ∈ I, c ∈ Ci } is a small set of compact generators of DLqc (X• ). In particular, the category DLqc (X• ) is compactly generated. Proof. Let t ∈ DLqc (X• ) and assume that HomD(X• ) (LLi c, t) ∼ = HomD(Xi ) (c, ti ) = 0 for any i ∈ I and any c ∈ Ci . Then, ti = 0 for all i. This shows t = 0. It is easy to see that LLi c is compact, and C is small. So C is a small set of compact generators. As DLqc (X• ) has coproducts, it is compactly generated. ⊓ ⊔ Lemma 17.2. Let f• : X• → Y• be a concentrated morphism in P(I, Sch/S). Then R(f• )∗ : DLqc (X• ) → DLqc (Y• ) preserves coproducts. Proof. Let (tλ ) be a small family of objects in DLqc (X• ), and consider the canonical map R(f• )∗ tλ → R(f• )∗ ( tλ ). λ

λ

For each i ∈ I, apply (?)i to the map. As (?)i obviously preserves coproducts and we have a canonical isomorphism (?)i R(f• )∗ ∼ = R(fi )∗ (?)i , J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

401

402

17 The Right Adjoint of the Derived Direct Image for Diagrams

the result is the canonical map R(fi )∗ (tλ )i → R(fi )∗ ( (tλ )i ), λ

λ

which is an isomorphism by Lemma 14.15. Hence, R(f• )∗ preserves coproducts. ⊓ ⊔ By Theorem 14.10, we have the ﬁrst (original) theorem of these notes: Theorem 17.3. Let I be a small category, S a scheme, and f• : X• → Y• a morphism in P(I, Sch/S). If X• and f• are concentrated, then R(f• )∗ : DLqc (X• ) → DLqc (Y• ) has a right adjoint f•× . (17.4) Let I be a small category. Let S be the category of concentrated I op diagrams of schemes. Note that any morphism of S is concentrated (follows easily from [15, (1.2.3), (1.2.4)]). For X• ∈ S, set R(X• )∗ := DLqc (X• ). For a morphism f• of S, set R(f• )∗ be the derived direct image. Then R(?)∗ is a covariant almost-pseudofunctor on S. Thus its right adjoint (?)× is a contravariant almost-pseudofunctor on S, and (R(?)∗ , (?)× ) is an opposite adjoint pair of Δ-pseudofunctors on S, see (1.18). For composable morphisms f• and g• , df• ,g• : f•× g•× → (g• f• )× is the composite u

c

ε

f•× g•× − → (g• f• )× R(g• f• )∗ f•× g•× − → (g• f• )× R(g• )∗ R(f• )∗ f•× g•× − → ε

→ (g• f• )× . (g• f• )× R(g• )∗ g•× − For X• ∈ S, f : id× X• → IdX•× is the composite e

ε

id× → R(idX• )∗ id× → Id . X• − X• − Lemma 17.5. Let S be a scheme, I a small category, and f• : X• → Y• and g• : Y•′ → Y• be morphisms in P(I, Sch/S). Set X•′ := Y•′ ×Y• X• . Let f•′ : X•′ → Y•′ be the ﬁrst projection, and g•′ : X•′ → X• the second projection. Assume that f• is concentrated, and g• is ﬂat. Then the canonical map θ(g• , f• ) : (g• )∗ R(f• )∗ → R(f•′ )∗ (g•′ )∗ is an isomorphism of functors from DLqc (X• ) to DLqc (Y•′ ). Proof. It suﬃces to show that for each i ∈ I, (?)i θ : (?)i (g• )∗ R(f• )∗ → (?)i R(f•′ )∗ (g•′ )∗ is an isomorphism. By Lemma 1.22, it is easy to verify that the diagram

17 The Right Adjoint of the Derived Direct Image for Diagrams c−1

gi∗ R(fi )∗ (?)i −−→ gi∗ (?)i R(f• )∗ ↓ θ(gi , fi ) R(fi′ )∗ (gi′ )∗ (?)i

θ

− →

θ

− →

c−1 R(fi′ )∗ (?)i (g•′ )∗ −−→

(?)i g•∗ R(f• )∗ ↓ (?)i θ(g• , f• )

403

(17.6)

(?)i R(f•′ )∗ (g•′ )∗

is commutative. As the horizontal maps and θ(gi , fi ) are isomorphisms by ⊓ ⊔ [26, (3.9.5)], (?)i θ(g• , f• ) is also an isomorphism.

Chapter 18

Commutativity of Twisted Inverse with Restrictions

(18.1) Let S be a scheme, I a small category, and f• : X• → Y• a morphism in P(I, Sch/S). Let J be an admissible subcategory of I. Assume that f• is concentrated. Then there is a natural map θ(J, f• ) : LLJ ◦ R(f• |J )∗ → R(f• )∗ ◦ LLJ

(18.2)

between functors from DLqc (X• |J ) to DLqc (Y• ), see [26, (3.7.2)]. (18.3) Let S, I and f• be as in (18.1). We assume that X• and f• are concentrated, so that the right adjoint functor f•× : DLqc (Y• ) → DLqc (X• ) of R(f• )∗ : DLqc (X• ) → DLqc (Y• ) exists. Let J be a subcategory of I which may not be admissible. We deﬁne the natural transformation ξ(J, f• ) : (?)J ◦ f•× → (f• |J )× ◦ (?)J to be the composite u

c−1

ε

(?)J f•× − →(f• |J )× R(f• |J )∗ (?)J f•× −−→(f• |J )× (?)J R(f• )∗ f•× − →(f• |J )× (?)J . By deﬁnition, ξ is the conjugate map of θ(J, f• ) in (18.2) if J is admissible. Do not confuse ξ(J, f• ) with ξ(f• , J) (see Corollary 6.26). Lemma 18.4. Let J2 ⊂ J1 ⊂ I be subcategories of I. Let S be a scheme, f• : X• → Y• be a morphism in P(I, Sch/S). Assume both X• and f• are concentrated. Then the composite map

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

405

406

18 Commutativity of Twisted Inverse with Restrictions ξ(J1 ,f• )

c

→ (?)J2 (?)J1 f•× −−−−−→(?)J2 (f• |J1 )× (?)J1 (?)J2 f•× − ξ(J2 ,f• |J )

c−1

1 −−−−−−− →(f• |J2 )× (?)J2 (?)J1 −−→ (f• |J2 )× (?)J2

is equal to ξ(J2 , f• ). Proof. Straightforward (and tedious) diagram drawing.

⊓ ⊔

Lemma 18.5. Let S and f• : X• → Y• be as in Lemma 18.4. Let J be a subcategory of I. Assume that Y• has ﬂat arrows and f• is cartesian. Then ξ(J, f• ) : (?)J f•× → (f• |J )× (?)J is an isomorphism between functors from DLqc (Y• ) to DLqc (X• |J ). Proof. In view of Lemma 18.4, we may assume that J = i for an object i of I. Then, as i is an admissible subcategory of I and ξ(i, f• ) is a conjugate map of θ(i, f• ), it suﬃces to show that (?)j θ(i, f• ) is an isomorphism for any j ∈ ob(I). As Y• has ﬂat arrows, Li : Mod(Yi ) → Mod(Y• ) is exact. As f• is cartesian, Li : Mod(Xi ) → Mod(X• ) is also exact. By Proposition 6.23, the composite (?)j θ

c

(?)j Li R(fi )∗ −−−→(?)j R(f• )∗ Li − →R(fj )∗ (?)j Li agrees with the composite λi,j

(?)j Li R(fi )∗ −−→

⊕θ

Yφ∗ R(fi )∗ −−→

φ∈I(i,j)

R(fj )∗ Xφ∗

φ

⎛ ⎞ λ−1 i,j C Xφ∗ ⎠ −−→R(fj )∗ (?)j Li , − →R(fj )∗ ⎝ φ

where C is the canonical map. By Lemma 14.15, C is an isomorphism. As f• is cartesian and Y• has ﬂat arrows, θ : Yφ∗ R(fi )∗ → R(fj )∗ Xφ∗ is an isomorphism for each φ ∈ I(i, j) by [26, (3.9.5)]. Hence the second composite is an isomorphism. As the ﬁrst composite is an isomorphism and c is also an ⊓ ⊔ isomorphism, we have that (?)j θ(i, f• ) is an isomorphism. (18.6) Let I be a small category, S a scheme, and f• : X• → Y• a morphism in P(I, Sch/S). Assume that X• and f• are concentrated. Lemma 18.7. Let J be a subcategory of I. Then the following hold: 1 The composite map u

ξ(J,f• )

cJ,f

• × (?)J − →(?)J f•× R(f• )∗ −−−−→(f• |J )× (?)J R(f• )∗ −−−→(f • |J ) R(f• |J )∗ (?)J

agrees with u.

18 Commutativity of Twisted Inverse with Restrictions

407

2 The composite map cJ,f

ξ(J,f• )

ε

• × × (?)J R(f• )∗ f•× −−−→R(f →(?)J • |J )∗ (?)J f• −−−−→R(f• |J )∗ (f• |J ) (?)J −

agrees with ε. Proof. The proof consists in straightforward diagram drawings.

⊓ ⊔

(18.8) Let I be a small category. For i, j ∈ ob(I), we say that i ≤ j if I(i, j) = ∅. This deﬁnition makes ob(I) a pseudo-ordered set. We say that I is ordered if ob(I) is an ordered set with the pseudo-order structure above, and I(i, i) = {id} for i ∈ I. Lemma 18.9. Let I be an ordered small category. Let J0 and J1 be full subcategories of I, such that ob(J0 ) ∪ ob(J1 ) = ob(I), ob(J0 ) ∩ ob(J1 ) = ∅, and I(j1 , j0 ) = ∅ for j1 ∈ J1 and j0 ∈ J0 . Let X• ∈ P(I, Sch/S). Then, we have the following. 1 2 3 4 5 6 7 8 9

The unit of adjunction u : IdMod(X• |J1 ) → (?)J1 ◦ LJ1 is an isomorphism. (?)J0 ◦ LJ1 is zero. LJ1 is exact, and J1 is an admissible subcategory of I. For M ∈ Mod(X• ), MJ0 = 0 if and only if ε : LJ1 MJ1 → M is an isomorphism. The counit of adjunction (?)J0 ◦ RJ0 → IdMod(X• |J0 ) is an isomorphism. (?)J1 ◦ RJ0 is zero. RJ0 is exact and preserves local-quasi-coherence. For M ∈ Mod(X• ), MJ1 = 0 if and only if u : M → RJ0 MJ0 is an isomorphism. The sequence ε u → Id − →RJ0 (?)J0 → 0 0 → LJ1 (?)J1 − is exact, and induces a distinguished triangle in D(X• ).

Proof. 1 This is obvious by Lemma 6.15. (J op ֒→I op ) 2 The category Ij0 1 is empty, if j0 ∈ J0 , since I op (j0 , j1 ) = ∅ if j1 ∈ J1 and j0 ∈ J0 . It follows that (?)j0 ◦ LJ1 = 0 if j0 ∈ J0 . Hence, (?)J0 ◦ LJ1 = 0. 3 This is trivial by 1,2 and their proof. 4 The ‘if’ part is trivial by 2. We prove the ‘only if’ part. By assumption and 2, both MJ0 and (?)J0 LJ1 MJ1 are zero, and (?)J0 ε is an isomorphism. On the other hand, ((?)J1 ε)(u(?)J1 ) = id, and u is an isomorphism by 1. Hence, (?)J1 ε : (?)J1 LJ1 MJ1 → (?)J1 M is an isomorphism. Hence, ε is an isomorphism. The assertions 5,6,7,8,9 are similar, and we omit the proof.

⊓ ⊔

408

18 Commutativity of Twisted Inverse with Restrictions

Lemma 18.10. Let I, S, J1 and J0 be as in Lemma 18.9. Let f• : X• → Y• be a morphism in P(I, Sch/S). Assume that X• and f• are concentrated. Then we have that θ(J1 , f• ) and ξ(J1 , f• ) are isomorphisms. Proof. Note that J1 is admissible by Lemma 18.9, 3, and hence θ(J1 , f• ) and ξ(J1 , f• ) are deﬁned. Since we have ξ(J1 , f• ) is the conjugate of θ(J1 , f• ) by deﬁnition, it suﬃces to show that θ(J1 , f• ) is an isomorphism. It suﬃces to show that (?)i θ (?)i LLJ1 R(f• |J1 )∗ −−−→(?)i R(f• )∗ LLJ1 ∼ = R(fi )∗ (?)i LLJ1

is an isomorphism for any i ∈ I. If i ∈ J0 , then both hand sides are zero functors, and it is an isomorphism. On the other hand, if i ∈ J1 , then the map in question is equal to the composite isomorphism ci,f

|

• J1 (?)i LLJ1 R(f• |J1 )∗ ∼ = (?)i R(f• |J1 )∗ −−−−−→R(fi )∗ (?)i ∼ = R(fi )∗ (?)i LLJ1

by Proposition 6.23. Hence θ(J1 , f• ) is an isomorphism, as desired.

⊓ ⊔

(18.11) Let S be a scheme, I an ordered small category, i ∈ I, and X• ∈ P(I, Sch/S). Let J1 be a ﬁlter of ob(I) such that i is a minimal element of J1 (e.g., [i, ∞)), and set Γi := LI,J1 ◦ RJ1 ,i . Then we have (?)j Γi = 0 if j = i and (?)i Γi = Id. Hence Γi does not depend on the choice of J1 , and depends only on i. Note that Γi preserves arbitrary limits and colimits (hence is exact). Assume that Xi is concentrated. Then DQch (Xi ) is compactly generated, and the derived functor Γi : DQch (Xi ) → DLqc (X• ) preserves coproducts. It follows that there is a right adjoint Σi : DLqc (X• ) → DQch (Xi ). As Γi is obviously way-out left, we have Σi is way-out right by Lemma 14.13. (18.12) Let S be a scheme, I a small category, and X• ∈ P(I, Sch/S). We deﬁne D+ (X• ) (resp. D− (X• )) to be the full subcategory of D(X• ) consisting of F ∈ D(X• ) such that Fi is bounded below (resp. above) and has quasicoherent cohomology groups for each i ∈ I. For a plump full subcategory A of Lqc(X• ), we denote the triangulated subcategory of D+ (X• ) (resp. D− (X• )) + consisting of objects all of whose cohomology groups belong to A by DA (X• ) − (resp. DA (X• )). (18.13) Let P be an ordered set. We say that P is upper Jordan-Dedekind (UJD for short) if for any p ∈ P , the subset [p, ∞) := {q ∈ P | q ≥ p}

18 Commutativity of Twisted Inverse with Restrictions

409

is ﬁnite. We say that an ordered small category I is UJD if the ordered set ob(I) is UJD, and I(i, j) is ﬁnite for i, j ∈ I. Proposition 18.14. Let I be an ordered UJD small category. Let S be a scheme, and g• : U• → X• and f• : X• → Y• be morphisms in P(I, Sch/S). Assume that Y• is noetherian with ﬂat arrows, f• is proper, g• is an open immersion such that gi (Ui ) is dense in Xi for each i ∈ I, and f• ◦ g• is cartesian. Then g• is cartesian, and for any i ∈ I the composite natural map via θ −1

via ξ(i)

(?)i g•∗ f•× −−−−−→gi∗ (?)i f•× −−−−−→gi∗ fi× (?)i is an isomorphism between functors D+ (Y• ) → DQch (Ui ), where θ : gi∗ (?)i → (?)i g•∗ is the canonical isomorphism. Proof. Note that U• has ﬂat arrows, since Y• has ﬂat arrows and f• ◦ g• is cartesian. We prove that g• is cartesian. Let φ : i → j be a morphism in I. Then, the canonical map (Uφ , fj gj ) : Uj → Ui ×Yi Yj is an isomorphism by assumption. This map factors through (Uφ , gj ) : Uj → Ui ×Xi Xj , and it is easy to see that (Uφ , gj ) is a closed immersion. On the other hand, it is an image dense open immersion, as can be seen easily, and hence it is an isomorphism. So g• is cartesian. Set J1 := [i, ∞) and J0 := ob(I) \ J1 . By Lemma 18.10, ξ(J1 , f• ) is an isomorphism. By Lemma 18.4, we may replace I by J1 , and we may assume that I is an ordered ﬁnite category, and i is a minimal element of ob(I). Now + (Y• ). Since we have ob(I) is ﬁnite, it is easy we have D+ (Y• ) agrees with DLqc to see that R(f• )∗ is way-out in both directions. It follows that f•× is way-out right by Lemma 14.13. It suﬃces to show that gi∗ ξ(i) : gi∗ (?)i f•× → gi∗ fi× (?)i + + is an isomorphism of functors from DLqc (Y• ) to DQch (Ui ). As gi∗ R(gi )∗ ∼ = Id, ∗ it suﬃces to show that R(gi )∗ gi ξ(i) is an isomorphism. This is equivalent to say that for any perfect complex P ∈ C(Qch(Xi )), we have

R(gi )∗ gi∗ ξ(i) : HomD(Xi ) (P, R(gi )∗ gi∗ (?)i f•× ) → HomD(Xi ) (P, R(gi )∗ gi∗ fi× (?)i ) is an isomorphism. By [41, Lemma 2], this is equivalent to say that the canonical map n lim HomD(Y• ) (R(f• )∗ LLi (P ⊗•,L OXi J ), ?) −→ n → lim HomD(Y• ) (LLi R(fi )∗ (P ⊗•,L OXi J ), ?) −→

410

18 Commutativity of Twisted Inverse with Restrictions

induced by the conjugate θ(i, f• ) of ξ(i) is an isomorphism, where J is a deﬁning ideal sheaf of the closed subset Xi \ Ui in Xi . As I is ordered and ob(I) is ﬁnite, we may label ob(I) = {i = i(0), i(1), i(2), . . .} so that I(i(s), i(t)) = ∅ implies that s ≤ t. Let J(r) denote the full subcategory of I whose object set is {i(r), i(r + 1), . . .}. By descending induction on t, we prove that the map n via θ(i, f• ) : lim HomD(Y• ) (LJ(t) (?)J(t) R(f• )∗ LLi (P ⊗•,L OXi J ), ?) −→ n → lim HomD(Y• ) (LJ(t) (?)J(t) LLi R(fi )∗ (P ⊗•,L OXi J ), ?) −→

is an isomorphism. This is enough to prove the proposition, since LJ(1) (?)J(1) = Id. Since the sequence via ε

0 → LJ(t+1) (?)J(t+1) −−−→LJ(t) (?)J(t) → Γi(t) (?)i(t) → 0 is an exact sequence of exact functors, it suﬃces to prove that the map n via θ(i, f• ) : lim HomD(Y• ) (Γi(t) (?)i(t) R(f• )∗ LLi (P ⊗•,L OXi J ), ?) −→ n → lim HomD(Y• ) (Γi(t) (?)i(t) LLi R(fi )∗ (P ⊗•,L OXi J ), ?) −→

is an isomorphism by induction assumption and the ﬁve lemma. By Proposition 6.23, this is equivalent to say that the map via θ : lim HomD(Yi(t) ) (R(fi(t) )∗ −→

n LXφ∗ (P ⊗•,L OX J ), Σi (?)) i

φ

n → lim HomD(Yi(t) ) ( Yφ∗ R(fi )∗ (P ⊗•,L OXi J ), Σi (?)) −→ φ

is an isomorphism, where the sum is taken over the ﬁnite set I(i, i(t)). It suﬃces to prove that the map × n via ξ : lim HomD(Xi ) (P ⊗•,L OXi J , R(Xφ )∗ fi(t) Σi (?)) −→ × n → lim HomD(Xi ) (P ⊗•,L OXi J , fi R(Yφ )∗ Σi (?)) −→ × induced by the map ξ : R(Xφ )∗ fi(t) → fi× R(Yφ )∗ , which is conjugate to

θ : Yφ∗ R(fi )∗ → R(fi(t) )∗ LXφ∗ ,

18 Commutativity of Twisted Inverse with Restrictions

411

× Σi , and fi× R(Yφ )∗ Σi is an isomorphism for φ ∈ I(i, i(t)). Since Σi , R(Xφ )∗ fi(t) are way-out right, it suﬃces to show the canonical map × gi∗ ξ(Yφ fi(t) = fi Xφ ) : gi∗ R(Xφ )∗ fi(t) → gi∗ fi× R(Yφ )∗ + + (Ui ). Let X ′ := (Yi(t) ) to DQch is an isomorphism between functors from DQch ′ ′ Xi ×Yi Yi(t) , p1 : X → Xi be the ﬁrst projection, p2 : X → Yi(t) the second projection, and π : Xi(t) → X ′ be the map (Xφ , fi(t) ). It is easy to see that ξ(Yφ fi(t) = fi Xφ ) equals the composite map ε × ∼ ∼ × →R(p1 )∗ p× R(Xφ )∗ fi(t) = R(p1 )∗ Rπ∗ π × p× 2 = fi R(Yφ )∗ . 2−

Note that the last map is an isomorphism since Yφ is ﬂat. As we have Ui(t) → Ui ×Yi Yi(t) ∼ = Ui ×Xi X ′ is an isomorphism and gi is an open immersion by assumption, the canonical map gi∗ R(p1 )∗ → R(Uφ )∗ (π ◦ gi(t) )∗ is an isomorphism. So it suﬃces to prove that via ε

(π ◦ gi(t) )∗ Rπ∗ π × −−−→(π ◦ gi(t) )∗ is an isomorphism. Consider the ﬁber square gi(t)

Ui(t) −−→ Xi(t) ↓ id σ ↓π π◦gi(t)

Ui(t) −−−−→ X ′ . ∗ By [41, Theorem 2], ζ0 (σ) : gi(t) π × → (π ◦ gi(t) )∗ is an isomorphism (in [41], schemes are assumed to have ﬁnite Krull dimension, but this assumption is not used in the proof there and unnecessary). By deﬁnition (16.7), ζ0 is the composite map u

∼ =

ε

∗ ∗ π× − →id× Rid∗ gi(t) π× − →id× (π ◦ gi(t) )∗ Rπ∗ π × − →(π ◦ gi(t) )∗ . gi(t)

Since the ﬁrst and the second maps are isomorphisms, the third map is an isomorphism. This was what we wanted to prove. ⊓ ⊔ Corollary 18.15. Under the same assumption as in the proposition, we have g•∗ f•× (D+ (Y• )) ⊂ D+ (U• ). + + Proof. This is because gi∗ fi× (DQch (Yi )) ⊂ DQch (Ui ) for each i ∈ I.

Chapter 19

Open Immersion Base Change

(19.1)

Let S be a scheme, I a small category, and g′

• X• X•′ −→ ↓ f•′ σ ↓ f• g• Y•′ −→ Y•

a ﬁber square in P(I, Sch/S). Assume that X• and f• are concentrated, and g• is ﬂat. By Lemma 17.5, the canonical map θ(g• , f• ) : g•∗ R(f• )∗ → R(f•′ )∗ (g•′ )∗ is an isomorphism of functors from DLqc (X• ) to DLqc (Y•′ ). We deﬁne ζ(σ) = ζ(g• , f• ) to be the composite map θ −1

u

ε

ζ(σ) : (g•′ )∗ f•× − →(f•′ )× R(f•′ )∗ (g•′ )∗ f•× −−→(f•′ )× g•∗ R(f• )∗ f•× − →(f•′ )× g•∗ . Lemma 19.2. Let σ be as above, and J a subcategory of I. Then the diagram θ −1

ξ

ξ

θ −1

(?)J (g•′ )∗ f•× −−→ (g•′ |J )∗ (?)J f•× − → (g•′ |J )∗ (f• |J )× (?)J ↓ζ ↓ζ (?)J (f•′ )× g•∗ − → (f•′ |J )× (?)J g•∗ −−→ (f•′ |J )× (g• |J )∗ (?)J is commutative. Proof. Follows immediately from Lemma 18.7 and the commutativity of (17.6). ⊓ ⊔ Lemma 19.3. Let σ be as above. Then the composite u

ζ

θ

(g•′ )∗ − → (g•′ )∗ f•× R(f• )∗ − → (f•′ )× g•∗ R(f• )∗ − → (f•′ )× R(f•′ )∗ (g•′ )∗ is u. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

413

414

19 Open Immersion Base Change

Proof. Follows from the commutativity of the diagram u

(g•′ )∗

/ (g ′ )∗ f × R(f• )∗ •

•

u

GF

u

(f•′ )× R(f•′ )∗ (g•′ )∗

@A

u

/ (f ′ )× R(f ′ )∗ (g ′ )∗ (f• )× R(f• )∗ •

•

•

θ −1

θ −1

id

u

/ (f ′ )× g ∗ R(f• )∗ f × R(f• )∗ (f•′ )× g•∗ R(f• )∗ • • • UUUU UUUU id UUUU ε UUUU U* / (f ′ )× R(f ′ )∗ (g ′ )∗ o (f•′ )× g•∗ R(f• )∗ . • • • θ

⊓ ⊔ Theorem 19.4. Let S be a scheme, I an ordered UJD small category, and j•

i′

• V• −→ U•′ − → X•′ U p• ↓ σ ↓ pX •

i

q•

• U• − → X• −→ Y•

be a diagram in P(I, Sch/S). Assume the following. 1 2 3 4 5

Y• is noetherian with ﬂat arrows. j• , i′• and i• are image dense open immersions. U q • , pX • and p• are proper. U X ′ p• i• = i• p• . q• i• and pU • j• are cartesian.

Then σ is a ﬁber square, and × × ∗ U × ∗ × j•∗ ζ(σ)q•× : j•∗ (i′• )∗ (pX • ) q• → j• (p• ) i• q•

is an isomorphism of functors from D+ (Y• ) to D+ (V• ). Proof. The square σ is a ﬁber square, since the canonical map U•′ → U• ×X• X•′ is an image dense closed open immersion, and is an isomorphism. To prove the theorem, it suﬃces to show that the map in question is an isomorphism after applying (?)i for any i ∈ I. By Proposition 18.14, Lemma 19.2, and [26, (3.7.2), (iii)], the problem is reduced to the ﬂat base change theorem (in fact open immersion base change theorem is enough) for schemes [41, Theorem 2], and we are done. ⊓ ⊔

Chapter 20

The Existence of Compactiﬁcation and Composition Data for Diagrams of Schemes Over an Ordered Finite Category

(20.1) Let I be an ordered ﬁnite category which is non-empty. Let A denote the category of noetherian I op -diagrams of schemes as its objects and morphisms separated of ﬁnite type as its morphisms. Let P denote the class of proper morphisms in Mor(A). Let I denote the class of image dense open immersions in Mor(A). Deﬁne D(X• ) := DLqc (X• ) for X• ∈ ob(A). Deﬁne a pseudofunctor (?)# on AP to be (?)× , where X•# = D(X• ) for X• ∈ ob(AP ). Deﬁne a pseudofunctor (?)♭ on AI to be (?)∗ , where X•♭ = D(X• ) for X• ∈ ob(AI ). For a pi-square σ, deﬁne ζ(σ) to be the natural map deﬁned in (19.1). Lemma 20.2. Let the notation be as above. Conditions 1–6 and 8–14 in Deﬁnition 16.1 are satisﬁed. Moreover, any pi-square is a ﬁber square. Proof. This is easy.

⊓ ⊔

Proposition 20.3. Let the notation be as in (20.1). Then the condition 7 in Deﬁnition 16.1 is satisﬁed. That is, for any morphism f• in A, there is a factorization f• = p• j• with p• ∈ P and j• ∈ I. Proof. Label the object set ob(I) of I as {i(1), · · · , i(n)} so that I(i(s), i(t)) = ∅ if s > t. Set J(r) to be the full subcategory of I with ob(J(r)) = {i(1), . . . , i(r)}. By induction on r, we construct morphisms j• (r) : X• |J(r) → Z• (r) and p• (r) : Z• (r) → Y• |J(r) such that 1 j• (r) is an open immersion whose scheme theoretic image is Z• (r) (i.e., for any j, j• (r)j is an open immersion whose scheme theoretic image is Z• (r)j ). In particular, j• (r) is an image dense open immersion. 2 p• (r) is proper. 3 p• (r)j• (r) = f• |J(r) . 4 Z• (r)|J(j) = Z• (j), j• (r)|J(j) = j• (j), p• (r)|J(j) = p• (j) for j < r. The proposition follows from this construction for r = n. We may assume that the construction is done for j < r. J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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20 The Existence of Compactiﬁcation and Composition Data

First, consider the case where i(r) is a minimal element in ob(I). By Nagata’s compactiﬁcation theorem [34] (see also [27]), there is a factorization k

p

→Z− → Yi(r) Xi(r) − such that p is proper, k is an open immersion whose scheme theoretic image is Z, and pk = fi(r) . Now deﬁne Z• (r)i(r) := Z and Z• (r)idi(r) = idZ . Deﬁning the other structures after 4, we get Z• (r), since I(i(r), i(s)) = ∅ = I(i(s), i(r)) for s < r and I(i(r), i(r)) = {id} by assumption. Deﬁne j• (r)i(r) := k and by 4, we get a morphism j• (r) : X• |J(r) → Z• (r) by the same reason. Similarly, p• (r)i(r) := p and 4 deﬁne p• (r) : Z• (r) → Y• |J(r) . and 1–4 are satisﬁed by the induction assumption. So this case is OK. Now assume that i(r) is not minimal so that j
Note that each Yr Yφ ×Yj Zj is proper over Yr , and hence W is proper over Yr . There is a unique Yr -morphism h : Xr → W induced by (fr , jj ◦ Xφ ) : Xr → Yr Yφ ×Yj Zj . Since h is separated of ﬁnite type, there is a factorization k

p

Xr − →Z− →W such that k is an open immersion whose scheme theoretic image is Z, p is proper, and pk = h. Now deﬁne Zi(r) = Z, Zidi(r) = idZ , and Zφ to be the composite p

projection

Z− → W −−−−−−→ Zj for j < r and φ ∈ I(i(j), i(r)). We deﬁne Z• (r) by these data and by 4. Set jr = j• (r)r = k, and pr = p• (r)r to be the composite p

Z− → W → Yr , where the second map is the structure map of W as a Yr -scheme. Note that Zψ jj = jj ′ Xψ and Yψ pj = pj ′ Zψ hold for 1 ≤ j ′ ≤ j ≤ r and ψ ∈ I(i(j ′ ), i(j)). Indeed, this is trivial by induction assumption if j < r, also trivial if ψ = idi(r) , and follows easily from the construction if j ′ < j = r. In particular, j• (r) and p• (r) are deﬁned so that 4 is satisﬁed and

20 The Existence of Compactiﬁcation and Composition Data

417

are morphisms of diagrams of schemes provided that Z• (r) is a diagram of schemes. We need to check that Z• (r) is certainly a diagram of schemes. To verify this, it suﬃces to show that, for any j ′ < j < r and any φ ∈ I(i(j), i(r)) and ψ ∈ I(i(j ′ ), i(j)), Zφψ = Zψ Zφ holds. Let A be the locus in Z such that Zφψ and Zψ Zφ agree. Note that the diagrams Xr

Xφ

jj

jr (a)

Zr

Zφ

/ Zj

Yφ

/ Xj ′ jj ′

(b) Zψ

/ Zj ′ pj ′

pj (d)

pr (c)

Yr

Xψ

/ Xj

/ Yj

Yψ

/ Yj ′

Xr jr

Zr

Xφψ

/ Xj ′ jj ′

(e) Zφψ

/ Zj ′ pj ′

pr (f)

Yr

Yφψ

/ Yj ′

are commutative. Since (c), (d) and (f) are commutative and Yφψ = Yψ Yφ , we have that pj ′ Zφψ = pj ′ Zψ Zφ . Since there is a cartesian square A ↓

−−−−−→

Zj ′ ↓∆

(Zφψ ,Zψ Zφ )

Zr −−−−−−−−→ Zj ′ ×Yj ′ Zj ′ and pj ′ : Zj ′ → Yj ′ is separated, we have that A is a closed subscheme of Zr . Since the scheme theoretic image of the open immersion jr : Xr ֒→ Zr is Zr , it suﬃces to show that jr factors through A. That is, it suﬃces to show that jr Zφψ = jr Zψ Zφ . But this is trivial by the commutativity of (a), (b), and (e), and the fact Xφψ = Xψ Xφ . So Z• (r) is a diagram of schemes, and j• (r) and p• (r) are morphisms of diagrams of schemes. The conditions 1–4 are now easy to verify, and the proof is complete. ⊓ ⊔ Theorem 20.4. Let the notation be as in (20.1). Set F to be the subcategory of A whose objects are objects of A with ﬂat arrows, and whose mor+ (X• ). phisms are cartesian morphisms in A. Deﬁne D+ by D+ (X• ) := DLqc + # ♭ Then (A, F, P, I, D, D , (?) , (?) , ζ) is a composition data of contravariant almost-pseudofunctors. Proof. Conditions 1–14 in Deﬁnition 16.1 have already been checked. 15 follows from Lemma 7.16. 16 is trivial. Since I is ﬁnite, the deﬁnition of D+ (X• ) is consistent with that in (18.12). Hence 17 is Corollary 18.15. 18 is Theorem 19.4. ⊓ ⊔ (20.5) We call the composite of (?)# and (?)♭ deﬁned by the composition data in the theorem the equivariant twisted inverse almost-pseudofunctor, and denote it by (?)! .

Chapter 21

Flat Base Change

Let the notation be as in Theorem 20.4. Let f• : X• → Y• be a morphism in F, and J a subcategory of I. Let f• = p• i• be a compactiﬁcation. Lemma 21.1. The composite map θ −1

via Υ

ξ

via Υ

∗ × ! (?)J f•! −−−→(?)J i∗• p× →i∗J p× • −−→iJ (?)J p• − J (?)J −−−→f• |J (?)J

is independent of choice of compactiﬁcation f• = p• i• , where Υ’s are the independence isomorphisms. The proof utilizes Lemma 16.3, and left to the reader. We denote by ξ¯ = ¯ f• ) the composite map in the lemma. ξ(J, Lemma 21.2. Let f• : X• → Y• be a morphism in F, and K ⊂ J ⊂ I be subcategories. Then the composite map ¯

¯

ξ ξ ! →(?)K f• |!J (?)J − →f• |!K (?)K (?)J ∼ (?)K (?)K f•! ∼ = fK = (?)K (?)J f•! −

¯ agrees with ξ(K, f• ). Proof. Follows easily from Lemma 18.4.

⊓ ⊔

Lemma 21.3. Let f• : X• → Y• be a morphism in F, and J a subcategory ¯ f• ) is an isomorphism. of I. Then ξ(J, ¯ f• ) is an isomorphism for any i ∈ Proof. It suﬃces to show that (?)i ξ(J, ob(J). By Lemma 21.2, we have ¯ f• ). ¯ f• |J ) ◦ ((?)i ξ(J, ¯ f• )) = ξ(i, ξ(i, ¯ f• |J ) and ξ(i, ¯ f• ) are isomorphisms. Hence By Proposition 18.14, we have ξ(i, ¯ f• ) is also an isomorphism. ⊓ ⊔ the natural map (?)i ξ(J,

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21 Flat Base Change

Lemma 21.4. Let f : X → Y be a ﬂat morphism of locally noetherian schemes, and U a dense open subset of Y . Then f −1 (U ) is a dense open subset of X. Proof. The question is local both on Y and X, and hence we may assume that both Y = Spec A and X = Spec B are aﬃne. Let I be the radical ideal of A deﬁning the closed subset Y \ U . By assumption, I is not contained in any minimal prime of A. Assume that f −1 (U ) is not dense in X. Then, there is a minimal prime P of B which contains IB. As we have I ⊂ IB ∩ A ⊂ P ∩ A and P ∩ A is minimal by the going-down theorem (see [30, Theorem 9.5]), this is a contradiction. ⊓ ⊔ (21.5)

Let the notation be as in Theorem 20.4. Let f′

• Y•′ X•′ −→ X g• ↓ σ ↓ g•

f•

X• −→ Y• be a diagram in P(I, Sch) such that 1 2 3 4

All objects lie in F; f• and f•′ are morphisms in F; σ is a ﬁber square; g• is ﬂat (not necessarily a morphism of A). By assumption, there is a diagram i′

p′

i•

p•

• • X•′ − → Z•′ −→ Y•′ X Z ↓ g• σ 1 ↓ g• σ 2 ↓ g•

(21.6)

X• − → Z• −→ Y• such that f• = p• i• is a compactiﬁcation, σ1 and σ2 are ﬁber squares, and the whole rectangle σ1 σ2 equals σ. By Lemma 21.4, we have that f•′ = p′• i′• is a compactiﬁcation. Lemma 21.7. The composite map Υ

d

ζ

Υ

(g•X )∗ f•! − →(g•X )∗ i∗• p× →(i′• )∗ (g•Z )∗ p× →(i′• )∗ (p′• )× g•∗ − →(f•′ )! g•∗ •− •− is independent of choice of the diagram (21.6), and depends only on σ, where Υ’s are independence isomorphisms. Proof. Obvious by Lemma 16.3. ¯ We denote the composite map in the lemma by ζ¯ = ζ(σ).

⊓ ⊔

21 Flat Base Change

421

Theorem 21.8. Let the notation be as above. Then we have: 1

Let J be a subcategory of I. Then the diagram θ −1

(?)J (g•X )∗ f•! −−→ (gJX )∗ (?)J f•! ¯ ↓ (?)J ζ(σ) ξ¯

ξ¯

− →

(gJX )∗ fJ! (?)J ¯ J )(?)J ↓ ζ(σ

(21.9)

−1 ∗ θ

(?)J (f•′ )! (g• )∗ − → (fJ′ )! (?)J (g• ) −−→ (fJ′ )! (gJ )∗ (?)J 2

is commutative. ¯ ζ(σ) is an isomorphism.

Proof. 1 is an immediate consequence of Lemma 19.2 and [26, (3.7.2)]. 2 Let i be an object of I. By Lemma 21.3, the horizontal arrows in the diagram (21.9) for J = i are isomorphisms. By Verdier’s ﬂat base change ¯ i ) is an isomorphism. Hence, we theorem [41, Theorem 2], we have that ζ(σ ¯ have that (?)i ζ(σ) is an isomorphism for any i ∈ I by 1 applied to J = i, and the assertion follows. ⊓ ⊔

Chapter 22

Preservation of Quasi-Coherent Cohomology

(22.1)

Let the notation be as in (20.1). Let F be as in Theorem 20.4.

Lemma 22.2. Let f : X → Y be a separated morphism of ﬁnite type between + + ! noetherian schemes. If F ∈ DCoh(Y ) (Mod(Y )), then f F ∈ DCoh(X) (Mod(X)). Proof. We may assume that both Y and X are aﬃne. So we may assume that f is either smooth or a closed immersion. The case where f is smooth is obvious by [41, Theorem 3]. The case where f is a closed immersion is also obvious by Proposition III.6.1 and Theorem III.6.7 in [17]. ⊓ ⊔ Proposition 22.3. Let f• : X• → Y• be a morphism in F, and φ : i → j a morphism in I. Then the composite map ξ¯

αφ

→fj! (?)j Xφ∗ (?)i f•! −→(?)j f•! − agrees with the composite map ζ¯

ξ¯

αφ

Xφ∗ (?)i f•! − →Xφ∗ fi! (?)i − →fj! Yφ∗ (?)i −→fj! (?)j . Proof. By Lemma 21.2, we may assume that I is the ordered category given by ob(I) = {i, j} and I(i, j) = {φ}. Then it is easy to see that there is a compactiﬁcation p•

i

• X• − →Y• →Z •−

of f• such that p• is cartesian. Note that i• is cartesian, and Z• has ﬂat arrows. ¯ it suﬃces to prove that the composite map By the deﬁnition of ξ¯ and ζ, θ −1

d

ξ

ζ

αφ

× ∗ × ∗ × ∗ ∗ ∗ × ∗ × Xφ (?)i i∗ −→Xφ ii (?)i p× →i∗ →i∗ →i∗ − →i∗ • p• − • − j Zφ (?)i p• − j Zφ pi (?)i − j pj Yφ (?)i − j pj (?)j

agrees with

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22 Preservation of Quasi-Coherent Cohomology θ −1

αφ

ξ

∗ × ∗ × Xφ∗ (?)i i∗• p× →i∗j p× • −→(?)j i• p• −−→ij (?)j p• − j (?)j .

By the “derived version” of (6.31), the composite map θ −1

αφ

d

∗ ∗ × ∗ × Xφ∗ (?)i i∗• p× →i∗j Yφ∗ (?)i p× • −−→Xφ ii (?)i p• − • −→ij (?)j p•

agrees with θ −1

αφ

∗ × ∗ × Xφ∗ (?)i i∗• p× • −→(?)j i• p• −−→ij (?)j p• .

Hence it suﬃces to prove the map αφ

ζ

ξ

× ∗ Zφ∗ (?)i p× →p× →Zφ∗ p× •− j Yφ (?)i −→pj (?)j i (?)i −

agrees with αφ

ξ

× Zφ∗ (?)i p× →p× • −→(?)j p• − j (?)j .

Now the proof consists in a straightforward diagram drawing utilizing Lemma 19.3 and the derived version of Lemma 6.20. ⊓ ⊔ Corollary 22.4. Let f• : X• → Y• be a morphism in F. Then we have + + + + (Y• )) ⊂ DCoh (X• ). (Y• )) ⊂ DQch (X• ) and f•! (DCoh f•! (DQch Proof. Let φ : i → j be a morphism in I. By the ﬂat base change theorem, Lemma 21.3, and the proposition, we have αφ : Xφ∗ (?)i f•! → (?)j f•! is an isomorphism if αφ : Yφ∗ (?)i → (?)j is an isomorphism. So f•! preserves equivariance of cohomology groups, and the ﬁrst assertion follows. On the other hand, by Lemma 22.2 and Proposition 18.14, f ! preserves local coherence of cohomology groups. Hence it also preserves the coherence of cohomology groups, by the ﬁrst paragraph. ⊓ ⊔

Chapter 23

Compatibility with Derived Direct Images

(23.1) Let the notation be as in (21.5). Consider the diagram (21.6). Lipman’s theta θ(σ2 ) : g•∗ R(p• )∗ → R(p′• )∗ (g•Z )∗ induces the conjugate map ξ(σ2 ) : R(g•Z )∗ (p′• )× → p× • R(g• )∗ . As σ2 is a ﬁber square, θ(σ2 ) is an isomorphism. Hence ξ(σ2 ) is also an isomorphism. Note that θ : i∗• R(g•Z )∗ → R(g•X )∗ (i′• )∗ is an isomorphism, since σ1 is a ﬁber square. We deﬁne ξ¯ : R(g•X )∗ (f•′ )! → f•! R(g• )∗ to be the composite θ −1

Υ

ξ

Υ

Z ′ × × R(g•X )∗ (f•′ )! − →R(g•X )∗ (i′• )∗ (p′• )× −−→i∗ →i∗ →f•! R(g• )∗ . • R(g• )∗ (p• ) − • p• R(g• )∗ .−

As the all maps in the composition are isomorphisms, we have Lemma 23.2. ξ¯ is an isomorphism. Lemma 23.3. For any subcategory J, the composite ξ

ξ

c

(?)J R(g•Z )∗ (p′• )× − →(?)J p× →(p• |J )× (?)J R(g• )∗− →(p• |J )× R(g• |J )∗ (?)J • R(g• )∗− agrees with the composite ξ

c

(?)J R(g•Z )∗ (p′• )× − →R(g•Z |J )∗ (?)J (p′• )× − → ξ

R(g•Z |J )∗ (p′• |J )× (?)J − →(p• |J )× R(g• |J )∗ (?)J . Proof. Follows from Lemma 18.7.

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

⊓ ⊔

425

Chapter 24

Compatibility with Derived Right Inductions

(24.1) Let I be a ﬁnite category, and X• ∈ P(I, Sch). Assume that X• has (J→I) concentrated arrows. Let J be a subcategory of I. For i ∈ I, Ii is ﬁnite, since I is ﬁnite. So for any M ∈ Lqc(X• |J ), we have RJ M ∈ Lqc(X• ) by (6.14). So RJ : Lqc(X• |J ) → Lqc(X• ) is a right adjoint of (?)J : Lqc(X• ) → Lqc(X• |J ). Lemma 24.2. Let I be a ﬁnite category, and X• ∈ P(I, Sch). Assume that X• is noetherian. If I ∈ Lqc(X• ) is an injective object, then it is injective as an object of Mod(X• ). Proof. Let J be the discrete subcategory of I such that ob(J) = ob(I). Let IJ ֒→ J be the injective hull in Lqc(X• |J ). Since (?)J is faithful, the composite I → RJ (?)J I ֒→ RJ J is a monomorphism, and hence it splits. ∼ For

each i ∈ I, Ji is injective as an object of Qch(Xi ), since Lqc(X• |J ) = Qch(X ) in a natural way. So it is also injective as an object of Mod(X i i) i by [17, (II.7)]. So J is injective as an object of Mod(X• |J ). Since RJ preserves injectives, RJ J is an injective object of Mod(X• ). Hence its direct summand I is also injective in Mod(X• ). ⊓ ⊔ Corollary 24.3. Let I and X• be as in the lemma. Then for any subcategory J ⊂ I, RRJ : D(X• |J ) → D(X• ) + + takes DLqc (X• |J ) to DLqc (X• ), and RRJ is right adjoint to (?)J + + DLqc (X• ) → DLqc (X• |J ).

:

Proof. By the way-out lemma, it suﬃces to prove that for a single object M ∈ Lqc(X• |J ), Rn RJ M ∈ Lqc(X• ). Let M → I be an injective resolution in the category Lqc(X• |J ), which exists by Lemma 11.9. Then I is also an injective resolution in Mod(X• |J ) by the lemma. So Rn RJ M ∼ = H n (RJ I) lies ⊓ ⊔ in Lqc(X• ).

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24 Compatibility with Derived Right Inductions

(24.4) Let the notation be as in Theorem 20.4. Let f• : X• → Y• be a morphism in A. Let J be a subcategory of I. As c : (?)J R(f• )∗ → R(f• |J )∗ (?)J + + (X• |J ), its conjugate map (X• ) to DLqc is an isomorphism of functors from DLqc

c′ : RRJ (f• |J )× → f•× RRJ is also an isomorphism. Let g• : U• → X• be a cartesian image dense open immersion in A. Let μ = μ(g• , J) be the canonical map ξ −1

u

ε

→g•∗ RRJ R(g• |J )∗ (g• |J )∗ −−→g•∗ R(g• )∗ RRJ (g• |J )∗ − →RRJ (g• |J )∗ , g•∗ RRJ − where ξ : R(g• )∗ RRJ → RRJ R(g• |J )∗ is the conjugate of the isomorphism θ : (g• )|∗J (?)J → (?)J g•∗ . Lemma 24.5. Let the notation be as above. Then μ : g•∗ RRJ → RRJ (g• |J )∗ is an isomorphism of functors from D(X• |J ) to D(U• ). Proof. As g•∗ , (g• |J )∗ , RJ , (g• )∗ , and (g• |J )∗ have exact left adjoints, it sufﬁces to show that (?)i μ : (?)i g•∗ RJ I → (?)i RJ (g• |J )∗ I is an isomorphism for any K-injective complex in C(Mod(X• |J )) and i ∈ ob(I). This map agrees with −1

θ (?)i g•∗ RJ I−−→gi∗ (?)i RJ I ∼ (X ) I ∼ lim g ∗ (X ) I = gi∗ lim ←− φ ∗ j = ←− i φ ∗ j θ θ → lim(Uφ )∗ (?)j (g• |J )∗ I ∼ − → lim(Uφ )∗ gj∗ Ij − = (?)i RJ (g• |J )∗ I, ←− ←− (J→I)

which is obviously an isomorphism, where the limit is taken over φ ∈ Ii

. ⊓ ⊔

(24.6) Let f• : X• → Y• be a morphism in F, and f• = p• i• a compactiﬁcation. We deﬁne c¯ : f•! RRJ → RRJ (f• |J )! to be the composite Υ

c′

μ

Υ

f•! RRJ − →i∗• p× →i∗• RRJ (p• |J )× − →RRJ (i• |J )∗ (p• |J )× − →RRJ (f• |J )! . • RRJ − By Lemma 24.5, we have Lemma 24.7. c¯ : f•! RRJ → RRJ (f• |J )! is an isomorphism of functors from + + (X• ). (Y• |J ) to DLqc DLqc

Chapter 25

Equivariant Grothendieck’s Duality

Theorem 25.1 (Grothendieck’s duality). Let f : X → Y be a proper mor+ phism of noetherian schemes. For F ∈ DQch (X) and G ∈ DQch (Y ), The canonical map H

Θ(f ) : Rf∗ R Hom•OX (F, f × G) −→ R Hom•OY (Rf∗ F, Rf∗ f × G) ε

− → R Hom•OY (Rf∗ F, G) is an isomorphism. Proof. As pointed out in [35, section 6], this is an immediate consequence of the open immersion base change [41, Theorem 2]. ⊓ ⊔ Theorem 25.2 (Equivariant Grothendieck’s duality). Let I be a small category, and f• : X• → Y• a morphism in P(I, Sch/Z). If Y• is noetherian with ﬂat arrows and f• is proper cartesian, then the composite H

× × Θ(f• ) : R(f• )∗ R Hom• →R Hom• Mod(X• ) (F, f• G)− Mod(Y• ) (R(f• )∗ F, R(f• )∗ f• G) ε

− →R Hom• Mod(Y• ) (R(f• )∗ F, G)

is an isomorphism for F ∈ DQch (X• ) and G ∈ D+ (Y• ). Proof. It suﬃces to show that (?)i Θ(f• ) is an isomorphism for i ∈ ob(I). By Lemma 1.39 and Lemma 18.7, 2, it is easy to see that the composite c

→R(fi )∗ (?)i R Hom•Mod(X• ) (F, f•× G) (?)i R(f• )∗ R Hom•Mod(X• ) (F, f•× G)− H

ξ

i • × −→R(f →R(fi )∗ R Hom•Mod(Xi ) (Fi , fi× Gi ) i )∗ R HomMod(Xi ) (Fi , (?)i f• G)−

Θ(fi )

−−−→R Hom•Mod(Yi ) (R(fi )∗ Fi , Gi )

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25 Equivariant Grothendieck’s Duality

agrees with the composite (?)i Θ(f• )

(?)i R(f• )∗ R Hom•Mod(X• ) (F, f•× G) −−−−−−→ (?)i R Hom•Mod(Y• ) (R(f• )∗ F, G) H

c

i → R Hom•Mod(Yi ) (R(fi )∗ Fi , Gi ). −−→ R Hom•Mod(Yi ) ((?)i R(f• )∗ F, Gi ) −

Consider the ﬁrst composite map. By Lemma 13.9, Hi is an isomorphism. By Lemma 18.5, ξ is an isomorphism. By Theorem 25.1, Θ(fi ) is an isomorphism. Hence the ﬁrst composite is an isomorphism, and so is the second. Consider the second composite map. By Lemma 8.7, R(f• )∗ F ∈ DQch (Y• ). So the second map Hi is an isomorphism by Lemma 13.9. So the ﬁrst map (?)i Θ(f• ) must be an isomorphism. This is what we wanted to prove. ⊓ ⊔

Chapter 26

Morphisms of Finite Flat Dimension

(26.1) Let ((?)∗ , (?)∗ ) be a monoidal adjoint pair of almost-pseudofunctors over a category S. For a morphism f : X → Y in S, we deﬁne the projection morphism Π = Π(f ) to be the composite u

∆

ε

f∗ a ⊗ b− →f∗ f ∗ (f∗ a ⊗ b)− →f∗ (f ∗ f∗ a ⊗ f ∗ b)− →f∗ (a ⊗ f ∗ b), where a ∈ X∗ and b ∈ Y∗ (see Chapter 1 for the notation). Lemma 26.2. Let the notation be as above, and f : X → Y and g : Y → Z be morphisms in S. For x ∈ X∗ and z ∈ Z∗ , the composite Π(g)

c

Π(f )

(gf )∗ x ⊗ z − →g∗ (f∗ x) ⊗ z −−−→g∗ (f∗ x ⊗ g ∗ z)−−−→g∗ f∗ (x ⊗ f ∗ g ∗ z) c−1

d

−−→(gf )∗ (x ⊗ f ∗ g ∗ z)− →(gf )∗ (x ⊗ (gf )∗ z) agrees with Π(gf ). ⊓ ⊔

Proof. Left to the reader.

(26.3) Let I be a small category, S a scheme, and f• : X• → Y• a morphism in P(I, Sch/S). Lemma 26.4 (Projection Formula). Assume that f• is concentrated. Then the natural map •,L ∗ Π = Π(f• ) : (Rf• )∗ F ⊗•,L OY• G → (Rf• )∗ (F ⊗OX• Lf• G)

is an isomorphism for F ∈ DLqc (X• ) and G ∈ DLqc (Y• ). Proof. For each i ∈ ob(I), the composite Π(fi )

θ(f• ,i)

•,L ∗ ∗ R(fi )∗ Fi ⊗•,L −−−→R(fi )∗ (Fi ⊗•,L OY Gi −−−→R(fi )∗ (Fi ⊗OX Lfi Gi )− OX (?)i Lf• G) i

i

m

i

c

∗ ∗ − →R(fi )∗ (?)i (F ⊗•,L →(?)i R(f• )∗ (F ⊗•,L OX• Lf• G)− OX• Lf• G)

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

431

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26 Morphisms of Finite Flat Dimension

is an isomorphism by [26, (3.9.4)] and Lemma 8.13, 1. On the other hand, it is straightforward to check that this composite isomorphism agrees with the composite c−1

m

R(fi )∗ Fi ⊗L −→(?)i R(f• )∗ F ⊗L →(?)i (R(f• )∗ F ⊗•,L OY Gi − OY Gi − OY• G) i

i

(?)i Π(f• )

∗ −−−−−−→(?)i R(f• )∗ (F ⊗•,L OX• Lf• G).

It follows that (?)i Π(f• ) is an isomorphism for any i ∈ ob(I). Hence, Π(f• ) is an isomorphism. ⊓ ⊔ (26.5) Let f• : X• → Y• be a morphism in P(I, Sch/S), and assume that both X• and f• are concentrated. Deﬁne χ = χ(f• ) to be the composite u

∗ ∗ f•× F ⊗•,L →f•× R(f• )∗ (f•× F ⊗•,L OX• Lf• G− OX• Lf• G) Π(f• )−1

ε

−−−−−→f•× (R(f• )∗ f•× F ⊗•,L →f•× (F ⊗•,L OY• G)− OY• G), where F, G ∈ DLqc (Y• ). Utilizing the commutativity as in the proof of Lemma 26.4 and Lemma 18.7, it is not so diﬃcult to show the following. Lemma 26.6. Let f• : X• → Y• be as in (26.5). For a subcategory J of I, the composite ξ⊗θ −1

(?)J f•× F ⊗•,L OX

• |J

(?)J Lf•∗ G−−−−→(f• |J )× FJ ⊗•,L OX

• |J

χ(f• |J )

L(f• |J )∗ GJ

m

−−−−→(f• |J )× (FJ ⊗•,L →(f• |J )× (?)J (F ⊗•,L OY | GJ )− OY• G) • J

agrees with (?)J f•× F ⊗•,L OX

• |J

m

∗ (?)J Lf•∗ G− →(?)J (f•× F ⊗•,L OX• Lf• G) χ(f• )

ξ

−−−→(?)J f•× (F ⊗•,L →(f• |J )× (?)J (F ⊗•,L OY• G)− OY• G). Lemma 26.7. Let f• : X• → Y• be as in (26.5). The composite α−1

∆

•,L •,L ∗ ∗ ∗ f•× F ⊗•,L →f•× F ⊗•,L OX• Lf• (G ⊗OY• H)− OX• (Lf• G ⊗OX• Lf• H) −−→ χ

χ

•,L •,L ∗ ∗ ∗ (f•× F ⊗•,L →f•× (F ⊗•,L → OX• Lf• G) ⊗OX• Lf• H− OY• G) ⊗OX• Lf• H− α

•,L •,L → f•× (F ⊗•,L f•× ((F ⊗•,L OY• G) ⊗OY• H) − OY• (G ⊗OY• H))

agrees with χ.

26 Morphisms of Finite Flat Dimension

433

Lemma 26.8. Let S, I and σ be as in (19.1). For F ∈ DLqc (Y• ), the composite ∆−1

′ ∗ ∗ ∗ (g•′ )∗ f•× F ⊗•,L −−→(g•′ )∗ (f•× F ⊗•,L OX ′ (g• ) Lf• G− OX• Lf• G) •

ζ(σ)

χ

•,L ′ × ∗ − →(g•′ )∗ f•× (F ⊗•,L OY• G)−−→(f• ) g• (F ⊗OY• G)

agrees with ζ(σ)⊗d

•,L ′ ∗ ∗ ′ × ∗ ′ ∗ ∗ (g•′ )∗ f•× F ⊗•,L OX ′ (g• ) Lf• G−−−−→(f• ) g• F ⊗OX ′ L(f• ) g• G •

•

χ

− →(f•′ )× (g•∗ F

⊗•,L OY ′

•

∆−1 g•∗ G)−−−→(f•′ )× g•∗ (F

⊗•,L OY• G).

Lemma 26.9. Let f• : X• → Y• and g• : Y• → Z• be morphisms in P(I, Sch/S). Assume that X• , Y• and g• are concentrated. Then the composite χ(f• )

•,L •,L ∗ ∗ ∗ × × ∗ ∼ × × (g• f• )× F ⊗•,L OX• (g• f• ) G = f• g• F ⊗OX• f• g• G−−−→f• (g• F ⊗OY• g• G) χ(g• )

•,L × ∼ −−−→f•× g•× (F ⊗•,L OZ• G) = (g• f• ) (F ⊗OZ• G)

agrees with χ(g• f• ). The proof of the lemmas above are left to the reader. (26.10) Let the notation be as in Theorem 20.4. Let i• : U• → X• be a morphism in I, and p• : X• → Y• a morphism in P. We deﬁne χ ¯ = χ(p ¯ • , i• ) to be the composite d−1

•,L •,L ∗ ∗ ∗ × ∗ χ ¯ : i∗• p× • F ⊗OU• L(p• i• ) G−−→i• p• F ⊗OU• i• Lp• G ∆−1

i∗ χ

•,L •,L • ∗ −−−→i∗• (p× − →i∗• p× • F ⊗OX• Lp• G)− • (F ⊗OY• G).

Lemma 26.11. Let f• be a morphism in F, and f• = p• i• = q• j• an independence square. Then the composite χ ¯

Υ(p• i• =q• j• )

•,L •,L ∗ i∗• p× →i∗• p× −−−−−−−−→j•∗ q•× (F ⊗•,L • F ⊗OU• L(p• i• ) G− • (F ⊗OY• G)− OY• G)

agrees with Υ⊗1

χ ¯

•,L ∗ ∗ i∗• p× −−→j•∗ q•× F ⊗•,L →j•∗ q•× (F ⊗•,L • F ⊗OU• L(p• i• ) G− OU• L(q• j• ) G− OY• G).

Proof. As Υ is constructed from ζ and d by deﬁnition, the assertion follows easily from Lemma 26.8 and Lemma 26.9. ⊓ ⊔

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26 Morphisms of Finite Flat Dimension

(26.12) Let f• : X• → Y• be a morphism in F. We deﬁne χ(f ¯ • ) to be χ(p ¯ • , i• ), where f• = p• i• is the (ﬁxed) compactiﬁcation of f• . By Lemma 26.11, χ(f ¯ • ) is an isomorphism if and only if there exists some ¯ • , j• ) is an isomorphism. compactiﬁcation f• = q• j• such that χ(q Lemma 26.13. Let the notation be as in Theorem 20.4, and f• : X• → Y• and g• : Y• → Z• morphisms in F. Then the composite ¯ •) ! ! •,L •,L ∗ ∗ ∗ χ(f ∗ ∼ ! ! (g• f• )! F ⊗•,L OX• L(g• f• ) G = f• g• F ⊗OX• Lf• Lg• G−−−→f• (g• F ⊗OY• Lg• G) χ(g ¯ •) •,L ! ∼ −−−→f•! g•! (F ⊗•,L OZ• G) = (g• f• ) (F ⊗OZ• G)

agrees with χ(g ¯ • f• ). Theorem 26.14. Let the notation be as in Theorem 20.4, and f• : X• → Y• a morphism in F. If f• is of ﬁnite ﬂat dimension, then •,L ∗ ! χ(f ¯ • ) : f•! F ⊗•,L OY• Lf• G → f• (F ⊗OX• G) + (Y• ). is an isomorphism for F, G ∈ DLqc

Proof. Let f• = p• i• be a compactiﬁcation of f• . It suﬃces to show that χ(p ¯ • , i• ) is an isomorphism. In view of Lemma 26.7, we may assume that F = OY• . Then in view of Proposition 18.14 and Lemma 26.6, it suﬃces to show that •,L •,L ∗ ∗ × χ(f ¯ j ) : i∗j p× j OYj ⊗OX Lfj Gi → ij pj (OYj ⊗OY Gj ) j

j

is an isomorphism for any j ∈ ob(I). So we may assume that I = j. By the ﬂat base change theorem and Lemma 26.8, the question is local on Yj . Clearly, the question is local on Xj . Hence we may assume that Yj and Xj are aﬃne. Set f = fj , Y = Yj , and X = Xj . Note that f is a closed immersion deﬁned by an ideal of ﬁnite projective dimension, followed by an aﬃne n-space. By Lemma 26.13, it suﬃces to prove that χ(f ¯ ) is an isomorphism if f is a closed immersion deﬁned by an ideal of ﬁnite projective dimension or an aﬃne n-space. Both cases are proved easily, using [35, Theorem 5.4] (note that an aﬃne n-space is an open subscheme of a projective n-space). ⊓ ⊔

Chapter 27

Cartesian Finite Morphisms

(27.1) Let I be a small category, S a scheme, and f• : X• → Y• a morphism in P(I, Sch/S). Let Z denote the ringed site (Zar(Y• ), (f• )∗ (OX• )). Assume that Y• is locally noetherian. There are obvious admissible ringed continuous functors i : Zar(Y• ) → Z and g : Z → Zar(X• ) such that gi = f•−1 . If f• is aﬃne, then g# : Mod(Z) → Mod(X• ) is an exact functor, as can be seen easily. Lemma 27.2. If f• is affine, then the counit ε : g# Rg # F → F + is an isomorphism for F ∈ DLqc (X• ).

Proof. The construction of ε is compatible with restrictions. So we may assume that f• = f : X → Y is an aﬃne morphism of single schemes. Further, the question is local on Y , and hence we may assume that Y = Spec A is aﬃne. As f is aﬃne, X = Spec B is aﬃne. By Lemma 14.3, we may assume that F = FX G for some G ∈ D(Qch(X)). In view of Lemma 14.6, it suﬃces to show that ε : g# g # G → G is an isomorphism if G is a K-injective complex in Qch(X). To verify this, it suﬃces to show that ε : g# g # M → M is an isomorphism for M ∈ Qch(X). By Lemma 7.19, f∗ = i# g # on Qch(X) respects coproducts and is exact. Since i# respects coproducts and is faithful exact, g # respects coproducts and is exact. So g# g # : Qch(X) → Qch(X) respects coproducts and is exact. Since X is aﬃne, there is an exact sequence of the form (J)

(I)

OX → OX → M → 0. So we may assume that M = OX . But this case is trivial.

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

⊓ ⊔

435

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27 Cartesian Finite Morphisms

(27.3) Let I, S, f• : X• → Y• , Z, g, and i be as in (27.1). Assume that f• is ﬁnite cartesian. We say that an OZ -module M is locally quasi-coherent (resp. quasicoherent, coherent) if i# M is. The corresponding full subcategory of Mod(Z) is denoted by Lqc(Z) (resp. Qch(Z), Coh(Z)). Lemma 27.4. Let the notation be as above. Then an OZ -module M is locally quasi-coherent if and only if for any j ∈ ob(I) and any affine open subscheme U of Yj , there exists an exact sequence of ((OZ )j )|U -modules (((OZ )j )|U )(T ) → (((OZ )j )|U )(Σ) → Mj |U → 0. Proof. As we assume that f• is ﬁnite cartesian, OZ is coherent. Hence the existence of such exact sequences implies that M is locally quasi-coherent. We prove the converse. Let j ∈ ob(I) and U an aﬃne open subset of Yj . Set C := Γ(U, (OZ )j ) = Γ(fj−1 (U ), OXj ) and M := Γ(U, Mj ). There is ˜ ) → Mj |U , where M ˜ is the quasi-coherent a canonical map (gj |f −1 (U ) )# (M j sheaf over Spec C ⊂ Xj associated with the C-module M . When we apply ˜ 0 → ((i# M)j )|U , where M0 is M viewed as a (ij |U )# to this map, we get M Γ(U, OYj )-module. This is an isomorphism, since (i# M)j |U is quasi-coherent ˜) ∼ and U is aﬃne. As (ij |U )# is faithful and exact, we have (gj |f −1 (U ) )# (M = j Mj |U . Take an exact sequence of the form C (T ) → C (Σ) → M → 0. Applying the exact functor (gj |f −1 (U ) )# ◦ ˜?, we get an exact sequence of the j desired type. ⊓ ⊔ Corollary 27.5. Under the same assumption as in the lemma, the functor g# preserves local quasi-coherence. Proof. As g# is compatible with restrictions, we may assume that I consists of one object and one morphism. Further, as the question is local, we may assume that Y• = Y is an aﬃne scheme. By the lemma, it suﬃces to show that g# OZ is quasi-coherent, since g# is exact and preserves direct sums. As g# OZ = g# g # OX ∼ ⊓ ⊔ = OX , we are done. Lemma 27.6. Let the notation be as above. The unit of adjunction u : F → + Rg # g# F is an isomorphism for F ∈ DLqc (Z). Proof. We may assume that I consists of one object and one morphism, and Y• = Y is aﬃne. By the way-out lemma, we may assume that F is a single quasi-coherent sheaf. Then by Corollary 27.5, g# F is quasi-coherent, and is g # -acyclic. So it suﬃces to show that u : M → g # g# M is an isomorphism for a quasi-coherent sheaf M on Z. Note that g # g# : Qch(Z) → Qch(Z)

27 Cartesian Finite Morphisms

437

respects coproducts. By Lemma 27.4 and the ﬁve lemma, we may assume that F = OZ = g # OX and it suﬃces to prove that ug # : g # OX → g # g# g # OX is an isomorphism. As id = (g # ε)(ug # ) and ε is an isomorphism, we are done. ⊓ ⊔ For N ∈ Mod(Y• ) and M ∈ Mod(Z), the sheaf HomOY• (M, N ) on Y• has a structure of OZ -module, and it belongs to Mod(Z). There is an obvious isomorphism of functors κ : i# HomOY• (M, N ) ∼ = HomOY• (i# M, N ). For M, M′ ∈ Mod(Z), there is a natural map υ : HomMod(Z) (M, M′ ) → HomOY• (M, i# M′ ). Note that the composite i# υ

κ

i# HomMod(Z) (M, M′ )−−→i# HomOY• (M, i# M′ )− → HomOY• (i# M, i# M′ ) agrees with H. (27.7) Let I, S, f• : X• → Y• , Z, g, and i be as in (27.1). Assume that f• is ﬁnite cartesian, and Y• has ﬂat arrows. Deﬁne f•♮ : D+ (Y• ) → D(X• ) by f•♮ (F) := g# R Hom•OY• (OZ , F). As f• is ﬁnite cartesian, OZ is coherent. By Lemma 13.10, i# R Hom•OY•

(OZ , F) ∈ D+ (Y• ). It follows that f•♮ (F) ∈ D+ (X• ), and f•♮ is a functor from D+ (Y• ) to D+ (X• ). Deﬁne ε : R(f• )∗ f•♮ → IdD+ (Y• ) by u−1

R(f• )∗ f•♮ F = i# Rg # g# R Hom•OY• (OZ , F)−−→i# R Hom•OY• (OZ , F) η κ − →R Hom•OY• (i# OZ , F) = R Hom•OY• ((f• )∗ OX• , F)− →R Hom•OY• (OY• , F) ∼ = F.

Deﬁne u : IdD+ (X• ) → f•♮ R(f• )∗ by ε−1

−→g# Rg # R Hom• F∼ = R Hom• OX (OX• , F)− OX (OX• , F) •

•

υ # # →g# R Hom• − →g# R Hom• OY• (OZ , R(f• )∗ F) Mod(Z) (g OX• , Rg F)− H

= f•♮ R(f• )∗ F.

Theorem 27.8. Let the notation be as above. Then f•♮ is right adjoint to R(f• )∗ , and ε and u deﬁned above are the counit and unit of adjunction, respectively. In particular, if, moreover, X• is quasi-compact, then f•♮ is isomorphic to f•× . Proof. It is easy to see that the composite

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27 Cartesian Finite Morphisms

• • ∼ R Hom• OY (OZ , F) = R HomMod(Z) (OZ , R HomOY (OZ , F)) •

•

κ

υ

# • • # →R Hom• − →R Hom• OY (OZ , i R HomOY (OZ , F))− OY (OZ , R HomOY (i OZ , F)) •

•

•

•

η • • ∼ − →R Hom• OY (OZ , R HomOY (OY• , F)) = R HomOY (OZ , F) •

•

•

is the identity. Utilizing this and Lemma 1.47, (f•♮ ε) ◦ (uf•♮ ) = id and (εR(f• )∗ ) ◦ (R(f• )∗ u) = id are checked directly. The last assertion is obvious, as the right adjoint functor is unique. ⊓ ⊔

Chapter 28

Cartesian Regular Embeddings and Cartesian Smooth Morphisms

(28.1) Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). An OX• -module sheaf M ∈ Mod(X• ) is said to be locally free (resp. invertible) if M is coherent and Mi is locally free (resp. invertible) for any i ∈ ob(I). A perfect complex of X• is a bounded complex in C b (Mod(X• )) each of whose terms is locally free. A point of X• is a pair (i, x) such that i ∈ ob(I) and x ∈ Xi . A stalk of a sheaf M ∈ AB(X• ) at the point (i, x) is deﬁned to be (Mi )x , and we denote it by Mi,x . A connected component of X• is an equivalence class with respect to the equivalence relation of the set of points of X• generated by the following relations. 1 (i, x) and (i′ , x′ ) are equivalent if i = i′ and x and x′ belong to the same connected component of Xi . 2 (i, x) and (i′ , x′ ) are equivalent if there exists some φ : i → i′ such that Xφ (x′ ) = x. We say that X• is d-connected if X• consists of one connected component (note that the word ‘connected’ is reserved for componentwise connectedness). If X• is locally noetherian, then a connected component of X• is a closed open subdiagram of schemes in a natural way. If this is the case, the rank function (i, x) → rankOXi ,x Fi,x of a locally free sheaf F is constant on a connected component of X• . Lemma 28.2. Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Let F be a perfect complex of X• . Then we have 1 The canonical map HJ : (?)J R Hom•OX• (F, G) → R Hom•OX• | (FJ , GJ ) J

is an isomorphism for G ∈ D(X• ).

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms

2 The canonical map •,L • R Hom•OX• (F, G) ⊗•,L OX• H → R HomOX• (F, G ⊗OX• H)

is an isomorphism for G, H ∈ D(X• ). Proof. 1 It suﬃces to show that HJ : (?)J Hom•OX• (F, G) → Hom•OX• | (FJ , GJ ) J

is an isomorphism of complexes if G is a K-injective complex in C(Mod(X• )), since FJ is K-ﬂat and GJ is weakly K-injective. The assertion follows immediately by Lemma 6.36. 2 We may assume that F is a single locally free sheaf. By 1, we may assume n that X = X• is a single scheme. We may assume that X is aﬃne and F = OX for some n. This case is trivial. ⊓ ⊔ (28.3) Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). An OX• -module M is said to be locally of ﬁnite projective dimension if Mi,x is of ﬁnite projective dimension as an OXi ,x -module for any point (i, x) of X• . We say that M has ﬁnite projective dimension if there exists some non-negative integer d such that proj.dimOX ,x Mi,x ≤ d for any point (i, x) of X• . i

Lemma 28.4. Let I be a small category, S a scheme, and X• ∈ P(I, Sch/S). Assume that X• has ﬂat arrows and is locally noetherian. If F is a complex in C(Mod(X• )) with bounded coherent cohomology groups which have ﬁnite projective dimension, then the canonical map •,L • R Hom•OX• (F, G) ⊗•,L OX• H → R HomOX• (F, G ⊗OX• H)

is an isomorphism for G, H ∈ D(X• ). Proof. We may assume that G = OX• . By the way-out lemma, we may assume that F is a single coherent sheaf which has ﬁnite projective dimension, say d. By Lemma 13.9, it is easy to see that ExtiOX• (F, G) = 0 (i > d) for G ∈ Mod(X• ). In particular, R Hom•OX• (F, ?) is way-out in both directions. On the other hand, as R HomOX• (F, OX• ) has ﬁnite ﬂat dimension, and hence

R HomOX• (F, OX• )⊗•,L OX• ? is also way-out in both directions. By the way-out lemma, we may assume that H is a single OX• -module. By Lemma 13.9, we may assume that X = X• is a single scheme. The question is local, and we may assume that X = Spec A is aﬃne. Moreover, we may assume that F is a complex of sheaves associated with a ﬁnite projective resolution of a single ﬁnitely generated module. As F is perfect, the result follows from Lemma 28.2. ⊓ ⊔

28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms

441

(28.5) Let S, I, and X• be as above. For a locally free sheaf F over X• , we denote HomOX• (F, OX• ) by F ∨ . It is easy to see that F ∨ is again locally free. If L is an invertible sheaf, then tr → HomOX• (L, L) ∼ OX• − = L∨ ⊗OX• L

are isomorphisms. (28.6) Let I be a small category, S a scheme, and i• : Y• → X• a closed immersion in P(I, Sch/S). Then the canonical map η : OX• → (i• )∗ OY• is an epimorphism in Lqc(X• ). Set I := Ker η. Then I is a locally quasi-coherent ideal of OX• . Conversely, if I is a given locally quasi-coherent ideal of OX• , then i• →X• Y• := Spec• OX• /I − is deﬁned appropriately, and i• is a closed immersion. Thus the isomorphism classes of closed immersions to X• in the category P(I, Sch/S)/X• and locally quasi-coherent ideals of OX• are in one-to-one correspondence. We call I the deﬁning ideal sheaf of Y• . Note that i• is cartesian if and only if (i• )∗ OY• is equivariant. If X• has ﬂat arrows, this is equivalent to say that I is equivariant. (28.7) Let X• be locally noetherian. A morphism i• : Y• → X• is said to be a regular embedding, if i• is a closed immersion such that ij : Yj → Xj is a regular embedding for each j ∈ ob(I), or equivalently, I is locally coherent and Ij,x is a complete intersection ideal of OXj ,x for any j ∈ ob(I) and x ∈ Xj . If this is the case, we say that I is a local complete intersection ideal sheaf. A cartesian closed immersion i• : Y• → X• with X• locally noetherian with ﬂat arrows is a cartesian regular embedding if and only if i∗• I is locally free and Ii,x is of ﬁnite projective dimension as an OXi ,x -module for any i ∈ I and x ∈ Xi . Note that i∗• I ∼ = I/I 2 , and we have htOXi ,x Ix = rankOYi ,y (i∗• I)i,y for any point (i, y) of Y• , where x = ii (y). We call these numbers the codimension of I at (i, y). Proposition 28.8. Let I be a small category, S a scheme, and i• : Y• → X• a morphism in P(I, Sch/S). Assume that X• is locally noetherian with ﬂat arrows and i• is a cartesian regular embedding. Let I be the deﬁning ideal of Y• , and assume that Y• has a constant codimension d. Then we have the following.

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28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms

1 ExtiOX• ((i• )∗ OY• , OX• ) = 0 for i = d. 2 The canonical map ExtdOX• ((i• )∗ OY• , OX• ) → ExtdOX• ((i• )∗ OY• , (i• )∗ OY• ) is an isomorphism. 3 The Yoneda algebra Ext•OY• ((i• )∗ OY• , (i• )∗ OY• ) :=

ExtjOY• ((i• )∗ OY• , (i• )∗ OY• )

j≥0

• ∗ ∨ is isomorphic to the exterior algebra (i• )∗ (i• I) as graded OX• algebras. 4 There is an isomorphism d ∗ ∨ i♮• OX• ∼ (i• I) [−d]. = 5 For F ∈ D+ (X• ), there is a functorial isomorphism i♮• F ∼ =

d

∗ (i∗• I)∨ ⊗•,L OY• Li• F[−d].

Proof. 1 is trivial, since Ii,x is a complete intersection ideal of the local ring OXi ,x of codimension d for any point (i, x) of X• . 2 Note that (i• )∗ OY• ∼ = OX• /I. From the short exact sequence 0 → I → OX• → OX• /I → 0, we get an isomorphism Ext1OX• ((i• )∗ OY• , (i• )∗ OY• ) ∼ = (i• )∗ (i∗• I)∨ . = HomOX• (I, OX• /I) ∼ The canonical map (i• )∗ (i∗• I)∨ ∼ = Ext1OX• ((i• )∗ OY• , (i• )∗ OY• ) ֒→ Ext•OX• ((i• )∗ OY• , (i• )∗ OY• ) is uniquely extended to an OX• -algebra map T• ((i• )∗ (i∗• I)∨ ) → Ext•OX• ((i• )∗ OY• , (i• )∗ OY• ), where T• denotes the tensor algebra. It suﬃces to prove that this map is an epimorphism, which induces an isomorphism • ((i• )∗ (i∗• I)∨ ) → Ext•OX• ((i• )∗ OY• , (i• )∗ OY• ). In fact, the exterior algebra is compatible with base change, and

28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms

•

((i• )∗ (i∗• I)∨ ) ∼ = (i• )∗ i∗•

443

•

((i• )∗ (i∗• I)∨ ) • ∗ • ∗ ∨ ∼ ((i• (i• )∗ )(i∗• I)∨ ) ∼ (i• I) . = (i• )∗ = (i• )∗

To verify this, we may assume that i• : Y• → X• is a morphism of single schemes, X• = Spec A aﬃne, and I = I˜ generated by an A-sequence. The proof for this case is essentially the same as [19, Lemma IV.1.1.8], and we omit it. 4 Let Z denote the ringed site (Zar(X• ), (i• )∗ OY• ), and g : Z → Zar(Y• ) the associated admissible ringed continuous functor. By 2–3, there is a sequence of isomorphisms in Coh(X• ) (i• )∗

d

(i∗• I)∨ ∼ = ExtdOY• ((i• )∗ OY• , (i• )∗ OY• ) ∼ = ExtdOX• ((i• )∗ OY• , OX• ).

In view of 1, there is an isomorphism d ∗ ∨ ∼ Rg # (i• I) = R Hom•OX• (OZ , OX• )[d] b (Z). Applying g# to both sides, we get in DCoh

d

(i∗• I)∨ ∼ = i♮• OX• [d].

5 is an immediate consequence of 4 and Lemma 28.4.

⊓ ⊔

(28.9) Let I and S be as in (28.6). Let f• : X• → Y• be a morphism in P(I, Sch/S). Assume that f• is separated so that the diagonal ΔX• /Y• : X• → X• ×Y• X• is a closed immersion. Deﬁne ΩX• /Y• := i∗• I, where I := Ker(η : OX• ×Y• X• → (ΔX• /Y• )∗ OX• ). Note that (ΔX• /Y• )∗ ΩX• /Y• ∼ = I/I 2 . Lemma 28.10. Let the notation be as above. Then we have 1 2 3

ΩX• /Y• is locally quasi-coherent. If f• is cartesian, then ΩX• /Y• is quasi-coherent. For i ∈ ob(I), there is a canonical isomorphism ΩXi /Yi ∼ = (ΩX• /Y• )i .

Proof. Easy.

⊓ ⊔

Theorem 28.11. Let I be a ﬁnite ordered category, and f• : X• → Y• a morphism in P(I, Sch). Assume that Y• is noetherian with ﬂat arrows, and f• is separated cartesian smooth of ﬁnite type. Assume that f• has a con+ (Y• ), there is a functorial stant relative dimension d. Then for any F ∈ DLqc isomorphism d ΩX• /Y• [d] ⊗•OX• f•∗ F ∼ = f•! F, where [d] denotes the shift of degree. Proof. In view of dTheorem 26.14, it suﬃces to show that there is an isomorΩX• /Y• [d]. Consider the commutative diagram phism f•! OY• ∼ =

444

28 Cartesian Regular Embeddings and Cartesian Smooth Morphisms p2 - X• X• Δ- X• ×Y• X• Q Q p1 f• idQ ? ? Q f• s X• Q - Y• .

By Lemma 7.17 and Lemma 7.16, the all morphisms in the diagrams are cartesian. As p1 is smooth of ﬁnite type of relative dimension d, Δ is a cartesian regular embedding of the constant codimension d. By Theorem 21.8, Theorem 27.8, and Proposition 28.8, we have OX• ∼ = f•∗ OY• ∼ = Δ! p!1 f•∗ OY• ∼ = Δ♮ p∗2 f•! OY• ∼ = d ∨ d ∨ •,L ∗ ∗ ! ! ΩX• /Y• [−d] ⊗OX• LΔ p2 f• OY• ∼ ΩX• /Y• [−d] ⊗•,L = OX• f• OY• . As

d

ΩX• /Y• is an invertible sheaf, we are done.

⊓ ⊔

Chapter 29

Group Schemes Flat of Finite Type

(29.1)

Let S be a scheme.

(29.2) Let F (resp. FM ) denote the subcategory of P((Δ), Sch/S) (resp. subcategory of P(ΔM , Sch/S)) consisting of noetherian objects with ﬂat arrows and cartesian morphisms separated of ﬁnite type. Let G be a ﬂat S-group scheme of ﬁnite type. Note that G is faithfully ﬂat over S. A G-scheme is an S-scheme with a left G-action by deﬁnition. Set AG to be the category of noetherian G-schemes and G-morphisms separated of ﬁnite type. For X ∈ AG , we associate a simplicial scheme BG (X) by BG (X)n = Gn ×X. For n ≥ 1, di (n) : Gn ×X → Gn−1 ×X is the projection p×1Gn−1 ×X if i = n, where p : G → S is the structure morphism. While di (n) = 1Gn−1 ×a if i = 0, where a : G × X → X is the action. If 0 < i < n, then di (n) = 1Gn−i−1 × μ × 1Gi−1 ×X , where μ : G × G → G is the product. For n ≥ 0, si (n) : Gn × X → Gn+1 × X is given by si (n)(gn , . . . , g1 , x) = (gn , . . . , gi+1 , e, gi , . . . , g1 , x), where e : S → G is the unit element. Indeed, BG (X) satisﬁes the relations (11op ), (12op ), and (13op ) in [29, (VII.5)]. Note that (BG (X)′ )|(∆) is canonically isomorphic to BG (G × X), where G × X is viewed as a principal G-action. Note also that there is an isomorphism from BG (X)′ to Nerve(p2 : G × X → X) given by BG (X)′n = Gn+1 × X → (G × X) ×X · · · ×X (G × X) = Nerve(p2 )n (gn , . . . , g0 , x) → ((gn · · · g0 , x), (gn−1 · · · g0 , x), . . . , (g0 , x)). Hence we have Lemma 29.3. Let G and X be as above. Then BG (X) is a simplicial S-groupoid with d0 (1) and d1 (1) faithfully ﬂat of ﬁnite type.

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29 Group Schemes Flat of Finite Type

M M We denote the restriction BG (X)|∆M by BG (X). Obviously, BG (X) is an S-groupoid with d0 (1) and d1 (1) faithfully ﬂat of ﬁnite type. For a morphism f : X → Y in AG , we deﬁne BG (f ) : BG (X) → BG (Y ) by (BG (f ))n = 1Gn × f . It is easy to check that BG is a functor from AG to M F. Thus BG is a functor from AG to FM . We deﬁne a (G, OX )-module to be an OBGM (X) -module. That is, an M object of Mod(BG (X)). So an equivariant (resp. locally quasi-coherent, quasi-coherent, coherent) (G, OX )-module is an equivariant (resp. locally M quasi-coherent, quasi-coherent, coherent) object of Mod(BG (X)). The category of G-linearized OX -modules in [32] is equivalent to that of our equivariant (G, OX )-modules. See also [6] and [19]. We denote the category of (G, OX )-modules by Mod(G, X). The category of equivariant (resp. locally quasi-coherent, quasi-coherent, coherent) (G, OX )-modules is denoted by EM(G, X) (resp. Lqc(G, X), Qch(G, X), Coh(G, X)). Note that EM(BG (X)) (resp. Qch(BG (X)), Coh(BG (X))) is equivalent to EM(G, OX ) (resp. Qch(G, X), Coh(G, X)) (Lemma 9.4). However, the M author does not know whether Mod(BG (X)) is equivalent to Mod(BG (X)). From our point of view, it seems that it is more convenient to work over ΔM , which is a ﬁnite ordered category, than (Δ). The discussion on derived categories of categories of sheaves over diagrams of schemes are interpreted to the derived categories of the categories of (G, OX )-modules. By Lemma 29.3 and Lemma 12.8, we have

Lemma 29.4. Let X ∈ AG . Then Qch(G, X) is a locally noetherian abelian category. M ∈ Qch(G, X) is a noetherian object if and only if M0 is coherent if and only if M ∈ Coh(G, X). Let M be a (G, OX )-module. If there is no danger of confusion, we may write M0 instead of M. For example, OX sometimes means OBGM (X) , since (OBGM (X) )0 = OX . This abuse of notation is what we always do when S = X = Spec k and G is an aﬃne algebraic group over k. A G-module and its underlying vector space are denoted by the same symbol. Similarly, an object M M (X)) and its restriction to D(BG (X)0 ) = D(X) are sometimes of D(BG denoted by the same symbol. Moreover, for a morphism f in AG , we denote M M for example R(BG (f ))∗ by Rf∗ , and BG (f )! by f ! . M D(BG (X)), or D(Mod(G, X)), is denoted by D(G, X) for short. Thus for + + example, DQch(G,X) (Mod(G, X)) is denoted by DQch (G, X). Thus, as a corollary to Theorem 25.2, we have Theorem 29.5 (G-Grothendieck’s duality). Let S be a scheme, and G a ﬂat S-group scheme of ﬁnite type. Let X and Y be noetherian S-schemes with G-actions, and f : X → Y a proper G-morphism. Then the composite

29 Group Schemes Flat of Finite Type

447 H

→R Hom•Mod(G,Y ) (Rf∗ F, Rf∗ f × G) Θ(f ) : Rf∗ R Hom•Mod(G,X) (F, f × G)− ε

− →R Hom•Mod(G,Y ) (Rf∗ F, G) + is an isomorphism in D(G, Y ) for F ∈ DQch (G, X) and G ∈ DQch (G, Y ).

Chapter 30

Compatibility with Derived G-Invariance

(30.1) Let S be a scheme, and G a ﬂat S-group scheme. Let X be an S-scheme with a trivial G-action. That is, a : G × X → X agrees with the M (X). second projection p2 . In other words, d0 (1) = d1 (1) in BG For an object M of Mod(G, X), we deﬁne the G-invariance of M to be the kernel of the natural map βd0 (1) − βd1 (1) : M0 → d0 (1)∗ M1 = d1 (1)∗ M1 , and we denote it by MG . (30.2)

˜ M (X) to be the augmented diagram Let X be as in (30.1). Deﬁne B G 1 ×a

G −−→

μ×1X G ×S X G ×S G ×S X −−→ p23

−−→

a

−−→ id → X. p2 X − −−→

˜ M (X) is an object of P(Δ+ , Sch/S). For an S-morphism f : X → Y Note that B G M ˜ M (f ) : B ˜ M (X) → B ˜ M (Y ) is between S-schemes with trivial G-actions, B G G G M M ˜ ˜ ˜M deﬁned by BG (f )n = 1Gn × f for n ≥ 0 and BG (f )−1 = f . Thus B G is a functor from the category of S-schemes (with trivial G-actions) to the M ˜M ˜M category P(Δ+ M , Sch/S) such that (?)|∆M BG = BG and (?)|−1 BG = Id. Lemma 30.3. The functor (?)G : Mod(G, X) → Mod(X) agrees with (?)−1 R∆M . Proof. Follows easily from (6.14).

⊓ ⊔

(30.4) We say that an object M of Mod(G, X) is G-trivial if M is equivariant, and the canonical inclusion MG → M0 is an isomorphism. Note that (?)∆M L−1 is the exact left adjoint of (?)G . Note also that M is G-trivial if and only if the counit of adjunction ε : (?)∆M L−1 MG → M is an isomorphism if ˜ M (X)). and only if M ∼ = N∆M for some N ∈ EM(B G J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

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30 Compatibility with Derived G-Invariance

M Let triv(G, X) denote the full subcategory of Mod(BG (X)) consisting of G G-trivial objects. Note that (?) : triv(G, X) → Mod(X) is an equivalence, whose quasi-inverse is (?)∆M L−1 . Assume that G is concentrated over S. If M is locally quasi-coherent, then + (G, X) → MG is quasi-coherent. Thus we get a derived functor R(?)G : DLqc + DQch (X).

Proposition 30.5. Let G be of ﬁnite type over S. Let X and Y be noetherian S-schemes with trivial G-actions, and f : X → Y an S-morphism, which is automatically a G-morphism, separated of ﬁnite type. Then there is a canonical isomorphism f ! R(?)G ∼ = R(?)G f ! + + between functors from DLqc (G, Y ) to DQch (X).

Proof. As (?)−1 is exact, we have R(?)G ∼ = (?)−1 RR∆M by Lemma 30.3. Thus, we have a composite isomorphism ¯−1

ξ c ¯ ˜ G (f ))! RR∆ − f ! R(?)G ∼ →(?)−1 RR∆M f ! ∼ = f ! (?)−1 RR∆M −−→(?)−1 (B = R(?)G f ! M M

by Lemma 21.3 and Lemma 24.7.

⊓ ⊔

Chapter 31

Equivariant Dualizing Complexes and Canonical Modules

(31.1) Let A be a Grothendieck category, and I ∈ D(A). We say that I has a ﬁnite injective dimension if R HomA (?, I) is way-out in both directions, see [17, (I.7)]. By deﬁnition, an object of C(A) or K(A) has a ﬁnite injective dimension if it does in D(A). F ∈ C(A) has a ﬁnite injective dimension if and only if there is a bounded complex J of injective objects in A and a quasi-isomorphism F → J. (31.2) Let I be a ﬁnite ordered category, S a scheme, and X• ∈ P(I, Sch/S). Lemma 31.3. Assume that X• has ﬂat arrows. Let I ∈ D(X• ). Then I has a ﬁnite injective dimension if and only if Ii has a ﬁnite injective dimension for any i ∈ ob(I). Proof. We prove the ‘only if’ part. Since (?)i is exact and has an exact left adjoint Li , and I has a ﬁnite injective dimension, Ii has a ﬁnite injective dimension for i ∈ ob(I). We prove the converse by induction on the number of objects of I. We may assume that I has at least two objects. Let i be a maximal element of ob(I). There is a triangle of the form u

→ RRi (?)i I → C → I[1]. I− Since Ii has a ﬁnite injective dimension and Ri has an exact left adjoint (?)i , it is easy to see that RRi (?)i I has a ﬁnite injective dimension. So it suﬃces to show that C has a ﬁnite injective dimension. Applying (?)i to the triangle above, it is easy to see that Ci = 0. Let J be the full subcategory of I such that ob(J) = ob(I) \ {i}. Then u : C → RJ CJ is an isomorphism by Lemma 18.9. On the other hand, by the only if part, which has already been proved, it is easy to see that Cj has a ﬁnite injective dimension for j ∈ ob(J). By induction assumption, CJ has a ﬁnite injective dimension.

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31 Equivariant Dualizing Complexes and Canonical Modules

So C ∼ = RJ CJ has a ﬁnite injective dimension, since RJ is exact and has an ⊓ ⊔ exact left adjoint (?)J . (31.4) Let the notation be as in Theorem 20.4. Let X• be an object of F (i.e., an I op -diagram of noetherian S-schemes with ﬂat arrows). We say that F ∈ D(X• ) is a dualizing complex of X• if F ∈ DCoh (X• ), F has a ﬁnite injective dimension, and the canonical map tr

OX• − →R Hom•OX• (F, F) is an isomorphism. A complex F ∈ C(Mod(X• )) is said to be a dualizing complex if it is as an object of D(X• ). (31.5) If there is a dualizing complex of X• , then it is represented by a bounded injective complex F ∈ C(Mod(X• )) with coherent cohomology groups such that tr → Hom•OX• (F, F) OX• − is a quasi-isomorphism. More is true. We may further assume that F ∈ C(Lqc(X• )). Indeed, we may replace F above by lqc F. Since F has coherent cohomology groups, it is easy to see that the canonical map lqc F → F is a quasi-isomorphism by Lemma 14.3, 3. Each term of lqc F is an injective object of Lqc(X• ), since lqc has an exact left adjoint. Note that each term of lqc F is still injective in Mod(X• ) by Lemma 24.2. Lemma 31.6. Let the notation be as in (31.4). An object F ∈ D(X• ) is a dualizing complex of X• if and only if F has equivariant cohomology groups and Fi ∈ D(Xi ) is a dualizing complex of Xi for any i ∈ ob(I). Proof. This is obvious by Lemma 13.9 and Lemma 31.3.

⊓ ⊔

Corollary 31.7. Let the notation be as in (31.4). If X• is Gorenstein with ﬁnite Krull dimension, then OX• is a dualizing complex of X• . Proof. This is clear by the lemma and [17, (V.10)].

⊓ ⊔

Lemma 31.8. Let the notation be as in (31.4). If X• has a dualizing complex F, then X• has ﬁnite Krull dimensions, and X• has Gorenstein arrows. Proof. As Fi is a dualizing complex of Xi for each i ∈ ob(I), X• has ﬁnite Krull dimensions by [17, Corollary V.7.2]. Let φ : i → j be a morphism of I. As Xφ is ﬂat, αφ : Xφ∗ Fi → Fj is an isomorphism of D(Xj ). As Xφ∗ Fi is a dualizing complex of Xj , Xφ is Gorenstein by [4, (5.1)]. ⊓ ⊔ Proposition 31.9. Let the notation be as above, and I a dualizing complex of X• . Let F ∈ DCoh (X• ). Then we have

31 Equivariant Dualizing Complexes and Canonical Modules

453

1 R Hom•OX• (F, I) ∈ DCoh (X• ). 2 The canonical map F → R Hom•OX• (R Hom•OX• (F, I), I) is an isomorphism for F ∈ DCoh (X• ). Proof. 1 As I has a ﬁnite injective dimension, R Hom•OX• (?, I) is way-out in both directions. Hence by [17, Proposition I.7.3], we may assume that F is bounded. This case is trivial by Lemma 13.10. 2 Using Lemma 13.9 twice, we may assume that X• is a single scheme. This case is [17, Proposition V.2.1]. ⊓ ⊔ Lemma 31.10. Let X be a noetherian scheme, and U = (Ui ) a ﬁnite open covering of X. Let I ∈ D(X). Then I is dualizing if and only if I|Ui is dualizing for each i. Proof. It is obvious that I has coherent cohomology groups if and only if I|Ui has coherent cohomology groups for each i. Assume that I is a bounded injective complex. Then I|Ui is a bounded injective complex, since (?)|Ui preserves injectives. Conversely, assume that I has coherent cohomology groups and I|Ui has a ﬁnite injective dimension for each i. Then by [17, (II.7.20)], there is an integer n0 such that for any i, any G ∈ Coh(Ui ), and any j > n0 , we have ExtjOU (G, I|Ui ) = 0. This shows i

that for any G ∈ Coh(X) and any j > n0 , ExtjOX (G, I) = 0, and again by [17, (II.7.20)], we have that I has a ﬁnite injective dimension. Let I be a bounded injective complex. Let C be the mapping cone of tr : OX → HomOX (I, I). C is exact (i.e., tr is a quasi-isomorphism) if and only if C|Ui is exact for each i. On the other hand, C|Ui is isomorphic to the mapping cone of the trace map OUi → HomOU (I|Ui , I|Ui ). Thus tr : OX → i HomOX (I, I) is a quasi-isomorphism if and only if OUi → HomOU (I|Ui , I|Ui ) i is a quasi-isomorphism for each i. Thus the lemma is obvious now. ⊓ ⊔ Lemma 31.11. Let the notation be as in (31.4). Let f• : X• → Y• be a morphism in F, and let I be a dualizing complex of Y• . Then f•! (I) is a dualizing complex of X• . Proof. By Corollary 22.4, f•! (I) has coherent cohomology groups. By Lemma 31.6 and Proposition 18.14, we may assume that f : X → Y is a morphism of single schemes. By Lemma 31.10, the question is local both on Y and X, So we may assume that both Y and X are aﬃne, and f is either an aﬃne n-space or a closed immersion. These cases are done in [17, Chapter V]. ⊓ ⊔ Lemma 31.12. Let the notation be as in (31.4), and I and J dualizing complexes on X• . If X• is d-connected and Xi is non-empty for some i ∈ ob(I), then there exist a unique invertible sheaf L and a unique integer n such that

454

31 Equivariant Dualizing Complexes and Canonical Modules •,L J∼ = I ⊗OX• L[n].

Such L and n are determined by L[n] ∼ = R Hom•OX• (I, J). Proof. Use [17, Theorem V.3.1]. Deﬁnition 31.13. Let the notation be as in (31.4), and I a ﬁxed dualizing complex of X• . For any object f• : Y• → X• of F/X• , we deﬁne the dualizing complex of Y• (or better, of f• ) to be f•! I. It is certainly a dualizing complex of Y• by Lemma 31.11. If Y• is d-connected and Yi is non-empty for some i ∈ ob(I), then we deﬁne the canonical sheaf ωY• of Y• (or better, f• ) to be H s (f•! I), where s is the smallest i such that H i (f•! I) = 0. If Y• is not d-connected, then we deﬁne ωY• componentwise. Lemma 31.14. Let S be a noetherian scheme, and G a ﬂat S-group scheme of ﬁnite type. Then G → S is a (ﬂat) local complete intersection morphism. That is, (it is ﬂat and) all ﬁbers are locally complete intersections. Proof. We may assume that S = Spec k, with k a ﬁeld. Then by [3, Theorem 1], we may assume that k is algebraically closed. First assume that the characteristic is p > 0. Then there is some r ≫ 0 such that the scheme theoretic image of the Frobenius map F r : G → G(r) (r) is reduced (or equivalently, k-smooth) and agrees with Gred . Note that the (r) induced morphism G → Gred is ﬂat, since the ﬂat locus is a G-stable open subset of G [19, Lemma 2.1.10], and the morphism is ﬂat at the generic point. As the group Gred acts on G transitively, it suﬃces to show that G is locally a complete intersection at the unit element e. So by [3, Theorem 2], it suﬃces to show that the rth Frobenius kernel Gr is a complete intersection. As Gr is ﬁnite connected, this is well known [46, (14.4)]. Now consider the case that G is of characteristic zero. We are to prove that G is k-smooth. Take a ﬁnitely generated Z-subalgebra R of k such that G is deﬁned. We may take R so that GR is R-ﬂat of ﬁnite type. Set H := (GR )red . We may take R so that H is also R-ﬂat. Then H is a closed subgroup scheme over R, since Spec R and H ×R H are reduced. As a reduced group scheme over a ﬁeld of characteristic zero is smooth, we may localize R if necessary, and we may assume that H is R-smooth. Let J be the deﬁning ideal sheaf

s of H in GR . There exists some s ≥ 0 such that J s+1 = 0. Note that G := i=0 J i /J i+1 is a coherent (H, OH )-module. M Applying Corollary 10.15 to the case that Y = Spec R and X• = BH (H), ∗ ˜ the coherent (H, OH )-module G is of the form f (V ), where V is a ﬁnite Rmodule, and f : H → Spec R is the structure map. Replacing R if necessary, we may assume that V ∼ = Ru . Now we want to prove that u = 1 so that H = GR , which implies G is k-smooth.

31 Equivariant Dualizing Complexes and Canonical Modules

455

There exists some prime number p > u and a maximal ideal m of R such that R/m is a ﬁnite ﬁeld of characteristic p. Let κ be the algebraic closure of

R/m, and consider the base change (¯?) :=? ⊗R κ. Note that G¯ = i J i /J i+1 r (recall that R is Z-ﬂat and H is R-ﬂat). Let I [p ] denote the deﬁning ideal ¯ R . By [46, (14.4)] again, dimk (OG¯ /I [pr ] )e of the rth Frobenius kernel of G R is a power of p, say pv(r) . Similarly, the k-dimension of the coordinate ring r ¯ is a power of p, say (OG¯ R /(J¯ + I [p ] ))e of the rth Frobenius kernel of H w(r) p . Note that r ¯ e = pw(r) u < pw(r)+1 . pw(r) ≤ pv(r) ≤ dimk (OG¯ R /I [p ] ⊗OG¯ R G) r

[p ] Hence w(r) ≤ v(r) < w(r) +1, and we have J¯e ⊂ Ie for any r. By Krull’s pr ¯ R is reduced at ¯ intersection theorem, Je ⊂ r (Ie ) = 0. This shows that G ¯ e, which shows that GR is κ-smooth everywhere. So the nilpotent ideal J¯ must be zero, and this shows u = 1. ⊓ ⊔

(31.15) Let S be a scheme, G a ﬂat S-group scheme of ﬁnite type, and X a noetherian G-scheme. By deﬁnition, a G-dualizing complex of X is a M (X). Let us ﬁx X and a G-dualizing complex I. For dualizing complex of BG (f : Y → X) ∈ AG /X, we deﬁne the G-dualizing complex of Y (or better, of f ) to be f ! (I). It is certainly a G-dualizing complex of Y . The canonical M sheaf of BG (Y ) is called the G-canonical sheaf of Y , and is denoted by ωY . Lemma 31.16. Let f : X → Y be a Gorenstein ﬂat morphism of ﬁnite type between noetherian schemes. If I is a dualizing complex of Y , then f ∗ (I) is a dualizing complex of X. Proof. Since Y has a dualizing complex, Y has ﬁnite Krull dimension [17, Corollary V.7.2]. Since X is of ﬁnite type over Y , X has ﬁnite Krull dimension. By [17, Proposition V.8.2], it suﬃces to show that f ∗ (I) is pointwise dualizing. So we may assume that X = Spec B and Y = Spec A are aﬃne, both A and B are local, and f is induced by a local homomorphism from A to B. Then the assertion follows from [4, (5.1)]. ⊓ ⊔ Lemma 31.17. Let S, G, and X be as in (31.15). Then I ∈ D(G, X) is a G-dualizing complex of X if and only if I has equivariant cohomology groups and I0 ∈ D(X) is a dualizing complex of X. Proof. The ‘only if’ part is obvious by Lemma 31.6. To prove the converse, it suﬃces to show that Ii ∈ D(Gi × X) is dualizing M for i = 1, 2 by the same lemma. Since BG (X) has ﬂat arrows and I has equivariant cohomology groups, αρi (i) : ri (i)∗ I0 → Ii is an isomorphism in D(Gi × X). Since ri (i) is Gorenstein ﬂat of ﬁnite type, the assertion follows from Lemma 31.16. ⊓ ⊔ Lemma 31.18. Let R be a Gorenstein local ring of dimension d, and S = M (S) is Gorenstein of ﬁnite Krull dimension. In particular, Spec R. Then BG

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31 Equivariant Dualizing Complexes and Canonical Modules

OS [d] is a G-dualizing complex of S (i.e., OBGM (S) [d] is a dualizing complex M (S)). of BG Proof. As S = Spec R is Gorenstein by assumption and G is Gorenstein over S by Lemma 31.14, the assertions are trivial. ⊓ ⊔ (31.19) When R, S and d are as in the lemma, then we usually choose and ﬁx the G-dualizing complex OS [d] of S. Thus for an object X ∈ AG = AG /S, the G-dualizing complex of X is f ! (OS [d]), where f is the structure morphism X → S of X. The G-canonical sheaf is deﬁned accordingly. Lemma 31.20. Let R, S and d be as in Lemma 31.18. Let X ∈ AG , and assume that G acts on X trivially. Then the dualizing complex IX := f ! (OS [d]) has G-trivial cohomology groups, where f : X → S is the structure map. In particular, ωX is G-trivial. Proof. By Proposition 18.14, M ˜G (f )! (OB˜ M (S) ))[d]. f ! (OS [d]) ∼ = f ! ((OB˜ M (S) )∆M )[d] ∼ = (?)∆M (B G

G

˜ M (f )! (O ˜ M ) has coherent cohomology groups. Hence, By Corollary 22.4, B G BG (S) f ! (OS [d]) has G-trivial cohomology groups. ⊓ ⊔

Chapter 32

A Generalization of Watanabe’s Theorem

Lemma 32.1. Let R be a noetherian commutative ring, and G a ﬁnite group which acts on R. Set A = RG , and assume that Spec A is connected. Then G permutes the connected components of Spec R transitively. Proof. Since Spec R is a noetherian space, Spec R has only ﬁnitely many connected components, say X1 , . . . , Xn . Then R = R1 × · · · × Rn , and each Ri is of the form Rei , where ei is a primitive idempotent. Note that E := {e1 , . . . , en } is the set of primitive idempotents of R, and G acts on E. Let E1 be an orbit of this action. Then e = ei ∈E1 ei is in A. As A does not have any nontrivial idempotent, e = 1. This shows that G acts on E transitively, and we are done. ⊓ ⊔ Lemma 32.2. Let R be a noetherian commutative ring, and G a ﬁnite group which acts on R. Set A = RG , and assume that the inclusion A ֒→ R is ﬁnite. If p ∈ Spec A, then G acts transitively on the set of primes of R lying over p. Moreover, the going-down theorem holds for the ring extension A ֒→ R. Proof. Note that A is noetherian by Eakin-Nagata theorem [30, Theorem 3.7]. Let A′ be the pAp -adic completion of Ap , and set R′ := A′ ⊗A R. As A′ is Aﬂat, A′ = (R′ )G . It suﬃces to prove that G acts transitively on the maximal ideals of R′ . But R′ is the direct product i Ri′ of complete local rings Ri′ . Consider the corresponding primitive idempotents. Since A′ is a local ring, G permutes these idempotents transitively by Lemma 32.1. It is obvious that this action induces a transitive action on the maximal ideals of R′ . We prove the last assertion. Let p ⊃ q be prime ideals of A, and P be a prime ideal of R such that P ∩ A = p. By the lying over theorem [30, Theorem 9.3], there exists some prime ideal Q′ of R such that Q′ ∩ A = q. By the going-up theorem [30, Theorem 9.4], there exists some prime P ′ ⊃ Q′ such that P ′ ∩ A = p. Then there exists some g ∈ G such that gP ′ = P . ⊓ ⊔ Letting Q := gQ′ , we have that Q ⊂ P , and Q ∩ A = q.

J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

457

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32 A Generalization of Watanabe’s Theorem

(32.3) Let k be a ﬁeld, and G a ﬁnite k-group scheme. Let S = Spec R be an aﬃne k-scheme of ﬁnite type with a left G-action. It gives a k-algebra automorphism action of G on R. Let A := RG be the ring of invariants. Proposition 32.4. Assume that G is linearly reductive (i.e., any G-module is semisimple). Then the following hold. 1 If R satisﬁes Serre’s (Sr ) condition, then the A-module R satisﬁes (Sr ), and A satisﬁes (Sr ). 2 If R is Cohen-Macaulay, then R is a maximal Cohen-Macaulay A-module, and A is also a Cohen-Macaulay ring. G ∼ 3 If R is Cohen-Macaulay, then ωR = ωA as A-modules. 4 Assume that R is Gorenstein and ωR ∼ = R as (G, R)-modules. Then A = RG is Gorenstein and ωA ∼ = A. Proof. Note that the associated morphism π : S = Spec R → Spec A is ﬁnite surjective. To prove the proposition, we may assume that Spec A is connected. ¯ where k¯ is the algebraic closure of k. ¯ and R ¯ = R ⊗k k, ¯ := G ⊗k k, Set G Let G0 be the identity component (or the Frobenius kernel for suﬃciently ¯ which is a normal high Frobenius maps, if the characteristic is nonzero) of G, ¯ Note that Spec R ¯ → Spec R ¯ G0 is ﬁnite and is a homesubgroup scheme of G. ¯ G0 contains omorphism, since G0 is trivial if the characteristic is zero, and R ¯ if the characteristic is positive. some suﬃciently high Frobenius power of R, ¯ acts on R ¯ = (G/G ¯ G0 , and the ¯ k) ¯ 0 )(k) On the other hand, the ﬁnite group G( ¯ By Lemma 32.2, for any prime ring of invariants under this action is A ⊗k k. ¯ G( ¯ acts transitively on the set of prime ideals of R ¯ k) ¯ (or ideal p of A ⊗k k, ¯ G0 ) lying over p. It follows that for any prime ideal p of A and a prime ideal R P of R lying over p, we have ht p = ht P. Let M be the sum of all non-trivial simple G-submodules of R. As G is linearly reductive, R is the direct sum of M and A as a G-module. It is easy to see that R = M ⊕ A is a direct sum decomposition as a (G, A)-module. 1 Since A is a direct summand of R as an A-module, it suﬃces to prove that the A-module R satisﬁes the (Sr )-condition. Let p ∈ Spec A and assume that depthAp Rp < r. Note that depthAp Rp = inf P depth RP , where P runs through the prime ideals lying over p. So there exists some P such that depth RP ≤ depthAp Rp < r. As R satisﬁes Serre’s (Sr )-condition, we have that RP is Cohen-Macaulay. So ht p = ht P = depth RP ≤ depthAp Rp ≤ depth Ap ≤ ht p, and all ≤ must be =. In particular, Rp is a maximal Cohen-Macaulay Ap module. This shows that the A-module R satisﬁes Serre’s (Sr )-condition. 2 is obvious by 1. We prove 3. We may assume that Spec A is connected. Note that π : S = Spec R → Spec A is a ﬁnite G-morphism. Set d = dim R = dim A. As A is Cohen-Macaulay and Spec A is connected, A is equidimensional of

32 A Generalization of Watanabe’s Theorem

459

dimension d. So ht m = d for all maximal ideals of A. The same is true of R, and hence R is also equidimensional. So ωR [d] and ωA [d] are the equivariant dualizing complexes of R and A, respectively. In particular, we have π ! ωA ∼ = ωR . By Lemma 31.20, ωA is G-trivial. By Theorem 29.5, we have isomorphisms in D(G, Spec A) ωR ∼ = R HomOSpec A (Rπ∗ OSpec R , ωA ). = Rπ∗ R HomOSpec R (OSpec R , π ! ωA ) ∼ As π is aﬃne, Rπ∗ OSpec R = R. As R is a maximal Cohen-Macaulay A-module and ωA is a ﬁnitely generated A-module which is of ﬁnite injective dimension, we have that ExtiA (R, ωA ) = 0 (i > 0). Hence ωR ∼ = R HomOSpec A (R, ωA ) ∼ = HomA (R, ωA ) in D(G, Spec A). As G is linearly reductive, there is a canonical direct sum decomposition R ∼ = RG ⊕ UR (as an (G, A)-module), where UR is the sum of all non-trivial simple G-submodules of R. As ωA is G-trivial, HomG (UR , ωA ) = 0. In particular, HomA (UR , ωA )G = 0. On the other hand, we have that G HomA (RG , ωA )G = HomA (A, ωA )G = ωA = ωA .

Hence G ∼ ωR = ωA . = HomA (UR , ωA )G ⊕ HomA (RG , ωA )G ∼ = HomA (R, ωA )G ∼

4 follows from 2 and 3 immediately.

⊓ ⊔

Corollary 32.5. Let k be a ﬁeld, G a linearly reductive ﬁnite k-group scheme, and V a ﬁnite dimensional G-module. Assume that the representation G → GL(V ) factors through SL(V ). Then the ring of invariants A := (Sym V )G is Gorenstein, and ωA ∼ = A. n Proof. Set R := Sym V . As R is k-smooth, we that ωR ∼ ΩR/k ∼ = = R⊗ n nhave ∼ V , where n = dimk V . By assumption, V = k, and we have that ωR ∼ = R, as (G, R)-modules. By the proposition, A is Gorenstein and ωA ∼ ⊔ = A. ⊓ Although it has nothing to do with the twisted inverse, we give some normality results on invariant subrings under the action of group schemes. For a ring R, let R⋆ denote the set of nonzerodivisors of R. Lemma 32.6. Let S be a ﬁnite direct product of normal domains, R a commutative ring, and F a set of ring homomorphisms from S to R. Assume that f (s) ∈ R⋆ for any f ∈ F and s ∈ S ⋆ . Then A := {a ∈ S | f (a) = f ′ (a) for f, f ′ ∈ F } is a subring of S, and is a ﬁnite direct product of normal domains.

460

32 A Generalization of Watanabe’s Theorem

Proof. We may assume that F has at least two elements. It is obvious that A is closed under subtraction and multiplication, and 1 ∈ A. So A is a subring of S. We prove that A is a ﬁnite direct product of normal domains. Let h : A → R be the restriction of f ∈ F to A, which is independent of choice of f . Let e1 , . . . , er be the primitive idempotents of A. Replacing S by Sei , A by Aei , R by R(h(ei )), and F by {f |Sei | f ∈ F }, we may assume that A = 0 and that A does not have a nontrivial idempotent. Indeed, if sei ∈ (Sei )⋆ , then sei + (1 − ei ) ∈ S ⋆ as can be seen easily. So we have f (sei ) + (1 − f (ei )) ∈ R⋆ , and hence h(ei )f (sei ) = f (sei ) ∈ R(h(ei ))⋆ . Assume that a ∈ A \ {0} is a zerodivisor of S. Then there is a nontrivial idempotent e of S such that ae = a and 1 − e + a ∈ S ⋆ . Then for f ∈ F , h(a)f (e) = h(a), and 1 − f (e) + h(a) ∈ R⋆ . So for any f, f ′ ∈ F , f (e)(1 − f ′ (e)) = 0, since (1 − f (e) + h(a))f (e)(1 − f ′ (e)) = h(a)(1 − f ′ (e)) = h(a) − h(a) = 0. Similarly we have f ′ (e)(1 − f (e)) = 0, and hence f (e) = f (e)(1 − f ′ (e) + f ′ (e)) = f (e)f ′ (e) = (1 − f (e) + f (e))f ′ (e) = f ′ (e). This shows that e ∈ A, and this contradicts our additional assumption. Hence any nonzero element a of A is a nonzerodivisor of S. In particular, A is an integral domain, since the product of two nonzero elements of A is a nonzerodivisor of S and cannot be zero. Let K = Q(A) be the ﬁeld of fractions of A, and L = Q(S) be the total quotient ring of S. By the argument above, A ֒→ S ֒→ L can be extended to a unique injective homomorphism K ֒→ L. We regard K as a subring of L. As f (S ⋆ ) ⊂ R⋆ , f ∈ F is extended to the map Q(f ) : L = Q(S) → Q(R). Set B := {α ∈ L | Q(f )(α) = Q(f ′ )(α) in Q(R) for f, f ′ ∈ F }. Then B is a subring of L. Note that K ⊂ B. As R → Q(R) is injective, A = B ∩ S. If α ∈ K is integral over A, then it is an element of B ⊂ L which is integral over S. This shows that α ∈ B ∩ S = A, and we are done. ⊓ ⊔ Corollary 32.7. Let Γ be an abstract group acting on a ﬁnite direct product S of normal domains. Then S Γ is a ﬁnite direct product of normal domains. Proof. Set R = S and F = Γ, and apply the lemma.

⊓ ⊔

Corollary 32.8. Let H be an affine algebraic k-group scheme, and S an H-algebra which is a finite direct product of normal domains. Then S H is also a finite direct product of normal domains.

32 A Generalization of Watanabe’s Theorem

461

Proof. Set R = S ⊗ k[H], and F = {i, ω}, where i : S → R is given by i(s) = s ⊗ 1, and ω : S → R is the coaction. Since both i and ω are ﬂat, the lemma is applicable. ⊓ ⊔

Chapter 33

Other Examples of Diagrams of Schemes

We deﬁne an ordered ﬁnite category K by ob(K) = {s, t}, and o u K(s, t) = {u, v}. Pictorially, K looks like t o v s . Let p be a prime number, and X an Fp -scheme. We deﬁne the Lyubeznik diagram Ly(X) of X to be an object of P(K, Sch/Fp ) given by (Ly(X))s = (Ly(X))t = X, Ly(X)u = idX , and Ly(X)v = FX , where FX denotes the absolute Frobenius morphism of X. Thus Ly(X) looks like (33.1)

idX

X

FX

/ /X.

We deﬁne an F -sheaf of X to be a quasi-coherent sheaf over Ly(X). It can be identiﬁed with a pair (M, φ) such that M is a quasi-coherent OX ∗ module, and φ : M → FX M is an isomorphism of OX -modules. Indeed, if N ∈ Qch(Ly(X)), then letting M := Ns and setting φ to be the composite −1

αv αu ∗ ∗ M = Ns ∼ N s = FX M, = id∗X Ns = Ly(X)∗u Ns −−→ Nt −−→ Ly(X)∗v Ns = FX

(M, φ) is such a pair. Thus if X = Spec R is aﬃne, then the category Qch(Ly(X)) of F -sheaves of X is equivalent to the category of F -modules deﬁned by Lyubeznik [28]. Note that Ly(X) is noetherian with ﬂat arrows if and only if X is a noetherian regular scheme by Kunz’s theorem [24]. Let f : X → Y be a morphism of noetherian Fp -schemes. Then Ly(f ) : Ly(X) → Ly(Y ) is deﬁned in an obvious way. (33.2) For a ring A of characteristic p, the Frobenius map A → A (a → ap ) e is denoted by F = FA . So FAe (a) = ap . Let k be a perfect ﬁeld of characteristic p. For a k-algebra u : k → A, we deﬁne a k algebra A(r) as follows. As a ring, A(r) = A, but the k-algebra structure of A(r) is given by F −r

u

→ A. k −−k−→ k − J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics 1960, c Springer-Verlag Berlin Heidelberg 2009

463

464

33 Other Examples of Diagrams of Schemes

For e ≥ 0, FAe : A(r+e) → A(r) is a k-algebra map. We sometimes denote an element a ∈ A, viewed as an element of A(r) , by a(r) . Thus e F e (a(r) ) = (ap )(r−e) . For a k-scheme X, the k-scheme X (r) is deﬁned e : X (r) → X (r+e) is a k-morphism. similarly, and the Frobenius morphism FX This notation is used for k = Fp for all rings of characteristic p. Lemma 33.3. Let k be a ﬁeld of characteristic p, and K a ﬁnitely generated extension ﬁeld of k. Then the canonical map ΦRA : k ⊗k(1) K (1) → K (the Radu-Andr´e homomorphism) given by ΦRA (α ⊗ β (1) ) = αβ p is an isomorphism if and only if K is a separable algebraic extension of k. Proof. We prove the ‘if’ part. Note that k ⊗k(1) K (1) is a ﬁeld. If d = [K : k], then both k ⊗k(1) K (1) and K have the same k-dimension d. Since ΦRA is an injective k-algebra map, it is an isomorphism. We prove the ‘only if’ part. Since k ⊗k(1) K (1) is isomorphic to K, it is a ﬁeld. So K/k is separable. Let x1 , . . . , xn be a separable basis of K over k. Then K, which is the image of ΦRA , is a ﬁnite separable extension of k(xp1 , . . . , xpn ). If n ≥ 1, then x1 is both separable and purely inseparable over k(xp1 , . . . , xpn ). Namely, x1 ∈ k(xp1 , . . . , xpn ), which is a contradiction. So n = 0, that is, K is separable algebraic over k. ⊓ ⊔ Lemma 33.4. Let A be a noetherian ring, and ϕ : F → F ′ an A-linear map between A-ﬂat modules. Then ϕ is an isomorphism if and only if ϕ ⊗ 1κ(p) : F ⊗A κ(p) → F ′ ⊗A κ(p) is an isomorphism for any p ∈ Spec A. Proof. Follows easily from [19, (I.2.1.4) and (I.2.1.5)].

⊓ ⊔

Lemma 33.5. Let f : X → Y be a morphism locally of ﬁnite type between locally noetherian Fp -schemes. Then the diagram X

FX

/ X (1) f (1)

f

Y

(33.6)

FY

/ Y (1)

is cartesian if and only if f is ´etale. Proof. Obviously, the question is local on both X and Y , so we may assume that X = Spec B and Y = Spec A are aﬃne. We prove the ‘only if’ part. By Radu’s theorem [38, Corollaire 6], A → B is regular. In particular, B is A-ﬂat. The canonical map A ⊗A(1) B (1) → B is an isomorphism. So for any P ∈ Spec B, κ(p) ⊗κ(p)(1) (κ(p) ⊗A BP )(1) → κ(p) ⊗A BP is an isomorphism, where p = P ∩ A. Let K be the ﬁeld of fractions of the regular local ring κ(p)⊗A BP . Then κ(p)⊗κ(p)(1) K (1) → K is an isomorphism. By Lemma 33.3, K is a separable algebraic extension of κ(p). Since κ(p) ⊂

33 Other Examples of Diagrams of Schemes

465

κ(p) ⊗A BP ⊂ K, we have that κ(p) ⊗A BP is a separable algebraic extension ﬁeld of κ(p). So f is ´etale at P . As P is arbitrary, B is ´etale over A. We prove the ‘if’ part. By Lemma 33.3, κ(p) ⊗κ(p)(1) (κ(p) ⊗A BP )(1) → κ(p)⊗A BP is an isomorphism for P ∈ Spec B, where p = P ∩A. Then it is easy to see that κ(p) ⊗κ(p)(1) (κ(p) ⊗A B)(1) → κ(p) ⊗A B is a isomorphism for p ∈ Spec A. By Lemma 33.4, A ⊗A(1) B (1) → B is an isomorphism, as desired. ⊓ ⊔ By the lemma, for a morphism f : X → Y of noetherian Fp -schemes, Ly(f ) is cartesian of ﬁnite type if and only if f is ´etale. (33.7) Let I be a small category, and R• a covariant functor from I to the category of (non-commutative) rings. A left R• -module is a collection M = ((Mi )i∈ob(I) , (βφ )φ∈Mor(I) ) such that Mi is a left Ri -module for each i ∈ ob(I), and for φ ∈ I(i, j), βφ : Mi → Mj is an Ri -linear map, where Mj is viewed as an Ri -module through the ring homomorphism Rφ : Ri → Rj . Moreover, we require the following conditions. 1 2

For i ∈ ob(I), βidi = idMi . For φ, ψ ∈ Mor(I) such that ψφ is deﬁned, βψ βφ = βψφ .

For φ ∈ Mor(I), let s(φ) = i and t(φ) = j if φ ∈ I(i, j). i (resp. j) is the source (resp. target) of φ. Set A(R• ) := φ∈Mor(I) Rt(φ) φ. We deﬁne (bψ)(aφ) = (b · Rψ a)(ψφ) if ψφ is deﬁned, and (bψ)(aφ) = 0 otherwise. Then A(R• ) is a ring possibly without the identity element. If ob(I) is ﬁnite, then i∈ob(I) idi is the identity element of A(R• ). We call A(R• ) the total ring of R• .

Let M = ((Mi )i∈ob(I) , (βφ ) φ∈Mor(I) ) be an R• -module. Then M = i Mi is an A(R• )-module by (aφ)( j mj ) = aβφ ms(φ) for φ ∈ Mor(I), a ∈ Rt(φ) , and mj ∈ Mj . It is a unitary module if ob(I) is ﬁnite. From now on, assume that ob(I) is ﬁnite. Then a (unitary) A(R• )-module M yields an R• -module. Set Mi = idi M . Then Mi is an Ri -module via r(idi m) = (ridi m) = idi (ridi m). For φ ∈ I(i, j), βφ : Mi → Mj is deﬁned by βφ (m) = φm. Thus an R• -module ((Mi ), (βφ )) is obtained. Note that the category of R• -modules and the category of A(R• )-modules are equivalent. Now consider the case that each Ri is commutative. Then R• yields X• = Spec• R• ∈ P(I, Sch). By (4.10), the category of R• -modules is equivalent to Lqc(X• ). Lemma 33.8. Let I be a ﬁnite ordered category, and R• a covariant functor from I to the category of commutative rings. If Ri is regular with ﬁnite Krull dimension for each i ∈ ob(I), and Rφ is ﬂat for each φ ∈ Mor(I), then A(R• ) has a ﬁnite global dimension. Proof. Follows easily from Lemma 31.3.

⊓ ⊔

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Vol. 1947: D. Abramovich, M. Mari˜no, M. Thaddeus, R. Vakil, Enumerative Invariants in Algebraic Geometry and String Theory. Cetraro, Italy 2005. Editors: K. Behrend, M. Manetti (2008) Vol. 1948: F. Cao, J-L. Lisani, J-M. Morel, P. Musé, F. Sur, A Theory of Shape Identification (2008) Vol. 1949: H.G. Feichtinger, B. Helffer, M.P. Lamoureux, N. Lerner, J. Toft, Pseudo-Differential Operators. Quantization and Signals. Cetraro, Italy 2006. Editors: L. Rodino, M.W. Wong (2008) Vol. 1950: M. Bramson, Stability of Queueing Networks, Ecole d’Eté de Probabilités de Saint-Flour XXXVI-2006 (2008) Vol. 1951: A. Moltó, J. Orihuela, S. Troyanski, M. Valdivia, A Non Linear Transfer Technique for Renorming (2008) Vol. 1952: R. Mikhailov, I.B.S. Passi, Lower Central and Dimension Series of Groups (2008) Vol. 1953: K. Arwini, C.T.J. Dodson, Information Geometry (2008) Vol. 1954: P. Biane, L. Bouten, F. Cipriani, N. Konno, N. Privault, Q. Xu, Quantum Potential Theory. Editors: U. Franz, M. Schuermann (2008) Vol. 1955: M. Bernot, V. Caselles, J.-M. Morel, Optimal Transportation Networks (2008) Vol. 1956: C.H. Chu, Matrix Convolution Operators on Groups (2008) Vol. 1957: A. Guionnet, On Random Matrices: Macroscopic Asymptotics, Ecole d’Eté de Probabilités de SaintFlour XXXVI-2006 (2008) Vol. 1958: M.C. Olsson, Compactifying Moduli Spaces for Abelian Varieties (2008) Vol. 1959: Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties (2008) Vol. 1960: J. Lipman, M. Hashimoto, Foundations of Grothendieck Duality for Diagrams of Schemes (2009)

Recent Reprints and New Editions Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999 – Corr. 3rd printing (2007) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 – 2nd corr. and augmented edition (2007) Vol. 1693: S. Simons, From Hahn-Banach to Monotonicity (Minimax and Monotonicity 1998) – 2nd exp. edition (2008) Vol. 470: R.E. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. With a preface by D. Ruelle. Edited by J.-R. Chazottes. 1975 – 2nd rev. edition (2008) Vol. 523: S.A. Albeverio, R.J. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integral. 1976 – 2nd corr. and enlarged edition (2008) Vol. 1764: A. Cannas da Silva, Lectures on Symplectic Geometry 2001 – Corr. 2nd printing (2008)

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Additional technical instructions, if necessary, are available on request from: [email protected]. 4. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files (and also the corresponding dvi-, pdf- or zipped ps-file) together with the final printout made from these files. The LaTeX source files are essential for producing the full-text online version of the book (see www.springerlink.com/content/110312 for the existing online volumes of LNM). The actual production of a Lecture Notes volume takes approximately 12 weeks. 5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3% on the price of Springer books purchased for their personal use, if ordering directly from Springer. 6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor J.-M. Morel, CMLA, ´ Ecole Normale Sup´erieure de Cachan, 61 Avenue du Pr´esident Wilson, 94235 Cachan Cedex, France E-mail: [email protected] Professor F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands E-mail: [email protected] Professor B. Teissier, Institut Math´ematique de Jussieu, ´ UMR 7586 du CNRS, Equipe “G´eom´etrie et Dynamique”, 175 rue du Chevaleret 75013 Paris, France E-mail: [email protected] For the “Mathematical Biosciences Subseries” of LNM: Professor P.K. Maini, Center for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LP, UK E-mail: [email protected] Springer, Mathematics Editorial I, Tiergartenstr. 17 69121 Heidelberg, Germany, Tel.: +49 (6221) 487-8259 Fax: +49 (6221) 4876-8259 E-mail: [email protected]

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