Chiral algebras

Contents Introduction

1

Chapter 1. Axiomatic Patterns

11

1.1. Pseudo-tensor categories

11

1.2. Complements

18

1.3. Compound tensor categories

24

1.4. Rudiments of compound geometry

32

Chapter 2. Geometry of D-schemes

53

2.1. D-modules: Recollections and notation

53

2.2. The compound tensor structure

67

2.3. DX -schemes

79

2.4. The spaces of horizontal sections

89

∗

2.5. Lie algebras and algebroids

95

2.6. Coisson algebras

113

2.7. The Tate extension

117

2.8. Tate structures and characteristic classes

129

2.9. The Harish-Chandra setting and the setting of c-stacks

143

Chapter 3. Local Theory: Chiral Basics

157

3.1. Chiral operations

157

3.2. Relation to “classical” operations

163

3.3. Chiral algebras and modules

164

3.4. Factorization

172

3.5. Operator product expansions

194

3.6. From chiral algebras to associative algebras

200

3.7. From Lie∗ algebras to chiral algebras

212

3.8. BRST, alias semi-infinite, homology

227

3.9. Chiral differential operators

238

3.10. Lattice chiral algebras and chiral monoids

257

Chapter 4. Global Theory: Chiral Homology

275

1

2

CONTENTS

4.1. The cookware

275

4.2. The construction and first properties

296

4.3. The BV structure and products

313

4.4. Correlators and coinvariants

325

4.5. Rigidity and flat projective connections

230

4.6. The case of commutative DX algebras

341

4.7. Chiral homology of the de Rham-Chevalley algebras

347

4.8. Chiral homology of chiral envelopes

353

4.9. Chiral homology of lattice chiral algebras

357

Bibliography

363

Index and Notation

369

Introduction V nastowe$i rabote ne soderats svedeni, kotorye mogli by sostavit~ predmet izobreteni ili otkryti. Anon., “Akt kspertizy”, ok. 1956.† 0.0. This book is an exposition of basic (local and global) aspects of chiral algebra theory. Chiral algebras have their origin in mathematical physics; they lie at the heart of conformal field theory1 ; see [BPZ]. Mathematicians, since the pioneering work of Borcherds [B1], usually look at them through the formalism of vertex algebras incorporated into representation theory of infinite-dimensional algebras. We follow a different approach which tastes more of algebraic geometry than of representation theory. In the introduction we outline the principal structures involved and their interrelations. More specific information, together with bibliographical comments and references, can be found in brief introductions to sections. 0.1. Chiral algebras are “quantum” objects. Let us describe first the corresponding “classical” objects; we call them coisson algebras (“coisson” may be considered as an abbreviation for “chiral Poisson” or “compound Poisson”, the word “compound” being related to the notion of compound tensor category which we discuss in 0.4). In fact, “coisson algebra” is a new name for a well-known class of objects. Informally, a coisson algebra is defined by a local Poisson bracket on a space of classical fields. In the simplest case where “classical field” means “a function u(x)” a mathematical physicist would define a local Poisson bracket by a formula like (0.1.1)

{u(x), u(y)} =

n X

ϕi (x, u(x), u0 (x), . . . )δ (i) (x − y),

ϕi ∈ A.

i=1

Here A denotes the algebra of functions of x, u, u0 , . . . , u(k) (k is not fixed). Of course in the left-hand side of (0.1.1) u(x) is understood as a functional u 7→ u(x) on the space of classical fields. The bracket should be skew-symmetric and satisfy the Jacobi identity, so the collection of functions ϕi should satisfy certain differential equations. For f ∈ A and every x we have the functional `f,x on the space of † “This article does not contain any information that could be the subject of an invention or a discovery.”, Anon., “Expert Certification”, circa 1956. (The signing of this Patriot Act was a precondition for publication of a mathematical paper in yonder Russia.) 1 More precisely, of its purely holomorphic sector which is “holomorphic quantum mechanics”.

1

2

INTRODUCTION

classical fields defined by `f,x (u) = f (x, u(x), u0 (x), . . . ). It follows from (0.1.1) that X f,g (0.1.2) {`f,x , `g,y }(u) = ϕi (x, u(x), u0 (x), . . . )δ (i) (x − y) i

for some (0.1.3)

ϕf,g i

∈ A. Formula (0.1.2) has a more symmetric version: X f,g {`f,x , `g,y }(u) = ∂xi ∂yi {ψij (x, u(x), u0 (x), . . . )δ(x − y)},

f,g ψij ∈ A.

i,j f,g Of course the ψij are not uniquely determined by ϕf,g i ; we must take into account the relations

(0.1.4)

(∂x + ∂y )(aδ) = (∂x a) · δ,

a ∈ A,

where δ = δ(x − y). Now from the algebraic point of view the primary object is the algebra A rather than the “space of classical fields”. For instance, one can take A = C[x, u, u0 , u00 , . . . ]. More generally, A can be any commutative differential unital algebra over C[x], and a “classical field” is a homomorphism of differential unital C[x]-algebras A → B where B is, e.g., the algebra of C ∞ -functions of x ∈ R. A choice of a set of generators and defining relations for the differential C[x]-algebra A identifies classical fields with collections of functions satisfying certain systems of differential equations. Let us try to formulate in terms of A what is a “local Poisson bracket on the space of classical fields”. A glance at (0.1.4) and the right-hand side of (0.1.3) shows that such an object is defined by a mapping (0.1.5)

A⊗A → V C

where V is the module over C[∂x , ∂y ] generated by the symbols aδ, a ∈ A, with the defining relations (0.1.4). We prefer another interpretation of V : V is the module over C[x, y, ∂x , ∂y ] generated by the symbols aδ, a ∈ A, with the defining relations (0.1.4) and also the following ones: (0.1.6)

f (x, y) · (aδ) = (f (x, x)a) · δ,

f ∈ C[x, y], a ∈ A.

The mapping (0.1.5) should satisfy certain properties. First of all, it should beh a morphism of C[x, y, ∂x , ∂y ]-modules (notice that A ⊗ A is a module over i ∂ C x, ∂x ⊗ C[x, ∂y ] = C[x, y, ∂x , ∂y ]). Before discussing the other properties, let us rewrite (0.1.5) in geometric terms. An algebra A over C[x] is the same as a quasi-coherent OX -algebra A where X := Spec C[x] is the affine line. Differential C[x]-algebras correspond to DX -algebras (see 0.2). The DX×X -module corresponding to the C[x, y, ∂x , ∂y ]-module V from (0.1.5) is canonically isomorphic to ∆∗ A where ∆: X → X × X is the diagonal embedding. So (0.1.5) is equivalent to a DX×X -module morphism (0.1.7)

A  A → ∆∗ A.

We denote by  the external tensor product, i.e., A  A := p∗1 A ⊗ p∗2 A where p1 and p2 are the projections X × X → X.

INTRODUCTION

3

Notice that (0.1.7) makes sense for an arbitrary smooth variety X. In 0.6 we will define a coisson algebra on X as a commutative (= commutative associative unital) DX -algebra A with a D-module morphism (0.1.7) satisfying certain properties which actually mean that (0.1.3) is a Poisson bracket on the “space(s) of classical fields”. To formulate these properties in a natural way, we need some polylinear algebra in the category of DX -modules (see 0.2 – 0.5). Remarks. (i) The notions of Poisson OX -algebra or Poisson DX -algebra are inadequate for expressing the idea of “Poisson bracket on the space(s) of classical fields”: a Poisson bracket on A is a morphism A ⊗OX A → A while (0.1.7) is an object of a different nature. The reason is that Spec A is the space of “jets of classical fields”, not the space of classical fields. (ii) The reader can compare our notion of coisson algebra with other ways of formalizing the notion of “Poisson bracket on the space of classical fields” (see [DG1], [DG2], [Ku], [KuM], [Ma] and references therein). As far as we understand, our approach is essentially equivalent to that of [KuM] and [Ma]. 0.2. Let X be a smooth complex algebraic variety. Consider the category M` (X) of left D-modules on X (see, e.g., [Ber], [Ba], or [Kas2]). The OX -tensor product ⊗ makes it a tensor category with unit object OX . A commutative algebra in M` (X) (commutative DX -algebra) is just a quasi-coherent commutative OX algebra A` equipped with a flat connection along X. We call an X-scheme equipped with a flat connection along X a DX -scheme; so the spectra of commutative DX algebras are DX -schemes affine over X. A standard example: for any X-scheme Y → X the space J(Y /X) of ∞-jets of sections of Y /X is a DX -scheme. Horizontal sections of J(Y /X) are the same as arbitrary sections of Y /X. One may view the (closed) DX -subschemes of J(Y /X) as systems of differential equations on sections of Y /X (see [G]). What makes the tensor category M` (X) substantially different from, say, the tensor category of OX -modules is the absence of duals. Precisely, for F ∈ M` (X) the dual object in the tensor category sense (see, e.g., [D1]) exists iff F is coherent as an OX -module, i.e., is a vector bundle with an integrable connection. For example, for a group DX -scheme G one has its Lie coalgebra CoLie(G) but we cannot dualize it to get the Lie algebra (unless G is finite dimensional as a usual scheme). We will see that the category M` (X) carries a richer structure (that of compound tensor category) which remedies the above flaw. 0.3. Consider the category M(X) of right D-modules on X. Let I be a finite non-empty set; denote by ∆(I) : X ,→ X I the diagonal embedding. For an I-family of D-modules Li ∈ M(X) and M ∈ M(X) set (0.3.1)

(I)

PI∗ ({Li }, M ) := Hom( Li , ∆∗ M ). (I)

Elements of PI∗ ({Li }, M ) are called ∗ I-operations; they are X-local (since ∆∗ M sits on the diagonal). The ∗ operations compose in a natural way,2 just as polylinear maps between vector spaces do (i.e., for a surjective map π: J → I, a J-family {Kj } of D-modules, and ϕ ∈ PI∗ ({Li }, M ), ψi ∈ Pπ∗−1 (i) ({Kj }, Li ) we have ϕ(ψi ) ∈ PJ∗ ({Kj }, M )). The composition is associative; if |I| = 1, then PI∗ = Hom. We call such data of operations (or “polylinear maps”) between the objects of a category 2 We

switched to right D-modules to make the “sign rule” for the ∗ operations obvious.

4

INTRODUCTION

a pseudo-tensor structure. Thus M(X) carries a canonical pseudo-tensor structure; M(X) equipped with this structure is denoted by M(X)∗ . One may view pseudo-tensor categories as a straightforward generalization of operads: an operad is just a pseudo-tensor category with single object. The notions of algebras, modules over them, etc., make perfect sense in any pseudo-tensor category. For example, a Lie algebra in M(X)∗ (or simply a Lie∗ algebra on X) is a D-module L together with a binary ∗ operation [ ] ∈ P2∗ ({L, L}, L) which is skew-symmetric and satisfies the Jacobi identity. For M ∈ M(X) set h(M ) := M ⊗ OX (the sheaf of middle de Rham cohomolDX

ogy). There is a canonical map PI∗ ({Li }, M ) → Hom(⊗h(Li ), h(M )) compatible with composition of operations. Therefore h sends Lie∗ algebras to the sheaves of usual Lie algebras. In fact, many important Lie algebras (including the Virasoro and affine Kac-Moody algebras) arise naturally from Lie∗ algebras. 0.4. Let us identify the categories of left and right D-modules by the usual ∼ −1 ⊗−1 equivalence M(X) −→ M` (X), M 7→ M ` = M ωX = M ⊗ ωX (here ωX = OX

X ). Therefore M(X) has the tensor product M ⊗! L := (M ` ⊗ L` )ωX . Now Ωdim X the ! tensor product of two ∗ operations is defined according to the following pattern. Let I, J be finite sets, i0 ∈ I, j0 ∈ J; denote by I ∨ J the disjoint union of I, J with i0 , j0 identified. Then there is a natural map ! ∗ ∗ ∗ ! ⊗I,J i0 ,j0 : PI ({Mi } , L) ⊗ PJ ({Nj } , K) → PI∨J ({Mi , Nj , Mi0 ⊗ Nj0 } i6=i0 , L ⊗ K). j6=j0

If I = {i0 }, J = {j0 } this is the usual tensor product of morphisms. A compound tensor structure on a category is the data of a tensor and a pseudotensor structure related as above by the ⊗I,J i0 ,j0 maps which are associative and commutative in the obvious sense. Thus M(X) (and M` (X)) is a compound tensor category. 0.5. The compound tensor structure makes it possible to do basic differential geometry in M(X) implementing a complementarity principle: the “functions” multiply according to ⊗! format, while “operators” (such as infinitesimal symmetries) act in the ∗ sense. In other words, the geometric objects we consider are DX -schemes, the Lie algebras are actually Lie∗ algebras, and one defines easily what the action of a Lie∗ algebra on a DX -scheme means. Note that an action of a Lie∗ algebra L on a DX -scheme Y yields an action on Y of the usual Lie algebra h(L). For example, let G be a group DX -scheme such that CoLie(G) is a locally free DX -module of finite rank. Set Lie(G) := CoLie(G)◦ = HomDX (CoLie(G), DX ) (the usual duality of D-modules theory). Then Lie(G) is a Lie∗ algebra. A G-action on a DX -scheme Y yields a CoLie(G)-coaction on OY (in the ⊗! sense). Dualizing, we get a Lie(G)-action on Y. In fact, for locally free DX -modules of finite rank the duality yields an identi∼ fication PI∗ ({Mi }, L) −→ Hom(L◦ , ⊗Mi◦ ). So, on an appropriate derived category level, the ∗ operations can be recovered from the usual tensor structure and the duality functor ◦ . This is a very awkward thing to do however, when dealing with both ! and ∗ operations simultaneously (as happens in coisson algebras), and it precludes chiral quantization (see 0.8 and 0.9).

INTRODUCTION

5

0.6. Now a coisson algebra is simply a Poisson algebra in the compound setting. Namely, this is a commutative DX -algebra A` together with a Lie∗ bracket on A (coisson bracket) such that the adjoint action is compatible with the multiplicative structure on A` . From this we get an action of h(A) on A` in the usual sense. Derivations of A` that come from h(A) are called hamiltonian vector fields. Coisson brackets localize nicely so one knows what is a coisson structure on any DX -scheme. 0.7. Let Y = Spec A` be an affine DX -scheme. The space of horizontal sections X → Y is an ind-affine ind-scheme hYi = hYi(X) = Spf hAi; if X is compact, this is actually a scheme. This construction also makes sense locally. For example, for x ∈ X the space of horizontal sections of Y over the formal punctured disc at x is an ind-affine scheme Spf Aas x . Suppose that X is compact; set Ux := X r{x}. The evaluation map which assigns to a global section on Ux its restriction to the punctured disc is a closed embedding of the ind-schemes of sections hYi(Ux ) ,→ SpfAas x ; its ideal is generated by the image of a certain canonical map rx : Γ(Ux , h(A)) → Aas x . We also have an embedding Yx ,→ Spf Aas whose image consists of horizontal secx tions that extend to x, so hYi(X) = hYi(Ux ) ∩ Yx . If A is a coisson algebra finitely generated as a DX -algebra, then Aasx is a topological Poisson algebra, Γ(Ux , h(A)) is a Lie algebra, and rx commutes with brackets. Therefore rx is a hamiltonian action of Γ(Ux , h(A)) on Spf Aas x , and hYi(Ux ) is the zero fiber of the momentum map. When we pass to quantization and chiral homology (see 0.9 and 0.10), the zero fiber changes into the Hamiltonian reduction. The algebra hAi is also denoted by H0ch (X, A). The derived version of this construction yields a graded commutative superalgebra H·ch (X, A). 0.8. Now let us pass to chiral algebras. There are two complementary (equivalent) approaches to this notion: Lie algebra style and commutative algebra style.3 We begin with the “Lie algebra” approach; for the “commutative algebra” picture see 0.12. From now on we assume that dim X = 1. Let j (I) : U (I) ,→ X I be the complement to the diagonal divisor. Set (0.8.1)

(I)

(I)

PIch ({Li }, M ) := Hom(j∗ j (I)∗  Li , ∆∗ M ).

These are chiral I-operations. One defines their composition in the obvious manner. We get the chiral pseudo-tensor structure on M(X); the corresponding pseudotensor category is denoted by M(X)ch . Now a chiral algebra on X is simply a Lie algebra in M(X)ch with an additional property (existence of unit). We refer to the corresponding Liech bracket as the chiral product. 0.9. Let us explain why the notion of chiral algebra is a quantization of that of coisson algebra. For every DX -module A we have a canonical exact sequence (0.9.1)

0 → Hom(A`⊗2 , A` ) → P2ch ({A, A}, A) → P2∗ ({A, A}, A).

The right arrow assigns to every chiral product µ on A a Lie∗ bracket [ ]µ . If the latter vanishes (we say then that our chiral algebra is commutative), then µ can be considered as a binary operation on A` with respect to ⊗. This way one 3A

poetically-minded reader may call them “dynamic” and “static” points of view.

6

INTRODUCTION

identifies commutative chiral algebra structures on A with commutative DX -algebra structures on A` . Now assume we have a family At of chiral algebras that
depends on a parameter is a coisson bracket t ∈ C such that A0 is commutative. Then { } := t−1 [ ]µt t=0 on A`0 . Thus chiral algebras are quantizations of coisson algebras as promised. In fact, the whole chiral pseudo-tensor structure can be considered as a quantization of the compound tensor structure (see 3.2). The problem of quantization of coisson algebras is fairly interesting. We do not know how to solve it in general. In the main body of this work we treat the simplest cases of linear brackets. In particular, we construct chiral enveloping algebras of Lie∗ algebras and Lie∗ algebroids (chiral algebras of twisted differential operators). We also discuss (in 3.9.10) quantizations modt2 . The general theory of deformations of chiral algebras was recently developed in [Tam1]. In a sense, in the “chiral world” chiral algebras are parallel to associative algebras while Lie∗ algebras and commutative DX -algebras play, respectively, the roles of Lie and commutative algebras. 0.10. The constructions from 0.7 generalize to arbitrary chiral algebras. For a chiral algebra A one can define its chiral homology Hich (X, A) (see 0.12). Here H0ch (X, A) is what mathematical physicists usually call “the space of conformal blocks”. For any x ∈ X we have an associative algebra Aas x ; the fiber Ax is natuch rally a quotient of Aas x modulo a left ideal. H0 (X, A) identifies canonically with a quotient of Ax modulo a right ideal generated by the image of a Lie algebra morphism rx : Γ(Ux , h(A)) → Aas x . In the classical limit we return to the objects from 0.7. 0.11. Let us sketch now the “commutative algebra” style description of chiral algebras. This approach is essential for certain subjects, in particular, for the definition of chiral homology. Consider the space R(X) of finite non-empty subsets of X; for such a subset S ⊂ X we denote the corresponding point of R(X) by [S]. So R(X) carries a natural filtration R(X)n ; the open stratum R(X)◦n is the space of configurations of n (distinct) points on X. Remarkably enough, R(X) is contractible. Informally, a factorization algebra on X is an O-module A`R(X) on R(X) such that for every two non-intersecting subsets S, T ⊂ X there is a canonical identification of fibers A`[S∪T ] = A`[S] ⊗ A`[T ] which is associative and commutative in the obvious sense. We also demand the existence of a unit section (the definition is left to the reader). Such A`R(X) provides an O-module A` on X = R(X)1 . In fact, A` is automatically a left DX -module, and A`R(X) amounts to a certain structure on A` of local origin (referred to as factorization algebra structure). Now a factorization algebra structure on A` amounts to a chiral algebra structure on A. Namely, the chiral product corresponding to a factorization algebra structure is the composition j∗ j ∗ A  A = j∗ j ∗ AX×X → ∆∗ ∆! AX×X = ∆∗ A. Here AX×X is the pull-back of AR(X) by the obvious map X × X → R(X)2 , ∆ : X ,→ X × X the diagonal, j : U ,→ X × X its complement, the equalities are structure identifications, and the arrow is a canonical morphism. The O-tensor product of factorization algebras is evidently a factorization algebra, so chiral algebras form a tensor category.

INTRODUCTION

7

0.12. Therefore chiral algebras can be considered as geometric objects on R(X). In this vein, the chiral homology of A is defined as the de Rham homology of R(X) with coefficients in A`R(X) . The chiral homology functor has many remarkable properties; e.g., it commutes with tensor products. In particular, higher chiral homology of the unit chiral algebra vanishes (which also follows from contractibility of R(X)). Chiral homology is naturally defined for DG chiral algebras and it is preserved by quasi-isomorphisms. In the exposition we do not pave a road across the morass of homotopy theory of chiral algebras, but we resort to an unworthy (yet solid) path of functorial resolutions. Namely, chiral homology can be realized as homology of certain functorial chiral chain complexes C ch (X, A)PQ which depend on appropriate auxiliary resolutions P, Q of OX (see 4.2.12).4 These complexes resemble Chevalley homology complexes of Lie algebras: for example, they carry a canonical BV (Batalin-Vilkovisky) algebra structure. In the classical limit it becomes an odd Poisson bracket. The comeuppance is the lack of understanding of the structure of the homotopy category of chiral algebras (see Remarks in 3.3.13). By the way, the latter lies outside of Quillen’s model category framework due to the absence of cofibrant objects (e.g., the chiral algebra freely generated by DX ∈ M(X) does not exist). 0.13. Physicists usually describe a chiral algebra structure in terms of operator product expansions, ope for short (see, e.g., [BPZ]). The same approach is common in the literature on vertex algebras; see 0.15. To explain what ope is, one needs to consider some non-quasi-coherent Dˆ ∗ A` be a sheaf of left DX×X -modules modules. Thus, for a left DX -module A` , let ∆ supported on the diagonal X ,→ X × X which is I-adically complete and satisfies ˆ ∗ A` /I ∆ ˆ ∗ A` = A` (here I ⊂ OX×X is the ideal of the diagonal). Such ∆ ˆ ∗ A` ∆ ` exists and is unique. Note that for any local section a ∈ A there is a unique ˆ ∗ A` such that a(1) is horizontal along the second variable and section a(1) ∈ ∆ (1) ` ˆ ∗ A = a. Therefore, if t is a local coordinate on X, one may write any a mod I ∆ ˆ ∗ A` . ˆ ∗ A` as a formal power series P a(1) (t2 −t1 )i , ai = 1 ∂ti ϕ mod I ∆ section ϕ of ∆ i i! 2 i≥0

˜ ∗ A` be the localization of ∆ ˆ ∗ A` with respect to the equation of the Now let ∆ P (1) diagonal. Its section is a Laurent power series ai (t2 − t1 )i ; i.e., we can write i−∞

˜ ∗ A` as A` ((t2 − t1 )). Notice that ∆ ˜ ∗ A` /∆ ˆ ∗ A` = ∆∗ A` . ∆ ˜ ∗ A` . Composing Now an ope is a morphism of DX×X -modules ◦ : A`  A` → ∆ ` it with the above projection to ∆∗ A , we get a binary chiral operation µ = µ◦ on A. One can explain what it means for ◦ to be commutative and associative. In fact, ◦ 7→ µ◦ is a bijective correspondence between the set of commutative and associative ope and that of chiral algebra structures on A. From the point of view of factorization algebras, our ope is the gluing isomorphism that reconstructs A`X×X from its restriction to U (which is A`  A` |U ) and ˆ ∗ A` ). to the formal neighbourhood of the diagonal (which is ∆ ∗ Notice also that the Lie bracket of a chiral algebra is just the the polar part of the ope. 4 A poetically-minded reader may say that P resolves ultraviolet problems of A, while Q takes care of its infrared behavior.

8

INTRODUCTION

0.14. To see an example of a non-commutative chiral algebra, let us describe in geometric terms the chiral enveloping algebra of a Kac-Moody Lie algebra. Let G be an algebraic group. For x ∈ X consider the set of pairs (F, α), where F is a G-bundle on X and α is a trivialization of F on X r {x}. This is the set of points of a formally smooth ind-scheme GRx ; a choice of a parameter tx at x identifies GRx with G(C((t)) )/G(C[[t]]). Our GRx are fibers of an ind-scheme GRX over X called the affine Grassmannian. This is a DX -ind-scheme: when x varies infinitesimally, X r {x}, hence GRx , do not change. The trivialized G-bundle (F, α) on X defines the section e: X → GR (which is the only horizontal section of GR). Denote by A`x the vector space of distributions on GRx supported at e(x); i.e., A`x is the (topological) dual to the formal completion of the local ring OGRx ,e(x) . When x varies, the A`x form a left DX -module A`X . Let us show that A is naturally a chiral algebra. As in 0.11 we have to define a factorization algebra A`R(X) . We have a D-ind-scheme GRR(X) over R(X) with fibers GR[S] equal to the space of pairs (F, α) where F is a G-bundle on X and α is a trivialization of F on X r S. There is a canonical horizontal section e of GRX n , and we set A`[S] to be the vector space of distributions on GR[S] supported at e. The factorization property for A`R(X) follows from the similar property of GRR(X) itself: there is a canonical identification GR[S∪T ] = GR[S] × GR[T ] if S ∩ T = ∅. To get the chiral envelope of a Kac-Moody algebra of a non-zero level, one must twist A` by an appropriate canonical line bundle on GR. 0.15. Chiral algebras are related to (various versions of) vertex algebras as follows. The category of vertex algebras in the sense of [B1] and [K] is equivalent to the category of chiral algebras on X = A1 equivariant with respect to the group T of translations.5 Namely, the vertex algebra VA corresponding to a T -equivariant chiral algebra A is the vector space of T -invariant sections of A` with the vertex algebra structure given by the operator product expansion (see 0.13); of course, VA identifies naturally with the fiber A`x at any point x ∈ A1 . In the same manner the category of VA -modules is identified with that of (weakly) T -equivariant A-modules or, more conveniently, with that of A-modules supported at x.6 Replacing T on the chiral side by the group Af f of affine transformations, we get on the vertex side essentially an object called the “vertex algebra” in [FBZ], “Z-graded vertex algebra” in [GMS2], and “graded vertex algebra” in [K].7 Adding to the structure a Virasoro vector, a.k.a. stress-energy tensor, which is a morphism from the Virasoro Lie∗ algebra (of some central charge) to A compatible with the T - or Af f -action, we get essentially a “conformal vertex algebra” from [K] and [FBZ] or the “vertex operator algebra” of [FLM], [FHL], [DL], [Hu]. Similarly, a chiral algebra on the “coordinate disc” Spec C[[t]] equivariant with respect to the action of the group ind-scheme Aut C[[t]] is the same as the “quasiconformal vertex algebra” from [FBZ]. The equivariance means that such an object yields a chiral algebra on any non-coordinate formal disc, and, in fact, by patching, 5 Here “equivariant” means that the group T of translations acts on A as on the O -module, X and the action of each t ∈ T is compatible with the chiral algebra structure. In particular, A is a weakly T -equivariant D-module. 6 One identifies a weakly T -equivariant A-module M with the A-module (j j ∗ M )/M supx∗ x ported at x; here jx is the embedding A1 r {x} ,→ A1 . 7 Here “essentially” means that we discard the varying finiteness conditions of the references.

INTRODUCTION

9

on any curve (see [FBZ] 18.3.3 and [HL2]). In this way it amounts to a “universal” chiral algebra, i.e., a rule that assigns to any curve a chiral algebra on it, in a way compatible with the ´etale localization When one is interested in local questions, such as the usual representation theory, chiral algebras are essentially equivalent to vertex algebras. In practice, however, chiral algebras are considerably more flexible: for example, twists of vertex algebras (see 3.4.17) and constructions like that of Aas x (see 3.6.4) are painful in the pure vertex algebra setting. Let us mention that all kinds of vertex/chiral algebras can be seen as chiral algebras on appropriate c-stacks (see 2.9, 3.1.16, 3.3.14), the above-mentioned functors and equivalences being mere base change. In the exposition we stick to the usual D-module setting. 0.16. We started this work with the modest intent of understanding the ingenious formal power series manipulations that haunt the books on vertex operator algebras. For a VOA insider, untrammelled by algebro-geometric affections, the mode of the output might resemble though zadopasomy$i vol’s8 features. A challenging problem is to define chiral algebras on higher-dimensional X (see [B2] and [Tam2] in this respect). The definitions from 0.8 and 0.11 formally work also for dim X > 1, but to make them sensible, one must plunge at once into the homotopy setting which we had no courage to do (see 0.12). Notice that coisson algebras live in any dimension, so it may be reasonable to look first for the corresponding quasi-classical objects. Let us also mention that while the general format of the classical setting is covered nicely by the compound tensor category axiomatics (see 1.4), chiral algebras proper defy such treatment. We consider chiral algebras on usual curves, while some applications demand the setting of super curves. Presumably, the rendition should not be difficult. 0.17. The book consists of four chapters. The first one discusses some relevant abstract nonsense. The classical (coisson) story occupies the second chapter. The third chapter deals with the local theory of chiral algebras proper (some readers may find the exposition of the textbooks [K], [FBZ] livelier,9 and we do recommend parallel reading). It divides into two parts: the first considers basic structures and their interrelations while the second deals with elementary methods of constructing chiral algebras. The final chapter treats global theory, i.e., the formalism of chiral homology. It contains an exposition of the general machinery and some results 8 “V barbari e sut~ voly glagolts epistony, zane ne pasutc obyqa$ino kak vol mesta sego, iduwe vpered, no onye pasuts nazad, potomu qto rogi u nih sut~ krkovaty i sklonts napered ... utykatc rogi ih v zeml”, iz “Damaskina arhiere Studita sobrani ot drevnih filosofov o nekih sobstvah estestva ivotnyh”. “In Barbary there are oxen called epistomi for

they do not graze walking forward in the usual manner of the ox of our land, but move backwards, since horns of theirs are crooked and bent forward ... sticking into the earth,” from “A collection (of excerpts) from ancient philosophers on certain properties of the nature of animals” by Damaskin, the high priest of the Studiy monastery. 9 “U nas dvady dva toe qetyre, da vyhodit kak-to bo$ iqee.” M. Gasparov, “Zapiski i Vypiski”; kommentari$i k ponti “Narodnost~”. “At

ours two by two is also four, yet it comes livelier”, from M. Gasparov, “Notes and Excerpts”, a comment on the notion narodnost’ (which has no adequate English analog).

10

INTRODUCTION

showing that the chiral homology functor transforms the constructions of chiral algebras from Chapter 3 to parallel constructions for BV algebras. It is known that distilled axioms are pretty indigestible. We strongly suggest the reader follow the complicated alembics of Chapter 1 simultaneously with respective sections of more wholesome Chapters 2 and 3. 0.18. We were particularly motivated by the observation that chiral algebras provide a natural tool for tackling geometric automorphic forms in the D-module setting. This can be seen already in the oper construction at the critical level (see [BD]). One hopes that general automorphic D-modules come as (higher) chiral homology of twisted chiral Hecke algebras (where the moduli of G-bundles and de Rham G∨ -local systems are parameters of the twists). A related local problem is to describe a “spectral decomposition” of the category of representations of an affine Kac-Moody algebra (at negative integral level) with spectral parameters being moduli of G∨ -local systems on the punctured formal disc (see [Be]). The chiral Hecke algebras arise from the global geometry of the affine Grassmannian; when G is a torus, they amount to lattice Heisenberg algebras. We will return to these subjects elsewhere. A general matter of special interest is the factorization property of chiral homology for degenerating families of curves. At the moment, it is to some extent understood only for H0ch of rational field theories (the Verlinde rules). Presumably, there should be an interesting theory beyond the rational situation with Verlinde’s summation over the finite set of irreducibles replaced by “integration” over an appropriate “compact space”. The (twisted) chiral Hecke algebra may provide an example of such a situation. 0.19. Our sincere gratitude is due to Sasha Belavin and Pierre Deligne. Apart from an obvious influence of their ideas, it was in their apartments in 1992–1995 where the prime part of the work was done. During the (all too long) course of writing, we were greatly helped by many mathematicians; we are very grateful to S. Arkhipov, J. Bernstein, R. Bezrukavnikov, P. Bressler, B. Feigin, E. Frenkel, V. Ginzburg, V. Hinich, Y.-Z. Huang, M. Kapranov, D. Kazhdan, J. Lepowski, Yu. Manin, B. Mazur, V. Schechtman, I. Shapiro, G. Segal, J. Wiennfield, G. Zuckerman, and, especially, D. Gaitsgory, for their interest, inspiration, and correction of mistakes. The first author would also like to thank his wife and children10 for their ability to endure the rewritings of the draft. We are grateful to Dottie Phares of IAS and Richard Lloyd of MIT for the careful typing of the first version of the first part of the manuscript back in 1994–1995, and to Sergei Gelfand and Arlene O’Sean of the AMS for the editing of the manuscript. The authors were partially supported by NSF grant DMS-0100108. 0.20. A few words about notation commonly used in the book. “x ∈ Y ” means that either x is an element of a set Y or x is an object of a category Y or x is a local section of a sheaf Y . For a category M we denote by M◦ the dual category; Sets is the category of all sets. For a smooth variety X we denote by ΘX and DX the tangent sheaf and the algebra of differential operators. 10 “I slonovu e oustraxaet rost~ malaa kviqawaa svinaa porosta”, iz “Pohvaly Bogu o sotvorenii vse$i tvari Georgi Pisidy.” “And the

elefant rage is terrified by tiny squealing swine piggies,” from “A praise unto the Lord for the creation of all living creatures” by George Pisida.

CHAPTER 1

Axiomatic Patterns Da porazit t puwe groma Uasna, sil~na aksioma. A. A. Nahimov, “Pot i Matematik”, Har~kovski$i Demokrit, 1816

†

1.1. Pseudo-tensor categories The notion of pseudo-tensor category is an immediate generalization of that of operad in the same way that the notion of category generalizes that of monoid. On the other hand, the pseudo-tensor structure appears naturally on any (not necessary closed under the tensor product) subcategory of a tensor category,1 hence the adjective “pseudo-tensor”. For example, you can play with Lie algebras and their representations, but the enveloping algebras are out of reach. Pseudo-tensor categories were considered sporadically and under various names for quite a while; see [La] (“multicategories”) or [Li], [B2] (“multilinear categories”). Operads were introduced by P. May [May] in the topological context; today they are very popular: see, e.g., [GK], [H], [Kap1], [KrM], [L], [J], and [MSS]. Pseudo-tensor categories are defined in 1.1.1, pseudo-tensor functors and adjunction are in 1.1.5, examples are in 1.1.6, the linear setting is considered in 1.1.7, the tensor product of pseudo-tensor categories is defined in 1.1.9. A funny lemma in 1.1.10 will be used later to identify commutative chiral algebras with commutative DX -algebras; its explanation in 1.1.11 is due to V. Ginzburg. In 1.1.13 we explain that a pseudo-tensor structure on the category of R-modules, R an associative algebra, can be naturally defined the moment one knows the operad of endooperations of R. In 1.1.16 we fix “super” conventions. 1.1.1. Denote by S the category of finite non-empty sets and surjective maps. For a morphism π: J  I in S and i ∈ I set Ji := π −1 (i) ⊂ J; let · ∈ S be the one element set. Definition. A pseudo-tensor category is a class of objects M together with the following datum: †

Let thee be smitten as by a thunderbolt By a dreadful, potent axiom. A. A. Nakhimov, “A Poet and a Mathematician”, Kharkovski Democritus, 1816 1 In fact, every pseudo-tensor category appears in this manner in a universal way; see Remark in 1.1.6(i).

11

12

1.

AXIOMATIC PATTERNS

(a) For any I ∈ S, an I-family of objects Li ∈ M, i ∈ I, and an object M ∈ M one has the set PIM ({Li }, M ) = PI ({Li }, M ); its elements are called Ioperations. (b) For any map π: J  I in S, families of objects {Li }i∈I , {Kj }j∈J , and an object M one has the composition map Y (1.1.1.1) PI ({Li }, M ) × PJi ({Kj }, Li ) → PJ ({Kj }, M ), (ϕ, (ψi )) 7→ ϕ(ψi ). I

Here in the notation PJi ({Kj }, Li ) we assume that j ∈ Ji . The following properties should hold: (i) The composition is associative: if H  J is another surjective map, {Fh } an H-family of objects, χj ∈ PHj ({Fh }, Kj ), then ϕ(ψi (χj )) = (ϕ(ψi ))(χj ) ∈ PH ({Fh }, M ). (ii) For any M ∈ M there is an element idM ∈ P· ({M }, M ) such that for any ϕ ∈ PI ({Li }, M ) one has idM (ϕ) = ϕ(idLi ) = ϕ. Q We often use notation PJ/I ({Kj }, {Li }) = Pπ ({Kj }, {Li }) for PJi ({Kj }, Li ) I

to write the composition map as PI × PJ/I → PJ . We also write Pn := P{1,...,n} . We denote our pseudo-tensor category simply by M by abuse of notation. 1.1.2. Let M be a pseudo-tensor category. For L, M ∈ M set Hom(L, M ) = P1 ({L}, M ). Composition of morphisms comes from (b), so we have a usual category with the same objects as M; we denote it also by M. The composition with morphisms makes PI a functor (M◦ )I × M → Sets. Therefore, we may consider the pseudo-tensor categories as usual categories equipped with extra pseudo-tensor structure formed by a collection of functors PI , |I| ≥ 2, together with composition morphisms. Every category admits a trivial pseudo-tensor structure with PI = ∅ for |I| > 1. 1.1.3. Assume that a family {Li }i∈I has the property that the functor M 7→ PI ({Li }, M ) is representable. We denote the corresponding object of M as ⊗ Li I

and call it the pseudo-tensor product of Li ’s. If the pseudo-tensor product exists for any family of objects, then we call our pseudo-tensor structure representable. In other words, a representable pseudo-tensor structure on a category M is a rule that assigns to any I ∈ S a functor ⊗: MI → M and to any map π: J  I a natural I

compatibility morphism (1.1.3.1)

επ : ⊗ Kj → ⊗(⊗ Kj ). J

I Ji

The morphisms επ must behave naturally with respect to composition of the maps π, and ⊗ is the identity functor. ·

If all the compatibility morphisms επ are isomorphisms, then we arrive at the usual notion of tensor (= symmetric monoidal) category. 1.1.4. A pseudo-tensor category B with single object is the same as an operad. Namely, such B amounts to a collection of sets Bn = PnB , n ≥ 1, equipped with the action of the symmetric group Σn , together with appropriate “composition of operations” maps between them, and our axioms coincide here with those of an operad.

1.1. PSEUDO-TENSOR CATEGORIES

13

Examples. (i) The operad Σ of linear orders: ΣI = {linear orders on I}; the composition is defined by the lexicographic rule. Notice that an operad over Σ, i.e., an operad B equipped with a morphism of operads B → Σ, is the same as “colored operad”. (ii) The operad of Q projections which assigns to I the same set I; the composition map for J  I is I × Ji → J, (i0 , (ji )) 7→ ji0 . (iii) The operad of trees which assigns to I the set TI of (homeomorphism classes of) trees with I ingoing and one outgoing edge; the composition is the gluing of the corresponding outgoing and ingoing edges. Notice that TI is naturally ordered (one tree is larger than another if the latter can be obtained from the former by contracting some connected subgraphs to points); the composition preserves the order. 1.1.5. Let M, N be pseudo-tensor categories. A pseudo-tensor functor τ : N → M is a law that assigns to any object G ∈ N an object τ (G) ∈ M and to {Fi }i∈I , G ∈ N, a map τI : PIN ({Fi }, G) → PIM ({τ (Fi )}, τ (G)) so that τI are compatible with composition and τ· (idG ) = idτ (G) . One defines a morphism between pseudo-tensor functors in the obvious way. So, if N is equivalent to a small category, pseudo-tensor functors N → M form a category. The composition of pseudo-tensor functors is a pseudo-tensor functor in the obvious way. Any pseudo-tensor functor τ defines an usual functor between the corresponding usual categories. We call τ a pseudo-tensor extension of this usual functor. ν

An adjoint pair is a pair of pseudo-tensor functors M  N together with idenτ

∼

tifications a: PIN ({ν(Mi )}, G) −→ PIM ({Mi }, τ (G)) for Mi ∈ M, G ∈ N, compatible with composition of operations.2 We say that ν is left adjoint to τ and τ is right adjoint to ν. The corresponding usual functors are obviously adjoint. Exercise. Show that for a given pseudo-tensor functor its left (resp. right) adjoint pseudo-tensor functor is determined uniquely up to a unique isomorphism. 1.1.6. Examples. (i) A pseudo-tensor subcategory of a pseudo-tensor category M is given by a subclass of objects T ⊂ M together with subsets PIT ({Li }, M ) ⊂ PIM ({Li }, M ) for Li , M ∈ T such that PIT are closed under the composition of operations, and idM ∈ P.T ({M }, M ) for any M ∈ T. Then (T, P T ) is a pseudo-tensor category, and T ,→ M is a pseudo-tensor functor. If PIT ({Li }, M ) = PIM ({Li }, M ) for any Li , M ∈ P, then we call T a full pseudo-tensor subcategory of M. Any subclass of objects of M may be considered as a full pseudo-tensor subcategory of M. In particular, for any M ∈ M we have the full pseudo-tensor subcategory with single object M . We denote by P (M ) = P M (M ) the corresponding operad. Remark. Any pseudo-tensor category M can be realized as a full pseudotensor subcategory of a tensor category. Here is a universal construction of such an embedding M ,→ M⊗ . By definition, an object of M⊗ is a collection {Mi }i∈I of objects of M labeled by some finite non-empty set I; we denote it by ⊗Mi = ⊗I Mi . A morphism ϕ : ⊗J Nj → ⊗I Mi is a collection (π, {ϕi }i∈I ) where π : J  I and ϕi ∈ PJi ({Nj }, Mi ); we write ϕ = ⊗ϕi = ⊗π ϕi . The composition of morphisms is defined using composition of operations. The tensor structure on M⊗ and the embedding M ,→ M⊗ are the obvious ones. 2 I.e.,

we demand that for ϕ ∈ PIN ({ν(Mi )}, G), ψi ∈ PJM ({Lj }, Mi ) one has a(ϕ(ν(ψi ))) = i

N ({G }, H), ϕ ∈ P N ({ν(M )}, G ) one has a(χ(ϕ )) = τ (χ)(a(ϕ )). a(ϕ)(ψi ) and for χ ∈ PK i k k k k k I k

14

1.

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(ii) Assume that M, N have representable pseudo-tensor structures. Then a pseudo-tensor functor N → M is the same as a usual functor τ : N → M together with natural morphisms νI : ⊗ τ (Ni ) → τ (⊗ Ni ) for any finite families of objects I

I

Ni ∈ N, such that νI commute with the compatibility morphisms επ (i.e., for any π: J  I and a J-family Kj ∈ N the diagram ⊗τ (Kj ) J  M yεπ (1.1.6.1)

ν

−−−J−→

τ (⊗Kj ) J

   N τ (επ ) y

⊗(⊗τ (Kj )) I Ji  ⊗νJi yI ν

⊗τ (⊗Kj ) −−−I−→ τ (⊗(⊗Kj )) I

Ji

I Ji

commutes). In particular, any tensor functor between tensor categories is a pseudotensor functor. (iii) If B is an operad, then we call a pseudo-tensor functor B → M a B algebra in M; we denote the category of B algebras in M as B(M). A B algebra in M amounts to an object L ∈ M together with a morphism of operads B → P M (L). For example, a Σ algebra in M is just a monoid, i.e., an object L equipped with an associative product · ∈ P2 ({L, L}, L) = P2M (L). An algebra for a “trivial” 1-point operad is a commutative monoid. (iv) For any family {Mj } of pseudo-tensor categories one defines their product ΠMj in the obvious way. A pseudo-tensor functor N → ΠMj is the same as a collection of pseudo-tensor functors N → Mj . (v) If A is a tensor category, then the product functor A × A → A is a pseudotensor functor in the obvious way. Let M be a pseudo-tensor category. An A-action on M is a pseudo-tensor functor ⊗: A × M → M together with an identification of ∼ two pseudo-tensor functors A × A × M → M, (A1 ⊗ A2 ) ⊗ M −→ A1 ⊗ (A2 ⊗ M ), that satisfies the obvious compatibilities. In concrete terms, such an action is given by an action of A on M as on a usual category together with natural maps PIM ({Mi }, N ) → PIM ({Ai ⊗ Mi }, (⊗Ai ) ⊗ N ) for Mi , N ∈ M, Ai ∈ A. (vi) We are in situation of (v), so we have a pseudo-tensor functor ⊗ : A⊗M → M. Let P be a commutative monoid in A. Then M → A × M, M 7→ (P, M ), is naturally a pseudo-tensor functor (see (iv) above). Thus M → M, M 7→ P ⊗ M , is a pseudo-tensor functor. Similarly, if F is a commutative monoid in M, then A → M, A 7→ A ⊗ F , is naturally a pseudo-tensor functor. So if a A is a B algebra in A and M is a B algebra in M, then P ⊗ M , A ⊗ F are B algebras in M. 1.1.7. The above definitions make sense when operations PI ({Li }, M ) are not sets but objects of a certain fixed tensor (or even pseudo-tensor!) category A.3 We call such thing a pseudo-tensor A-category. The common example is A = the tensor category of k-modules, where k is a commutative ring; this is the case of pseudo-tensor k-categories. Other examples (super setting, DG categories, etc· ) are discussed in 1.1.16. 3 So

the composition maps are morphisms PI ⊗ (⊗I PJi ) → PJ in A, etc.

1.1. PSEUDO-TENSOR CATEGORIES

15

Explicitly, a pseudo-tensor k-category is a pseudo-tensor category M together with a k-module structure on the sets PI such that the composition map is kpolylinear. For example, if M consists of a single object, then we have a k-operad. A pseudo-tensor k-category M is additive if it is additive as a usual k-category, and M is abelian if it is abelian as a usual k-category and the polylinear functors PI are left exact. A pseudo-tensor functor F : N → M between the pseudo-tensor k-categories is k-linear if the maps FI are k-linear. If B is a k-operad, we get a notion of a B k-algebra in M (we usually call them simply B algebras). For example, we have the favorite k-operads Ass, Com, Lie, Pois which give rise, respectively, to associative, commutative and associative, Lie, and Poisson algebras. Note that Ass, Com are k-envelopes of the corresponding set-theoretic operads: Ass = k [Σ], Com = k. To avoid problems with skew-symmetry, we always assume that 1/2 ∈ k when speaking about Lie or Poisson algebras. 1.1.8. Remarks. (i) Let B be any k-operad. For I ∈ S denote by FB(I) the free B algebra in the tensor category k mod of k-modules with generators ei labeled by elements i ∈ I. Then the morphism of k-modules BI → FB(I), ϕ 7→ ϕ(⊗ ei ), I

is injective, so one may consider BI as the k-submodule of FB(I) generated by “multiplicity free monomials of degree I”. (ii) The operad Pois carries a natural grading such that the product and the Poisson bracket have degrees, respectively, 0 and 1. The operad Ass carries a natural decreasing filtration such that gr Ass = Pois. 1.1.9. Let Nj , j ∈ J, be a finite family of pseudo-tensor k-categories. Their tensor product ⊗Nj is a pseudo-tensor category defined as follows. Its objects are collections {Nj }, Nj ∈ Nj ; we denote this object of ⊗Nj by ⊗Nj . One has N PI ({⊗Nji } , ⊗Kj ) := ⊗ PI j ({Nji }, Kj ); the composition maps are tensor prodkJ

ucts of those in Nj . Note that for any pseudo-tensor k-category M a k-linear pseudo-tensor functor ⊗Nj → M is the same as a k-polylinear pseudo-tensor functor ΠNj → M. 1.1.10. Let M be any pseudo-tensor k-category. Then M ⊗ Lie coincides with M as a usual k-category, and one has PIM⊗Lie ({Li }, M ) = PIM ({Li }, M ) ⊗ LieI . The following funny lemma will be of use: Lemma. Lie(M ⊗ Lie) = Com(M). Proof. We have to show that Lie algebras in M ⊗ Lie are the same as commutative algebras in M. Let M be an object in M. One has P2M⊗Lie (M ) = P2M (M ) ⊗ Lie2 , so a skew-symmetric binary M ⊗ Lie operation [ ]M on M amounts to a symmetric binary M operation ·M . Let us identify Lien with the space of Lie polynomials of variables e1 , . . . , en , so [ ]M = ·M ⊗ [e1 , e2 ]. Since Lie3 is generated by [e1 , [e2 , e3 ]], [e3 , [e1 , e2 ]], [e2 , [e3 , e1 ]] with the only relation [e1 , [e2 , e3 ]] + [e3 , [e1 , e2 ]] + [e2 , [e3 , e1 ]] = 0, we see that [ ]M satisfies the Jacobi relation if and only if ·M is associative.  1.1.11. The material of this subsection will not be used in the sequel. The above lemma is a particular case of a general duality statement for quadratic operads. To formulate it, we need some notions from §2 of [GK]. Assume for simplicity that k is a field. Let A be a quadratic k-operad (i.e., A is a quotient

16

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of the free k-operad generated by a finite-dimensional vector space with Σ2 -action A2 modulo relations in A3 ). Its quadratic dual A! is defined in [GK] 2.1.9 (this is a quadratic operad with A!2 equal to the vector space dual to A2 , the Σ2 -action is twisted by sgn). If B is another quadratic operad, then we have a quadratic operad A • B defined in [GK] 2.2.5 (one has (A • B)2 = A2 ⊗ B2 with the Σ2 -action twisted by sgn) . Notice that for any k-operad P the map Hom(A, P) → Hom(A2 , P2 ) is injec= P2 and relations coming from tive. Let Pquad be a quadratic operad with Pquad 2 ∼ P3 . Then the obvious morphism Pquad → P yields a bijection Hom(A, Pquad ) −→ Hom(A, P). Lemma. For any quadratic operads A, B and a pseudo-tensor k-category M one has (1.1.11.1)

A(M ⊗ B) = (A • B! )(M) = B! (M ⊗ A! ).

Proof. Since • is symmetric, it suffices to define the first identification. We may assume that M has a single object. So we have a k-operad P, and we want to ∼ define a canonical bijection Hom(A, P ⊗ B) −→ Hom(A • B! , P). It comes from the obvious identification Hom(A2 , (P ⊗ B)2 ) = A!2 ⊗ P2 ⊗ B2 = Hom((A • B! )2 , P2 ). To check the relations, we may replace P by Pquad , P ⊗ B by (P ⊗ B)quad = Pquad ◦ B (see [GK] 2.2.3), and we are done by [GK] 2.2.6(b).  If A = B = Com, then A! = B! = Lie (see [GK] 2.1.11) and the above lemma becomes the lemma from 1.1.10. 1.1.12. Remark. Suppose that our M is actually a tensor category with tensor product ⊗. Let us show that the (pseudo) tensor structure of M reconstructs uniquely from that of M ⊗ Lie. Note that the pseudo-tensor structure of M ⊗ Lie is representable; denote by ⊗Lie the corresponding tensor product. One has ⊗Lie = Lie∗I ⊗ ⊗. So one I

I

has a canonical identification M1 ⊗ M2 = M1 ⊗Lie M2 , and the commutativity constraints differ by sign. It remains to recover the associativity constraint. For ∼ M1 , M2 , M3 ∈ M consider the composition ⊗ Lie Mi → M1 ⊗Lie (M2 ⊗Lie M3 ) −→ {1,2,3}

M1 ⊗ (M2 ⊗ M3 ); here the first arrow is the compatibility morphism for ⊗Lie (see 1.1.3), and the second isomorphism comes from the ordering of our indices ∼ (e.g., Lie{2,3} −→ k, [e2 , e3 ] 7→ 1). Let ⊗ Lie Mi → M1 ⊗ (M2 ⊗ M3 ) ⊕ M3 ⊗ {1,2,3}

(M1 ⊗ M2 ) ⊕ M2 ⊗ (M3 ⊗ M1 ) be the sum of these maps for cyclic transposition of indices; denote by K its cokernel. The projection maps each of the 3 terms to K isomorphically, and the associativity constraint for ⊗ coincides with composition ∼ ∼ M1 ⊗ (M2 ⊗ M3 ) −→ K −→ M3 ⊗ (M1 ⊗ M2 ) = (M1 ⊗ M2 ) ⊗ M3 . 1.1.13. One can use operads to define pseudo-tensor structures on some categories of modules and their dual categories. Let us explain how to do this in the k-linear setting. Let R be an associative k-algebra; denote by Rmod, modR, the category of left (right) R-modules. Let B be a strict R-operad, i.e., a k-operad endowed with ∼ an isomorphism of k-algebras R −→ B1 . So for any I ∈ S the k-module BI is actually an (R − R⊗I )-bimodule (here ⊗ is the tensor power over k), and for J 

1.1. PSEUDO-TENSOR CATEGORIES

17

I the composition map may be written as a morphism of (R − R⊗J )-bimodules BI ⊗ BJ/I → BJ . R⊗I

Such B defines the pseudo-tensor structures P ∗ = P ∗B and P ! = P !B on modR and Rmod◦ , respectively. Namely, for Li , M ∈ modR set   (1.1.13.1) PI∗ ({Li }, M ) := HommodR⊗I ⊗Li , M ⊗ BI . R

The composition of ∗ operations comes from the composition law in B. Similarly, for Ci , D ∈ Rmod set   ! (1.1.13.2) PI (D, {Ci }) := HomRmod D, BI ⊗ (⊗Ci ) . R⊗I I

Here we write the operations in Rmod◦ as operations on R-modules with reversed order of arguments, so that P.! (D, {C}) = HomRmod (D, C) (= HomRmod◦ (C, D)). The composition of ! operations comes in an obvious manner from the composition law in B. Remarks. (i) B = P ∗ (R) = P ! (R); here we consider R as right and left R-module, respectively. (ii) The ∗ and ! structures are abelian (see 1.1.7) iff BI are flat as, respectively, R- and R⊗I -modules. 1.1.14. Example. Assume that R has a cocommutative Hopf algebra struc⊗I , and the composition ture. It defines a strict R-operad BR . Namely, BR I := R (i) (i) R R (Ji ) map BI ⊗ BJ/I → BJ is (⊗ri ) ⊗ (⊗(⊗rj )) 7→ ⊗(∆ (ri ) · (⊗rj )) ∈ ⊗R⊗Ji = I

I Ji

I

Ji

I

R⊗J . Here ∆(Ji ) : R → R⊗Ji is the Ji -multiple coproduct. Note that the ! pseudotensor structure coincides with the standard tensor category structure on Rmod. 1.1.15. The above notions easily sheafify. Namely, let X be a topological space (or a Grothendieck topology) and M a sheaf of categories on X (we assume that both morphisms and objects of M have local nature, i.e., satisfy the gluing property). A pseudo-tensor structure on M assigns to any open U a pseudo-tensor structure on M(U ) and to any j: V → U a pseudo-tensor extension of the pullback functor j ∗ : M(U ) → M(V ). We assume that this extension is compatible with the composition of j’s and the pseudo-tensor operations have local nature (i.e., for any Li , M ∈ M(U ), the prehseaf P I ({Li }, M ) on U , P I ({Li }, M )(V ) = PI ({Li }, M ), is actually a sheaf). We call such M a sheaf of pseudo-tensor V V categories. All the notions we discuss in Chapter 1 have straightforward sheaf versions. We will tacitly assume them for the next sections. 1.1.16. Super conventions. Sometimes we need to consider super objects in a given category. Let M be a k-category. Its super (:= Z/2-graded) objects form a k-category Ms . In other words, Ms is the tensor product of M and the category of finitedimensional super vector spaces. Ms is also a “super” category: for M, N ∈ Ms the k-module of all M-morphisms Hom(M, N ) has an obvious Z/2-grading, and the morphisms in Ms are the even part in Hom(M, N ). If M is a tensor category, then Ms is automatically a tensor (super) category according to the Koszul rule of signs. A Z-graded super object M in M is a super object in M equipped with an extra Z-grading. For such M we set M [1] := k[1] ⊗ M where k[1] is an odd copy

18

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of k placed in degree −1; i.e., M [1] is the Z-graded super object obtained from M by simultaneous shift of Z- and Z/2-gradings by 1. The shift is an automorphism of the category of Z-graded super objects, so we have M [a] for every a ∈ Z. We consider M a as a subobject of M . The category MsZ of Z-graded super objects has an obvious “Z-graded super structure”: for M, N ∈ MsZ we have a Z-graded super k-module Hom(M, N ) with components Hom(M, N )i = ΠHom(M a , N a+i ). If M is a tensor category, then so is MsZ (the signs come from the “super” grading, so the forgetting of the Z-grading is a tensor functor). A DG super object, or super complex, in M is a Z-graded super object equipped with an action of the commutative graded Lie super algebra k[−1], which is the same as an odd endomorphism d of degree 1 with square 0. They form a DG super category CMs in the obvious manner. This is a tensor DG super category if M is a tensor category. We identify the category of Z-graded objects in M with a full subcategory in MsZ of those objects for which the Z-degree mod 2 coincides with the super degree. In the same manner the category of complexes CM is a full subcategory of CMs . If M is a tensor category, then these are embeddings of tensor subcategories. If M is a pseudo-tensor k-category, then Ms is automatically a pseudo-tensor super k-category (see 1.1.7). Namely, for Li , M ∈ Ms the k-module PI ({Li }, M ) carries the obvious (Z/2)I × Z/2-grading; hence it is a super k-module with respect to the total Z/2-grading. The composition takes the Koszul sign rule into account. Thus for any Li , M ∈ Ms and free super k-modules Vi , U of finite rank one has a canonical identification of super k-modules PI ({Vi ⊗ Li }, U ⊗ M ) = Hom(⊗ Vi , U ) ⊗ PI ({Li }, M ).4 One can also view Ms as a plain pseudo-tensor I

k-category considering only even operations. Similarly, MsZ is a pseudo-tensor Z-graded super k-category and CMs is a pseudo-tensor DG super k-category. One has PI ({Li [ai ]}, M [b]) = P ({Li }, M )[b − Σai ]. Notation. For L ∈ CM (or CMs ) we set L† := Cone(idL ), or, equivalently, L† = k† ⊗ L. Notice that k† is naturally a comutative DG k-algebra (indeed, we can write k† = k[η], d† = ∂η , for an odd variable η of Z-degree −1). So, by 1.1.6(vi),5 for a pseudo-tensor M the functor L 7→ L† is a DG pseudo-tensor endofunctor of CM or CMs and the canonical morphism L → L† is a morphism of DG pseudo-tensor functors. In particular, if L is a B algebra, then so is L† and L → L† is a morphism of B algebras. 1.2. Complements This section is a continuation of 1.1. We consider inner objects of operations in 1.2.1, augmented pseudo-tensor structures in 1.2.4–1.2.7, augmented pseudotensor functors in 1.2.8, modules over “operadic” algebras in 1.2.11–1.2.15, the corresponding monoids (or associative algebras) in 1.2.16, and the action of an augmentation functor on algebras and modules in 1.2.17–1.2.18. 1.2.1. Inner Hom. Let M be a pseudo-tensor category, I a finite set, {Li }i∈I , and M objects in M. Assume we have an object X ∈ M and an element ε ∈ PIe({Li , X}, M ) where Ie := I t · . Then for any J ∈ S and a J-family of objects 4 And

this identification is compatible with composition of operations. to A = C(k) (the category of k-complexes), P = k† .

5 Applied

1.2. COMPLEMENTS

19

{Aj } we get a natural map (1.2.1.1)

κ = κJ (ε) : PJ ({Aj }, X) → PJtI ({Aj , Li }, M ),

κ(ϕ) := ε(ϕ, idLi ). Here “natural” means that for any H  J, an H-family of objects {Bh } and χj ∈ PHj ({Bh }, Aj ) one has κ(ϕ(χj )) = κ(ϕ)(χj , idLi ). A pair (X, ε) with the property that all the maps κ are isomorphisms is uniquely defined (if its exists); we denote it by (PI ({Li }, M ), εI,{Li },M ) and call it the inner P object. Note that Pφ (M ) = M , and P1 (L, M ) = Hom(L, M ) is the “inner Hom” object. If well defined, inner P objects depend on their arguments in a functorial manner: for a surjection E  I, ϕ ∈ Hom(M, N ), and ψi ∈ PEi ({Ke }, Li ) (as usual Ei := π −1 (i)) there is an obvious morphism PI ({Li }, M ) → PE ({Ke }, N ) which is compatible with the composition of ϕ’s and ψ’s. 1.2.2. Here is an “inner” version of the functoriality property. Let π: E → I be any map of finite sets, {Ke } an E-family of objects. Assume that the objects PI ({Li }, M ), PE ({Ke }, M ), PEi ({Ke }, Li ) exist. Then one has a canonical “composition” element (1.2.2.1)

c ∈ PIe({PI ({Li }, M ), PEi ({Ke }, Li )}, PE ({Ke }, M ))


defined as the image of the composition εI,{Li },M idPI ({Li },M ) , εEi ,{Ke },Li by the canonical isomorphism κ−1 . Note that if E = ∅, then c = ε. One may check that the composition is associative in the obvious sense. Example. Consider L ∈ M such that End L = Hom(L, L) exists. Then the composition c ∈ P2 ({End L, End L}, End L) is associative; hence it makes End L a monoid. In the k-linear setting End L is an associative k-algebra. 1.2.3. Example. Suppose we are in situation 1.1.13, so M = modR and the pseudo-tensor structure is given by the functors P ∗ defined by the operad B. For Li , M as above set X := HomR⊗I (⊗Li , M ⊗ BIe). This is a right R-module in the I

R

obvious way, and we have the canonical operation ε ∈ PIe∗ ({Li , X}, M ). The maps κJ (ε) are isomorphisms for |J| = 1; i.e., X necessarily coincides with PI ({Li }, M ) if the latter object exists. This happens in the following situations: (i) Assume that the composition maps BI ⊗ BJ/I → BJ are isomorphisms. R⊗I

Then X = PI ({Li }, M ) if the Li are projective modules of finite rank. (ii) Suppose (i) holds and all BI are flat R-modules. Then X = PI ({Li }, M ) if the Li are finitely presented R-modules. ˆ the category 1.2.4. Augmented pseudo-tensor structures. Denote by S ˆ If in 1.1.1 of arbitrary finite sets and maps between them, so S is a subcategory of S. ˆ we replace S by S, then we get the definition of augmented pseudo-tensor category. An augmented pseudo-tensor category with single object is called an augmented operad. Examples. (i) A tensor category with a unit object 1 is an augmented tensor category: set PI ({Li }, M ) = Hom(⊗Li , M ). We assume that ⊗ = 1. I

φ

(ii) The linear orders operad Σ (see 1.1.4) extends in the obvious way to an ∧ ∧ augmented operad Σ with Σ∅ = {point}. We also have the “trivial” augmented ∧ ∧ ∧ one-point operad, and their k-linear envelopes Ass = k[Σ ] and Com = k.

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1.2.5. One may consider an augmented pseudo-tensor category as a usual pseudo-tensor category equipped with some extra structure. Let us describe this structure explicitly. We need the following definition. Let M be any pseudo-tensor category. Suppose we have a functor h: M → Sets,

(1.2.5.1) and for any finite set I of order (1.2.5.2)

≥2

and i0 ∈ I we have natural maps

hI,i0 : PI ({Li }, M ) × h(Li0 ) → PIr{i0 } ({Li }, M )

(here, as usual, {Li } is an I-family of objects, M ∈ M). We call these data an augmentation functor if the following compatibilities (i), (ii) hold: (i) Let J be a finite set of order ≥ 2, π: J  I a surjective map, j0 ∈ J. Then for ϕ ∈ PI ({Li }, M ), ψi ∈ PJi ({Kj }, Li ), a ∈ h(Kj0 ) one has hJ,j0 (ϕ(ψi ), a) = ϕ(ψi0 , hJi0 ,j0 (ψi0 , a)) (here i0 = π(j0 ), i0 ∈ I r {i0 }) if |Ji0 | ≥ 2, and hJ,j0 (ϕ(ψi ), a) = hI,i0 (ϕ, ψi0 (a))(ψi0 ) if |Ji0 | = 1. (ii) Assume that |I| ≥ 2; let i0 , i1 ∈ I be two distinct elements. Then for ϕ ∈ PI ({Li }, M ), ai0 ∈ h(Li0 ), ai1 ∈ h(Li1 ), one has hIr{i0 },i1 (hI,i0 (ϕ, a0 ), a1 ) = hIr{i1 },i0 (hI,i1 (ϕ, a1 ), a0 ) ∈ PIr{i0 ,i1 } ({Li }, M ) if |I| > 2, and hI,i0 (ϕ, a0 )a1 = hI,i1 (ϕ, a1 )a0 ∈ h(M ) if |I| = 2. One defines morphisms of augmentation functors in the obvious way. Remark. Any subfunctor of an augmentation functor is itself an augmentation functor. The functor h(M ) ≡ ∅ is an augmentation functor. Definition. We say that our augmentation functor is non-degenerate if all the maps PI ({Li }, M ) → Maps(h(Li0 ), PIr{i0 } ({Li }, M )) coming from (1.2.5.2) are injective. ∧

1.2.6. Now let M be an augmented pseudo-tensor category, M the corresponding usual pseudo-tensor category. Then M carries an augmentation functor h = P∅ ; hI,i0 is the composition map for the embedding I r {i0 } ,→ I. Lemma. The above construction yields an 1-1 correspondence between augmented pseudo-tensor categories and usual pseudo-tensor categories equipped with an augmentation functor.  1.2.7. Note that any augmentation functor h: M → Sets is automatically a pseudo-tensor functor (we consider Sets as a tensor category with ⊗ = Π); i.e., we have natural maps (1.2.7.1)

hI : PI ({Li }, M ) → Hom(⊗h(Li ), h(M )). I

1.2. COMPLEMENTS

21

To see this, we can assume, by the above lemma, that h came from an augmented pseudo-tensor structure. Then hI is the composition map for ∅ → I. Equivalently, for ϕ ∈ Pn ({Li }, M ), ai ∈ h(Li ) one has (here hn : = h{1,...,n} ) hn (ϕ)(a1 ⊗ · · · ⊗ an ) = h{n−1,n},n−1 (. . . (h{1,...,n},1 (ϕ, a1 ) . . . ), an−1 )an . In the situation of 1.2.1 there is a canonical map (1.2.7.2)

¯ : h(PI ({Li }, M )) → PI ({Li }, M ), h

λ 7−→ hI,· e (εI,{Li },M , λ).

¯ arrows intertwine h e(c) and the usual composition map for For c as in 1.2.2 the h I P operations. We say that PI ({Li }, M ) is an inner P object in the augmented sense if (1.2.7.2) is an isomorphism. This amounts to the fact that the canonical morphism (1.2.2.1) (for X = PI ({Li }, M )) is an isomorphism for J = ∅ as well. ∧

∧

∧

1.2.8. One defines an augmented pseudo-tensor functor τ : N → M between augmented pseudo-tensor categories in the obvious way. Such τ ∧ amounts to a usual pseudo-tensor functor τ : N → M together with a morphism of augmentation functors hN → hM τ that are compatible in the obvious manner. For example, given ∧ ∧ an augmented operad B , we have the notion of a B algebra in any augmented ∧ pseudo-tensor category M . ∧

∧

Examples. Σ algebras in M are semigroups; algebras for the trivial aug∧ mented operad are commutative semigroups. In the k-linear setting Ass algebras ∧ are unital associative algebras, Com algebras are unital commutative algebras. If A is a unital associative algebra, then the image 1A ∈ h(A) of 1 ∈ k = Ass∅ is called the unit element in h(A). Remark. 1A is the unit of the associative algebra h(A). So a unital associative ∧ algebra in M is the same as an associative algebra in M such that the associative algebra h(A) is unital and the left and right product with the unit element of h(A) is the identity endomorphism of A. 1.2.9. Following the pattern of 1.1.13, one may use augmented operads to define augmented pseudo-tensor structures on some categories of modules and their duals. ∧ Let R be an associative k-algebra and B an augmented strict R-operad; i.e., ∧ ∧ ∼ B is an augmented k-operad equipped with an isomoprhism R −→ B1 . Such B is the same as a usual strict R-operad B together with a left R-module h = B0 and a system of R⊗Ir{i} -linear morphisms hI,i0 : BI ⊗ B0 → BIr{i0 } (here |I| ≥ 2, i0 ∈ I, R

and BI is considered as a right R-module via the i0 th action) that satisfy obvious compatibilities. According to 1.1.13, B defines the ∗ and ! pseudo-tensor structures on modR and Rmod◦ , respectively. Now h defines the augmentation functors h∗ , h! for these structures. Namely, for M ∈ modR, D ∈ Rmod set (1.2.9.1)

h∗ (M ) := M ⊗ h, R

h! (D) := HomRmod (D, h).

∗B The structure morphisms h∗I,i0 : PI∗B ({Li }, M ) ⊗ h∗ (Li0 ) → PIr{i ({Li }, M ) and 0} ! !B ! ! hI,i0 : PI (D, {Ci })⊗h (Ci0 ) → PIr{i0 } (D, {Ci }) come from the hI,i0 maps. Com∧

patibilities (i) and (ii) in 1.2.5 are clear. Notice that B = P ∗∧ (R) = P !∧ (R).

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1.2.10. Example. Assume that R has a cocommutative Hopf algebra structure with counit ε : R → k. It defines an augmented strict R-operad BR ∧ . Namely, we already defined BR in 1.1.14. Set h = k (the R-module structure is defined by ∼ ε), and let hI,i0 be the obvious isomorphisms R⊗I ⊗ k = R⊗Ir{i0 } ⊗ (R ⊗ k) −→ R

R

R⊗Ir{i0 } . We leave it to the reader to check the compatibilities. Note that k is a unit object for the standard tensor structure on Rmod, and ! augmented structure is the corresponding augmented pseudo-tensor structure (see example (i) in 1.2.4). 1.2.11. Let A be a pseudo-tensor category and P a plain category. An Aaction on P is a pseudo-tensor category C together with an identification of A with a full pseudo-tensor subcategory of C and P with a full subcategory of C such that (a) As a plain category, C is equivalent to a disjoint union of A and P. (b) If PIC ({Ci }, B) is non-empty, then either B and every Ci are in A, or B and exactly one of Ci are in P. To define such a structure, one has to present for every P, Q in P and {Ai }i∈I in A, where I ∈ S, a set PI˜({Ai , P }, Q) which behaves naturally with respect to morphisms of P and Q and operations of Ai ’s. The definition can be rewritten in the k-linear setting in the obvious way. Remark. If A is actually a tensor category, then its action on P as of a tensor category can be considered as an A-action in the above sense: just set PI˜({Ai , P }, Q) := Hom((⊗Ai ) ⊗ P, Q). If both A and P have single object, then we refer to C as an A module operad. The A and P objects of C are denoted, respectively, by • and ∗. 1.2.12. Examples. (i) Let A be any pseudo-tensor category. There is a canonical A-action on A, considered as a plain category. The corresponding C is denoted by Am . It is equipped with a faithful pseudo-tensor functor Am → A m ∼ left inverse to both structure embeddings A ⊂ Am such that P A −→ P A unless prohibited by condition (b). This determines Am uniquely. (ii) Consider operad Σ (see 1.1.4). The Σ module operad Σm contains two Σ m m m module suboperads Σm ` , Σr ⊂ Σ . Namely, ΣI := ΣIt{∗} = set of linear orders m m on I t {∗}, and Σ`I (resp. ΣrI ) is the subset of orders such that ∗ is the maximal (resp. the minimal) element. (iii) Let Bi be operads, and let Ci be Bi module operads, i ∈ I. Denote by Π0 Ci ⊂ Π Ci the full pseudo-tensor subcategory with objects • := (• i ), ∗ := (∗i ). I

I

This is a Π Bi module operad. I

1.2.13. Let B be an operad, C a B module operad, and M a pseudo-tensor category. Let τ : C → M be a pseudo-tensor functor. Then L := τ (• ) is a B algebra in M. We call M := τ (∗) an L-module with respect to C, or simply an L-module (if C is fixed). Explicitly, a structure of an L-module on an object M ∈ M is given by a system of compatible “action” maps CI → PIe({Li , M }, M ). Denote by C(M) the category of pseudo-tensor functors τ : C → M (see 1.1.5); its objects are pairs (L, M ), where L is a B algebra and M is an L-module. If L is a fixed B algebra, then we denote by M(L) = M(L, C) ⊂ C(M) the subcategory of L-modules: its objects are pairs (L, M ); the morphisms are identity on L. Example. Let L be an object of a pseudo-tensor category M such that End L ∈ M exists. Then L is an End L-module with respect to Σm ` .

1.2. COMPLEMENTS

23

Variant. More generally, suppose we have B, C, M as above, and a category P equipped with an M-action D. Denote by C(D) the category of pseudo-tensor functors τ : C → D which map • to M, ∗ to P. Then L := τ (• ) is a B algebra in M and P := τ (∗) is an L-module in P (with respect to C, D). For fixed L we get the category P(L) = P(L, C, D) of L-modules in P; its objects are pairs (L, P ) ∈ C(D), and the morphisms are morphisms in C(D) which are identity on L. The situation described in the beginning of this subsection corresponds to P = M, D = Mm (see 1.2.12(i)). 1.2.14. The above definitions render to the k-linear setting in the obvious way. Lemma. If M is an abelian pseudo-tensor k-category, then M(L, C) is an abelian k-category.  A similar statement for P(L, C, D) is left to the reader. 1.2.15. Examples. (i) B = Lie; C = Bm . Then M(L) is the usual category of modules over the Lie algebra L: its object is M ∈ M equipped with a pairing · ∈ P2 ({L, M }, M ) that satisfies the condition `1 · (`2 · m) − `2 · (`1 · m) = [`1 , `2 ] · m. (ii) B = Ass = k[Σ]. Let L be an associative algebra. If C = Bm = k[Σm ], m then M(L, C) is the category of L-bimodules. If C = k[Σm ` ] (resp. C = k[Σr ]), then M(L, C) is the category of left (resp. right) L-modules. 1.2.16. Let B be an operad, C a B module operad, and L a B algebra in the tensor category Sets (with ⊗ = Π; see 1.2.7). We have the obvious “forgetting of the L-module structure” functor o: Sets(L, C) → Sets. Denote by C(L) the monoid of endormorphisms of this functor. Here is an explicit description of C(L). It is generated by elements ((`i ), ϕ), where I is a finite set, `i ∈ L is an I-family of elements of L, ϕ ∈ CI . The relations for C(L) come from surjective maps π: Je → Ie such that π(·) = · and collections ψi ∈ BJi , i ∈ I. The relation corresponding to π, (ψi ) is (1.2.16.1)

((`j ), ϕ1 (ϕ2 , ψi )) = ((ψi (`j )j∈Ji , ϕ1 ) · ((`j )j∈J. , ϕ2 )).

Here (`j ) is a J-family of elements of L, ϕ1 ∈ CI , ϕ2 ∈ CJ. , where J. := J ∩ π −1 (·). The element 1 = (∅, id.) is the unit in C(L). Denote by C(L)mod the category of sets equipped with C(L)-action. The functor o lifts to the functor o˜: SetsLC → C(L)mod which is an equivalence of categories. More generally, one may define a C-action of L on an object of any category, and this is the same as a C(L)-action. The above construction has its k-linear version. Namely, for a k-operad B, a B module operad C, and a B algebra L in kmod (or in the category of sheaves of k-modules on some space) we get an associative k-algebra with unit C(L) (or a sheaf of k-algebras) such that the category of L-modules coincides with the one of C(L)-modules. We call C(L) the enveloping algebra of L. In situation 1.2.15(i) it coincides with the usual enveloping algebra. ∧

1.2.17. Let now M = (M, h) be an augmented pseudo-tensor category, B, C as above. Let L be a B algebra in M, and let M ∈ M(L, C) be an L-module. Then h(L) is a B algebra in Sets (since h is a pseudo-tensor functor), and M (considered as an object of the usual category M) carries a canonical C(h(L))-action. Explicitly, a generator ((`i ), ϕ) ∈ C(h(L)) acts as hI (aI (ϕ))(`i ) ∈ End M . Therefore we have a canonical “identity” functor (1.2.17.1)

M(L, C) → C(h(L))-modules in M.

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1.2.18. Example. Let M be a pseudo-tensor k-category, and let L be a Lie algebra in M. Let {Mi }i∈I , N be a finite collection of L-modules. We say that an P operation ϕ ∈ PI ({Mi }, N ) is L-compatible if ·N (idL , ϕ) = ϕ(·Mi0 , idMi )i6=i0 ∈ i0 ∈I

PIe({L, Mi }, N ).6 Let PLI ({Mi }, N ) ⊂ PI ({Mi }, N ) be the k-submodule of Lcompatible operations. It is easy to see that composition of L-linear operations is L-linear. Therefore PLI is a pseudo-tensor structure on M(L). If M was an abelian pseudo-tensor category, then M(L) is also an abelian pseudo-tensor category (see 1.2.14). If h is a k-linear augmentation functor on M, then h(L) is a Lie k-algebra, and for every M ∈ M(L), h(M ) is an h(L)-module. The functor hL of h(L)invariants, M 7→ hL (M ) := h(M )h(L) , is an augmentation functor on M(L). h(L) If {Fj }j∈J , G are h(L)-modules in M, then the subspace PJ ({Fj }, G) ⊂ PJ ({Fj }, G) of h(L)-linear operations is defined in the obvious way. Therefore h(L)-modules form a pseudo-tensor k-category. The “identity” functor M(L) → h(L)-modules in M

(1.2.18.1)

is naturally a pseudo-tensor functor. For a k-operad B a B algebra in M(L) is the same as a B algebra A in M on which L acts by derivations. According to (1.2.18.1) such an A carries a canonical h(L)-action by derivations. 1.3. Compound tensor categories The notion of pseudo-tensor category, as opposed to that of tensor category, is not self-dual. Thus “algebraic geometry” in a general pseudo-tensor k-category is very poor: one knows what are commutative algebras, but there are no coalgebras, hence no group schemes, etc. The notion of compound pseudo-tensor category restores self-duality which makes it possible to develop algebro-geometric basics on these grounds as we will see in the next section. We do recommend the reader to consider the basic example of 2.2 below before, or while, reading this section. He may also restrict himself to compound tensor categories (see 1.3.12–1.3.15) instead of general compound pseudo-tensor categories (see 1.3.4–1.3.7), but in this case the self-duality 1.3.8(i) will be lost. The notion of transversality for quotient sets of a given finite set is considered in 1.3.1–1.3.3. We define compound pseudo-tensor structure in 1.3.6–1.3.7, compound pseudo-tensor functors in 1.3.9, augmented version in 1.3.10. We consider duality in 1.3.11 and representable setting in 1.3.12–1.3.13. The notion of a compound category is treated in 1.3.14–1.3.16. An approach to the construction of compound tensor structures parallel to 1.1.13 is in 1.3.18–1.3.19; for an example see 1.3.20. 1.3.1. We have to fix some notation. For a finite non-empty set I denote by Q(I) the set of all equivalence relations on I. We identify an element of Q(I) with the corresponding quotient map πS : I → S and denote it by (S, πS ) or simply S by abuse of notation; as usual for s ∈ S we set Is := πS−1 (s). The set Q(I) is ordered: we write S1 ≤ S2 iff S1 is a quotient of S2 . It is a lattice: for any S, T ∈ Q(I) there exist inf{S, T } ∈ Q(I) and sup{S, T } ∈ Q(I). For any S ∈ Q(I) we have the obvious identifications Q(I) ≤ S := {S 0 ∈ Q(I): S 0 ≤ S} = Q(S), Q(I) ≥ S := {S 00 ∈ Q(I): S 00 ≥ S} = ΠQ(Is ). S

6 Here

·N ∈ P2 ({L, N }, N ), ·Mi ∈ P2 ({L, Mi }, Mi ) are the L-actions.

1.3. COMPOUND TENSOR CATEGORIES

25

We say that S, T ∈ Q(I) are transversal if |S| + |T | = |I| + | inf{S, T }|; S and T are called complementary if they are transveral and | inf{S, T }| = 1. 1.3.2. Proposition. (i) Let V be a non-zero vector space. Let us assign to S ∈ Q(I) the corresponding “diagonal” subspace V S ⊂ V I . Then S, T are transversal iff the subspaces V S , V T ⊂ V I are (i.e., V S + V T = V I ), and S, T are complementary iff, in addition, V S ∩ V T = V · ⊂ V I . (ii) Consider the graph G with vertices labeled by S t T and edges labeled by I, such that an edge i ∈ I connects πS (i) and πT (i). Then S and T are complementary (resp. transversal) iff G is a tree (resp. forest). Proof. (i) Note that |S| = dim V S / dim V , and for any S, T one has V inf{S,T } = V ∩ V T , S = T iff V S = V T . Therefore |S| + |T | ≤ | sup{S, T }| + | inf{S, T }|, and the equality holds iff V sup{S,T } = V S + V T . Hence |S| + |T | ≤ |I| + | inf{S, T }|, and equality holds iff V I = V S + V T . (ii) Note that | inf{S, T }| coincides with the number of components of G, and compute the Euler characteristic of G.  S

1.3.3. Corollary. (a) Suppose S, T ∈ Q(I) are transversal. Then (i) for any α ∈ inf{S, T } the equivalence relations induced by S, T on Iα are complementary; (ii) for any J ∈ Q(I) such that J ≥ S the pair T, J ∈ Q(I) is transversal as well as inf{T, J}, S ∈ Q(J). The map Q(I) ≥ S → Q(T ), J 7→ inf{T, J} is injective. (b) Suppose S, P ∈ Q(I) are such that inf{S, P } = ·. Then P = inf {T } where T is the set of all T ∈ Q(I) such that T

T ∈T

≥P

and T , S are complementary.



1.3.4. Let M be a category. For any functors F : M → Sets, G: M◦ → Sets we denote by hF, Gi the quotient of the disjoint union of sets F (X) × G(X), X ∈ M, modulo the relations (ϕ(f ), g) = (f, t ϕ(g)), where f ∈ F (X), g ∈ G(Y ), ϕ ∈ Hom(X, Y ) (and X, Y are objects in M). We tacitly assume that hF, Gi is a well-defined set; to assure this, it suffices to know that isomorphism classes of objects in M form a set, or that either F or G is a representable functor (if, say, F = Hom(XF , ·), then hF, Gi = G(XF )). The above construction generalizes to the setting of 1.1.7 in the obvious way. In particular, in the k-linear situation (so M is a k-category and F , G are klinear functors with values in k-modules), the k-module hF, Gik is the quotient of ⊕ F (X) ⊗ G(X) modulo the relations ϕ(f ) ⊗ g = f ⊗ t ϕ(g). We have an obvious X∈M

k

map of sets hF, Gi → hF, Gik which is bijective if either F or G is representable. 1.3.5. Now assume that both categories M and M◦ carry the pseudo-tensor structures given by functors PI∗ and PI! , respectively. It will be convenient to write the variables for PI! in reverse order, i.e., as PI! (M, {Li }) so that P.! (M, L) = HomM (M, L). For any I, J ∈ S and I- and J-families of objects {Li }, {Mj }, respectively, set

 ∗! (1.3.5.1) PI,J ({Li }, {Mj }) := PI∗ ({Li }, ·), PJ! (·, {Mj }) . ∗! Our PI,J is a functor; moreover, the ∗ and ! operations act on P ∗! . Precisely, for any surjective maps H  J, G  I and families of objects {Nh }, {Kg } we have the composition map ! ∗! ∗ ∗! PH/J ({Mj }, {Nh }) × PI,J ({Li }, {Mj }) × PG/I ({Kg }, {Li }) → PG,H ({Kg }, {Nh }).

26

1.

AXIOMATIC PATTERNS

∗! ∗! Note that PI,. = PI∗ , P.,J = PJ! .

1.3.6. Assume that for any I ∈ S and a complementary pair S, T ∈ Q(I) we have a natural map (1.3.6.1)

h

I

∗ ! ∗! ({Li }, {Ms }) × PI/T ({Nt }, {Li }) → PT,S iS,T : PI/S ({Nt }, {Ms }), I

((ϕs ), (ψt )) 7→ h(ϕs ), (ψt )iS,T . Here {Li }, {Ms }, {Nt } are arbitrary families of objects parametrized by I, S, T , respectively, and “natural” means that h

I

iS,T is

I iS,T

functorial with respect to Ms and Nt variables. Note that h extends uniquely to the family of maps Y Y I ∗! h iS,T : PI∗!s ,Hs ({Li }, {Bh }) × PG∗!t ,It ({Ag }, {Li }) → PG,H ({Ag }, {Bh }) S

T

labeled by surjecitons H  S, G  T , in a way compatible with composition of ∗ and ! operations (acting on Ag and Bh , respectively). 1.3.7. Definition. The data (P ∗ , P ! , h i) form a compound pseudo-tensor structure on M if the following compatibility axioms hold (here I, S, T , {Li }, {Ms }, {Nt } are as above): (i) Take H ∈ Q(I) such that H ≥ S; set V := inf{H, T }. Then for any H-family of objects {Kh } the following diagram commutes: ! ∗ ! ∗ ∗ PH/S ({Kh }, {Ms })×PI/H ({Li }, {Kh })×PI/T ({Nt }, {Li }) −−→ PI/S ({Li }, {Ms })×PI/T ({Nt }, {Li }) ?Q ? ? h iIv ? ? ?h iI Hv ,Tv S,T yV y ∗ PH/S ({Kh }, {Ms }) ×

Q V

PT∗!v ,Hv ({Nt }, {Kh })

−−→

PT∗!,S ({Nt }, {Ms })

Here the upper horizontal arrow is the product of composition maps for Is  Hs , the lower one is h iH S,V . (ii) Take G ∈ Q(I) such that G ≥ T ; set U := inf{S, G}. Then for any G-family of objects {Rg } the following diagram commutes: ∗ ! ! ∗ ! PI/S ({Li }, {Ms })×PI/G ({Rg }, {Li })×PG/T ({Nt }{Rg }) −−→ PI/S ({Li }, {Ms })×PI/T ({Nt }, {Li }) ?Q ? ? h iIu ? ? ?h iI Su ,Gu S,T yU y

Q U

∗! ! PG ({Rg }, {Ms }) × PG/T ({Nt }, {Rg }) u ,Su

−−→

PT∗!,S ({Nt }, {Ms })

Here the upper horizontal arrow is the product of composition maps for It  Gt , the lower one is h iG U,T . I

(iii) If S = I, T = . , Mi = Li , then h(idLi ), ψiI, . = ψ for any ψ ∈ PI! (N, {Li }). I

(iv) If S = . , T = I, Ni = Li , then hϕ, (idLi )i. ,I = ϕ for any ϕ ∈ PI∗ ({Li }, M ). We call a category M equipped with a compound pseudo-tensor structure a compound pseudo-tensor category. We usually denote it by M∗! , while M∗ and Mo! denote M and Mo considered as plain pseudo-tensor categories (with the operations P ∗ and P ! , respectively).

1.3. COMPOUND TENSOR CATEGORIES

27

1.3.8. Remarks. (i) The axioms (i)–(ii) and (iii)–(iv) are mutually dual. Therefore a compound pseudo-tensor structure on M amounts to that on M◦ . (ii) Any tensor structure on M defines a compound pseudo-tensor structure: set I PI∗ ({Li }, M ) = Hom(⊗Li , M ), PI! (N, {Li }) = Hom(N, ⊗Li ), and h(ϕs ), (ψ)t iS,T = I

I

∗! (⊗ϕs ) ◦ (⊗ψt ) (note that PT,S ({Nt }, {Ms }) = Hom(⊗Nt , ⊗Ms )). S

T

T

S

1.3.9. A compound pseudo-tensor functor τ ∗! : N∗! → M∗! between the compound pseudo-tensor categories is a pair of pseudo-tensor functors τ ∗ : N∗ → M∗ , τ ! : No! → Mo! that give rise to the same usual functor τ : N → M, such that all the diagrams ∗ ! PI/S ({Li }, {Ms }) × PI/T ({Nt }, {Li })   τ ∗ ×τ ! y

< >IS,T

−−−−−→

∗! PT,S ({Nt }, {Ms })   τ ∗! y

< >IS,T

∗ ! ∗! PI/S ({τ Li }, {τ Ms }) × PI/T ({τ Nt }, {τ Li }) −−−−−→ PT,S ({τ Nt }, {τ Ms })

commute. Here the arrow τ ∗! is defined in the obvious way using τ ∗ and τ ! . 1.3.10. Let us explain what an augmented compound pseudo-tensor category is. Let M∗! be a compound tensor category and h an augmentation functor on M∗ . We say that h is compatible with the compound structure, or h is a ∗-augmentation functor on M∗! , if it satisfies the following condition. Let S, T ∈ Q(I) be a complementary pair and i0 ∈ I an element such that πT−1 (t0 ) = {i0 }, |πS−1 (s0 )| ≥ 2 where s0 = πS (i0 ), t0 = πT (i0 ). Set I 0 := I r {i0 }, T 0 := T r {t0 }. We may consider S and T 0 as a complementary pair in Q(I 0 ). Our condition is commutativity of the diagram ∗ ! PI/S ({Li }, {Ms }) × PI/T ({Nt }, {Li }) × h(Nt0 ) −−→ PI∗0 /S ({Li }, Ms ) × PI! 0 /T 0 (Nt , {Li }) ? ? ? ? ?< >I ?< >I 0 S,T y y S,T 0

PT∗!,S ({Nt }, {Ms }) × h(Nt0 )

−−→

PT∗!0 ,S ({Nt }, {Ms })

∗! Here the lower horizontal map is the obvious extension of hT,t0 to PT, . ; the upper one is ((ϕs ), (ψt ), a) 7→ ((ϕs , hIs0 ,i (ϕs0 , ψt0 (a))), ψt ). 0 0 One spells out the compatibility property of a !-augmentation functor in a similar (dual) way. We call a compound pseudo-tensor category endowed with both ∗- and !-augmentation functors an augmented compound pseudo-tensor category.

The above definitions extend to the setting of 1.1.7 in the obvious way. In particular, one has the k-linear version (replace product of sets by tensor product of k-modules, etc.). 1.3.11. Duality. One can use the augmentation functors to relate the pseudotensor categories M∗ and Mo! . For example, consider a !-augmentation functor h! on M∗! . Assume that h! is representable by an object 1, so h! (M ) = HomM (M, 1), and for any M ∈ M the object M ◦ := HomM∗ (M, 1) exists (here HomM∗ means inner Hom with respect to ∗ structure; see 1.2.1).

28

1.

AXIOMATIC PATTERNS

Lemma. The functor M◦ → M, M 7→ M ◦ , admits a canonical pseudo-tensor extension M◦! → M∗ .

(1.3.11.1)

Proof. We need to define the maps PI! (M, {Li }) → PI∗ ({L◦i }, M ◦ ). Let I 1 , I 2 be two copies of I. Set J := I 1 t I 2 , Ie = I t ·, and let πI : J  I, πIe: J  Ie be the projections defined by formulas πI (i1 ) = πI (i2 ) = i, πIe(i1 ) = i, πIe(i2 ) = ·; here i ∈ I, and i1 ∈ I 1 , i2 ∈ I 2 are the copies of i. Then I, Ie ∈ Q(J) is a complementary pair (e.g., by 1.3.2(ii)). Consider a J-family of objects {Aj }, Ai1 = L◦i , Ai2 = Li , e an I-family of objects {1i } (copies of 1), and an I-family of objects which are Li for i ∈ I ⊂ Ie and the · one is M . The desired map is the composition ∼

∗! ◦ ∗ ◦ PI! (M, {Li }) → PI,I → PI∗ ({L◦i }, M ◦ ). e ({Li , M }, {1i }) → PIe ({Li , M }, 1) −

J Here the first arrow sends ψ ∈ PI! (M, {Li }) to (κi ), (ψ, idL◦i ) I,Ie (where κi ∈ P2∗ ({Ai1 , Ai2 }, 1i ) = P2∗ ({L◦i , Li }, 1) is the canonical pairing), the second arrow is ! ∗! λ 7→ h!I (λ)(id⊗I in the obvious manner), and 1 ) (for hI see 1.2.7; we extend it to P the last one is the canonical isomorphism (see 1.2.1). We leave it to the reader to check that the construction is compatible with the composition of operations.  Remark. Assume in addition that we have a ∗-augmentation functor h on M∗! . Then M → h(M ◦ ) is an augmentation functor on M◦! . We have a canonical morphism h(M ◦ ) → h! (M )

(1.3.11.3)

defined as restriction of the structure map P2∗ ({M ◦ , M }, 1) × h(M ◦ ) → Hom(M, 1) (see (1.2.5.2)) to κ × h(M ◦ ) where κ is the canonical pairing. We leave it to the reader to check that (1.3.11.3) is a morphism of augmentation functors on M◦! . 1.3.12. Assume that the ! pseudo-tensor structure is representable (see 1.1.3). I ∗ Then PI,J ({Li }, {Mj }) = PI∗ ({Li }, ⊗! Mj ), and we may rewrite the h iS,T data as J

a collection of canonical morphisms (the compound tensor product maps) (1.3.12.1)

∗ ⊗IS,T : PI/S ({Li }, {Ms }) =

Y

PI∗s ({Li }, Ms ) → PT∗ ({⊗! Li }, ⊗! Ms ), It

S

S

⊗IS,T := h·, (id⊗! Li )iIS,T . The axioms (i), (ii) of 1.3.7 can be rewritten as commuIt

tativity of the following diagrams (where I, S, T, . . . have the same meaning as in 1.3.2):

(i)

∗ ∗ PH/S ({Kh }, {Ms }) × PI/H ({Li }, {Kh })  Q Iv ⊗H y S,V × V ⊗Hv ,Tv

∗ −−−−→ PI/S ({Li }, {Ms })  ⊗I y S,T

PV∗ ({ ⊗ ! Kh }, ⊗! Ms ) × PT∗/V ({⊗! Li }, ⊗ ! Kh ) −−−−→ PT∗ ({⊗! Li }, ⊗! Ms ) Hv

S

It

Hv

It

S

1.3. COMPOUND TENSOR CATEGORIES

29

(the horizontal arrows are composition maps) ⊗IS,T

∗ PI/S ({Li }, {Ms }) Q  ⊗Iu y U Su ,Gu

(ii)

−−−−→

PT∗ ({⊗! Li }, ⊗! Ms ) It S   y

∗ PG/U ({⊗! Li }, {⊗ ! Ms }) −−−−→ PT∗ ({⊗ ! (⊗! Li )}, ⊗! Ms ) Ig

Gt

Su

Ig

S

Here the right vertical arrow is the ∗ composition map with compatibility morphisms ⊗ ! (⊗! Li ) → ⊗! Li ; the lower horizontal arrow is the composition of ⊗G U,T Gt

It

Ig

and the map of the PT∗ arising from ⊗! (⊗ ! Ms ) → ⊗! Ms . U

Su

S

1.3.13. The simplest non-trivial ⊗IS,T operations are those with |S| = 2 (the binary compound tensor products). They look as follows. If S = {1, 2}, then the projection I  S amounts to a non-trivial decomposition I = I1 t I2 . The complementarity of T , S means that the only non-trivial T -equivalence class It has two elements i1 ∈ I1 , i2 ∈ I2 ; i.e., T coincides with I1 ∨ I2 = the union of I1 , I2 with i1 , i2 identified. We call the map ⊗ = ⊗Ii11,i,I22 := ⊗IS,T : PI∗1 ({Li }, M1 ) ⊗ PI∗2 ({Li }, M2 )

i1 ,i2

→ PI∗1 ∨I2 ({Li , Li1 ⊗! Li2 }, M1 ⊗! M2 ), ϕ1 , ϕ2 7→ ϕ1 ⊗ ϕ2 , the binary tensor product at i1 , i2 (here the objects Li under i1 ,i2

the PI∗1 ∨I2 sign are those for i ∈ I r {i1 , i2 }). 1.3.14. Assume that M◦! is actually a tensor category; we call such M∗! a compound tensor category. One may consider ⊗! as a tensor product on M itself; denote this tensor category as M! (so M! is the tensor category dual to Mo! ). If we are in the k-linear situation, then we call M∗! abelian if M is an abelian k-category, the PI∗ are left exact functors, and ⊗! is right exact. A compound tensor functor τ ∗! : N∗! → M∗! between the compound tensor categories is a compound pseudo-tensor functor such that all the canonical morphisms νI : τ (⊗! Ni ) → ⊗ τ (Ni ) from 1.1.6(ii) are isomorphisms; i.e., τ ! : N! → M! is a tensor I

I

functor. Such τ ∗! amounts to a pair (τ ∗ , τ ! ) where τ ∗ : N∗ → M∗ , τ ! : N! → M! are, respectively, the pseudo-tensor and tensor extensions of the same functor τ : N → M which commute with ⊗IS,T maps. Remark. If M is a compound tensor k-categrory, then the category of super objects in M is a compound tensor category in the obvious way, as well as the categories of graded super objects and DG super objects (see 1.1.16 for the terminology). 1.3.15. If M∗! is a compound tensor category, then the right vertical arrow in diagram (ii) in 1.3.12 is an isomorphism. Therefore all the morphisms ⊗IS,T are completely determined by binary ones (use this diagram to compute ⊗IS,T inductively). The binary tensor products are functorial, commutative and associative. Let us spell out these properties explicitly. Functoriality means that the binary products

30

1.

AXIOMATIC PATTERNS

are compatible with composition of ∗ operations in the obvious sense, and ⊗Ii11,i,I22 coincides with the usual ! tensor product of morphisms if I1 = {i1 }, I2 = {i2 }. Commutativity means that ⊗Ii11,i,I22 = ⊗Ii22,i,I11 . To spell out the associativity property, assume that we have three sets I1 , I2 , I3 and the elements i1 ∈ I1 , i12 , i32 ∈ I2 , i3 ∈ I3 ; denote by I1 ∨ I2 ∨ I3 the disjoint union of Ij ’s modulo the relation i1 = i12 , i32 = i3 . Take ϕj ∈ PI∗j ({Li }, Mj ). The associativity says that (ϕ1 ⊗ ϕ2 ) ⊗ ϕ3 = ϕ1 ⊗ (ϕ2 ⊗ ϕ3 ) i1 ,i12

i1 ,i12

i32 ,i3

∈

PI∗1

∨

I2

i32 ,i3

∨

I3 ({Li , Li1

⊗! Li12 , Li32 ⊗! Li3 }, M1 ⊗! M2 ⊗! M3 )

if i12 6= i32 , and (ϕ1 ⊗ ϕ2 ) ⊗ ϕ3 = ϕ1 ⊗ (ϕ2 ⊗ ϕ3 ) = (ϕ1 ⊗ ϕ3 ) ⊗ ϕ2 i1 ,i2

i2 ,i3

i1 ,i2

∈

i2 ,i3

PI∗1 ∨I2 ∨I3 ({Li , Li1

i1 ,i3

!

i1 ,i2

!

⊗ Li2 ⊗ Li3 }, M1 ⊗! M2 ⊗! M3 )

if i12 = i32 = i2 . It is easy to check that for a category M equipped with a tensor structure ⊗! and a pseudo-tensor structure P ∗ the data of functorial, commutative and associative binary tensor products amount to the whole data of compound tensor products, i.e., to a compound tensor structure. Remark. (We use terminology from 1.1.6(v).) The obvious action of the tensor category M! on M extends canonically to an M! -action on M∗ defined by the maps PI∗ ({Mi }, N ) → PI∗ ({Ai ⊗! Mi }, (⊗Ai ) ⊗! N ), ϕ 7→ ϕ ⊗ {idAi }. 1.3.16. Let M∗! be a compound tensor category such that M! has unit object 1 (as a tensor category). Then h! (M ) := HomM (M, 1) is a !-augmentation functor on Mo! . We say that 1 is a unit object in M∗! if h! is compatible with the compound structure (see 1.3.10). Explicitly, this means that for any I-family of objects {Li } and an object M the binary tensor product at i0 ∈ I, End 1 × PI∗ ({Li }, M ) → PI∗ ({Li , 1 ⊗! Li0 }i6=i0 , 1 ⊗! M ) = PI∗ ({Li }, M ), sends id1 × ϕ to ϕ for any ϕ ∈ PI∗ ({Li }, M ). Recall that in any tensor category F = End 1 is an abelian semigroup (a commutative k-algebra in the k-linear situation) that acts canonically on any object M (the action F → End M sends f ∈ F to f ⊗! idM ∈ End(1 ⊗! M ) = End M ). The above compatibility implies that the F -action on PI∗ ({Li }, M ) through M coincides with the action through any Li . If we are in the k-linear situation this means that the PI∗ are canonically F -modules and M∗! is actually a compound tensor F -category. We will call a compound tensor category M∗! which has a unit and is endowed with a ∗-augmentation functor h simply an augmented compound tensor category. We leave the definition of an augmented compound tensor functor between such categories to the reader. 1.3.17. Assume we are in the k-linear situation. We say that our unit object 1 is a strong unit object in M∗! if for any I ∈ S and Li , M ∈ M one has PIe∗ ({Li , 1}, M ) = 0.

1.3. COMPOUND TENSOR CATEGORIES

31

1.3.18. A compound operad is the same as a compound pseudo-tensor category with single object. One may use compound operads to define compound tensor structures on some categories of modules (cf. 1.1.13). As in 1.1.13 we explain this in the k-linear setting. Let R be an associative k-algebra, and let B∗! be a compound strict R-operad, ∼ i.e., a compound k-operad B∗! together with isomorphism of k-algebras R −→ B1 . Explicitly, such B∗! is a pair (B∗ , B! ), where B∗ is a strict R-operad, and B! is a strict Ro -operad (Ro is the opposite algebra, so B∗I is an (R − R⊗I )-bimodule and B!I is an (R⊗I − R)-bimodule), together with the pairings h

(1.3.18.1)

I

iS,T : (⊗ B∗Is ) ⊗ (⊗ B!It ) → B!S ⊗ B∗T S

R⊗I T

R

defined for any complementary pair S, T ∈ Q(I). These maps are morphisms of (R⊗S − R⊗T )-bimodules, and the axioms in 1.3.7 just say that the diagram (here I, S, T, H, V are as in (i) in 1.3.7) (⊗ B∗Hs ) ⊗ (⊗ B∗Ih ) ⊗ (⊗ B!It ) −−−−→ (⊗ B∗I3 ) ⊗ (⊗ B!It ) S S R⊗H H R⊗I T R⊗I T   y   (⊗ B∗HS ) ⊗ (⊗ B!Hv ) ⊗ (⊗ B∗Tv ) y S R⊗H V R⊗V V   y

(1.3.18.2)

B!S ⊗ B∗V ⊗ (⊗ B∗Tv )

B!S ⊗ B∗T

−−−−→

R⊗V V

R

R

I

commutes as well as the corresponding “dual” diagram, and h iS,T coincides with idB∗T , idB!S if | S |= 1, resp. | T |= 1. Now any such B∗! defines a compound pseudo-tensor structure on modR. Namely, set (see 1.1.13) PI∗ ({Li }, M ) = HommodR⊗I (⊗ Li , M ⊗ B∗I ), I R
PI! (N, {Li }) = HommodR N, (⊗ Li ) ⊗ B!I .

(1.3.18.3)

R⊗I

I

The composition of operations comes from composition laws in B∗ and B! . Note !

that the ! pseudo-tensor structure is representable (one has ⊗ Li = (⊗ Li ) ⊗ B!I ), I

I

R⊗I

so we may use the format of 1.3.12. The compound tensor product !

⊗IS,T : ⊗ PI∗s ({Li }, Ms ) → PT∗ ({⊗ Li }, ⊗! Ms )

(1.3.18.4)

S

It

S

sends ⊗ϕs ∈ ⊗ HommodR⊗Is (⊗ Li , Ms ⊗ B∗Is ) to the composition S

Is

R


⊗ϕs ⊗ (⊗ Li ) ⊗ B!It = (⊗ Li ) ⊗ (⊗ B!It ) −−→ (⊗ Ms ) ⊗ (⊗ B∗Is ) ⊗ (⊗ B!It ) T

It

R⊗It

I

R⊗I T

S

h

R⊗S S

R⊗I T

iIS,T

−−−−−→ (⊗ Ms ) ⊗ B!S ⊗ B∗T S

R⊗S

R

32

1.

AXIOMATIC PATTERNS

We leave it to the reader to check the commutativity of diagrams (i) and (ii) in 1.3.12. Note that ⊗! is a tensor product (i.e., we have a compound tensor structure) iff for any J  I the composition map (⊗ B!Ji ) ⊗ B!I − → B!J is an isomorphism. R⊗I

I

1.3.19. Now suppose that B∗! is a ∗-augmented compound operad (see 1.3.10). So we have a left R-module B∗0 and the maps B∗I ⊗ B∗0 → BI\{i0 } , i0 ∈ I, that are R

compatible with h i in the sense explained in 1.3.10. According to 1.2.9, we get an augmentation functor h∗ for the ∗ structure on modR, h∗ (M ): = M ⊗ B∗0 . We R

leave it to the reader to check that h∗ is a ∗-augmentation functor for our compound pseudo-tensor structure (see 1.3.10). Similarly, a !-augmentation for B∗! defines a !-augmentation h! on modR by the formula h! (M ) = HommodR (M, B!0 ). 1.3.20. Example. Assume we have a cocommutative Hopf algebra structure on R. It yields a similar structure on Ro (the coproduct map for Ro coincides with that for R). By 1.1.14 we have the strict R- and Ro -operads B∗ = BR o and B! = BR , respectively. Let us show that B∗ and B! form a compound I strict R-operad. We need to define the maps h iS,T . For S ∈ Q(I) denote by 4IS : R⊗S → R⊗I the morphism of k-algebras ⊗ 4(Is ) (recall that 4(Is ) : R → R⊗Is S

is the Is -multiple coproduct). Let us consider R⊗I as a (R⊗S − R⊗T )-bimodule with respect to the action (⊗rs )(⊗ri )(⊗rt ): = 4IS (⊗rs ) · (⊗ri ) · 4IT (⊗rt ). We have the obvious indentifications of (R⊗S − R⊗T )-bimodules (⊗ B∗Is ) ⊗ (⊗ B!It ) = S

(⊗ R S

⊗Is

) ⊗ (⊗ R

⊗It

) = (R

R⊗I T

sider the morphism of (R

⊗S

⊗I

) ⊗ (R R⊗I ⊗T

−R

⊗I

) = R

⊗I

,

B!S

⊗ B∗T R

= R

⊗S

R⊗I T ⊗T

⊗R

. Con-

R

)-bimodules

ν: R⊗S ⊗ R⊗T → R⊗I , R


ν (⊗rs ) ⊗ (⊗rt ) : = (⊗rs )1(⊗rt ) = 4IS (⊗rs ) · 4IT (⊗rt ). If S, T are complementary, I then ν is an isomorphism (use 1.3.2(ii)), and our h iS,T is ν −1 . We leave it to the reader to check that B∗! = (B∗ , B! , h i) is a compound operad. If our Hopf algebra has counit, then the operads B∗ and B! are augmented in a natural way (see 1.2.10); it is easy to see that (B∗ˆ, B!ˆ, h i) is an augmented compound operad. 1.4. Rudiments of compound geometry This section treats some basic algebro-geometric constructions in compound setting. We consider matrix algebras in 1.4.2, Lie∗ algebras and their cohomology in 1.4.4–1.4.5, commutative! algebras in 1.4.6, reliable augmentation functors in 1.4.7, differential ∗ operations in 1.4.8, Lie∗ algebroids and modules over them in 1.4.11–1.4.12, Lie coalgebroids and the de Rham-Chevalley complexes in 1.4.14, the formal groupoid of a Lie coalgebroid in 1.4.15, the tangent Lie∗ algebroid in 1.4.16, connections in 1.4.17, coisson and odd coisson algebras in 1.4.18, hamiltonian reduction in 1.4.19, coisson modules in 1.4.20, the BRST reduction in 1.4.21–1.4.26. Finally, we explain in 1.4.27 how to encode a compound tensor structure by a single pseudo-tensor structure; the construction looks ad hoc at the moment, but in 3.2 we will see that it represents the “classical limit” of the chiral structure.

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33

The reader is invited to attend to the continuation of the list. For usual Lie algebroids see the textbook [M], written in the differential geometry setting, or a brief exposition in 2.9.1 below. For the BRST in the usual (non-compound) setting see, e.g., [KS]. Statement 1.4.24(iii) is a version of a theorem of Akman [A]. In this section M∗! is an abelian augmented compound tensor k-category (see 1.3.14 and 1.3.16); everything adapts immediately to the sheafified setting of 1.1.15. We abbreviate ⊗! to ⊗. There are no assumptions about exactness of the ∗augmentation functor h. 1.4.1. B∗ and B! algebras. For M ∈ M set Γ(M ) := Hom(1, M ). Then Γ is an augmentation functor on M! ; in particular Γ: M! → kmod is a pseudo-tensor functor (see 1.2.4 and 1.2.7). For a k-operad B we denote by B(M∗ ) the category of B algebras in M∗ ; we call them simply B∗ algebras. In particular we have the category Lie(M∗ ) of Lie∗ ∧ ∧ algebras. If B is an augmented k-operad, then B algebras in the augmented ∧ ∗ ∗ category M are called B algebras. In particular, we have the category of unital associative∗ algebras (see 1.2.8). Similarly, we refer to B algebras in M! as B! algebras. As follows from Remark in 1.3.15 (and 1.1.6(v)), if E is a B∗ algebra and R a 0 B ! algebra, then E ⊗! R is a (B⊗B0 )∗ algebra. In particular, if R is a commutative! algebra, then E ⊗ R is a B∗ algebra. 1.4.2. Matrix algebras and non-degenerate ∗ pairings. Assume we have V, V 0 ∈ M and h i ∈ P2∗ ({V 0 , V }, 1) a ∗ pairing. Then E = V ⊗ V 0 carries a canonical structure of associative∗ algebra which acts on V from the left and on V 0 from the right. Namely, the product operation · ∈ P2∗ ({E, E}, E) is the image of idV ⊗ h i ⊗ idV 0 ∈ End(V ) ⊗ P2∗ ({V 0 , V }, 1) ⊗ End(V 0 ) by the ⊗IS,T map, where I = {1, 2, 3, 4}, and S, T are equivalence relations {2 = 3} and {1 = 2, 3 = 4}, respectively. The E-action on V is the image of idV ⊗ h i by the binary map {1},{2,3} ⊗1,2 : End(V )⊗P2∗ ({V 0 , V }, 1) → P2∗ ({E, V }, V ) (see 1.3.13 for notation). We leave the definition of the right E-action on V 0 , as well as verification of associativity, to the reader. One often wants to know if the E-action on V identifies E with End V . The following condition on h i assures this: Definition. A ∗ pairing h i ∈ P2∗ ({V 0 , V }, 1) is left non-degenerate if for every finite set I and objects Ai , B ∈ M, i ∈ I, the canonical map (1.4.2.1)

PI∗ ({Ai }, B ⊗ V 0 ) → PI˜∗ ({Ai , V }, B),

ϕ 7→ (idB ⊗ h i)(ϕ, idV ),

is an isomorphism.7 We say that h i is right non-degenerate if the transposed pairing is left non-degenerate; h i is non-degenerate if it is both left and right nondegenerate. If h i is left non-degenerate, then for every B ∈ M one has (1.4.2.2)

∼

B ⊗ V 0 −→ Hom(V, B),

∼

h(B ⊗ V 0 ) −→ Hom(V, B)

(see 1.2.1 and 1.2.2, and (1.4.2.1) for I = ∅). In particular, taking B = 1, we see that ∼ V 0 = Hom(V, 1); i.e., V 0 equals the ∗ dual V ◦ to V , and E = V ⊗ V 0 −→ End V , 7 Here

idB ⊗ h i ∈ P2∗ ({B ⊗ V 0 , V }, B).

34

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∼

h(E) −→ End V . Our E is a unital associative∗ algebra: the unit element8 is idV ∈ End V = h(E). Notice that V 0 is ⊗-flat; if h is right exact, then V is a projective object. Exercise. The above definitions make sense if we replace M by the DG category of complexes CM (or CMs ; see 1.1.16). Show that a complex L admits a dual L◦ if and only if every Ln admits a dual (then (L◦ )n = (L−n )◦ , dL◦ = −t dL ). A pairing h i ∈ P2∗ ({L0 , L}, 1) is non-degenerate if and only if L is supported in finitely many degrees and every component ∈ P2∗ ({L0n , L−n }, 1) is non-degenerate. 1.4.3. We will often use the fact that ⊗ behaves naturally with respect to ∗ actions on multiples. Precisely, assume we have a map I → J of finite sets and families of objects {Li }i∈I , {Mj }j∈J , {Nj }j∈J . Then there is a canonical map (1.4.3.1)

⊗ItJ : ⊗ PI˜∗j ({Li , Mj }, Nj ) → PI˜∗ ({Li , ⊗ Mj }, ⊗ Nj ) , J,I˜ J

J

J

⊗ϕj 7→ ⊗ ϕj , J

compatible with composition of operations. Example. Assume we have a family {Bj }j∈J of k-operads together with the corresponding Bj module operads Cj . Let Lj be a B∗j algebra, Mj an Lj -module in M∗ with respect to Cj . Then ⊗ Mj carries canonical Lj -actions that mutually J

commute. Namely, for j0 ∈ J the j0 ’s action map Cj0 I → PI˜∗ ({Lj0 I , ⊗ Mj }, ⊗ Mj ) J

J

sends c ∈ Cj0 I to cMj0 ⊗ ( ⊗ idMj ), where cMj0 ∈ PI˜∗ ({Lj0 I , Mj0 }, Mj0 ) is the j6=j0

c-action on Mj0 . Recall that h(Lj ) is a Bj algebra and Mj is automatically an h(Lj )-module (see 1.2.17). It is easy to see that the action of h(Lj0 ) on ⊗ Mj that comes from J

the above Lj0 -action on ⊗ Mj coincides with the action a 7→ aMj0 ⊗ (⊗idMj )j6=j0 . J

∗

1.4.4. Lie algebras and modules. Let L be a Lie∗ algebra. Denote by M(L) the category of L-modules in M∗ . Lemma. M(L) is naturally an abelian augmented compound tensor k-category. We denote it by M(L)∗! . Proof. According to 1.2.18, M(L) carries a canonical abelian pseudo-tensor structure PL∗ . If {Mj }j∈J is a finite family of L-modules, then L acts on ⊗ Mj via J

the sum of the above J commuting actions (i.e., by Leibniz rule). This is the ! tensor product of L-modules. One easily checks that the maps ⊗IS,T send L-linear operations to L-linear ones; this provides the maps ⊗IS,T for M(L). Compatibility axioms 1.3.12 are clear. The object 1 with trivial L-action is a unit object in M∗! L . For an L-module M the L-action on M yields the morphism of k-modules h(M ) → HomM (L, M ).  Denote by hL (M ) its kernel. hL is a ∗-augmentation functor on M(L)∗! . Remark. A morphism L0 → L of Lie∗ algebras yields the “restriction of action” augmented compound tensor functor M(L)∗! → M(L0 )∗! . 8 See

1.2.8.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

35

1.4.5. The Chevalley complex. The constructions below (except Remark (ii)) use only ∗ pseudo-tensor structure on M. For a Lie∗ algebra L and an L-module M we have the Chevalley complex C(L, M ) which is a complex with terms C n (L, M ) := Pn∗ ({L, . . . , L}, M )sgn (the skew-coinvariants of the action of the symmetric group Σn on operations of n variables) and differential given by the standard formula (see below). Set H · (L, M ) = H · C(L, M ); these are the cohomology of L with coefficients in M . As in the case of usual Lie algebras, the Chevalley complex is naturally defined in the more general super setting (see 1.1.16; below we often drop the adjective “super”). Namely, let L be a super Lie∗ algebra, M a super L-module. Consider the graded super k-module with components Pn∗ ({L[1], . . . , L[1]}, M )Σn (coinvariants of the symmetric group action on super operations of n variables). It carries a canonical odd differential δ of degree 1 coming from the morphism ∗ Pn∗ ({L[1], . . . , L[1]}, M )[−1] → Pn+1 ({L[1], . . . , L[1]}, M ), ϕ 7→ ·M (idL[1] , ϕ) − 1 ϕ([ ], id , . . . , id ) where · ∈ P2 ({L, M }, M ) is the action of L on M . M L[1] L[1] 2 One has δ 2 = 0, so we have a DG super k-module denoted by C(L, M ). Let us describe its natural symmetries. Consider the DG Lie∗ algebra L† (see 1.1.16 for notation). The DG Lie algebra h(L† ) = h(L)† acts on C(L, M ) in a canonical way. Namely, h(L) ⊂ h(L† ) acts via the adjoint action on copies of L[1] and the L-module structure on M , and the action of its complement h(L)[1] comes from the map h(L[1]) ⊗ Pn∗ ({L[1] . . . .L[1]}, M ) → ∗ Pn−1 ({L, . . . , L}, M ) which is minus the sum of the canonical “convolution” maps for each of the n arguments. The above constructions are functorial in the obvious manner. Remarks. (i) If for every n the inner P object P∗n ({L, . . . , L}, M ) is well defined (see 1.2.1), then one has the inner Chevalley complex C(L, M ) defined in the obvious way. It carries a canonical ∗ action of L† . There is a canonical morphism of DG h(L† )-modules h(C(L, M )) → C(L, M ) which is an isomorphism if our P’s are P objects in the augmented sense (see 1.2.7). (ii) Let us assume that the ∗ dual L◦ := Hom∗ (L, 1) is well defined and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Then for every n ≥ 0 and every N ∈ M one has N ⊗ L◦⊗n = P∗n ({L, . . . , L}, N ) and h(N ⊗ ∼ L◦⊗n ) −→ Pn∗ ({L, .., L}, N ) (see 1.2.1 and 1.2.7). Thus the inner Chevalley complex is well-defined (for an L-module M one has C(L, M )n = M ⊗ Symn (L◦ [−1])), and we have a canonical isomorphism of complexes (1.4.5.1)

∼

h(C(L, M )) −→ C(L, M ).

Consider now the super DG setting (see 1.1.16), so L is a DG Lie∗ super algebra, M a DG super L-module. The Chevalley super Q complex C(L, M ) is defined then as follows. Its components are C a (L, M ) := Pn∗ ({L[1], . . . , L[1]}, M )a (the n

operations of degree a with respect to the Z-gradings). Its differential is the sum of two components: the Chevalley component considered above and that coming from differentials of L, M by transport of structure. As above, the DG Lie super algebra h(L† ) acts on C(L, M ). ∗ Our C(L, M ) carries a decreasing filtration F n = C ≥n (L, M ) formed by P≥n . We equip L† with a decreasing filtration F −1 = L† , F 0 = L, F 1 = 0; then the h(L† )-action on C(L, M ) is compatible with the F -filtrations.

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1.4.6. Commutative! algebras and modules. We can do some commutative algebra in M! as in any abelian tensor k-category. Denote by Comu(M! ) the category of commutative and associative algebras with unit in M! . An object A ∈ Comu(M! ) (we call it a commutative! algebra) is an object A ∈ M together with a commutative and associative product · = ·A : A ⊗ A → A and a morphism 1 = 1A : 1 → A which is a unit for ·. For a finite family {Ai } of commutative! algebras the tensor product ⊗ Ai has an obvious structure of commutative! algebra; I

this is the coproduct of {Ai } in Comu(M! ). Therefore ⊗ defines a tensor category structure on Comu(M! ); the object 1 endowed with an obvious product is a unit in Comu(M! ). For A ∈ Comu(M! ) we denote by M(A) the category of unital A-modules in M! ; this is an abelian k-category. The usual tensor product ⊗ of A-modules is A

right exact, so M(A) is an abelian tensor k-category with unit object A. The “forgetting of A-action” functor M(A) → M admits a left adjoint “induction” functor M → M(A), M 7→ A ⊗ M , which is a tensor functor. Below for an Amodule N we denote by ·N : A ⊗ N → N the action map. Lemma. M(A) is naturally an abelian augmented compound tensor k-category. We denote it by M(A)∗! . Proof. The tensor structure has been discussed already, so it remains to de∗ , the compound tensor product maps ⊗IS,T , and the ∗fine the ∗ operations PAI augmentation functor. Take I ∈ S and objects {Li }i∈I , M of M. Assume that some Li0 , i0 ∈ I, and M are A-modules. We say that an operation ϕ ∈ PI∗ ({Li }, M ) is A-linear with respect to the i0 th variable if the operations ·M (ϕ ⊗i0 idA ), ϕ(idLi , ·Li0 )i6=i0 ∈ PI∗ ({Li , A ⊗ Li0 }i6=i0 , M ) coincide. Here ϕ ⊗i0 idA ∈ PI∗ ({Li , A ⊗ Li0 }i6=i0 , A ⊗ M ). ∗ ({Li }, M ) ⊂ PI∗ ({Li }, M ) the kIf all Li are A-modules, then we denote by PAI submodule of A-polylinear (i.e., A-linear with respect to each variable) operations. ∗ The composition of A-polylinear operations is again A-polylinear, so PAI form a ∗ pseudo-tensor structure on M(A). The functors PAI are left exact. ∗ ∗ ({ ⊗ Li }, ⊗ Ms ), ({Li }, Ms ) → PAT To define the structure maps ⊗IS,T : ⊗ PAI s AIt

S

AS

∗ consider the embedding PAT ({ ⊗ Li }, ⊗ Ms ) ,→ PT∗ ({⊗Li }, ⊗ Ms ) defined by the AIt

AS

It

AS

∗ projections ⊗Li → ⊗ Li . The composition ⊗ PAI ({Lis }, Ms ) → ⊗ PI∗s ({Li }, Ms ) It

AIt

S

S

⊗IS,T

−−−→ PT∗ ({⊗Li }, ⊗Ms ) → PT∗ ({⊗Li }, ⊗ Ms ) takes values in the image of the It

S

It

AS

∗ embedding. Therefore we can consider it as a morphism ⊗ PAI ({Li }, Ms ) → s

PT∗ ({ ⊗ Lit }, ⊗ Ms ); this is the ⊗IS,T map for M(A)∗! . AIt

S

AS

Compatibility axioms 1.3.12 for M(A)∗! follow from those for M∗! . The object A, considered as an A-module, is a unit object in M(A)∗! . Note ∼ that the morphism 1: 1 → A yields a canonical isomorphism EndM(A) (A) −→ Γ(A) of commutative k-algebras. Therefore, by 1.3.16, M(A)∗! is actually a compound tensor Γ(A)-category. We define the ∗-augmentation functor on M∗! A simply as the restriction of h to M(A) (note that hI,i0 maps preserve A-polylinear operations). 

1.4. RUDIMENTS OF COMPOUND GEOMETRY

37

Remarks. (i) The “induction” functor M → M(A) extends naturally to an augmented compound tensor functor M∗! → M(A)∗! . Namely, we already mentioned that the induction is a ! tensor functor. The map ∗ PI∗ ({Fi }, G) → PAI ({A ⊗ Fi }, A ⊗ G)

(1.4.6.1)

∗ is the composition PI∗ ({Fi }, G) → PI∗ ({A ⊗ Fi }, A⊗I ⊗ G) → PAI ({A ⊗ Fi }, A ⊗ G) ItI where the first arrow sends ϕ to ⊗I,I ˜ (ϕ, {idA }i∈I ) and the second arrow is the

composition with ·I ⊗ idG where ·I : A⊗I → A is the I-fold product. We leave it to the reader to check the compatibility with ⊗IS,T maps (see 1.3.14). The arrow h(F ) → h(A ⊗ F ) is h(1A ⊗ idF ). More generally, any morphism of commutative! algebras f : B → A yields a compound tensor functor (base change) f ∗ : M(B)∗! → M(A)∗!

(1.4.6.2)

such that for M ∈ M(B) one has f ∗ M = A ⊗ M . To define f ∗ , you just replace B

M∗! by M(B)∗! in the above considerations. The base change is compatible with the composition of f ’s in the obvious way. (ii) The induction pseudo-tensor functor M∗ → M(A)∗ is left adjoint (see 1.1.5) to the obvious forgetful pseudo-tensor functor M(A)∗ → M∗ . To see this, consider ∗ ∗ ({Fi , Lj }, M ) ({A⊗Fi , Lj }, M ) → PItJ for M ∈ M(A) and Fi , Lj ∈ M the map PItJ sending an operation ψ to its composition with 1A ⊗idFi : Fi → A ⊗Fi . This projection identifies the subspace of operations A-linear with respect to I variables with ∗ ∗ PItJ ({Fi , Lj }, M ). The inverse map sends ϕ ∈ PItJ ({Fi , Lj }, M ) to the compo⊗I ∗ sition of ϕ ⊗I (⊗idA ) ∈ PItJ ({A ⊗ Fi , Lj }, A ⊗ M ) with the product morphism A⊗I ⊗ M → M . If the Lj are A-modules, then our maps preserve operations Alinear with respect to J-variables, so PA∗ ItJ ({A ⊗ Fi , Lj }, M ) identifies with the ∗ ({Fi , Lj }, M ) that consists of operations A-linear with respect to subspace of PItJ the J variables. In the setting of (1.4.6.2) the pseudo-tensor functor f ∗ : M(B)∗ → M(A)∗ is left adjoint to the obvious forgetful pseudo-tensor functor M(A)∗ → M(B)∗ .9 (iii) If a ∗ pairing h i ∈ P2∗ ({L0 , L}, 1) in M is non-degenerate (see 1.4.2), then so is the corresponding pairing h iA ∈ P2∗ ({A ⊗ L0 , A ⊗ L}, A) in M(A). 1.4.7. Reliable augmentation functors. We know that h defines a pseudotensor functor on M∗ with values in the tensor category of k-modules. Usually this pseudo-tensor functor is not fully faithful. Let us formulate a property of h which is an effective remedy for this nuisance. ˜ For M ∈ M let h(M ) : Comu(M! ) → kmod be the functor R 7→ h(R ⊗ M ) (so we consider R as a variable “test” algebra). Since h is a pseudo-tensor functor on each M(R)∗ , we see that every ∗ operation ϕ ∈ PI∗ ({Mi }, N ) yields a morphism of ˜ ˜ i ) → h(N ˜ ) on Comu(M! ). functors10 h(ϕ) : ⊗ h(M I

9 To

see this, replace M by M(B) in the above argument. ˜ the functor ⊗ h(M i ) is R 7→ ⊗ h(R ⊗ Mi ).

10 Here

I

I

38

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Definition. We say that our augmentation functor h is reliable if for ev˜ : P ∗ ({Mi }, N ) → ery non-empty family of objects {Mi } and N ∈ M the map h I 11 ˜ ˜ Mor(⊗ h(Mi ), h(N )) is bijective. I

Therefore if h is reliable, then for a k-operad B and an object M ∈ M a B∗ algebra structure on M amounts to a rule that assigns to every commutative! algebra R a B algebra structure on h(R ⊗ M ) such that for every morphism of commutative! algebras R1 → R2 the corresponding morphism h(R1 ⊗M ) → h(R2 ⊗ M ) is a morphism of B algebras. Remarks. (i) Assume that the category Φ of all functors F : Comu(M! ) → kmod, R 7→ FR , makes sense (which happens if M is equivalent to a small category). ˜ : M∗ → Φ⊗ is a Then Φ is a tensor category (one has (⊗Fi )R := ⊗(FiR )) and h ˜ pseudo-tensor functor. Reliability of h just means that h is a fully faithful pseudotensor embedding. (ii) If we are playing with sheaves of categories, then in the above definition kmod should be replaced by the tensor category of sheaves of k-modules. (iii) If h is reliable, then the corresponding augmentation functor on every M(A)∗ is also reliable. 1.4.8. Differential ∗ operations. Many important ∗ operations between A-modules are not A-polylinear but belong to a larger class of A-differential ∗ operations we are going to define. Let I ⊂ A ⊗ A be the ideal of the diagonal, i.e., the kernel of the product map A ⊗ A → A. For an A-module L consider A ⊗ L as an A ⊗ A-module and set L(n) := A ⊗ L/In+1 (A ⊗ L) = A(n) ⊗ L. Now assume we have M ∈ M(A) and A

Li ∈ M. Pick some i0 ∈ I and identify PI∗ ({Li }, M ) with the subspace of operations in PI∗ ({Li , A ⊗ Li0 }, M ) A-linear with respect to the i0 th variable (see Remark (ii) in 1.4.6). Assume that Li0 is also an A-module. The operations vanishing on In+1 (A ⊗ Li0 ) form a subspace of PI∗ ({Li }, M ); its elements are differential operations of order ≤ n with respect to the i0 th variable. If n = 0, these are the same as A-linear operations. If all the Li are A-modules, then the ∗ operations which are differential operations of some order with respect to each variable are called A-polydifferential S (n) ∗ operations of finite order. The space of such operations is PA∗ I ({Li }, M ) ⊂ n

PA∗ I ({A ⊗ Li }, M ) = PI∗ ({Li }, M ). They are stable with respect to composition, so A-polydifferential ∗ operations form another pseudo-tensor structure on M(A). 1.4.9. Action of Lie∗ algebras on commutative! algebras. If L is a Lie∗ algebra, then an action of L (or an L-action by derivations) on a commutative! algebra A is an L-module structure τ ∈ P2∗ ({L, A}, A) on A (as on an object of M∗ ) such that ·A : A ⊗ A → A, 1A : 1 → A are morphisms of L-modules. Let τ be a fixed action of L on A. Then for M ∈ M(A) a τ -action (or a τ -compatible action) of L on M is an L-module structure on M (as on an object of M∗ ) such that ·M : A ⊗ M → M is a morphism of L-modules. The A-modules equipped with τ -actions form an abelian category M(A, L, τ ). 11 Here

˜ ˜ Mor(⊗ h(M i ), h(N )) denotes the collection of morphisms of functors. I

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39

Lemma. M(A, L, τ ) is naturally an abelian augmented compound tensor kcategory. We denote it by M(A, L, τ )∗! . ∗ One defines the ∗ operations in M(A, L, τ ) as PA,L,I ({Mi }, N ) := ∗ ∗ ∩ PAI ({Mi }, N ) (the intersection is taken in PI ({Mi }, N )). Notice that ⊗ Mi is a quotient L-module of ⊗Mi , so we define the ! tensor product on

Proof.

∗ PLI ({Mi }, N ) AI

I

M(A, L, τ ) as ⊗. The ⊗IS,T operations all come from M(A)∗! . The unit object of A

M(A, L, τ ) is A. The ∗-augmentation functor on M(A, L, τ )∗! is hL (see 1.4.4).  1.4.10. The inner Chevalley complex. Let L be a Lie∗ algebra such that the ∗ dual L◦ := Hom∗ (L, 1) exists and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Below we give another interpretation of the inner Chevalley complexes from Remark (ii) in 1.4.5. We deal with the general super DG setting (skipping the adjectives). Consider first the graded commutative! algebra Sym(L◦ [−1]). It carries a canonical action of the graded Lie∗ algebra L† (we forget about the differentials at the moment). Namely, L ⊂ L† acts according to the coadjoint action, and its complement L[1] ⊂ L† acts on the generators L◦ [−1] by − h i ∈ P2∗ ({L[1], L◦ [−1]}, 1). Now the graded algebra Sym(L◦ [−1]) admits a unique differential δ (odd of degree 1; see 1.1.16) such that the action of L† is compatible with the differentials. ν Explicitly, the restriction of δ to L◦ [−1] is equal to the composition L◦ [−2] − → −1/2·

L◦ [−1] ⊗ L◦ [−1] −−−→ Sym2 (L◦ [−1]) where ν is the morphism dual to the Lie∗ bracket.12 Our DG commutative! algebra carries a decreasing filtration by the DG ideals ◦ ≥i ◦ Sym≥i (L◦ [−1]). The topological DG algebra lim ←− Sym(L [−1])/Sym (L [−1]) is denoted by C(L); it carries a natural (continuous) L† -action. The above filtration yields a filtration C≥i (L) on C(L), and the L† -action is compatible with the filtrations.13 Remark. Here “topological algebra” means simply the (formal) projective limit of algebras. We pass to the completion in order to assure that the functor C preserves quasi-isomorphisms. Let M be any L-module. Consider M as an L† -module (we forget about the differential on L† at the moment) via the projection L† → L† /L[1] = L. Then L† acts ◦ ≥i ◦ ˆ on the topological C(L)-module M ⊗C(L) := ← lim − M ⊗(Sym(L [−1])/Sym (L [−1])) ˆ via the tensor product of the our actions. As above, M ⊗C(L)) admits a unique differential such that the actions of L† and C(L) are compatible with the differentials. Explicitly, on the C(L)-generators M the differential is the sum of the differential on M and the morphism M → M ⊗ L◦ dual to the L-action. We denote this topological DG C(L)-module equipped with the L† -action by C(L, M ). It carries a natural filtration by DG submodules C≥i (L, M ) compatible with the filtration C≥i (L). We leave it to the reader to check that C(L) coincides with the inner Chevalley complex from Remark (ii) of 1.4.5. ˆ If L acts on a commutative! algebra A by derivations, then L† acts on A⊗C(L) also by derivations, and the Chevalley differential is a derivation. Thus C(L, A) if h i2 ∈ P3∗ ({L◦ ⊗ L◦ , L, L}, 1) is the tensor product of two copies of h i, then ν is uniquely determined by property h i2 (ν, idL , idL ) = h i (idL◦ , [ ]) ∈ P3∗ ({L◦ , L, L}, 1). 13 Here L is equipped with the filtration defined at the end of 1.4.5. † 12 So

40

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is a topological DG commutative! algebra equipped with a DG L† -action. It is a C(L)-deformation of A = C(L, A)/C≥1 (L, A). If M is an A-module equipped with a compatible L-action (see 1.4.9), then C(L, M ) is a topological DG C(L, A)-module equipped with a compatible L† -action. Remark. The morphism ν : L◦ → L◦ ⊗ L◦ is a Lie cobracket. The duality ∼ isomorphism P2∗ ({L, M }, M ) −→ Hom(M, L◦ ⊗ M ) identifies ∗ actions of L on M ◦ with ! coactions of L on M . So the category of L-modules in M∗ coincides with the category of L◦ -comodules in M! . If M is an L-module, then C(L, M ) is the usual Chevalley complex of the Lie coalgebra L◦ with coefficients in L◦ -comodule M in the tensor category M! . 1.4.11. Lie∗ algebroids. Let A be any commutative unital algebra in M! . Definition. A Lie∗ A-algebroid is an A-module L together with a Lie∗ bracket [ ] ∈ P2∗ ({L, L}, L) and an action τ = τL ∈ P2∗ ({L, A}, A) of L (considered as a Lie∗ algebra) on A by derivations. This datum should satisfy the following properties: (i) τ is A-linear with respect to the L-variable; (ii) the adjoint action adL is a τ -action of L (as a Lie∗ algebra) on L (as an A-module). Notice that the structure ∗ operations (a), (b) are A-polydifferential (see 1.4.8). ∗ . Note that Lie(M(A)∗ ) coincides Lie∗ A-algebroids form a category LieAlgA ∗ with the full subcategory of LieAlgA that consists of L with τ = 0. ∗ 1.4.12. For L ∈ LieAlgA an L-module M is an A-module M equipped with a τ -action τM ∈ P2∗ ({L, M }, M ) of L (considered as a plain Lie∗ algebra) such that τM is A-linear with respect to L-variable. L-modules form a full subcategory14 M(A, L) ⊂ M(A, L, τ ) closed under subquotients and ⊗; clearly A ∈ M(A, L). A

∗ Define the ∗ operations in M(A, L) as PA,L ; i.e., let us consider M(A, L) as a full augmented compound tensor subcategory M(A, L)∗! in M(A, L, τ )∗! .

Lemma. M(A, L)∗! is an abelian augmented compound tensor k-category.



Remark. If our compound tensor structure is a mere tensor one (see 1.3.8(ii)), then Lie∗ algebroids are the same as Lie algebroids. The above L-modules are the same as left L-modules for L considered as a usual Lie algebroid (see 2.9.1). There seems to be no adequate compound analog for the notion of right module over a usual Lie algebroid. 1.4.13. Let L be any Lie∗ algebra, τ an action of L on A. Lemma. (i) The induced A-module LA := A ⊗ L has a unique structure of a Lie∗ A-algebroid such that the obvious morphism 1A ⊗ idL : L → LA is a morphism of Lie∗ algebras compatible with their actions on A. The functor L 7→ LA from the category of Lie∗ algebras acting on A to that of Lie∗ A-algebroids is left adjoint to the “forgetting of the A-action” functor. (ii) The functor M(A, LA )∗! → M(A, L, τ )∗! assigning to an LA -module M the same M considered as an L-module is an equivalence of augmented compound tensor categories.  14 In

the notation M(A, L, τ ) we consider L as a plain Lie∗ algebra acting on A.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

41 ∼

For a Lie∗ A-algebroid L we refer to an isomorphism of Lie∗ algebroids LA −→ L as (L, τ )-rigidification of L. 1.4.14. The de Rham-Chevalley complex. As in 1.4.5, 1.4.10, we consider the super setting. (i) Let L be a Lie∗ A-algebroid. For M ∈ M(A, L) the A-polylinear operations form a subcomplex CA (L, M ) ⊂ C(L, M ) closed under the h(L† )-action. This is the de Rham–Chevalley complex of L with coefficients in M . Set CA (L) := CA (L, A). Below we discuss an “inner” version of this complex. (ii) The next definitions make sense in any unital abelian tensor k-category M! (we do not use the compound structure at the moment). Let A be a commutative unital algebra, L◦ an A-module. A Lie A-coalgebroid structure on L◦ is an odd differential d = dL◦ of degree 1 and square 0 on the algebra SymA (L◦ [−1]). We denote this commutative DG algebra by CA (L◦ ); this is the de Rham-Chevalley DG algebra of L◦ . Let L◦ be a Lie A-coalgebroid. For an A-module M a L◦ -action, or L◦ module structure, on M is a dL◦ -action on SymA (L◦ [−1]) ⊗ M , i.e., a differential A

of SymA (L◦ [−1]) ⊗ M that makes it a DG CA (L◦ )-module. Such a differential is A

uniquely determined by its restriction M → L◦ ⊗ M . We denote this DG CA (L◦ )A

module by CA (L◦ , M ). The category of L◦ -modules is denoted by M(A, L◦ ). This is a tensor category: ◦ if L acts on (finitely many) A-modules Mα , then it acts naturally on ⊗Mα = ⊗ Mα so that CA (L◦ , ⊗M ) =

⊗

CA (L◦ )

CA (L◦ , Mα ).

A

(iii) Let us return to the compound tensor category setting. Suppose that an A-module L admits a dual L◦ and the canonical ∗ pairing ∗ h i ∈ PA2 ({L◦ , L}, A) is non-degenerate (in M(A)∗! ; see 1.4.2). Lemma. There is a natural bijective correspondence between Lie∗ A-algebroid structures on L and Lie A-coalgebroid structures on L◦ . Sketch of a proof. Suppose we have a Lie∗ A-algebroid structure on L. There is a unique action ad◦L of the Lie∗ algebra L on L◦ (called the coadjoint action) such that h i ∈ P2∗ ({L◦ , L}, A) is L-invariant (here L acts on L by the adjoint action, on A by τ ). Indeed, the operation τ (idL , h i) − h[ ], idL◦ i ∈ P3∗ ({L, L, L◦ }, A) is A-linear with respect to the second argument L, so, since h i is non-degenerate (use Remark (ii) from 1.4.6) there is a unique ad◦L ∈ P2∗ ({L, L◦ }, L◦ ) such that the above operation equals h i (idL , ad◦L ). By the universality property of h i, our ad◦L is a τ -action. Consider the graded commutative! algebra SymA (L◦ [−1]). The Lie∗ algebra L acts on it (via τ and ad◦L ). This action extends to an action of the graded Lie∗ algebra L† (see 1.1.16); we forget about the differential on L† at the moment. Namely, L[1] ⊂ L† kills A and acts on L◦ [−1] by h i. We leave it to the reader to show that L◦ carries a unique Lie A-coalgebroid structure such that the L† -action on SymA (L◦ [−1]) is compatible with the differentials and to check that this map is, indeed, bijective.  For a Lie∗ A-algebroid L as above we call L◦ the dual Lie coalgebroid and CA (L) = CA (L, A) := CA (L◦ ) the de Rham-Chevalley DG algebra of L.

42

1.

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Remark. This DG algebra can be seen as the algebra of functions on the quotient space of Spec A modulo the action of L. Lemma. For an A-module M there is a natural bijection between L- and L◦ module structures on M . This bijection is compatible with tensor products, so we get a natural equivalence of tensor categories (1.4.14.1)

∼

M(A, L) −→ M(A, L◦ ).

Sketch of a proof. The subspace of operations φ ∈ P2∗ ({L, M }, M ) which are A-linear with respect to the first variable identifies naturally with Hom(M, L◦ ⊗ M ) A

(see 1.4.2). We leave it to the reader to check that φ is an L-module structure on M if and only if the morphism M → L◦ ⊗ M is an L◦ -action on M .  A

For M ∈ M(A, L) we usually denote the DG CA (L)-module CA (L◦ , M ) by CA (L, M ); this is the inner de Rham-Chevalley complex of L with coefficients in ∼ M . One has h(CA (L, M )) −→ CA (L, M ) (cf. (1.4.5.1)). Remarks. (i) For M ∈ M(A, L) consider the graded SymA (L◦ [−1])-module M ⊗ SymA (L◦ [−1]). It carries a natural action of the graded Lie∗ algebra L† : A

namely, L ⊂ L† acts according to the tensor product of the L-actions, and L[1] ⊂ L† acts via the above action on SymA (L◦ [−1]). Now the differential in CA (L, M ) is characterized by the property that the L† -action is compatible with the differentials. (ii) In the situation of 1.4.13 the above constructions amount to that of 1.4.10: one has CA (LA ) = C(L, A) and CA (LA , M ) = C(L, M ). 1.4.15. Lie coalgebroids and formal groupoids. In this subsection the compound tensor structure is not used, so M! can be any unital abelian k-category (⊗ is right exact, and k is of characteristic 0); for simplicity, we assume that M! has many flat objects. We have the category of commutative unital algebras in M! and its dual category of affine schemes in M! . Many notions of affine algebraic geometry generalize in a straightforward manner to such a setting. In particular, we know what are the affine formal schemes in M! . Let A be a commutative unital algebra in M! , Y := Spec A the corresponding affine scheme. Let G = Spf E be a formal groupoid on Y . Thus we have the structure projection G → Y ×Y and the “identity morphism” embedding i : Y ,→ G; let I ⊂ E be the ideal of i. We say that G is flat if E is an I-adic topological algebra, I/I 2 is ∼ P a flat A-module, and SymA (I/I 2 ) −→ I n /I n+1 . We say that a Lie A-coalgebroid P is flat if P is flat as an A-module. Proposition. The category of flat formal groupoids on Spec A is naturally equivalent to the category of flat Lie A-coalgebroids. Sketch of a proof. (a) Let P be a flat Lie A-coalgebroid, C := CA (P ) its de Rham-Chevalley DG algebra. Let us construct the corresponding formal groupoid G = Spf E on Y . Idea of the construction: Spec C is a DG version of the orbit space of G; thus G is fiber square of Y over Spec C with its evident groupoid structure. Of course, the fiber square should be taken in the DG sense and appropriately completed. We consider C as a filtered topological pro-algebra: C = ← lim − Cn where Cn := C/F n+1 and F n := C ≥n is the “stupid” filtration, so grF C = SymA (P [−1]). Set

1.4. RUDIMENTS OF COMPOUND GEOMETRY

43

L

Tn := A ⊗ A; one computes it using a Cn -flat resolution of A. One has the projecCn

tions . . . → C2 → C1 → C0 = A and . . . → T2 → T1 → T0 = A. Set En := H 0 Tn ; these are commutative algebras. The natural morphisms A = A ⊗ Cn → Tn ← Cn

Cn ⊗ A = A show that Tn and En are A ⊗ A-algebras. Cn

Lemma. One has H >0 Tn = 0 for every n. The projections . . . → E2 → E1 → E0 = A are P surjective. Set In := Ker(En → A); then Ker(En → Em ) = Inm+1 for m < n, and Ina /Ina+1 = SymA≤ n P for n ≥ 1. Proof of Lemma. Cn is filtered by the stupid filtration, and we can compute Tn using a filtered resolution of A. One gets a filtration on Tn such that grF Tn = L

≤n A ⊗ A. Then H >0 grF Tn = 0 = H <0 grF≤ n Tn and H 0 grF · Tn = SymA P by a

grF Cn

usual Koszul resolution computation, which implies the lemma.  Set E := ← lim E . By the lemma, this is an I-adic topological A ⊗ A-algebra, n − P n n+1 where I := lim I /I = SymA P . ←− In , and It remains to define on G := Spf E the structure of a formal groupoid on Y . To do this, we consider for every a ≥ 0 a formal Y a+1 -scheme Ga = Spf E (a) defined as the appropriately completed DG fiber product of a + 1 copies of Y over Spec C (i.e., the DG pull-back of the (Spec C)a+1 -scheme Y a+1 by the diagonal embedding Spec C ,→ (Spec C)a+1 ). Thus G0 = Y , G1 = G. Now the G· form a formal simplicial scheme in the obvious way, and each standard projection Ga → G × . . . × G (a times) Y

Y

is an isomorphism. Therefore G is a groupoid on Y (so that G· is the “homotopy quotient” of Y modulo G, i.e., the “classifying space” BG of G). The details are left to the reader. Remark. The formal groupoid G carries a canonical coaction of the Lie A ⊗ Acoalgebroid P1 := (P ⊗A)×(A⊗P ). Namely, for n ≥ 0 let CnF be a copy of Cn conL

sidered as a filtered Cn -algebra (with filtration F · ). Set Φn := CnF ⊗ CnF . This is a Cn

·

filtered DG Cn -algebra, and gr Φn = gr

·

CnF

·

⊗ gr

Cn ⊗2

Therefore ⊕ H a gra Φn = (Sym≤n A (P [−1])) a

L

CnF

=

⊗2 (Sym≤n A (P [−1]))

⊗ Tn .

A⊗A

⊗ En . Being the first term of the

A⊗A

spectral sequence for the filtration on Φn , this graded algebra carries a natural differential. Passing to the projective limit by n, we see that (Sym·A⊗A (P1 [−1])) ⊗ E A⊗A

carries a natural differential, which is the promised P1 -coaction on G. This coaction is compatible with the groupoid structure on G in an evident way. In fact, each Y a+1 -scheme Ga carries a natural action of the Lie coalgebroid Pa := the product of copies of P acting along each of the coordinates, and the structure projections are compatible with these actions. (b) Suppose we have a flat formal groupoid G = Spf E. The corresponding Lie A-coalgebroid is P := I/I 2 . We will identify SymA (P [−1]) with the de Rham algebra of invariant forms on G; the Lie coalgebroid structure on P is given then by the de Rham differential. (1) Let DRG/Y be the de Rham complex of G relative to the first structure pro(1)

jection G → Y . Similarly, let DRG2 /G be the de Rham complex of G2 := G × G Y

44

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relative to the first projection G × G → G. The second projection and the product Y

(1)

(1)

map G × G → G yield the pull-back morphisms of DG algebras DRG/Y → DRG2 /G . Y

(`)

Their equalizer is the DG algebra DRG/Y of left invariant forms on G. Forgetting about the differential, we have an evident morphism of graded A(`) (1) algebras DRG/Y → i∗ DRG/Y = SymA (P [−1]); one checks immediately that this is an isomorphism. We have defined the promised Lie A-coalgebroid structure on P . (r)

Remark. We can also consider the DG algebra DRG/Y of right invariant differential forms (which lies in the relative de Rham complex for the second structure ∼ (`) (r) projection G → Y ). There is a canonical identification DRG/Y −→ DRG/Y coming from the involution g 7→ g −1 of G. (c) It remains to show that the functors defined in (a), (b) are mutually inverse: Suppose we have a flat Lie coalgebroid P ; let G = Spf E be the corresponding formal groupoid as in (a), and P 0 its Lie coalgebroid as in (b). To define a canonical ∼ isomorphism P −→ P 0 , consider the canonical P1 -coaction on G from Remark in (a). Its A⊗P -component is a differential on the graded algebra E ⊗ Sym(P [−1]); denote A (1)

this formal DG algebra by DRG,P . One checks that the canonical morphism of DG (1)

(1)

algebras DRG/Y → DRG,P is an isomorphism. Its inverse identifies Sym(P [−1]) (`)

with DRG/Y . This identification is compatible with the differential; this is the ∼

promised isomorphism of Lie coalgebroids P −→ P 0 . Now suppose we have a flat formal groupoid G = Spf E. Let P be the corresponding Lie A-coalgebroid, C its de Rham-Chevalley DG algebra as in (b), and G0 = Spf E 0 the formal groupoid defined by P as in (a). Let us define a canonical ∼ isomorphism G −→ G0 . The “identity morphism” embedding Y ,→ G yields a morphism of algebras (1) (1) DRG/Y  A. Its kernel is a DG ideal; denote by F n the filtration on DRG/Y by powers of this ideal. By construction, we have an embedding of DG alge(1) bras C ,→ DRG/Y compatible with the F · filtrations. Notice that each projection (1)

(1)

DRG/Y /F n+1  DRG/Y /F 1 = A is a quasi-isomorphism. It yields a morphism of L

(1)

A ⊗ A-algebras En0 := H 0 (A ⊗ A) → (DRG/Y /F n+1 ) ⊗ A = E/I n+1 =: En . One Cn

Cn

checks that this is an isomorphism; passing to the limit, we get an isomorphism ∼ of formal Y × Y -schemes G −→ G0 . We leave it to the reader to check that it is compatible with the groupoid composition law.  Remark. Here is another (equivalent) way to define the Lie coalgebroid of a formal groupoid G. Consider the formal simplicial scheme G· = BG , Ga = G × . . . × G Y

Y

(a times) as in (a). Let Spec Q· ⊂ G· be the maximal simplicial subscheme such that Q1 = E/I 2 . Thus Q· is a cosimplicial algebra; our C is the corresponding normalized complex. This is a DG algebra in the usual way; one has C 0 = A, C 1 = P . We leave it to the reader to check that C is strictly commutative, equals SymA (P [−1]) as a mere graded algebra, and the corresponding Lie coalgebroid structure on P coincides with one from (b).

1.4. RUDIMENTS OF COMPOUND GEOMETRY

45

Exercise. Let P be an A-flat A-coalgebroid and G the corresponding formal groupoid. Show that for any A-module a P -action on it amounts to a G-action. Thus M(A, P ) coincides with the category of G-equivariant A-modules. Returning to the compound tensor category setting, consider a Lie∗ A-algebroid L as in 1.4.14(iii). Its dual Lie A-coalgebroid L◦ is A-flat (see 1.4.2). We get the corresponding formal groupoid G = GL . By Exercise and (1.4.14.1), L-modules are the same as GL -equivariant A-modules. 1.4.16. The tangent algebroid. For M ∈ M(A) we have the k-module of derivations (with respect to the ! tensor structure) Der(A, M ) ⊂ HomM (A, M ). The functor Der(A, ·) is representable: there is a universal derivation d: A → ΩA . More generally, for any finite collection of objects {Ni }i∈I of M we have a k-submodule Der(A, M )({Ni })I ⊂ PI˜∗ ({Ni , A}, M ) of operations ϕ that are derivations with respect to A, i.e., satisfy ϕ(idNi , ·A ) = ·M (ϕ ⊗1 idA ) + ·M (ϕ ⊗2 idA ) ∈ PI˜∗ ({Ni , A ⊗ A}, M ). Here ϕ ⊗1 idA , ϕ ⊗2 idA ∈ PI˜∗ ({Ni , A ⊗ A}, M ) are the operations defined in 1.4.3; the subscripts 1 and 2 show which copy of A in A ⊗ A we insert in ϕ (the transposition of factors symmetry of A ⊗ A interchanges them). Note that for any Ni -valued ∗ operations χi the composition ϕ(χi , idA ) is again a derivation with respect to A, and in case I = ∅ we have our old Der(A, M ). By Remark (ii) in 1.4.6 (and left exactness of P ∗ ), the map PI˜∗ ({Ni , ΩA }, M ) → ∗ PI˜ ({Ni , A}, M ), ψ 7→ ψ(idNi , d), identifies the subspace of A-linear with respect to ΩA operations with Der(A, M )({Ni })I . If the Ni are also A-modules, then this map identifies PA∗ I˜({Ni , ΩA }, M ) with the subspace of derivations A-linear with respect to Ni . Suppose that the object Der(A, M ): = HomM(A)∗ (ΩA , M ) ∈ M(A) exists (see 1.3.12). By the above and Remark (ii) in 1.4.6 for any finite collection {Gi }, i ∈ I,
one has PI∗ {Gi }, Der(A, M ) = Der(A, M )({Gi })I ⊂ PI˜∗ ({Gi , A}, M ). Let us show that ΘA := Der(A, A) (if it exists) is a Lie∗ A-algebroid in a natural way; this is the tangent algebroid of A. Namely, the action τ ∈ P2∗ ({ΘA , A}, A) is the universal derivation (that corresponds to idΘA under the above identification), and [ ] ∈ P2∗ ({ΘA , ΘA }, ΘA ) is uniquely defined by the condition that τ is a Lie∗ action of ΘA on A. Compatibilities (i) and (ii) in 1.4.11 are clear. ∗ there Remarks. (i) ΘA is a universal Lie∗ A-algebroid: for any L ∈ LieAlgA ∗ is a unique morphism of Lie A-algebroids τ : L → ΘA called the anchor morphism. Notice that L♦ := Ker τ is the maximal Lie∗ A-subalgebra of L. (ii) If the canonical ∗ pairing ∈ PA∗ ({ΘA , ΩA }, A) is non-degenerate, then the de Rham-Chevalley DG algebra CA (ΘA ) equals the de Rham complex DRA , DRi = ΩiA := Λi ΩA . (iii) Let M be an A-module. One may look for a universal Lie∗ A-algebroid E(M ) acting on M . If E(M ) exists, then E(M )♦ ⊂ E(M ) is the Lie∗ A-algebra that corresponds to the associative∗ A-algebra End M ; we denote it by gl(M ).

1.4.17. Connections. The notion of connection on an A-module M is clear (it uses only the tensor category structure); we know what the curvature of a connection is (this is a morphism M → Ω2A ⊗ M ), etc. Let us explain what the connections in the context of Lie∗ algebroids are.

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Let L be a Lie∗ A-algebroid as in 1.4.14(ii), so we have the de Rham-Chevalley DG commutative! algebra C = CA (L). Let P be a Lie∗ A-algebra, and E a P extension of L. So E is an A-algebroid, and we have a short exact sequence 0 → i π P − →E− → L → 0 where i and π are morphisms of Lie∗ A-algebroids. An L-connection on E is an A-linear section ∇ : L → E of π. Its curvature is c(∇) := [∇, ∇] − ∇([ ]) ∈ P2∗ ({L, L}, P ) ⊂ P2∗ ({L, L}, E). One checks that c(∇) is skew-symmetric and A-bilinear. Thus, by (1.4.2.1), one can write c(∇) ∈ h(C2 ⊗P ). Consider the Lie∗ C· -algebra C ⊗ P . Then ∇ defines a dC -derivation d∇ of the C-module C⊗P such that d∇ : P → C1 ⊗P corresponds to [∇, idP ] ∈ P2∗ ({L, P }, P ) via (1.4.2.1). If non-empty, the set of L-connections on E is a torsor for Hom(L, P ) = h(C1 ⊗ P ). For ν ∈ h(C1 ⊗ P ) one has 1 c(∇ + ν) = c(∇) + d∇ (ν) + [ν, ν], d∇+ν = d∇ + adν . 2 Remark. We call ΘA -connections simply connections (here π is the canonical projection). If M is an A-module such that E(M ) is well defined (see Remark (iii) in 1.4.16), then a connection on E(M ) is the same as a connection on M ; the identification Hom(M, Ω2A ⊗ M ) = h(Ω2A ⊗ gl(M )) identifies the corresponding curvatures. (1.4.17.1)

The set of Lie∗ algebroids equipped with an L-connection can be described as follows. Namely, for a given Lie∗ A-algebra P consider the following two sets: (i) the set15 F(P ) of pairs (E, ∇) where E is a P -extension of L, ∇ a connection on E; (ii) the set D(P ) of pairs (d, c) where d is a dC -derivation of the Lie∗ C-algebra C ⊗ P ,16 and c ∈ h(C2 ⊗ P ) is an element such that d(c) = 0 and d2 = adc . We call such a pair (d, c) compatible. The group h(C1 ⊗ P ) acts on both sets: for ν ∈ h(C1 ⊗ P ) the ν-translations are (E, ∇) 7→ (E, ∇ + ν), (d, c) 7→ (d + adν , c + d(ν) + 21 [ν, ν]). Lemma. For any (E, ∇) ∈ F(P ) the pair (d∇ , c(∇)) is compatible. The map F(P ) → D(P ), (E, ∇) 7→ (d∇ , c(∇)), is bijective and compatible with the h(C1 ⊗ P )actions.  Remark. Consider a larger set D0 (P ) of pairs (d, c) which satisfy only the second compatibility d2 = adc from (ii). The h(C1 ⊗ P )-action extends to D0 (P ) by the same formula. The function (d, c) 7→ d(c) is constant on the h(C1 ⊗ P )-orbits. 1.4.18. Coisson and odd coisson algebras. A compound Poisson, or simply coisson, bracket on A ∈ Comu(M! ) is a Lie∗ bracket { } ∈ P2∗ ({A, A}, A) such that the adjoint action τ{ } is an action of A (considered as a Lie∗ algebra) on A (as on a commutative! algebra). In other words, { } satisfies the Leibniz rule with respect to either of the variables. Such pair (A, { }) is called a coisson algebra. By 1.4.13 (i) a coisson bracket { } yields a Lie∗ A-algebroid structure on A ⊗ A. One checks easily that the kernel of the projection A ⊗ A → ΩA , a ⊗ b 7→ adb, is an ideal in A ⊗ A, so ΩA inherits the Lie∗ A-algebroid structure. In other words, ΩA carries a unique Lie∗ A-algebroid structure such that d: A → ΩA is a morphism of Lie∗ algebras and τΩA (d, idA ) = { }. 15 Pairs 16 I.e.,

(E, ∇) are rigid, so we make no distinction between (E, ∇) and its isomorphism class. d is a dC -derivation for the C-module structure and a derivation for the Lie∗ bracket.

1.4. RUDIMENTS OF COMPOUND GEOMETRY

47

Conversely, a structure of a Lie∗ A-algebroid on ΩA yields a binary ∗ operation { }: = τΩA (d, idA ) on A. If { } is skew-symmetric and d{ } = [d, d] ∈ P2∗ ({A, A}, ΩA ), then { } is a coisson bracket. Therefore coisson brackets on A are the same as Lie∗ A-algebroid structures on ΩA that satisfy the above conditions. The category of coisson algebras Cois(M∗! ) is a tensor category. Namely, for a finite family (AP i , { }i ), i ∈ I, of coisson algebras their tensor product (A, { }) is ∼ ∗ A = ⊗Ai , { } = { }∼ i where for j ∈ I the bracket { }j ∈ P2 ({A, A}, A) is the i∈I

⊗ Ai -linear extension of { }j (see Remark (ii) in 1.4.6). The algebra 1 with zero Ir{j}

coisson bracket is a unit object for this tensor structure. Remark. The evident morphisms Ai → A are morphisms of coisson algebras which mutually coisson commute in the obvious sense. This datum is universal: for any coisson algebra B a morphism of coisson algebras A → B amounts to a collection (ϕi ) of mutually coisson commuting morphisms ϕi : Ai → B. Examples. (i) If L is a Lie∗ algebra, then the symmetric! algebra Sym L carries a unique coisson bracket { } such that L = Sym1 L ,→ Sym L is a morphism of Lie∗ algebras; this is the Kostant-Kirillov bracket. The functor Lie(M∗ ) → Cois(M∗ ), L 7→ (Sym L, { }), is left adjoint to the “forgetting of commutative! algebra structure” functor Cois(M∗! ) → Lie(M∗ ). (ii) More generally, if A is a commutative! algebra and L is a Lie∗ A-algebroid, then the symmetric! algebra Sym L := SymA L of the A-module L carries a unique coisson bracket { } such that {Symp L, Symq L} ⊂ Symp+q−1 L, and { } coincides with [ ] on Sym1 × Sym1 and with τL on Sym1 × Sym0 . For example, Sym ΘA is the algebra of functions on the cotangent bundle to Spec A and { } is a canonical coisson structure on it. (iii) The above constructions often come in a twisted version. Let L be a Lie∗ algebra and L[ a central extension of L by 1. The twisted symmetric algebra Sym[ L is the quotient of Sym(L[ ) modulo the ideal generated by the image of the difference of embeddings 1 = Sym0 (L[ ) ,→ Sym(L[ ), 1 ⊂ L[ = Sym1 (L[ ) ,→ Sym(L[ ). This quotient algebra inherits the coisson bracket from Sym(L[ ). Similarly, in the setting of (ii), if L[ is a central extension of a Lie∗ algebroid L by A, then the twisted symmetric algebra Sym[A L inherits a coisson bracket from SymA (L[ ). The above definition, of course, makes sense in the super, or DG super, setting (see 1.1.16). For a commutative super, or DG super, algebra A we can also consider n-coisson brackets on A, which are Lie∗ brackets { } ∈ P2∗ ({A[−n], A[−n]}, A[−n]) on A[−n] such that the adjoint action of A[−n] on A satisfies the Leibniz rule. One calls (A, { }) an n-coisson algebra. For n = 0 this is the usual coisson algebra. As above, for n-coisson A the A-module ΩA [−n] is naturally a Lie∗ algebroid. Remarks. (i) For the plain super setting, it is only the parity of n that matters, so we speak of odd and even (= usual) coisson algebras. (ii) In (the particular case of) the mere tensor (non-compound) category setting, we talk of n-Poisson and odd or even Poisson algebras. Example. For any Lie∗ A-algebroid L the symmetric! algebra SymA (L[n]) is naturally an n-coisson algebra whose bracket is characterized by the property that it coincides with [ ] on Sym1 × Sym1 and with τL on Sym1 × Sym0 . In particular, if ΘA is well defined, then SymA (ΘA [1]) is an odd coisson algebra; its odd coisson

48

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bracket is called the Schouten-Nijenhuis bracket. More generally, SymA (ΘA [n]) is an n-coisson algebra. Exercise. Any ν ∈ HomA (L, A) yields a degree 1 odd derivation dν of the algebra SymA (L[1]) which equals ν on Sym1 . Show that it is compatible with the odd coisson bracket if and only if ν, considered as an element of CA (L)1 (see 1.4.14(i)), is closed. In particular, suppose that ΘA is well defined and the canonical ∗ pairing ∈ PA2 ({ΘA , ΩA }, A) is non-degenerate. Then ν ∈ h(Ω1A ) is closed if and only if dν is a derivation of the Schouten-Nijenhuis bracket. Let A be an n-coisson algebra. Suppose ΘA is well defined and the canonical ∗ pairing ∈ PA2 ({ΩA , ΘA }, A) is non-degenerate. Consider the de Rham-Chevalley algebra CA (ΩA [−n]) of the Lie∗ algebroid ΩA [−n] (see 1.4.14). As a mere graded algebra, it equals SymA (ΘA [n − 1]). Since ΘA is a Lie∗ algebroid, the latter is an n − 1-coisson algebra. One checks that the de Rham-Chevalley differential is compatible with the coisson bracket, so CA (ΩA ) is a DG n − 1-coisson algebra. In particular, for an odd coisson A, Ccois (A) is a coisson algebra. The n − 1-coisson bracket is compatible with the stupid filtration CA (ΩA )≥a , so ≥a the completion Ccois (A) := lim ←− CA (ΩA [−n])/CA (ΩA [−n]) is naturally a topologcois ical (n − 1)-coisson algebra. The spectrum of C (A) is “the space of symplectic leaves” of SpecA. 1.4.19. Hamiltonian reduction. The usual operations on Poisson algebras have obvious coisson versions. For example, if (A, { }) is a coisson algebra and I ⊂ A is an ideal such that {I, I} ⊂ I, then I acts on A/I as a Lie∗ algebra. Set B := (A/I)I (:= the maximal I-invariant subobject of A/I; we tacitly assume that it exists). Then B is a ! subalgebra of A/I, and B inherits a coisson bracket { } such that for b1 , b2 ∈ B and ˜bi ∈ A, ˜bi mod I = bi , one has {b1 , b2 } = {˜b1 , ˜b2 }mod I. In particular, for (A, { }) ∈ Cois(M∗! ), L ∈ Lie(M∗ ), and a morphism of Lie∗ algebras α: L → A, the commutative! algebra B: = (A/α(L)A)L carries a canonical coisson bracket { } (here α(L)A is the ideal generated by α(L) ⊂ A, so L acts on A via α and { }, A/α(L)A carries the quotient action, and ( )L is the subobject of L-invariants). We call (B, { }) the hamiltonian reduction of (A, { }). 1.4.20. Coisson modules. Let (A, { }) be a coisson algebra. A coisson Amodule, or (A, { })-module, is an A-module M (here A is considered as a plain commutative! algebra) together with a τ{ } -action of A on M (here A is considered (A) as a Lie∗ algebra) τM ∈ P2∗ ({A, M }, M ) which is a derivation with respect to the A-variable. The coisson modules form a category M(A, { }). In fact M(A,{ }) coincides with the category M(A, ΩA ) where ΩA is the Lie∗ ∼ A-algebroid from 1.4.18. The equivalence M(A, ΩA ) −→ M(A, { }) assigns to an (A) ΩA -module M the same M with τM coming from the morphism of Lie∗ algebras d : A → ΩA . Therefore M(A, { }) carries a canonical augmented compound tensor structure; we denote it M(A, { })∗! . 1.4.21. BRST reduction (coisson version). This is a refined version of the hamiltonian reduction; both are compared in 1.4.26. Below we tacitly deal with DG super objects, i.e., with the category CMs , following the conventions of 1.1.16. For A ∈ CMs , h(A) is a complex of super k-modules. We assume that k ⊃ Q. Suppose we have L, L0 ∈ CMs and a ∗ pairing h i ∈ P2∗ ({L0 , L}, 1). Consider L[1] ⊕ L0 [−1] as a commutative Lie∗ algebra. Let (L[1] ⊕ L0 [−1])[ = L[1] ⊕

1.4. RUDIMENTS OF COMPOUND GEOMETRY

49

L0 [−1] ⊕ 1 be its central extension by 1 with commutator ∈ P2∗ ({L0 [−1], L[1]}, 1) = P2∗ ({L0 , L}, 1) equal to h i. Set Clc := Sym[ (L0 [−1] ⊕ L[1]) (see Example (iii) in (·) 1.4.18).17 This is the Clifford coisson algebra. It carries an extra Z-grading Clc (−1) 0 (1) determined by the conditions L[1] ⊂ Cl , L [−1] ⊂ Cl . One has: (i) There are obvious embeddings Sym(L0 [−1]), Sym(L[1]) ,→ Clc ; the product morphism Sym(L0 [−1] ⊕ L[1]) = Sym(L0 [−1]) ⊗ Sym(L[1]) → Clc is an isomorphism of commutative! algebras. (ii) The coisson bracket vanishes on both L0 [−1] and L[1]; the { } pairing between L0 [−1] and L[1] is equal to h i. (0) (iii) The subobject L ⊗ L0 = L[1] ⊗ L0 [−1] ⊂ Clc is a Lie∗ subalgebra. The coisson bracket on it coincides with the bracket that comes from the associative∗ algebra structure as defined in 1.4.2. This subalgebra normalizes L[1], L0 [−1] ⊂ (±1) Clc , and the corresponding adjoint action coincides with the action from 1.4.2. We will need a simple lemma. For A ∈ CMs consider the coisson Clc -module A ⊗ Clc . Let Fc ⊂ h(A ⊗ Clc ) be the centralizer of L[1] ⊂ Clc . In other words, Fc is the kernel of the morphism ψ : h(A ⊗ Clc ) → Hom(L[1], A ⊗ Clc ) that comes from the adjoint action of L[1] on Clc . It is clear that Fc ⊃ h(A ⊗ Sym(L[1])). 1.4.22. Lemma. If the ∗ pairing h i is non-degenerate (see 1.4.2), then Fc = h(A ⊗ Sym(L[1])). (a)

Proof. We have h(A ⊗ Clc ) = ⊕ h(A ⊗ Symn (L0 [−1]) ⊗ Symn−a (L[1])). Take (a) any ν ∈ h(A ⊗ Clc ); let n be the largest number such that the nth component of ν is non-zero; denote it by ν¯ ∈ h(A ⊗ Symn (L0 [−1]) ⊗ Symn−a (L[1])). The morphism χ := ψ(ν) sends L[1] to the degree a−1 part of A⊗Sym≤n−1 (L0 [−1])⊗Sym(L[1]); let χ ¯ be its composition with the projection to A ⊗ Symn−1 (L0 [−1]) ⊗ Symn−a (L[1]). Thus (see 1.4.2) χ ¯ ∈ Hom(L[1], A ⊗ Symn−1 (L0 [−1]) ⊗ Symn−a (L[1])) = h(A ⊗ 0 n−1 (L [−1] ⊗ Sym (L0 [−1])) ⊗ Symn−a (L[1])). According to the Leibnitz formula the image of χ ¯ by the product map L0 [−1] ⊗ Symn−1 (L0 [−1]) → Symn (L0 [−1]) equals n¯ ν . If ν lies in Fc , i.e., χ = 0, then n¯ ν = 0, hence n = 0.  1.4.23. From now on L is a Lie∗ algebra. We assume that the ∗ dual L◦ := Hom(L, 1) exists and the canonical pairing h i ∈ P2∗ ({L◦ , L}, 1) is non-degenerate (see 1.4.2). Then the adjoint action of L on L, hence on L[1], yields a morphism of (0) Lie∗ algebras β : L → L ⊗ L◦ = L[1] ⊗ L◦ [−1] ⊂ Clc . We can consider β as an (0) ◦ element of h(Clc ⊗ L ). Denote by β˜c ∈ h(Clc [1]) its image by the product map Clc ⊗ L◦ [−1] → Clc . Let A be a coisson algebra, and let α : L → A be a morphism of Lie∗ algebras. Consider the graded coisson algebra A ⊗ Clc· . We have a morphism of mere graded Lie∗ algebras (1.4.23.1)

` : L† → A ⊗ Clc .

Here L† is a contractible DG Lie∗ algebra defined in 1.1.16 (we forget about its differential for the moment), and the components of ` are `(0) = α+β : L → A⊗Cl(0) and `(−1) := the composition L[1] ,→ Cl(−1) → A ⊗ Cl. 17 The

subscript “c” is for “coisson” or “classical.”

50

1.

AXIOMATIC PATTERNS

Let us rewrite α as α ˜ ∈ h(A ⊗ L◦ )even ⊂ h(A ⊗ Clc [1]). Let β˜ ∈ h(A ⊗ Clc [1]) ˜ be the image of βc by the morphism of algebras Clc → A ⊗ Clc . Set (1.4.23.2)

dc := α ˜+

1 ˜ β ∈ h(A ⊗ Clc [1]) = Hom(k[−1], h(A ⊗ Clc ); 2

this is the (classical) BRST charge. Its adjoint action is an odd derivation d of degree 1 of the graded coisson algebra A ⊗ Clc called the BRST differential. 1.4.24. Proposition. (i) One has {dc , dc } = 0, so d2 = 0. We denote the DG coisson algebra (A ⊗ Clc , d) by CBRST (L, A)c . (ii) ` commutes with the differentials, so we have a morphism of DG Lie∗ algebras (1.4.24.1)

` : L† → CBRST (L, A)c .

We refer to this as the BRST property of d or d. It determines d and d uniquely: (iii) dc is a unique even element of degree zero in h(A ⊗ Clc [1]) such that [dc , `(−1) ] equals `(0) , and d is a unique odd derivation of A ⊗ Cl of degree 1 such that d`(−1) = `(0) . Proof. (i), (ii) are straightforward computations. (iii) follows from the next lemma. Let Derc be the graded Lie algebra of derivations of the coisson algebra A ⊗ Clc . (a)

(a−1)

), 1.4.25. Lemma. The maps h(A ⊗ Clc ) → Der ac → Hom(L[1], A ⊗ Clc where the first arrow is the adjoint action and the second one is δ 7→ δ`(−1) , are injective for a > 0. Proof. (a) Let δ be a coisson derivation of A ⊗ Clc of degree a > 0 such that δ`(−1) = 0; i.e., δ kills L[1] ⊂ Clc . Then it sends L◦ [−1] and A to the centralizer of (a) (a+1) and A ⊗ Clc , which is zero by 1.4.22. Thus δ = 0. L[1] in, respectively, A ⊗ Clc (b) The kernel of the first arrow is contained in the centralizer of L[1]; by 1.4.22 it vanishes if a > 0.  Considered as an object of an appropriate homotopy category of coisson algebras, CBRST (L, A)c is called the BRST reduction of A with respect to α. Its · (L, A)c . The reduction is regucohomology is a graded coisson algebra HBRST 6=0 lar if HBRST (L, A)c = 0. Then the BRST reduction is a plain coisson algebra 0 HBRST (L, A)c . 1.4.26. Let us compare the BRST and hamiltonian reductions (see 1.4.19). Set A¯ := A/α(L)A. This is a commutative! algebra; the L-action on A (via ¯ Let I be the DG ideal of α and the coisson bracket) yields an L-action on A. (−1) (−1) CBRST (L, A)c generated by the image of ` : L[1] → A ⊗ Clc . The morphism ◦ of graded algebras A ⊗ Sym(L [−1]) → A ⊗ Clc /I is surjective; by 1.4.24 its kernel is the ideal generated by α(L). Thus CBRST (L, A)c /I = A¯ ⊗ Sym(L◦ [−1]). One ◦ ¯ checks immediately that the BRST differential yields on A⊗Sym(L [−1]) the usual Chevalley differential (see 1.4.10), so we have a canonical surjective morphism of DG commutative! algebras (1.4.26.1)

¯ CBRST (L, A)c  C(L, A).

1.4. RUDIMENTS OF COMPOUND GEOMETRY

51

Passing to zero cohomology, we get a canonical morphism of commutative algebras 0 HBRST (L, A)c → A¯L .

(1.4.26.2)

It is easy to see that it is compatible with the coisson brackets. 1.4.27. The compound tensor structure on M can be encoded in a single pseudo-tensor structure Mc . For the meaning of this construction see 3.2. So let us bring the sundry polylinear operations of a compound tensor category under one roof. For an I-family Li ∈ M, M ∈ M, and S ∈ Q(I) (see 1.3.1) set (1.4.27.1)

PIc ({Li } , M )S := PS∗ ({ ⊗ Li }, M ) ⊗ ( ⊗ LieIs ). i∈Is

s∈S

The k-module of c operations is (1.4.27.2)

PIc ({Li } , M ) :=

⊕ S∈Q(I)

PIc ({Li } , M )S .

The c operations compose in a natural manner. Namely, assume we have J  I and a J-family of objects Kj ∈ M. Take ϕ ∈ PIc ({Li } , M )S , ψi ∈ PJci ({Kj }, M )Ti (where Ti ∈ Q(Ji )). For R ∈ Q(J) the R-component ϕ(ψi )R ∈ PJc ({Kj } , M )R vanishes unless R ≤ T ,18 I and R are transversal in Q(T ), and inf{I, R} = S (see 1.3.1). If these conditions are satisfied, then ϕ(ψi )R = ϕ(γs ) where for s ∈ S one has19 γs := ⊗TIss,Rs ψi ∈ PR∗ s ({ ⊗ Kj }, ⊗ Li ) ⊗ ( ⊗ LieJt ) and the composition j∈Jr

i∈Is

t∈Ts

ϕ(γs ) is the tensor product of compositions of ∗ and Lie operations.20 We leave it to the reader to check its associativity. We have defined an abelian (see 1.1.7) pseudo-tensor structure Mc on M. Set PIc ({Li }, M )n :=

⊕ S∈Q(I,|I|−n)

PIc ({Li } , M )S where Q(I, m) ⊂ Q(I) con-

sists of quotients I  S, |S| = m. This is a natural Z-grading on P c compatible with the composition of c operations. One has PIc ({Li }, M )|I| = Hom(⊗ Li , M )⊗LieI , PIc ({Li }, M )0 = PI∗ ({Li }, M ). I

The corresponding embedding and projection define the maps P ! ⊗Lie → P c → P ∗ which are compatible with the composition of operations, so we have the pseudotensor functors (1.4.27.3)

M! ⊗ Lie → Mc → M∗

that extend the identity functor on M. Remark. By 1.1.12 the compound pseudo-tensor structure M∗! is uniquely determined by the pseudo-tensor structure Mc together with the Z-grading on P c . 1.4.28. Lemma. Lie algebras in Mc are the same as coisson algebras without unit in M∗! . Proof. A binary c operation µ ∈ P2c (M ) is the same as a pair (·, { }) where21 · ∈ Hom(M ⊗ M, M ) and { } ∈ P2∗ (M ). The skew-symmetry of µ amounts to the 18 Here

T := tTi ∈ Q(J). 1.3.12 for notation. 20 The composition of Lie operations is the tensor product of maps Lie Is ⊗ ( ⊗ LieJt ) → 19 See

∼

t∈Tr

LieJr labeled by r ∈ Rs . We use the fact that Tr −→ Is (the transversality condition). ∼ 21 We use the identification k − → Lie2 , 1 7→ [ ].

52

1.

AXIOMATIC PATTERNS

symmetry of · and the skew-symmetry of { }. The Jacobi identity for µ amounts to three identities (corresponding to the graded components): – the Jacobi identity for · ⊗ [ ] which, by 1.1.10, is the same as the associativity of ·, – the Jacobi identity for { }, – the identity which says that { } acts by derivations of ·. 

CHAPTER 2

Geometry of D-schemes . . . inogda e, udals~ v bliz~ leawu rowu, igral na fle$itravere, ostav druga molodago medu grobov odnogo, ko by dl togo, qto b izdali emu pritnee bylo sluxat~ muzyku. M. I. Kovalinski$i, “izn~ Grigori Skovorody. Pisana 1794 goda v drevnem vkuse.” †

2.1. D-modules: Recollections and notation The language of D-modules plays for us about the same role as the language of linear algebra for the usual commutative algebra and algebraic geometry. Good references are [Ber], [Ba], [Kas2] (we will not use the theory of holonomic Dmodules though). This section collects some information that will be of use; we recommend the reader to skip it, returning when necessary. We discuss left and right D-modules in 2.1.1, basic D-module functoriality in 2.1.2, Kashiwara’s lemma in 2.1.3, the exactness of the pull-back functor for locally complete intersection maps in 2.1.4, locally projective D-modules in 2.1.5, the sheafified middle de Rham cohomology functor h in 2.1.6, the induced D-modules in 2.1.8–2.1.10 (we refer to [S1] for all details) and the quasi-induced D-modules in 2.1.11, maximal constant quotients and de Rham homology in 2.1.12, modifications of a D-module at a point and completions of the local de Rham cohomology in 2.1.13–2.1.15, and a variant of the same subject when the point moves in 2.1.16. Finally, in 2.1.17 we prove an auxiliary result from commutative algebra which explains how the formalism of I-adic topology can be rendered to the setting of arbitrary (possibly infinitely generated) modules. All schemes and algebras are defined over a fixed base field k of characteristic zero. The reader may prefer to work all the way with super objects following conventions of 1.1.16. 2.1.1. Left and right D-modules; homotopical flatness. For a scheme X we denote by MO (X) the category of O-modules on X (: = quasi-coherent sheaves of OX -modules). If X is smooth, then Mr (X) (resp. M` (X)) denotes the category †

“. . . sometimes he played flute in a nearby grove, abandoning the young friend betwixt the tombs, as if the music were to be better appreciated from afar.” M. I. Kovalinski, “The life of Gregori Skovoroda, written in 1794 in ancient gusto.”

53

54

2.

GEOMETRY OF D-SCHEMES

of right (resp. left) D-modules on X (: = sheaves of DX -modules quasi-coherent as OX -modules). Notice that left D-modules are the same as O-modules equipped with an integrable connection. We mostly deal with right D-modules and often call them simply D-modules writing M(X) := Mr (X). One has the obvious forgetting of D-action functors or : M(X) → MO (X), o` : M` (X) → MO (X). If G, L are left D-modules, then G ⊗ L = G ⊗ L is a left D-module in a natural OX

way. Therefore M` (X) is a tensor category with a unit object OX , and o` is a tensor functor. If M is a right D-module and L a left D-module, then M ⊗ L is naturally OX

a right D-module (a field τ ∈ ΘX ⊂ DX acts as (m ⊗ `)τ = mτ ⊗ ` − m ⊗ τ `) X denoted by M ⊗ L. The sheaf ωX = Ωdim (we consider it as an even super line) X has a canonical right D-module structure, namely, ντ = −Lieτ (ν) for ν ∈ ωX . The functor M` (X) → M(X), L 7→ Lr := LωX = ωX ⊗ L, is an equivalence of −1 categories. The inverse equivalence is M 7→ M ` := M ⊗ ωX . We refer to complexes of left (right) D-modules as left (right) D-complexes on X; sometimes right D-complexes are called simply D-complexes. We call the identification of derived categories ∼

DM` (X) −→ DMr (X),

(2.1.1.1)

L 7→ LωX [dim X],

the canonical equivalence. So DM(X) carries two standard Mr and M` t-structures, o`

differing by shift by dim X. The composition DM(X) = DM` (X) −→ DMO (X) is denoted simply by o` . For details about the next definitions see [Sp].1 We say that a complex M of O-modules is homotopically OX -flat if for every acyclic complex F of O-modules the complex M ⊗ F is acyclic. A D-complex M ∈ CM(X) is homotopically OX -flat if it is homotopically flat as a complex of OX -modules. Equivalently, this means that for every acyclic L ∈ CM` (X) the complex M ⊗ L is acyclic. Similarly, M ∈ CM(X) is homotopically DX -flat if for every L as above the complex of sheaves of k-vector spaces M ⊗ L is acyclic. A D-complex M is homotopically DX -flat if and only DX

if it is homotopically OX -flat and for every P ∈ CM` (X) the canonical projection DR(M ⊗ P ) → h(M ⊗ P ) (see 2.1.7 below) is a quasi-isomorphism. 2.1.2. Functoriality. A morphism f : X → Y of smooth varieties yields canonical functors f∗

(2.1.2.1)

DM(X)  DM(Y ) f!

that satisfy standard compatibilities (the base change property, adjunction property for proper f , etc.). If f is quasi-finite, then f∗ is left exact, and if f is affine, then f∗ is right exact with respect to the Mr t-structure. 1 Our

“homotopically flat” is Spaltenstein’s “K-flat”.

2.1. D-MODULES: RECOLLECTIONS AND NOTATION

55

The diagram of functors Rf !

DMO (X) ←− ↑ or (2.1.2.2)

DM(X)

DMO (Y ) ↑ or

f!

←−

↓ o`

DM(Y ) ↓ o`

Lf ∗

DMO (X) ←−

DMO (Y )

commutes; here Rf ! , Lf ∗ are standard pull-back functors for O-modules (recall that for any F ∈ DMO (Y ) one has (Lf ∗ F ) ⊗ ωX [dim X] = Rf ! (F ⊗ ωY [dim Y ])). In particular, f ! is right exact with respect to the M` t-structure. We often denote the corresponding functor M` (Y ) → M` (X) by f ∗ ; this is the usual pull-back of an O-module equipped with a connection. If Xi , i ∈ I, is a finite collection of smooth schemes, then have the exterior Q we Q r r tensor product functors for both left and right D-modules M (X Xi ), i) → M ( Q ` Q M (Xi ) → M` ( Xi ), (Mi ) 7→ Mi . The canonical equivalence (2.1.1.1) identifies the corresponding derived functors.2 The functors f∗ ,Q f ! are compatible with : ifQfi : Xi → Yi are morphisms of smooth schemes, then ( fi )∗ (Mi ) = (fi∗ Mi ), ( fi )! (Ni ) = (fi! Ni ). Assume that all Xi coincide with X; let ∆(I) : X → X I be the diagonal embedding. For Li ∈ DM` (X) one has a canonical isomorphism (2.1.2.3)

L

⊗Li = ∆(I)! (Li ).

In particular, for Li ∈ M` (X) one has (2.1.2.4)

⊗Li = ∆(I)∗ (Li ).

2.1.3. If i: X ,→ Y is a closed embedding, then i∗ is exact with respect to the Mr t-structure, and its right adjoint i! is left exact. They define mutually inverse equivalences between M(X) and the full subcategory M(Y )X ⊂ M(Y ) that consists of DY -modules that vanish on Y \ X (Kashiwara’s lemma). Our functors can be described explicitly as follows. Let I ⊂ OY be the ideal of X. If N ∈ M(Y ), then i! N coincides, as an O-module, with the subsheaf of N that consists of sections killed by I. For a L ∈ M` (Y ) one has i∗ L = L/IL. For M ∈ M(X) one has (2.1.3.1)

i∗ M = i· (M ⊗ i∗ DY ). DX

Here we consider DY as a left DY -module, and the right DY -module structure on i∗ M comes from the right action of DY on DY (as on a left D-module). The ∼ identification M −→ i! i∗ M is m 7→ m ⊗ 1. For L ∈ M` (Y ) and M ∈ M(X) there is a canonical isomorphism (2.1.3.2)

∼

i∗ (M ⊗ i∗ L) −→ (i∗ M ) ⊗ L. ∼

2 There is a canonical isomorphism of super lines ω ΠXi [dim ΠXi ] −→ (ωXi [dim Xi ]); in particular, it does not depend on the ordering of I.

56

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GEOMETRY OF D-SCHEMES ∼

The corresponding isomorphism M ⊗i∗ L −→ i! ((i∗ M )⊗L) sends m⊗` to (m⊗1)⊗`. The reader can skip the next remark. Remark. We will use only D-modules on smooth algebraic varieties. However at some places (such as 2.1.4) the smoothness restriction on X is unnatural. Here is a short comment about the notion of D-module on an arbitraly singular scheme. One can assign to any k-scheme X of locally finite type an abelian k-category M(X) whose objects are called “D-modules on X” together with a conservative3 left exact functor o : M(X) → MO (X). If X is smooth, then M(X) is the category of right D-modules on X and o is the functor or from 2.1.1. The important point is that the notion of left D-module on an arbitrary singular X makes no sense. The standard functors from 2.1.2 exist in this setting4 and satisfy the usual properties. According to Kashiwara’s lemma, if one has a closed embedding i : X ,→ Y where Y is smooth, then i∗ identifies M(X) with the full subcategory of usual right Dmodules on Y which vanish on Y r X; for M ∈ M(X) one has o(M ) = i! (or i∗ M ). There exist two (different yet equivalent) ways to define M(X) (see [S2]5 and [BD] 7.10). The method of [S2] is to consider all possible closed embeddings iY : U ,→ Y where U ⊂ X is an open subset and Y is a smooth scheme; a D-module on X is a rule M which assigns to every such datum a D-module iY ∗ M on Y in a compatible manner. [BD] presents a crystalline approach: a D-module M on X is a rule that assigns to any U as above and an infinitesimal thickening6 iZ : U ,→ Z an O-module MZ = o(iZ∗ M ) on Z in a compatible (with respect to morphisms of Z’s) manner. 2.1.4. The pull-back functor. If f is a smooth morphism then f ∗ is exact with respect to the M` t-structure. Left D-modules satisfy the smooth descent property, so the categories M` (U ), U is a smooth X-scheme, form a sheaf of abelian categories on the smooth topology Xsm , which we denote by M` (Xsm ). One defines then the category M` (Z) for any smooth algebraic stack Z in the usual way. Remark. If our schemes are not necessarily smooth (so we are in the setting of Remark in 2.1.3) but f is a smooth morphism, then the functor f † := f ! [− dim X/Y ] : DM(Y ) → DM(X) is t-exact (i.e., f † (M(Y )) ⊂ M(X)), so we have a canonical exact functor M(Y ) → M(X) which we denote also by f † . The smooth descent property holds. Of course, if our schemes are smooth, then f † coincides with the pull-back functor f ∗ : M` (Y ) → M` (X).7 The above picture generalizes to the case of locally complete intersection morphisms. Namely, assume that f : X → Y makes X a locally complete intersection over Y of pure relative dimension d ≥ 0 (i.e., X can be locally represented as a closed subscheme of Y × An defined by n − d equations and the fibers of f have pure dimension d). Set f † := f ! [−d] : DM(Y ) → DM(X). Proposition. The functor f † is t-exact, i.e., f † M is a D-module for every D-module M on Y . We give two proofs. The first one makes sense even if D-modules are replaced, say, by `-adic perverse sheaves. The second one is based on the relation between D-modules and O-modules. 3 Which

means that o(M ) 6= 0 for M 6= 0. assume that our schemes are quasi-compact and separated. 5 M. Saito considers D-modules in the analytic setting; the algebraic picture is similar. 6 I.e., i is a closed embedding i : U ,→ Z defined by a nilpotent ideal. Z Z 7 We identify M with M` as in 2.1.1. 4 We

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57

First proof. For any f : X → Y with fibers of dimension ≤ d one has H i f ! M = 0 for i < −d.8 So it remains to show that H i f ! M = 0 for i > −d. We can assume that f = πi where π : Y˜ → Y is smooth of pure relative dimension n and i : X ,→ Y˜ is a closed embedding such that Y˜ \ i(X) is covered by n − d open subsets Uj affine over Y˜ . Now compute i∗ f ! M = i∗ i! π ∗ M [n] using the Uj ’s. The direct image with respect to an affine open embedding is an exact functor, so H i f ! M = 0 for i > −d, and we are done. Second proof. We can assume that there is a closed embedding X ,→ Y × An such that i(X) ⊂ Y × An is defined by n − d equations. One can find subschemes Xk ⊂ Y × An , k ∈ N, such that Xk ⊂ Xk+1 , (Xk )red = i(Xred ), each Xk is a locally complete intersection over Y , and every subscheme Z ⊂ Y ×An with Zred = i(Xred ) is contained in some Xk . The OY ×An -module corresponding to H p f ! M is the direct limit of the OXk -modules H p fk! MO , where fk is the morphism Xk → Y and MO is the OY -module corresponding to M . But Xk is Cohen-Macaulay over Y , so H p fk! MO = 0 for p 6= −d.  Remark. In the above proposition it is not enough to require X to be CohenMacaulay over Y (a counterexample: Y is a point and X is the scheme of 2 × 3 matrices of rank ≤ 1). 2.1.5. Vector D-bundles. If X is affine, then the projectivity of a DX module is a local property for the Zariski or ´etale topology of X (see 2.3.6 below where this fact is proven in a more general situation). For arbitrary X we call a locally projective right DX -module of finite rank a vector DX -bundle or a vector D-bundle on X. For a vector DX -bundle V its dual V ◦ is HomDX (V ` , DX ). So a section of V ◦ is a morphism of left DX -modules V ` → DX , and the right DX -module structure on V ◦ comes from the right DX -action on DX . It is clear that V ◦ is again a vector DX -bundle, and (V ◦ )◦ = V . For further discussion see 2.2.16. A vector DX -bundle need not be a locally free DX -module as the following example from [CH] (see also [BGK]) shows. Example. Assume that X is an affine curve, and let V ⊂ DX be a Dsubmodule such that V equals DX outside of a point x ∈ X. Then V is a projective DX -module (since the category of D-modules has homological dimension 1).9 By 2.1.13 below, such V ’s are the same as open k-vector subspaces of the formal completion Ox of the local ring Ox . If V is locally free, then (shrinking X if necessary) we can assume that it is generated by a single section φ; i.e., V = φDX ⊂ DX . Since φDX = DX on X r {x}, the differential operator φ has order 0; i.e., V = tn DX for some n ≥ 0 (where t is a parameter at x). Therefore locally free V ’s correspond to open ideals in Ox . 2.1.6. The middle de Rham cohomology sheaf. For M ∈ M(X) set h(M ): = M ⊗ OX = M/M ΘX . For a local section m ∈ M we denote by m its class DX

8 To prove this, choose a stratification of Y such that for each stratum Y α the morphism ! f ! M , ν : X ,→ X. Xα := (f −1 (Yα ))red → Yα is smooth and look at the sheaves H i να α α 9 In fact, it is enough to know that D /V has a projective resolution of length 1, and this X follows from Kashiwara’s lemma.

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in h(M ). We consider h(M ) as a sheaf on the ´etale topology of X. This will be relevant in Chapter 4; for the present chapter Zariski localization works as well. Notice that for any M ∈ M(X), L ∈ M` (X) one has M ⊗ L = h(M ⊗ L). DX

The reader can skip the next technical lemma returning to it when necessary. Lemma. (i) If M is a non-zero coherent DX -module then h(M ) 6= 0. (ii) If M is a coherent DX -module with no OX -torsion then for every DX module L the map Hom(L, M ) → Hom(h(L), h(M )) is injective. Proof. (i) (a) Using induction by the dimension of the support of M , we can assume, by Kashiwara’s lemma (see 2.1.3), that M is non-zero at the generic point of X. Replacing X by its open subset, we can assume that M is a free OX -module.10 (b) Let us show that for any (not necessarily coherent) non-zero DX -module M which is free as an OX -module one has h(M ) 6= 0. We use induction by dim X. Replacing X by its open subset, we can find a smooth hypersurface i : Y ,→ X such that the determinant of the normal bundle to Y is trivial. Let j : U ,→ X be its complement. Set MY := i! (j∗ j ∗ M/M ) = (i ∗ M ` )r ; this is a non-zero D-module on Y which is free as an OY -module by the condition on Y . Hence h(MY ) 6= 0 by the induction hypothesis. Notice that j· j · h(M ) = h(j∗ j ∗ M ) and h(i∗ MY ) = i· h(MY ). Since h is right exact and i∗ MY is a quotient of j∗ j ∗ M , we see that h(M ) 6= 0. (ii) (a) Since h is right exact, it suffices to show that for every non-zero L ⊂ M the map h(L) → h(M ) is non-zero. The condition on M assures that we can replace X by any open subset, so we can assume that L, M , and M/L are free OX -modules. (b) Let us show that for every (not necessarily coherent) non-zero DX -modules L ⊂ M such that L, M , M/L are free OX -modules, the map h(L) → h(M ) is non-zero. We use induction by dim X and follow notation from (i)(b) above. It suffices to show that the map h(j∗ j ∗ L) → h(j∗ j ∗ M ) is non-zero. It admits as a quotient the map h(LY ) → h(MY ) which is non-zero by induction by dim X.  2.1.7. The de Rham complex. Denote by DR(M ) the de Rham complex of M , so DR(M )i : = M ⊗ Λ−i ΘX . One has h(M ) = H 0 DR(M ); in fact, as an object OX

of the derived category of sheaves of k-vector spaces on XZar or X´et , DR(M ) equals L

to M ⊗ OX . In particular, if M is a locally projective DX -module (say, a vector DX

DX -bundle), then the projection DR(M ) → h(M ) is a quasi-isomorphism. Set ΓDR (X, M ) := Γ(X, DR(M )), RΓDR (X, M ) := RΓ(X, DR(M )). For a morphism f : X → Y of smooth schemes there is a canonical isomorphism ∼ Rf· (DR(M )) −→ DR(f∗ M ) in the derived category of sheaves on Y ; here f· is the sheaf-theoretical direct image, and Rf· is its derived functor. For a more precise setting for this quasi-isomorphism; see 2.1.11. We mention two particular cases: (a) The global de Rham cohomology coincides with the push-forward by pro· jection π: X → (point): one has HDR (X, M ) := H · (X, DR(M )) = H · π∗ M . (b) If i: X ,→ Y is a closed embedding, then the above quasi-isomorphism arises as a canonical embedding of complexes i· DR(M ) ,→ DRi∗ (M ) defined by the obvious embedding of OY -modules i· M ,→ i∗ M . Thus i· h(M ) = h(i∗ M ). 10 Indeed, choose a good filtration on M ; then, according to [EGA IV] 6.9.2, grM is O -free X over a non-empty open V ⊂ X.

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59

Notice also that in situation (b) for N ∈ DM(Y ) the obvious morphism DR(Ri! N ) → Ri! DR(N ) is a quasi-isomorphism.11 0 Lemma. If X is an affine curve, then the map HDR (X, M ) → Γ(X, h(M )) is surjective. d

Proof. Consider the de Rham complex M ` − → M ; set K := Ker(d), I := Im(d). Notice that H 2 (X, K) = 0 (since dim X = 1) and H 1 (X, M ` ) = 0 (since X is affine). Thus H 1 (X, I) = 0, which implies that Γ(X, M )  Γ(X, h(M )).  2.1.8. Induced D-modules. See [S1] for all the details. Let X be a smooth scheme. For an O-module B on X we have the induced right D-module BD : = B ⊗ DX ∈ M(X). The de Rham complex DR(BD ) is acyclic in OX

non-zero degrees, and the projection DR(BD )0 = B ⊗ DX → B, b ⊗ ∂ 7→ b∂(1), OX

∼

yields an isomorphism h(BD ) −→ B. In other words, there is a canonical quasiisomorphism νB : DR(BD ) → B.

(2.1.8.1)

If C is another O-module, then for any ϕ ∈ HomM(X) (BD , CD ) the corresponding map h(ϕ): h(BD ) → h(CD ) is a differential operator B → C. In fact, the map h: Hom(BD , CD ) → Diff(B, C),

(2.1.8.2)

where Diff is the vector space of all differential operators, is an isomorphism. The inverse map assigns to a differential operator ψ: B → C the morphism ψD : BD → CD , b ⊗ ∂ 7→ ψb ∂, where ψb ∈ C ⊗ DX = Diff(OX , C) is the differential operator ψb (f ) = ψ(bf ). Let Dif f (X) be the category whose objects are O-modules on X and where morphisms are differential operators. We see that the induction B 7→ BD defines a fully faithful functor Dif f (X) ,→ M(X). For N ∈ M` (X), B ∈ MO (X) there is a canonical isomorphism of D-modules ∼

(B ⊗ N )D −→ BD ⊗ N,

(2.1.8.3)

OX

(b ⊗ n) ⊗ ∂ 7→ (b ⊗ n)∂.

Therefore N, B 7→ B ⊗ N := B ⊗ N is an action of the tensor category M` (X) on OX

Dif f (X); the induction functor ·D commutes with M` (X)-actions.12 The “identity” functor from Dif f (X) to the category Sh(X) of sheaves of kvector spaces on X is faithful, so we know what a Dif f (X)-structure on P ∈ Sh(X) is. Explicitly, this is an equivalence class of quasi-coherent OX -module structures on P , where two OX -module structures P1 , P2 are said to be equivalent if idP : P1 → P2 is a differential operator. Remark. For an O-module B the structure action of OX on B yields an OX action on the D-module BD . Therefore BD is an (OX , DX )-bimodule. This bimod· ule satisfies the following property: the ideal of the diagonal Ker(OX ⊗ OX − → OX ) k

acts on BD in a locally nilpotent way. As follows from Kashiwara’s lemma, the 11 Use

the standard exact triangle i∗ Ri! N → N → Rj∗ j ∗ N → i∗ Ri! N [1] for j : Y ri(X) ,→ Y . M` (X) acts on M(X) by the usual tensor product.

12 Here

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functor B 7→ BD from the category of O-modules (and O-linear maps) to that of (OX , DX )-bimodules satisfying the above property is an equivalence of categories. Examples. (i) Let X ,→ Y be a closed embedding. Suppose that the formal neighborhood Yˆ of X in Y admits a retraction Yˆ → X. Let E be an OY -modules E supported (set-theoretically) on X. Then E, considered as a sheaf of k-vector spaces on X, admits a canonical Dif f (X)-structure. Indeed, every retraction π : Yˆ → X yields an OX -module structure on E, and its equivalence class does not depend on the choice of π. A particular situation we will use is the diagonal embedding X ,→ X n . Notice that for E as above the corresponding DX -module coincides with ∆! (E ⊗ DX n ). OX n

(ii) For a right D-module M its de Rham complex DR(M ) is a complex in Dif f (X). If L is a left D-module, then DR(L ⊗ M ) = L ⊗ DR(M ). (iii) Let i: X ,→ Y be a closed embedding. For B as above consider the canonical embeddings of OY -modules i· B ,→ i· (B ⊗ DX ) ,→ i∗ (BD ). The composition OX

induces a canonical morphism of DY -modules γ: (i· B)D → i∗ (BD ) which is an isomorphism. 2.1.9. For any M ∈ M(X) the de Rham complex DR(M ) is a complex in Dif f (X) (the de Rham differential is a first order differential operator), so we have the corresponding complex DR(M )D of D-modules. It is acyclic in degrees 6= 0, and the projection DR(M )0D = M ⊗ DX → M , m ⊗ ∂ 7→ m∂, defines an isomorphism ∼

OX

H 0 DR(M )D −→ M . In other words we have a canonical quasi-isomorphism (2.1.9.1)

µM : DR(M )D → M.

2.1.10. One may replace modules by complexes of modules to get the functors between the DG categories of complexes (2.1.10.1)

DR: CM(X) → CDif f (X),

·D : CDif f (X) → CM(X)

together with natural morphisms ν, µ as in (2.1.8.1) and (2.1.9.1). The category Dif f (X) is not abelian. Following Saito, a morphism ψ: B → C of complexes in Dif f (X) is called a quasi-isomorphism if ψD is. One defines the derived category DDif f (X) by inverting quasi-isomorphisms. Remark. A morphism ψ : B → C in CDif f (X) is called a naive quasiisomorphism if it is a quasi-isomorphism of complexes of plain sheaves of vector spaces. It is clear that ψ is a quasi-isomorphism if and only if for any O-flat left D-module N the morphism idN ⊗ ψ : N ⊗ B → N ⊗ C is a naive quasiisomorphism. If both B, C are bounded above complexes of coherent O-modules, then every naive quasi-isomorphism ψ is necessarily a quasi-isomorphism.13 For general quasi-coherent B, C this may be false (for example, take B = DR(M ) where M is a constant sheaf equal to the field of fractions of DX , C = 0). Both µ and ν are quasi-isomorphisms. They are compatible in the following sense: for every B · ∈ CDif f (X), M · ∈ CM(X) the morphisms DR(µM ), νDR(M ) : DR(DR(M )D ) → DR(M ) are homotopic; the same is true for the morphisms 13 Let M be the top non-zero cohomology of the cone of ψ . Then M is a coherent D-module D such that h(M ) = 0. We are done by the lemma from 2.1.6.

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(νB )D , µBD : DR(BD )D → BD . Therefore DR and ·D define mutually inverse equivalences between the derived categories: ∼

DM(X)↔DDif f (X).

(2.1.10.2)

For a morphism of schemes f : X → Y there is an obvious sheaf-theoretic pushforward functor Rf· : DDif f (X) → DDif f (Y ). Equivalence (2.1.10.2) identifies it with f∗ : there are canonical isomorphisms (2.1.10.3)

DR(f∗ M ) = Rf· DR(M ),

f∗ (BD ) = (Rf· (B))D .

For particular cases see 2.1.7 and Example (iii) in 2.1.8. 2.1.11. Quasi-induced D-modules and D-complexes. The full subcategory of induced D-modules, i.e., D-modules isomorphic to ND for some O-module N , does not look reasonable. For example, a direct summand of an induced Dmodule need not be induced (as the example from 2.1.5 shows); induced D-modules do not have a local nature. A more friendly category of quasi-induced D-modules is defined as follows. We say that M ∈ M(X) is quasi-induced if for every OX -flat L ∈ M` (X) DX one has T or>0 (M, L) = 0; i.e., the canonical projection DR(M ⊗ L) → h(M ⊗ L) is a quasi-isomorphism.14 Similarly, a DX -complex M ∈ CM(X) is homotopically quasi-induced if for every homotopically OX -flat L ∈ CM` (X) one has L

∼

M ⊗ L −→ M ⊗ L; i.e., the canonical projection DR(M ⊗ L) → h(M ⊗ L) is a DX

DX

quasi-isomorphism. Here is an intrinsic characterization of quasi-induced D-modules. Let η be a (not necessarily closed) point of X, iη : η ,→ X the embedding. For C ∈ DM(X) denote by i!η C the fiber of i!Y C at η, where Y is a non-empty open smooth subvariety of the closure of η and iY is the embedding Y ,→ X. So i!η C belongs to the derived category of modules over the ring of differential operators on η. Let us say that C is strongly non-negative if for every η ∈ X the Tor-dimension of i!η C is nonnegative. Equivalently, C is strongly non-negative if and only if for every η ∈ X the Tor-dimension of the stalk Cη of C at η is ≤ codim η. It is easy to see that strong non-negativity implies the usual one (i.e., H <0 C = 0) and that strong nonnegativity is preserved under direct images with respect to arbitrary morphisms. Lemma. M ∈ M(X) is quasi-induced if and only if it is strongly non-negative. Proof. Suppose M is quasi-induced. We can assume that X is affine. Take any η ∈ X and L ∈ M` (X). Let P be a left resolution of L such that Pa = 0 for a > codim η and P0 , . . . , Pcodim η−1 are projective D-modules. Localizing at L

η, we see that every Paη is a flat Oη -module. Thus (M ⊗ L)η = h(Pη ); hence DX

T oraDX (M, L)η = 0 for a > codim η; q.e.d. Suppose M is strongly non-negative. Then M ⊗ L is strongly non-negative for any OX -flat L ∈ M` (X). So it suffices to check that H a DR(M ) = 0 for every a < 0. Suppose this is not true. Let η be a point of smallest codimension such that DR(M )η has non-zero negative cohomology. Since DR(i!η M ) = i!η DR(M ), one has 14 This local property amounts to the corresponding global property which says that for every ∼ OX -flat L ∈ M` (X) one has RΓDR (X, M ⊗ L) −→ RΓ(X, h(M ⊗ L)).

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H a DR(i!η M ) = H a DR(M )η for a < 0. Therefore i!η M has negative Tor-dimension, and we are done.  Remarks. (i) Assume that k = C. A holonomic complex with regular singularities is strongly non-negative if and only if the complex of (usual, not perverse) sheaves corresponding to its Verdier dual is non-positive. (ii) An induced D-module is quasi-induced. The direct image of a quasi-induced D-module under an affine morphism is quasi-induced. The exterior tensor product of quasi-induced D-modules is quasi-induced. A holonomic D-module is quasiinduced if and only if its support has dimension 0. A tensor product of an OX -flat D-module and a quasi-induced D-module is quasi-induced. (iii) If X is a curve, then a D-module is quasi-induced if and only if its restriction to any open subset does not contain a non-trivial lisse submodule. 2.1.12. Maximal constant quotients and de Rham homology. Suppose that X is connected and dim X = n. −a (X, M [n]); this is the de Rham homolFor M ∈ M(X) set HaDR (X, M ) := HDR ogy of X with coefficients in M . Notice that HaDR (X, M ) vanish unless a ∈ [0, 2n]; if X is affine, the vanishing holds unless a ∈ [n, 2n]. The vector space H0DR (X, M ) vanishes if X is non-proper. We say that M ∈ M(X) is constant if M ` is generated by global horizontal sections. The category of constant D-modules is closed under subquotients. It identifies canonically with the category of vector spaces by means of the functors M 7→ Hom(OX , M ` ) = Γ∇ (X, M ` ), V 7→ V ⊗ ωX .15 Every D-module M admits the maximal constant submodule Γ∇ (X, M ` ) ⊗ ωX ⊂ M . For any M ∈ M(X) all its constant quotients form a directed projective system of constant D-modules Mconst . The functor M 7→ Mconst is right exact. Suppose X is proper. Then M admits the maximal constant quotient; i.e., Mconst is a plain DX -module.16 The adjunction formula for the projection from X to a point (see 2.1.2) yields ∼ ` a canonical identification Mconst −→ H0DR (X, M ) ⊗ OX . For any point s ∈ X πs the composition H 0 Ri!s M [n] = Ms` −→ (Mconst )s = H0DR (X, M ) comes from the ! canonical morphism is∗ Ris M → M . Let S ⊂ X be any finite non-empty subset, jS : U ,→ X its complement. The standard exact triangle ⊕ is∗ Ri!s M → M → RjS∗ jS∗ M → ⊕ is∗ Ri!s M [1] shows that S

S

the kernel of the morphism πS = ⊕πs : ⊕ Ms` → H0DR (X, M [n]) is equal to the S

n−1 image of the “residue” map ResS = ⊕Ress : HDR (U, M ) → ⊕ Ms` . S

2.1.13. Modifications at a point. Let x ∈ X be a (closed) point. Denote by ix the embedding {x} ,→ X; let jx : Ux := X r {x} ,→ X be its complement. For a D-module M on X let Ξx (M ) be the set of all submodules Mξ ⊂ M such that M/Mξ is supported at x. This is a subfilter in the ordered set of all submodules of M , so Ξx (M ) is a topology on M . 15 To prove that these functors are mutually inverse, notice that for a vector space V all sub-D-modules of V ⊗ OX are of the form W ⊗ OX , W ⊂ V ; this follows from the irreducibility of the D-module OX . 16 A direct way to see this: reduce to the case where M ` is coherent; then notice that dim HomD (M ` , OX ) < ∞.

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Consider the vector space h(M )x . For every Mξ ∈ Ξx (M ) the sequence 0 → −1 h(Mξ )x → h(M )x → h(M/Mξ ) → 0 is exact (since HDR (M/Mξ ) = 0). Therefore the map N 7→ h(N )x ⊂ h(M )x is a bijection between the set of D-submodules N ⊂ M containing Mξ and that of vector subspaces of h(M )x containing h(Mξ )x (since D-submodules of M/Mξ are the same as vector subspaces of h(M/Mξ )x ). The subspaces h(Mξ )x , Mξ ∈ Ξx (M ), form a topology on h(M )x called the Ξx topology. Thus Mξ 7→ h(Mξ )x is a bijection between Ξx (M ) and the set of open vector subspaces for the Ξx -topology. ˆ x (M ) the completion of h(M )x with respect to the Ξx -topology, Denote by h i.e., the projective limit of the system of vector spaces i!x (M/Mξ ) = h(M/Mξ )x , ˆ x as a functor from M(X) to the category of complete Mξ ∈ Ξx (M ). We consider h separated topological vector spaces.17 For any discrete vector space F one has18 (2.1.13.1)

ˆ x (M ), F ) = Hom(M, ix∗ F ), Homcont (h

ˆ x is right exact. so the functor h n Remarks. (i) Let Ox := ← lim − Ox /mx be the formal completion of the local ring at x. Set MOx := M ⊗ Ox ; this is a DOx -module. Then the above picture OX

depends only on MOx . Indeed, the map Mξ 7→ MξOx is a bijection between Ξx (M ) and the set of DOx -submodules of MOx with the quotients supported at x.19 So Ξx (M ) can be understood as the Ξx -topology on MOx . The vector space h(MOx ) = M ⊗ Ox carries the Ξx -topology formed by the subspaces h(MξOx ). For every DX

Mξ ∈ Ξx (M ) one has M/Mξ = MOx /MξOx ; hence h(M/Mξ ) = h(MOx /MξOx ) = ˆ x (M ) is the completion of h(MO ) with respect to h(MOx )/h(MξOx ). Therefore h x the Ξx -topology. ˆ x (M ) is a profinite-dimensional vec(ii) If M is a coherent D-module, then h tor space. Indeed, every M/Mξ is a coherent D-module supported at x; hence i!x (M/Mξ ) is a finite-dimensional vector space. We refer to any topology Ξ?x (M ) weaker than Ξx (M ) as a topology at x on M . As above, it amounts to a topology on h(M )x which is weaker than the Ξx -topology. ˆ ? (M ) equals lim i! (M/Mξ ), Mξ ∈ Ξ? (M ). As The corresponding completion h x x ←− x ˆ ? (M ) as the in Remark (i) we can consider Ξ?x (M ) as a topology on MOx and h x completion of h(MOx ). Examples. (i) Assume that M is an induced D-module, M = BD (see 2.1.8), so h(M ) = B. Denote by Ξx (B) the topology on B formed by all quasi-coherent O-submodules P ⊂ B such that B/P is supported set-theoretically at x. Then ˆ x (M ) is the Ξx (M ) is generated by submodules PD , P ∈ Ξx (B).20 Therefore h completion of B with respect to the topology Ξx (B). If B is O-coherent, then this is just the formal completion of B at x. We will prove in 2.1.17 that for arbitrary B the Ξx (B)-completion of B equals B ⊗ Ox . 17 By “topological vector space” we always mean the vector space equipped with a linear topology (there is a base of neighborhoods of 0 formed by linear subspaces). 18 We would denote h ˆ x (M ) by i∗ M but, unfortunately, the notation was already used in 2.1.2. x 19 Since the O-modules supported at x are the same as the O -modules supported at x. x 20 For M ∈ Ξ (M ) set P := M ∩ B. Then P ∈ Ξ (B) and M ⊃ P . x x ξ ξ ξ D

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ˆ x (M ), one can (ii) Assume that M is any coherent D-module. To compute h m n represent M as a cokernel of a morphism ϕ : DX → DX . We have seen that ˆ x (DX ) = Ox , so h ˆ x (M ) is the cokernel of the corresponding matrix of differential h 21 m operators ϕ : Ox → Oxn . 2.1.14. Lemma. If the restriction of M to Ux := X r {x} is a coherent D-module, then the Ξx -topology on h(MOx )22 is complete and separated; i.e., ∼ ˆ h(MOx ) −→ h x (M ).

(2.1.14.1)

ˆ x (M ) Proof. We want to show that the morphism of vector spaces h(MOx ) → h 0 is an isomorphism. Our M is an extension of a D-module M supported at x by a coherent D-module M 00 . We have already know the statement for M 0 (see Remark (i) in 2.1.13) and M 00 (see Example (ii) of 2.1.13). Since the sequences ˆ x (M 00 ) → h ˆ x (M ) → h ˆ x (M 0 ) → 0 → h(MO00 x ) → h(MOx ) → h(MO0 x ) → 0 and 0 → h 0 are exact, we are done.  ˆ x (M ) admits an open Remarks. (i) If M is a D-module as in 2.1.14, then h profinite-dimensional subspace.23 Such topological vector spaces are discussed in 2.7.9 under the name of Tate vector spaces. (ii) If M is an arbitrary D-module, then the canonical map φM : h(MOx ) → ˆ hx (M ) need not be an isomorphism. Here is an example. Assume that X = Spec R 0 is an affine curve, so R = HDR (X, DX ) = Ext1M(X) (ωX , DX ). Let M be the ˆ x (M ) = 0 universal extension of R ⊗ ωX by DX . Then M ⊗ Ox = Ox /R 6= 0 and h DX

ˆ x (M ) = 0. is the completion of Ox /R with respect to the quotient topology; i.e., h However φM is always an isomorphism if M is either an induced D-module (see Example (i) of 2.1.13 and see 2.1.17) or isomorphic to a direct sum of coherent D-modules. In fact, the argument used in the proof of the above lemma shows that it suffices to assume that the restriction of M to Ux satisfies either of these conditions. 2.1.15. Remark. Sometimes it is convenient to consider a finer object than ˆ x (M ). Namely, consider the directed set of all a plain topological vector space h ˜ x (M ) := lim h ˆ D-coherent submodules Mα ⊂ M and set h lim −→ x (Mα ) = − → Mα ⊗ Ox . DX

˜ x (M ) equals M ⊗ Ox as a plain vector space, but we consider it as an indThus h DX

object of the category of profinite-dimensional vector spaces. Recall that such an animal is the same as a left exact contravariant functor on the category of profinite˜ x (M ) “represents” the dimensional vector spaces with values in vector spaces. So h ˆ functor F 7→ lim −→ Hom(F, hx (Mα )), F a profinite-dimensional vector space. Notice ˜ that the dual object hx (M )∗ – which is a pro-object of the category of vector spaces – equals ← lim − Hom(Mα , ix∗ k). 21 We

use the fact that a morphism of profinite-dimensional vector spaces has closed image. Remark (i) in 2.1.13. 23 Namely, the image of h ˆ x (Mξ ) where Mξ ∈ Ξx (M ) is a coherent DX -module (as explained ˆ x (Mξ ) is profinite-dimensional). in 2.1.13, h 22 See

2.1. D-MODULES: RECOLLECTIONS AND NOTATION

65

2.1.16. Moving the point. We will also need a version of 2.1.13 in the case where x depends on parameters. So let Y be a scheme which we assume to be quasi-compact and quasi-separated,24 x ∈ X(Y ), ix : Y ,→ X × Y the graph of x. Denote by M(X × Y /Y ) the category of right DX×Y /Y := DX  OY -modules which are quasi-coherent as OX×Y -modules. Let M(X × Y /Y )x be the full subcategory of DX×Y /Y -modules supported (set-theoretically) at the graph of x. A version of Kashiwara’s lemma (see 2.1.3) says that the functor i!x identifies M(X × Y /Y )x with the category of quasi-coherent OY -modules. For M ∈ M(X × Y /Y ) we denote by Ξx (M ) the set of submodules Mξ ⊂ M such that M/Mξ ∈ M(X × Y /Y )x . This is a subfilter in the ordered set of all submodules of M . We have a Ξx (M )-projective system of quasi-coherent OY modules i!x (M/Mξ ) connected by surjective maps. For N ∈ M(X × Y /Y ) set h(N ) := N ⊗ OX×Y ; this is a p·Y OY -module.25 DX×Y /Y

Since i!x (M/Mξ ) = i·x h(M/Mξ ), the projective limit of i!x (M/Mξ ) equals the completion of i·x h(M ) with respect to the topology defined by the images of the i·x h(Mξ ). ˆ x (M ). We denote it by h If M is an induced module, M = B ⊗ DX×Y /Y where B is an OX×Y -module, ˆ x (M ) is the completion of B with respect to the topology then h(M ) = B and h formed by all quasi-coherent O-submodules P ⊂ B which coincide with B outside ˆ x (M ) is the formal of (the graph of) x. In particular, if B is O-coherent, then h completion of B at x. ˆ x (M ) with the As in 2.1.14 for a coherent DX×Y /Y -module M one identifies h DX×Y /Y -tensor product of M and the formal completion of OX×Y at x. Remarks. (i) The above construction is compatible with the base change by arbitrary morphisms f : Y 0 → Y . Namely, the evident morphism of pro-OY 0 ˆ xf ((f × idX )∗ M ) → f ∗ h ˆ x (M ) is an isomorphism. modules h (ii) If Y is a smooth variety and M is a right DX×Y -module, then the differential operators on Y act on M in a continuous way with respect to the Ξx -topology. ˆ x (M ) is a sheaf of topological right DY -modules. Similarly, if DY acts Therefore h ˆ x (M ) is a sheaf of topological left DY -modules. on M from the left, then h ˆ x (M ) → F is a continuous DY -morphism, (iii) If F is a D-module on Y and ϕ : h ˆ then it vanishes on the image of hx (Mξ ) where Mξ ∈ Ξx (M ) is a sub-DX×Y -module of M . In particular, assume that Y = X, x = idX , so ix = ∆, and M = L  OX for a D-module L on X. Then every morphism ϕ as above is trivial (if dim X 6= 0). 2.1.17. I-topology. In this section we prove a result in commutative algebra we referred to in Example (i) of 2.1.13 and Remark (ii) in 2.1.14. It will not be used in the rest of this work. Let A be a noetherian ring, I ⊂ A an ideal such that A is complete in the I-adic topology. Let C be the category of A-modules and C0 ⊂ C the full subcategory of A-modules M such that each element of M is annihilated by some power of I. The I-topology on an A-module M is the weakest topology such that every A-linear map from M to a module from C0 equipped with the discrete topology 24 The assumption is needed to assure that topological O -modules have a local nature, i.e., Y pro-objects in the category of quasi-coherent O-modules form a sheaf of categories on YZar , and our constructions are compatible with the localization of Y . 25 Here p : X × Y → Y is the projection. Y

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˜ ⊂ M such that M/M ˜ ∈ C0 is a is continuous. The set BM of all submodules M base of neighborhoods of 0 for the I-topology. If M is finitely generated, then the I-topology coincides with the I-adic topology. Theorem. (a) If N is a submodule of an A-module M then the I-topology on M induces the I-topology on N . (b) If A has finite Krull dimension then every A-module M is complete and separated for the I-topology. ˜ ∈ BN there is an M ˜ ∈ Proof. To prove (a), we have to show that for every N ˜ ∩N = N ˜ . Let P be the set of pairs (L, L) ˜ where L is a submodule BM such that M ˜ ∈ BL . Equip P with the following ordering: (L, L) ˜ ≤ (L0 , L ˜ 0 ) if of M and L 0 0 ˜ ˜ ˜ L ⊂ L and L = L ∩ L . By Zorn’s lemma there is a maximal (L, L) ∈ P such that ˜ ≥ (N, N ˜ ). It remains to show that L = M . Indeed, if x ∈ M , x ∈ (L, L) / L, then ˜ 0 ) > (L, L) ˜ by putting L0 := L + Ax, L ˜ 0 := L + U where one can construct (L0 , L ˜ ∩ Ax (U exists because U ⊂ Ax is an open submodule of Ax such that L ∩ U = L the I-topology on L ∩ Ax is induced by the I-topology on Ax). Using (a) we reduce the proof of separatedness of M to the well-known case where M is finitely generated (A need not have finite Krull dimension for this property). ˆ the completion of M with respect to the I-topology. Let C ? ⊂ C Denote by M ˆ is bijective. be the full subcategory of A-modules M such that the map M → M ? We will show that C = C if A has finite Krull dimension. Lemma. C ? has the following properties: (i) Finitely generated modules belong to C ? . (ii) Suppose that the sequence 0 → M 0 → M 00 → M → 0 is exact and M ∈ C ? . Then M 0 ∈ C ? ⇔ M 00 ∈ C ? . L (iii) If Mj ∈ C ? , j ∈ J, then j Mj ∈ C ? . S (iv) Suppose M = i Mi , i = 1, 2, . . . , Mi ⊂ Mi+1 . If Mi ∈ C ? for all i, then M ∈ C ?. Proof of Lemma. (i) is well known. ˆ , which is proved (ii) is a corollary of the left exactness of the functor M 7→ M ˆ2 → M ˆ 3 ) is as follows. If the sequence 0 → M1 → M2 → M3 is exact, then Ker(M ˆ the closure of M1 in M2 , i.e., the completion of M1 with respect to the topology induced from the I-topology on M2 . We have proved that the induced topology is ˆ2 → M ˆ 3) = M ˆ 1. the I-topology on M1 , so Ker(M L To prove (iii) it suffices to show that if M = j Mj , then the image of the ˆ →Q M ˆ j equals L M ˆ j . We only have to prove that natural embedding f : M j L ˆ Q jˆ 0 Imf ⊂ M . Assume that x = (x ) ∈ j j j j Mj and J := {j ∈ J|xj 6= 0} is infinite. Choose open submodules Uj ⊂ Mj so thatQ for every j ∈ J 0 the image of xj in Mj /Uj is non-zero. Then the image of x in j Mj /Uj does not belong to L / Imf . j Mj /Uj , so x ∈ ˆ belongs to the closure of To prove (iv), it suffices to show that every x ∈ M ˆ for some i (this closure equals M ˆ i = Mi ). If this is not true, then for every i Mi ⊂ M there is an open submodule Ui ⊂ M such that the image of x in M/(M S i +Ui ) is nonzero. We can assume that U1 ⊃ U2 ⊃ · · · . The submodule U := i (Mi ∩ Ui ) ⊂ M is open and U + Mi ⊂ Ui + Mi . So for every i the image of x in M/(Mi + U ) is

2.2. THE COMPOUND TENSOR STRUCTURE

67

non-zero; i.e., the image of x in M/U S does not belong to the image of Mi in M/U . This is impossible because M = i Mi .  We assume now that A has finite Krull dimension. It remains to show that if a full subcategory C ? ⊂ C satisfies properties (i)–(iv) from the lemma, then C ? = C. Let Ci ⊂ C be the full subcategory consisting of modules M such that Mp = 0 for every prime ideal p ⊂ A with dim A/p > i. We will prove by induction that Ci ⊂ C ? . Suppose we know that Ci−1 ⊂ C ? . Let M ∈ CQ i . Denote by Pi the set of all primes p ⊂ A such that dim A/p = i. The map M → p∈Pi Mp factors through P P p∈Pi Mp , and the kernel and cokernel of the morphism M → p∈Pi Mp belong to Ci−1 . By (ii),(iii), and the induction assumption, to prove that M ∈ C ? it suffices to show that Mp ∈ C ? for each p ∈ Pi . Let p ∈ Pi . Our Mp is an Ap -module such that each element of Mp is annihilated by a power of pAp . By (ii) and (iv) we are reduced to proving that vector spaces over Ap /pAp belong to C ? . By (iii) it suffices to show that Ap /pAp ∈ C ? . The morphism A/p → Ap /pAp is injective and its cokernel belongs to Ci−1 ⊂ C ? . So Ap /pAp ∈ C ? by (i) and (ii).  Remark. If A is a 1-dimensional local domain, then the proof of statement (b) of the above theorem is very simple. In this case every A-module M has a free submodule N such that M/N ∈ C0 , so we can assume that M is free. Then apply statement (iii) of the lemma. 2.2. The compound tensor structure In this section we describe a canonical augmented compound tensor structure on the category of DX -modules M(X) where X is a smooth scheme. We define the tensor structure ⊗! in 2.2.1, the pseudo-tensor structure P ∗ in 2.2.3, the compound tensor product maps ⊗IS,T in 2.2.6, and the augmentation functor h in 2.2.7; h is non-degenerate and reliable (see 2.2.8). We show in 2.2.9 that the translation functor M → M [− dim X] admits a canonical homotopy pseudo-tensor extension CM(X)! → CM(X)∗ . In 2.2.10 it is shown that DR is naturally a homotopy pseudo-tensor functor; another construction, that works in case dim X = 1, is presented in 2.2.11. We explain in 2.2.12 that our compound tensor structure can be encoded into a compound DX -operad, discuss duality and inner Hom in 2.2.15– 2.2.17, explain how ∗ operations can be seen on the level of polylinear operations between the h sheaves in 2.2.18, and establish their continuity properties in 2.2.19. 2.2.1. The ! tensor structure. The category M` is a tensor category (see ∼ −1 2.1.1). We use the standard equivalence M(X) −→ M` (X), M 7→ M ` : = M ωX , to ! ! ` define the tensor product ⊗ on M(X), so we have M1 ⊗ M2 = M1 ⊗ M2 . The OX

object ωX is a unit object for ⊗! . L

2.2.2. Consider the derived tensor product ⊗! on DM(X).26 According to (2.1.2.3) for Mi ∈ DM(X) one has L

⊗!I Mi = ∆(I)! ((Mi [dim X]))[− dim X] = ∆(I)! (Mi ) ⊗ λI⊗ dim X [(|I| − 1) dim X] 26 Notice

L

that ⊗! is identified with the tensor product on DM` (X) by the above “standard” t-exact equivalence, and not by the canonical equivalence (2.1.1.1).

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where λI := det(k I )[|I|] = (k[1])⊗I [−|I|] (the last equality comes since k[dim X]⊗I dim X = λ⊗ [|I| dim X]). Therefore for Mi ∈ M(X) there is a canonical isomorphism I ⊗!I Mi = H (|I|−1) dim X ∆(I)! (Mi ) ⊗ λI⊗ dim X .

(2.2.2.1)

!

⊗I I Example. Assume that Mi = ωX . One has ωX = ωX and ωX = ωX I ⊗ ⊗ dim X (|I|−1) dim X (I)! λI , so (2.2.2.1) comes from the identification ωX = H ∆ (ωX I ).

2.2.3. The ∗ pseudo-tensor structure. For an I-family Li of D-modules, I ∈ S, and a D-module M set (I)

PI∗ ({Li }, M ) := Hom(Li , ∆∗ M ).

(2.2.3.1)

The elements of PI∗ are called ∗ I-operations. They have ´etale local nature: we have the sheaf P ∗I ({Li }, M ), U 7→ PI∗ ({Li }, M ), on the ´etale topology of X. U U For a surjective map π: J  I the composition map PI∗ ({Li }, M ) ⊗ (⊗I PJ∗i ({Kj }, Li )) −→ PJ∗ ({Kj }, M )

(2.2.3.2)

sends ϕ ⊗ (⊗ψi ) to ϕ(ψi ) defined as the composition ψi

(Ji )

J Kj −−→ I ∆∗

∆(π) (ϕ)

(π)

(π)

(I)

(J)

∗ Li = ∆∗ (I Li ) −−− −−→ ∆∗ ∆∗ M = ∆∗ M

where ∆(π) = Π ∆(Ji ) : X I ,→ X J . The composition is associative, so PI∗ define on I

M(X) an abelian pseudo-tensor structure. We denote this pseudo-tensor category by M(X)∗ . The pseudo-tensor categories M(U )∗ , U ∈ X´et , form a sheaf of pseudotensor categories on the ´etale topology of X. For ϕ ∈ PI∗ ({Li }, M ) we will sometimes denote its image considered (by Kashiwara’s lemma) as a submodule of M by ϕ({Li }). Remarks. (i) The sheaf-theoretic restriction to the diagonal identifies our ∗ (I)· (I) I-operations with D⊗I ∆∗ M . X -module morphisms ⊗ Li → ∆ k

(ii) One has PI∗ ({Li }, M ) = Hom(⊗∗ Li , M ) where ⊗∗ Li := ∆(I)∗ (Li ) is the projective system of all the DX I -module quotients of Li supported on the diagonal, considered as a projective system of DX -modules via the ∆(I)! -equivalence. (iii) The category MO (X) of quasi-coherent OX -modules carries a natural M(X)∗ -action (see 1.2.11). Namely, the vector space of operations PI˜∗ ({Mi , F }, G), where F , G are (quasi-coherent) OX -modules and Mi , i ∈ I, are DX -modules, is ˜ (I)

˜ (I)

defined as PI˜∗ ({Mi , F }, G) := HomDI OX ((Mi )  F, ∆∗ G). Here ∆∗ G := X

˜ (I)

(∆· G) ⊗ (DI X  OX ). The composition of these operations with usual ∗ operO

˜ XI

ations between D-modules and morphisms of O-modules (as needed in 1.2.11) is clear. Consider the induction functor MO (X) → M(X), F 7→ FD (see 2.1.8). One has an obvious natural map PI˜∗ ({Mi , F }, G) ,→ PI˜∗ ({Mi , FD }, GD ). On the other hand, if P, Q are D-modules, then there is an obvious map PI˜∗ ({Mi , P }, Q) ,→ PI˜∗ ({Mi , PO }, QO ), where PO , QO are P, Q considered as O-modules. Therefore both induction and restriction functors MO (X)  M(X) are compatible with the M(X)∗ -actions. Here the M(X)∗ acts on M(X) in the standard way; see 1.2.12(i).

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69

2.2.4. Examples. (i) Let us describe the ∗ pseudo-tensor structure on the subcategory of induced modules (see 2.1.8). For O-modules {Fi }i∈I , G denote by DiffI ({Fi }, G) ⊂ Homk (⊗ Fi , G) the subspace of differential I-operations (those ϕ’s k

for which the maps Fi → G, fi 7→ ϕ(fi ⊗ ( ⊗ fj )), are differential operators for j6=i

any i and any fixed local sections fj ∈ Fj , j 6= i). Equivalently, DiffI ({Fi }, G) = (I) Diff(Fi , ∆· G). The differential operations are stable with respect to composition so they define a pseudo-tensor structure on the category Dif f (X); denote this pseudo-tensor category by Dif f (X)∗ . One has PI∗ ({FiD }, GD ) = Hom((Fi )D , (∆· G)D ) = DiffI ({Fi }, G) (see 2.1.8, in particular Example (iii) there); i.e., ·D extends to a fully faithful pseudo-tensor functor ·D : Dif f (X)∗ ,→ M(X)∗ .

(2.2.4.1)

(ii) Assume that |I| ≥ 2 and one of the D-modules Li is lisse (e.g., equal to ωX ). Then PI∗ ({Li }, M ) = 0. (iii) The de Rham DG algebra DR := (OX → Ω1X → · · · ) is a commutative DG algebra in Dif f (X)∗ , so the corresponding induced complex DRD is a commutative DG algebra in M(X)∗ . Notice that DRD is naturally a resolution of ωX [− dim X] (see, e.g., (2.1.9.1)), so ωX [− dim X] is naturally a homotopy commutative DG algebra in M(X)∗ . Exercise. Show that the cone of the product morphism DRD  DRD → ∆∗ DRD is naturally quasi-isomorphic to j! ωU [−2 dim X + 1] where j! is the usual functor on the derived category of holonomic D-modules. 2.2.5. Let i: X ,→ Y be an embedding of smooth schemes. The left exact functors H 0 i∗ , H 0 i! between M(X) and M(Y ) act on PI∗ in the obvious way; i.e., they are pseudo-tensor functors. If Y is a closed subscheme, then H 0 i! is right adjoint to i∗ = H 0 i∗ as pseudo-tensor functors (which means that PI∗ ({i∗ Ni }, M ) = PI∗ ({Ni }, H 0 i! M ) for any Ni ∈ M(X), M ∈ M(Y )), so i∗ is a fully faithful embedding of pseudo-tensor categories. 2.2.6. The tensor product maps. We have to define the canonical maps (see 1.3.12) (2.2.6.1)

⊗IS,T : ⊗S PI∗s ({Li }, Ms ) → PT∗ ({⊗!It Li }, ⊗!S Ms ).

According to the first formula in 2.2.2, one has canonical isomorphisms in DM(X T ) (here α := k[− dim X], so α−1 = k[dim X]): L

∆(πT )! (I Li ) = T (∆(It )! (It Li )) = T ((⊗ !It Li ) ⊗ α−1 ⊗ α⊗It ) L

= (T (⊗ !It Li )) ⊗ (α−1 )⊗T ⊗ α⊗I , (I )

(πS )

∆(πT )! (S (∆∗ s Ms )) = ∆(πT )! ∆∗

(T )

= ∆∗



(T )

(S Ms ) = ∆∗ ∆(S)! (S Ms )  L ! ⊗ S Ms ⊗ α−1 ⊗ α⊗S . (π )

In the second line we used the base change canonical isomorphism ∆(πT )! ∆∗ S = (T ) ∆∗ ∆(S)! (recall that X S , X T ⊂ X I are transveral and X S ∩X T = X; see 1.3.2(i)).

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Consider the diagonal embeddings ∆(S) : k ,→ k S , ∆(πT ) : k T ,→ k I . The em∼ bedding ∆(πS ) : k S → k I induces an isomorphism between the quotients k S /k −→ k I /k T . One has α−1 ⊗ α⊗S = det((k S /k)[− dim X]) and (α−1 )⊗T ⊗ α⊗I = det((k I /k)T [− dim X]), so we get the identification α−1 ⊗ α⊗S = (α−1 )⊗T ⊗ α⊗I . Now ⊗IS,T is the composition (I )

(T )

⊗S PI∗s ({Li }, Ms ) → Hom(I Li , S ∆∗ s Ms ) → Hom(T (⊗!It Li ), ∆∗ (⊗!S Ms )) where the first arrow is S and the second one is H (|I|−|T |) dim X ∆(πT )! . The commutativity of diagrams (i) and (ii) in 1.3.12 is clear. Therefore we have defined the compound tensor category structure on M(X). This compound tensor category M(X)∗! is abelian (see 1.3.14) with a strong unit object ωX (see 1.3.16, 1.3.17, 2.2.4(ii)). The above constructions are compatible with ´etale localization, so M(U )∗! , U ∈ X´et , form a sheaf of compound tensor categories M(X´et )∗! over X´et . 2.2.7. The augmentation functor. Let us show that the de Rham functor h (see 2.1.6) is a non-degenerate ∗-augmentation functor on M(X´et )∗! (see 1.2.5, 1.3.10). We have to define for a finite set I of order ≥ 2 and i0 ∈ I a canonical morphism of sheaves27 (2.2.7.1)

∗ hI,i0 : PI∗ ({Li }, M ) ⊗ h(Li0 ) → PIr{i ({Li }, M ), 0}

ϕ ⊗ ¯l 7→ ϕ¯l .

Namely, for ϕ ∈ PI∗ ({Li }, M ) and li ∈ Li one has     (2.2.7.2) ϕ¯li0  l i = T r i0 ϕ  l i . Ir{i0 }

(I) T ri0 : prI,i0 · ∆∗ M I Ir{i0 }

I

(Ir{i0 }) ∆∗ M

Here → is the trace morphism for the projection prI,i0 : X → X . It is easy to see that hI,i0 is well-defined by (2.2.7.2). Compatibilities (i), (ii) in 1.2.5 and 1.3.10 are straightforward, so h makes M(X)∗! an augmented compound tensor category. Let Sh(X)⊗ be the tensor category of sheaves of k vector spaces. According to 1.2.7, h extends to a pseudo-tensor functor (2.2.7.3)

h : M(X)∗ → Sh(X)⊗ .

The maps hI : PI∗ ({Li }, M ) → Hom(⊗I h(Li ), h(M )) are hI (ϕ)(⊗¯li ) = ϕ(li ); here (I) we consider ϕ(li ) as a section of ∆(I)· h(∆∗ M ) = h(M ) (see 2.1.7). If i: X ,→ Y is a closed embedding, then i∗ : M(X)∗ ,→ M(Y )∗ is compatible with the augmentation functors; i.e., it is an augmented pseudo-tensor functor (see 1.2.8). Remark. The map hI : PI∗ ({Li }, M ) → Hom(⊗I h(Li ), h(M )) need not be injective. However it is injective if M is either an induced D-module (e.g., if M is supported at a single point) or a coherent DX -module without OX -torsion.28 2.2.8. The next proposition shows that these difficulties disappear if we con˜ from 1.4.7 instead of h: sider h we consider P ∗ as a sheaf on the ´ etale topology of X. In the induced case by left exactness of h we can assume that the Li are also induced (replace Li by LiD  Li if necessary), and then use 2.2.4(i). In the coherent case use statement (ii) of the lemma from 2.1.6. 27 Here

28 Proof:

2.2. THE COMPOUND TENSOR STRUCTURE

71

Proposition. Our h is non-degenerate (see 1.2.5) 29 and reliable (see 1.4.7). Proof. Let us show that h is non-degenerate. Consider the map α : PI∗ ({Li }, M ) ∗ → Hom(h(Li0 ), PIr{i ({Li }, M )) coming from (2.2.7.1). We want to show that α 0} is injective. Take a non-zero ϕ ∈ PI∗ ({Li }, M ). Replacing M by the image of ϕ, we can assume that ϕ is surjective. To check that α(ϕ) 6= 0, we will show that the ∗ sum of images of all operations ϕ¯l ∈ PIr{i ({Li }, M ), l ∈ Li0 , equals M . Take 0} (Ir{i })

(I)

0 m ∈ ∆∗ M . Choose m ˜ ∈ ∆∗ M such ˜ Since ϕ is surjective, P Pthat m = T ri0 m. one can write m ˜ = ϕ(lia ). Then m = ϕ¯lia (  lia ); q.e.d.

a

a

0

i6=i0

Let us show that h is reliable. A commutative unital algebra R ∈ M(X)! is the same as a commutative unital algebra R` in the tensor category M` (X), i.e., a quasicoherent OX -algebra equipped with an integrable connection along X. We call (see 2.3.1) such R` a commutative DX -algebra and denote the corresponding category ˜ by Comu! (X). Every M ∈ M(X) yields a functor h(M ) : Comu! (X) → Sh(X), ` ` ` ˜ R 7→ h(M )R` := h(M ⊗ R ) = M ⊗ R . We want to show that the map (see DX

˜ : P ∗ ({Mi }, N ) → Mor(⊗ h(M ˜ i ), h(N ˜ )) is bijective. 1.4.7) h I I

˜ ˜ i ), h(N ˜ )) → P ∗ ({Mi }, N ) to h. Let us construct the inverse map κ : Mor(⊗h(M I ` ˜ ˜ Let ψ : ⊗h(Mi ) → h(N ) be a morphism of functors. Let A L be the commutative DX -algebra freely generated by I sections ai , so A` = Sym( DX · ai ) = Sym(D`I ) where D` is DX considered as a left DX -module. Notice that End A` ⊃ End(D`I ) = opp I ˜ i )A` → MatI (Dopp (the diagonal matrices), so the morphism ψA` : ⊗h(M X ) ⊃ DX I I 30 ˜ h(N )A` commutes with the right action of DX . The obvious Z -grading on A` ˜ A` ; since it comes from the action of diagonal matrices yields a ZI -grading on h(?) k I ⊂ EndA` , our ψA` is compatible with the grading. The component of degree ˜ )A` equals 1I = (1, . . . , 1) of A` is D⊗I , so the corresponding component of h(N ⊗I (I)· (I) 31 ˜ i )A` conN ⊗ D = ∆ ∆∗ N . The component of the same degree of ⊗h(M DX

tains ⊗ Mi : we identify ⊗mi with ⊗(mi ⊗ai ). Restricting ψ to this subsheaf, we get k

(I)

a morphism κ(ψ) : ⊗ Mi → ∆(I)· ∆∗ N . Notice that for every (∂i ) ∈ DIX ⊂ End A` , k

˜ i )A` and ∆(I)· ∆(I) ˜ the restriction of its action to ⊗ Mi ⊂ ⊗h(M ∗ N ⊂ h(N )A` coink

cides with the obvious action of ⊗∂i ∈ D⊗I X coming from the D-module structure on the Mi and N . Since κ(ψ) commutes with this action, one has κ(ψ) ∈ PI∗ ({Mi }, N ) according to Remark (i) in 2.2.3. ˜ So it remains to show that κ is injective; It is clear that κ is left inverse to h. i.e., ψ is uniquely determined by κ(ψ). Take any commutative DX -algebra R` ˜ i )R` → h(N ˜ )R` . Take a local section and consider the morphism ψR` : ⊗h(M ˜ i )R` ; let us compute ψR` (γ) ∈ h(N ˜ )R` in terms of κ(ψ). γ = ⊗(mi ⊗ ri ) ∈ h(M We can assume that the ri ’s are global sections of R` (replacing R` by j∗ j ∗ R` where j : U ,→ X is an open subset on which all ri ’s are defined).32 Consider 29 See

2.2.15 for a more precise statement. acts as a semigroup with respect to multiplication. latter equality comes from (2.1.3.1) for i = ∆(I) since ∆(I)∗ DX I = ∆(I)∗ D`I = D`⊗I by (2.1.2.4). See also 2.2.9 below. 32 We use the fact that the morphism of D -algebras R` → j j ∗ R` yields a morphism ∗ X ˜ ) ` → h(N ˜ ) ∗ ` which is an isomorphism over U . h(N R j∗ j R 30 DI X 31 The

72

2.

GEOMETRY OF D-SCHEMES

the morphism ν : A` → R` of DX -algebras which sends ai to ri . Then ψR` (γ) = νψA` (⊗(mi ⊗ ai )) = νκ(ψ)(mi ). We are done.  2.2.9. According to Remark in 1.3.15, the tensor category M(X)! acts on M(X)∗ , so the tensor product extends canonically to a pseudo-tensor functor (2.2.9.1)

⊗ : M(X)! ⊗ M(X)∗ → M(X)∗ .

Therefore, by 1.1.6(vi), any commutative algebra F ∈ M(X)∗ yields a pseudo-tensor functor M(X)! → M(X)∗ , M 7→ M ⊗ F . For example, DRD ∈ CM(X)∗ (see 2.2.4(iii)) yields a pseudo-tensor DG functor (2.2.9.2)

CM(X)! → CM(X)∗ ,

M 7→ M ⊗ DRD ,

which is a resolution of the functor M 7→ M [− dim X]. Similarly, by 1.1.6(vi), any commutative (not necessary unital) algebra P ∈ M(X)! yields a pseudo-tensor functor M(X)∗ → M(X)∗ , M 7→ M ⊗ P . This fact can be used as follows. 2.2.10. Enhanced de Rham complexes. The de Rham functor (see 2.1.7) DR : CM(X) → CSh(X) is not a pseudo-tensor functor, as opposed to h (see (2.2.7.3)). It is naturally a homotopy pseudo-tensor functor though. In fact, there is a natural family of mutually homotopically equivalent pseudo-tensor functors which are homotopically equivalent to DR as mere DG functors. Below we suppose that X is quasi-projective. Let P : P → OX be any homotopically DX -flat non-unital commutative! algebra resolution of OX . So P is a commutative DG algebra in the tensor category of left DX -modules, P a quasi-isomorphism of such algebras, and we assume that P is homotopically DX -flat as a mere complex of DX -modules (see 2.1.1). Example. If X is a curve, then a simplest P is a two term resolution of OX with P0 = Sym>0 DX , P sending 1 ∈ DX ⊂ P0 to 1 ∈ OX , and P−1 := KerP . Remark. For an arbitrary X one can take for P any resolution of OX which is isomorphic as a mere graded commutative! algebra to Sym>0 T , where T is an X-locally projective graded DX -module having degrees ≤ 0. For a quasi-projective X, it can be choosen so that T is isomorphic to a direct sum of D-modules of type DX ⊗ L where L is a line bundle (if needed, one can assume that all the L’s are negative powers of a given ample line bundle). By (2.2.7.3) and 2.2.9, we have a pseudo-tensor DG functor (2.2.10.1)

CM(X)∗ → CSh(X)⊗ ,

M 7→ h(M ⊗ P). 

P The canonical quasi-isomorphisms h(M ⊗ P) ← DR(M ⊗ P) −→ DR(M ) for M ∈ CM(X) make (2.2.10.1) a homotopy version of the de Rham functor.

The rest of the section can be skipped by the reader. Consider the category of all P = (P, P ) as above. It is a tensor category (without unit) in the obvious way. Lemma. Every functor from our category to a groupoid is isomorphic to a trivial one. Proof. Denote our functor by ¯. For two objects P, P0 let πP : P ⊗ P0 → P be the morphism πP (p ⊗ p0 ) := P0 (p0 )p.

2.2. THE COMPOUND TENSOR STRUCTURE

73

(i) Let ν, µ : P1 → P2 be two morphisms and let P be any object. Set νP := ν ⊗ idP , and the same for µP . Then ν¯ = µ ¯ if ν¯P = µ¯P . This follows since νπP1 = πP2 νP , µπP1 = πP2 µP . 0 0 (ii) Take any P. We have the morphisms πP , πP : P ⊗ P → P where πP := πP σ where σ is the transposition of multiples. These morphisms produce the same arrow 0 in the groupoid. Indeed, by (i) it suffices to check that the images of (πP )P , (πP )P : 0 (P ⊗ P) ⊗ P → P ⊗ P in the groupoid coincide, which is clear since πP (πP )P = 0 0 πP (πP )P . (iii) Let ζ, η : P1 → P2 be any two morphisms. Then ζ¯ = η¯. This follows from 0 0 (ii) since ζπP1 = πP2 (ζ ⊗ η) and ηπP = πP (ζ ⊗ η). 1 2 (iv) For any finite collection of objects {Pα } one can find P such that for every α there is a morphism φα : P → Pα . Namely, one can take P := ⊗Pα and πα := idPα ⊗ ( ⊗ Pα0 ). α0 6=α

(v) To finish the proof, let us show that for a pair of objects P0 , Pn and a chain ψn χn χ1 ψ1 χ2 ¯n ψ¯n−1 · · · ψ¯1−1 χ ¯1 : of morphisms P0 → P01 ← P1 → · · · → P0n ← Pn the composition χ ¯ ¯ P0 → Pn does not depend on the chain. To see this, choose φi : P → Pi as in (iv). Since χ ¯i φ¯i−1 = ψ¯i φ¯i by (iii), our composition equals φ¯n φ¯−1  0 , and we are done. Remarks. (i) It is clear that one can find P such that P>0 = 0 and each Pa is DX -flat. Since the homological dimension of DX equals dim X, for every such P the truncated DG algebra P− dim X /d(P− dim X−1 ) → P− dim X+1 → · · · → P0 satisfies the same conditions. So one can find P supported in degrees [− dim X, 0] and such that every Pa is DX -flat. (ii) The categories of these two kinds of P satisfy the above lemma as well (the proof does not change). 2.2.11. This section will not be used in the text. The reader can skip it. In case dim X = 1, there is another way to make DR a homotopy pseudo-tensor functor. It is convenient to consider DR as a functor with values in CDif f (X). Let us show that it extends naturally to a pseudo-tensor functor (see 2.2.4(i)) (2.2.11.1)

¯ → CDif f (X)∗ DR : CM(X)∗ ⊗ E

¯ is a certain DG operad homotopically equivalent to the unit k-operad Com. where E Consider the operad of projections (see 1.1.4(ii)) as an operad of groupoids with the contractible groupoid structure. Passing from groupoids to the classifying simplicial sets (see 4.1.1(ii)), we get a simplicial operad E. Let E := Norm k[E] be the ¯ := E/τ≤−1 E; this corresponding DG operad of normalized k-chains. Finally, set E ¯ is a quotient DG operad of E. Notice that both E and E are naturally resolutions of ¯ 0 is the k-linear envelope of the operad of projections (see Com. The operad E0 = E 0 1.1.4), the projection E → Com is E0I → k, ei 7→ 1, where ei are the base vectors ¯ is a length 2 resolution of Com; it is uniquely determined by of E0I := k[I]. Our E ¯ 0 . We define cii0 ∈ E ¯ −1 by the condition dcii0 = ei − ei0 . the above description of E I (I) For M ∈ CM(X), I ∈ S the complexes DR(M ) and DR(∆∗ M ) can be considered naturally as objects CDif f (X) (see 2.1.8, especially Example (i) there). (I) One has a canonical embedding DR(M ) ,→ DR(∆∗ M ) (see 2.1.7(b)) which is a quasi-isomorphism (see 2.1.10). We will define in a moment a natural morphism (I) (I) ¯ I → DR(M ) such that π(m ⊗ ei ) is the of complexes π = πM : DR(∆∗ M ) ⊗ E integral of m along the fibers of the ith projection X I → X. Such π is unique,

74

2.

GEOMETRY OF D-SCHEMES

since the de Rham differential DR(M )−1 → DR(M )0 is an injective morphism in Dif f (X). To construct π, consider first the case M = DX . We need to specify the (I) ¯ −1 → DR(DX )−1 , i.e., ∆∗(I) DX ⊗ E ¯ −1 → DX ⊗ ΘX , component DR(∆∗ DX )0 ⊗ E I I (I) of πDX . Since the differential in DR(DX ) is injective as a usual map of sheaves, we determine it from the formula dπ(m ⊗ c) = π(m ⊗ dc). Notice that DX is actually a DX -bimodule, and the above construction used only the right DX -module structure (I) on it, so πDX just defined is a morphism of left DX -modules. Now for arbitrary (I)

(I)

M one has DR(M ) = M ⊗ DR(DX ), DR(∆∗ M ) = M ⊗ DR(∆∗ DX ), and DX

(I)

DX

(I)

πM := idM ⊗ πDX .

(I)

One defines (2.2.11.1) as follows. For a ∗ operation ϕ : Li → ∆∗ M and ¯ I the operation DR(ϕ ⊗ c) : ⊗DR(Li ) → DR(M ) is ⊗`i 7→ π (I) ϕ(`i ) ⊗ c. c∈E M The compatibility with composition of operations is immediate. The rest of this section contains auxiliary technical material; we suggest the reader skip it, returning when necessary. 2.2.12. We return to arbitrary dim X. Let us show that our augmented compound tensor structure comes from a certain augmented compound strict DX operad B∗! in the way explained in 1.3.18, 1.3.19. Let us define B∗! = (B∗ , B! , < >IS,T ). For a finite set I let B∗I be the tensor product of I copies of DX considered as left D-modules. This is a left D-module on X, and the right DX -action on each copy of DX defines the right D⊗I X -action on B∗I (here D⊗I is the Ith tensor power of D considered as a sheaf of k-algebras). X X ⊗I ! ! ∗ So BI is a (DX − DX )-bimodule. Similarly, let BI be the ⊗ -product of I copies of DX considered as right DX -modules; this is a (D⊗I X − DX )-bimodule. -module, by the subsheaf OX = OX 1⊗I Note that B∗I is generated, as right D⊗I X ⊗I ⊗I ∗ on which OX ⊂ DX acts via the product map O⊗I X → OX . In fact BI = (1−|I|) (1−|I|) ⊗I ⊗I ⊗I ! OX ⊗ DX . Similarly, BI = DX ⊗ (ωX ) as left DX -modules where ωX O⊗I X

O⊗I X

is the OX -tensor power. For a map J → I the composition morphisms for our augmented operads B∗I ⊗ (B∗Ji ) → B∗J , (⊗B!Ji ) ⊗ B!I → B!J are the isomorphisms of, respectively, D⊗I X

D⊗I X

D⊗J X )-

(D⊗J X

(DX − and − DX )-bimodules, uniquely determined by the property (1−|J|) that fI ⊗ (⊗fJi ) 7→ fI ΠfJi ∈ OX ⊂ B∗J , (⊗νJi ) ⊗ νI 7→ (ΠνJi )νI ∈ ωX ⊂ B!J (1−|I|) for fI ∈ OX ⊂ B∗I , νI ∈ ωX ⊂ B!I , etc. I It remains to define < >S,T : (⊗ B∗Is ) ⊗ (⊗ B!It ) → B!S ⊗ B∗T . The arrow is, in S

T D⊗I X

DX

fact, an isomorphism, and we will construct its inverse. One has (⊗ B∗Is ) ⊗ (⊗ B!It ) S

=

O⊗S X

−1 ⊗I ⊗ (DX ωX ) ⊗ O⊗I O⊗I X X

⊗T ωX

where the

O⊗I X -module

structure on

T D⊗I X ⊗S OX ,

⊗T ωX

⊗S ⊗I ⊗T comes from the πS and πT product morphisms O⊗I X → OX , OX → OX . This is −|I| ⊗S ⊗T ⊗StT ⊗S ⊗T a (DX − DX )-bimodule containing an OX -submodule OX ⊗ ωX ⊗ ωX O⊗I X

O⊗I X

2.2. THE COMPOUND TENSOR STRUCTURE −|S|+1

= ωX

75

⊗T . On the other hand, B!S ⊗ B∗T is a (D⊗S X , DX )-bimodule freely generOX

−|S|+1

ated by its O⊗StT -module ωX X

. Therefore we get a morphism β : B!S ⊗ B∗T → OX

−1 ⊗I ⊗T (⊗ B∗Is ) ⊗ (⊗ B!It ). Notice that the action of ΘX on O⊗S ⊗ (DX ωX ) ⊗ ωX X T D⊗I X

S

O⊗I X

O⊗I X

(D⊗S X

by transport of structure coincides with the adjoint action for the − D⊗T X )⊗T bimodule structure of the diagonal embedded ΘX ,→ D⊗S X , DX . In particular, the −|S|+1 latter action of ΘX on ωX ⊂ (⊗ B∗Is ) ⊗ (⊗ B!It ) coincides with the evident T D⊗I X

S

action; hence β factors through

B!S

⊗

DX

B∗T

→ (⊗ B∗Is ) ⊗ (⊗ B!It ). A computation T D⊗I X

S

on the level of symbols shows that this arrow is an isomorphism; our < >IS,T is its inverse. The axioms of augmented compound operad are immediate. Let us check that the augmented compound tensor structure defined on M(X) by B∗! (see 1.3.18 and 1.3.19) coincides with that defined in 2.2.1 – 2.2.7. It is clear that ⊗!I Li = (⊗Li ) ⊗ B!I . The sheaf-theoretic restriction of the DX I -module D⊗I X

(I) ∆∗ M

⊗I to the diagonal is a D⊗I X -module. There is a unique isomorphism of DX modules ∼

(I)

M ⊗ B∗I −→ ∆(I)· ∆∗ M

(2.2.12.1)

DX

which induces the identity isomorphism between the subsheaves M = M ⊗ 1⊗I and (I) M = ∆(I)! ∆∗ M . This yields the isomorphism (2.2.12.2)

∼

HomD⊗I (⊗Li , M ⊗ B∗I ) −→ PI∗ ({Li }, M ). X

DX

We leave it to the reader to identify the maps <

>IS,T .

2.2.13. We finish this section with several remarks about the ∗ pseudo-tensor structure. The operad B∗I from 2.2.12 satisfies conditions (i) and (ii) of 1.2.3. Therefore, for any coherent D-modules Li and an arbitrary D-module M , the object PI ({Li }, M ) ∈ M(X) exists. Explicitly, one has (2.2.13.1)

PI ({Li }, M ) = HomD⊗I (⊗ Li , M ⊗ B∗Ie). X

k

DX

(OX , D⊗I X )-bimodules

Consider the morphism of B∗I → B∗Ie, b 7→ 1· ⊗ b (recall that Ie = { · } t I). The corresponding morphism DX ⊗ B∗I → B∗Ie of (DX , D⊗I X )OX

bimodules is an isomorphism. It yields the identification M ⊗ B∗I = M ⊗ B∗Ie; OX

DX

hence also (2.2.13.2)

PI ({Li }, M ) = HomD⊗I (⊗ Li , M ⊗ B∗I ). X

k

OX

Note that the right DX -action on M ⊗ B∗Ie (used to define the D-module strucOX

ture on the Hom sheaf) identifies with the right D-module structure on M ⊗ B∗I OX

that comes from the right DX -module structure on M and the left DX -module structure on B∗I (see 2.1.1).

76

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Remark. The functor P∗I obviously commutes with inverse images for ´etale morphisms; i.e., it is defined on the sheaf of categories M(X´et ). It also commutes with direct images for closed embeddings (as follows immediately from 2.2.5). 2.2.14. If we deal with induced DX -modules, i.e., Li = FiD , M = GD , where Fi are coherent O-modules, then the corresponding D-module P∗I can be described as follows. Consider the sheaf of differential I-operations DiffI ({Fi }, G)X .33 This is an object of Dif f (X) by Example (i) in 2.1.8, so we have the corresponding ˜ D-module DiffI ({Fi }, G)D = DiffI ({Fi }, G)XD . Now the canonical I-operation ∗ ∈ DiffI˜({DiffI ({Fi }, G)X , Fi }, G) = PI˜ ({DiffI ({Fi }, G)D , FiD }, GD ) yields an isomorphism34 ∼

DiffI ({Fi }, G)D −→ P∗I ({FiD }, GD ).

(2.2.14.1)

2.2.15. According to 2.2.13, for a DX -coherent L and any M ∈ M(X) we have the Hom DX -module Hom∗ (L, M ) = HomDX (L, M ⊗ DX ) ∈ M(X) equipped OX

with a universal “evaluation” ∗ pairing ε ∈ P2∗ ({Hom∗ (L, M ), L}, M ). If K is another coherent D-module, then the composition ∗ pairing (2.2.15.1)

c ∈ P2∗ ({Hom∗ (L, M ), Hom∗ (K, L)}, Hom∗ (K, M ))

is defined; the composition is associative and compatible with the evaluation (see 1.2.2). Here is a more detailed description. Recall that M ⊗ DX := M ⊗ DX carries OX

two commuting right DX -actions: the first one comes from the right DX -action on DX ; the second one comes from the right DX -module structure on M and the left DX -module structure on DX . Since35 M ⊗ DX = M ⊗ B∗{1,2} , one has a canonical DX

automorphism of M ⊗ DX interchanging the two D-module structures. Now a section of Hom∗ (L, M ) is a morphism of DX -modules L → M ⊗ DX with respect to the first D-module structure on M ⊗ DX ; the D-module structure on Hom∗ (L, M ) comes from the second D-module structure on M ⊗ DX . Here ε is just the usual evaluation morphism Hom(L, M ⊗ DX ) ⊗ L → M ⊗ DX . To describe k

c explicitly, it is convenient to understand HomDX (L, M ⊗ DX ) in a way different from that used above: the morphisms are understood with respect to the second Dmodule structure on M ⊗ DX , and the action of DX on HomDX (L, M ⊗ DX ) comes from the first one. Let us identify Hom∗ (L, M ) with HomDX (L, M ⊗ DX ) using the above symmetry of M ⊗ DX interchanging the D-module structures. Then the ∗ pairing c becomes the usual composition Hom(L, M ⊗ DX ) ⊗ Hom(K, L ⊗ DX ) → Hom(K, (M ⊗ DX ) ⊗ DX ) k

= Hom(K, M ⊗ DX ) ⊗ DX . 33 Which is the X-sheafified version of the vector space of differential I-operations from 2.2.4(i). 34 To check that our arrow is an isomorphism, one reduces (using the left exactness of our functors) to the case Fi = OX where the statement is obvious. 35 See 2.2.13 for notation.

2.2. THE COMPOUND TENSOR STRUCTURE

77

2.2.16. We see that for any coherent M ∈ M(X) the D-module End∗ (M ) = Hom∗ (M, M ) is an associative algebra in the ∗ sense acting on M . This action is universal: for any associative aglebra A in M(X)∗ an A-module structure on M is the same as a morphism of associative algebras A → End∗ M . Set M ◦ := Hom∗ (M, ωX ) = HomDX (M, ωX ⊗ DX ); this is again a coherent OX

module. We have a canonical pairing h i ∈ P2∗ ({M ◦ , M }, ωX ). If M is a vector DX -bundle (see 2.1.5), then h i is non-degenerate (see 1.4.2); ∼ in particular, M ⊗ M ◦ −→ End∗ M . Remarks. (i) Notice that ωX ⊗ DX considered as a right DX -module with OX

respect to the second D-module structure (see 2.2.15) equals (DX )r where we consider DX as a left D-module. So, by 2.2.15, one has M ◦ = HomDX (M ` , DX ). (ii) Let F be a vector bundle. It yields the corresponding left and right induced DX -modules D F := DX ⊗ F ∈ M` (X), FD := F ⊗ DX ∈ M(X). We see that36 OX

OX

(2.2.16.1)

◦

∗

(FD ) = (F ⊗ ωX )D ,

(D F )◦ = (F ∗ )D .

2.2.17. Global duality. Suppose X is proper of dimension n. Let V , V ◦ be mutually dual vector DX -bundles. The canonical pairing ∈ P2∗ ({V ◦ , V }, ωX ) together with the trace morphism tr : RΓDR (X, ωX ) → k[−n] yields a canonical pairing (2.2.17.1)

RΓDR (X, V ◦ ) ⊗ RΓDR (X, V ) → k[−n].

Lemma. This pairing is non-degenerate; i.e., RΓDR (X, V ◦ ) is dual to the complex RΓDR (X, V )[n]. Proof. Using appropriate resolutions, one is reduced to the induced situation where the above duality is the usual Serre duality.  If X is not proper, the above lemma holds if one replaces one of the cohomology groups by cohomology with compact supports. We do not need the statement in i (X, V ) = 0 whole generality; consider just the case when X is affine. Then HDR 0 0 for i 6= 0, and HDR (X, V ) = Γ(X, h(V )). Now the morphism HDR (X, V ) → Hom(V ◦ , ωX ), defined by the canonical pairing, is an isomorphism.37 One can rewrite it as a canonical isomorphism (see 2.1.12 for notation)38 (2.2.17.2)

∼

0 (V ◦ )const −→ HDR (X, V )∗ ⊗ ωX .

Remark. We can consider families of vector DX -bundles; i.e., locally projective F ⊗ DX -modules V of finite rank where F is any commutative k-algebra. For such V its dual V ◦ is HomF ⊗DX (V ` , F ⊗ DX ). The above statements (together with the proofs) remain valid with the following modifications: If X is projective, then RΓDR (X, V ) is a perfect F -complex, and the canonical pairing identifies RΓDR (X, V ◦ ) with the F -complex dual to RΓDR (X, V )[dim X]. If X is affine, then i 0 HDR (X, V ) = 0 for i 6= 0, HDR (X, V ) = Γ(X, h(V )) is a projective F -module, and 0 0 (2.2.17.2) holds with HDR (X, V )∗ := HomF (HDR (X, V ), F ). the second equality we identified left and right D-modules in the usual way. suffices to check this statement for V = DX where it is obvious. 38 Here H 0 (X, V )∗ is considered as a profinite-dimensional vector space. DR 36 In

37 It

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2.2.18. According to 1.3.11, the duality functor ◦ : M(X)◦coh → M(X)coh ex∗ tends canonically to a pseudo-tensor functor ◦ : M(X)◦! coh → M(X)coh . It induces an equivalence between the full pseudo-tensor subcategories of vector DX -bundles. ! On M(X)◦! coh we have a standard augmentation functor h (M ) := Hom(M, ωX ) ◦ ! and a canonical morphism h(M ) → h (M ) of augmentation functors; see (1.3.11.3). If M is a vector DX -bundle, then this is an isomorphism.39 ◦ Remark. If Fi , G are vector bundles on X, then the duality Hom(G◦D , ⊗! FiD ) ∗ −→ PI ({FiD }, GD ) is the composition of the following identifications (see 2.2.12) Hom( D G∗ , ⊗( D Fi∗ )) = HomOX (G∗ , B∗I ⊗ (⊗ Fi∗ )) = HomOX I (Fi , G ⊗ B∗I ) = ∼

(I) Hom(FiD , (∆· G)D )

=

OX

O⊗I X (I) Hom(FiD , ∆∗ GD ).

2.2.19. A remark about the augmentation functor. Take I, J ∈ S, the families of objects {Li }, {Aj }, M and consider the composition morphism for I ,→ I t J in our augmented pseudo-tensor structure P ∗ItJ ({Li , Aj }, M ) ⊗ ⊗ h(Aj ) → J

P ∗I ({Li }, M ). We can rewrite it as an embedding40

∗ hIJ : PItJ ({Li , Aj }, M ) ,→ Hom (⊗ h(Aj ), P ∗I ({Li }, M )) .

(2.2.19.1)

J

Lemma. The image of hIJ is the subspace of those ϕ: ⊗ h(Aj ) → PI∗ ({Li }, M ) J

aj )(⊗`i ), for which the map ϕ: e (⊗Li ) ⊗ (⊗Aj ) → M ⊗ B∗I , (⊗`i ) ⊗ (⊗aj ) 7→ ϕ(⊗¯ DX

is a differential operator with respect to any of the variables Aj (see Example (iii) in 2.1.8). In other words, ϕ satisfies the following condition (here I is the kernel of ⊗ItJ OX → OX ): (∗) For any local section s ∈ (⊗Li ) ⊗ (⊗Aj ) one has ϕ(I e N s) = 0 for N  0. ∗ Proof. Let T r: M ⊗ B∗ItJ → M ⊗ B∗ItJ ⊗ O⊗J X = M ⊗ BI be the trace DX

DX

along J-variables. The morphism M ⊗

DX

B∗ItJ

DX

D⊗J X

→

⊗J HomD⊗I (D⊗I X ⊗OX , M X

⊗ B∗I ) of

DX

D⊗ItJ -modules which assigns to b ∈ M ⊗ B∗ItJ the map α 7→ T r(bα) is injective. X DX

⊗J N ∗ ⊂ Its image is the subspace of those ψ: D⊗I X ⊗ OX → M ⊗ BI that kill some I DX

⊗ItJ ⊗I ∗ OX ⊂ DX ⊗ O⊗J X . This morphism maps PItJ ({Li , Aj }, M ) injectively to ⊗I ∗ HomD⊗ItJ ((⊗Li ) ⊗ (⊗Aj ), HomD⊗I (DX ⊗ O⊗J X , M ⊗ BI )) X

DX

X

= HomD⊗I ((⊗Li ) ⊗ (⊗h(Aj )), M ⊗ X

DX

B∗I )

= Hom(⊗h(Aj ), P ∗I ({Li }, M )).

The image coincides with the subspace defined by condition (∗). Since this map coincides with hI,J , we are done.  2.2.20. The following technical lemma will be of use. Let Li , M be D-modules on X, ϕ ∈ PI∗ ({Li }, M ), and let x ∈ X be a closed point. Set Ux := X r {x}. We use the terminology of 2.1.13. 39 It

suffices to check this statement for M = DX where it is obvious. h is non-degenerate by 2.2.8.

40 Our

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79

Lemma. (i) The polylinear map h(ϕ) : ⊗h(Li )x → h(M )x is Ξx -continuous with respect to each variable. (ii) Assume that the restriction of each Li to Ux is a countably generated Dmodule. Then h(ϕ) is Ξx -continuous. (iii) Assume that for certain i0 ∈ I every Li , but, possibly Li0 , is a coherent DX -module. Then for each Mξ ∈ Ξx (M ) there exists Li0 ξ ∈ Ξx (Li0 ) such that (I)

ϕ(Li0 ξ  (  Li )) ⊂ ∆∗ Mξ .

(2.2.20.1)

i6=i0

Proof. (i) Pick i0 ∈ I and l¯i ∈ h(Li )x for i 6= i0 . We have to show that the map h(Li0 )x → h(M )x , ¯li0 7→ h(ϕ)(¯li0 ⊗ (⊗¯li )), is Ξx -continuous. This is clear since this map comes from a morphism of D-modules Li0 → M that comes from ϕ and the ¯li ’s by the augmentation functor structure on h. (ii) An induction by |I| shows (use the augmentation functor structure on h) that to prove continuity of h(ϕ), it suffices to find for every Mξ ∈ Ξx (M ) some Liξ ∈ Ξx (Li ) such that h(ϕ)(⊗Liξ ) ⊂ Mξ . Shrinking X if necessary, we can assume that X is affine and we have functions ta , a = 1, . . . , n, on it which are local coordinates at x and have no other common zero. Let li1 , li2 , . . . be sections of Li that generate the restriction of Li to Ux as a DUx -module. The quotient (I) (I) ∆∗ M/∆∗ Mξ = ∆∗ (M/Mξ ) is a DX I -module supported at ∆(I) (x), so every its section is killed by sufficiently high power of each ta lifted from either copy of X. Therefore, using induction by b = 1, 2, . . . , we can choose integers n(a, b, i) ≥ 0 (I) (I) n(a,b ,i) such that h(ϕ)( ta i libi ) ∈ ∆∗ Mξ ⊂ ∆∗ M for every 1 ≤ bi ≤ b. Our Liξ is I

n(a,b,i)

lib }. the D-submodule of Li generated by {ta (iii) Choose for i 6= i0 a finite set {lij } of local sections of Li at x that generates (I) Li . Now take for Li0 ξ the intersection of the preimages of ∆∗ Mξ by all maps (I) Li0 → ∆∗ M , l 7→ ϕ(l  (lij )).  We see that in situation (ii) the morphism hI (ϕ): ⊗h(Li )x → h(M )x exˆ I (ϕ)x : ⊗ ˆ x (Li ) → h ˆ x (M ). Here ⊗ ˆ x (Li ) := bh bh tends by continuity to the map h lim ⊗h (L /L ). x i i ξi ←− ˜ x from 2.1.15 instead of plain topological If we decide to play with objects h vector spaces, then the countability condition of (ii) becomes irrelevant. 2.3. DX -schemes A commutative algebra in M! (X) can be considered as a “coordinate-free” version of a system of non-linear differential equations (just as D-modules provide a coordinate-free language for systems of linear differential equations). In this section we deal with basic algebro-geometric properties of these objects. The bulk of the literature on foundational subjects of the geometric theory of non-linear differential equations is huge (take [G] and [V] to estimate the span). The Euler-Lagrange equations (the classical calculus of variations) are treated in the nice review articles [DF], [Z]. We consider jet schemes in 2.3.2–2.3.3, quasi-coherent OY [DX ]-modules on a DX -scheme Y in 2.3.5, and prove in 2.3.6–2.3.9 that for affine Y projectivity of such modules is a local property, which is a D-version of a theorem of Raynaud-Gruson [RG]. The ∗ operations between quasi-coherent OY [DX ]-modules are considered

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in 2.3.11–2.3.12, the notion of smoothness for DX -schemes in 2.3.13–2.3.16. We show that smooth DX do not necessary admit local coordinate systems (see 2.3.17– 2.3.18), and discuss the notion of vector DX -scheme in 2.3.19. The setting of the calculus of variations is briefly described in 2.3.20. 2.3.1. A commutative DX -algebra is a commutative unital OX -algebra equipped with an integrable connection along X. Unless stated explicitly otherwise, our DX -algebras are assumed to be OX -quasi-coherent. A DX -scheme is an X-scheme equipped with an integrable connection along X. Denote by ComuD (X), SchD (X) the corresponding categories. We have a fully faithful functor Spec: ComuD (X)◦ ,→ SchD (X); its essential image is the category Af f SchD (X) of DX -schemes affine over X. Replacing schemes by algebraic spaces, we get the notion of an algebraic DX space. For a DX -algebra R` and an algebraic DX -space Y a morphism of algebraic spaces Spec R` → Y is sometimes referred to as a DX -algebra R` -point of Y. Remark. Let xa be a coordinate system on X. Assume we have a system of non-linear differential equations Pα (ui , ∂a ui , . . . ) = 0 where Pα are polynomial functions with coefficients in functions on X. It yields a DX -algebra A` defined as the quotient of the free DX -algebra Sym(⊕DX · ui ) modulo the relations Pα . A solution of our system with values in a (not necessarily quasi-coherent) DX algebra R` is a collection of φi ∈ R` such that Pα (φi , ∂a φi , . . . ) = 0. This is the same as a morphism of DX -algebras A → R, ui 7→ φi , i.e., a DX -algebra R` point of Spec A` . Therefore DX -algebras provide a “coordinate-free” language for non-linear differential equations. Set Comu! (X) := Comu(M! (X)) (we use the notation of 1.4.6). The equivalence ∼ −1 of tensor categories M! (X) −→ M` (X), A 7→ A` = AωX , yields the equivalence (2.3.1.1)

∼

Comu! (X) −→ ComuD (X).

2.3.2. Jet schemes. Denote by Comu(X) the category of commutative OX algebras quasi-coherent as OX -modules and by Sch(X) the category of X-schemes. We have the obvious forgetting of connection functors (2.3.2.1)

o: ComuD (X) → Comu(X),

SchD (X) → Sch(X).

They admit the left, resp. right, adjoint functors (2.3.2.2)

Comu(X) → ComuD (X),

Sch(X) → SchD (X).

We denote them both by J. For R ∈ Comu(X), JR is generated by R.Explicitly, it  is the DX -algebra quotient of the symmetric algebra Sym· DX ⊗ R modulo the  OX  2 ideal generated by elements ∂(1⊗r1 ·1⊗r2 −1⊗r1 r2 ) ∈ Sym DX ⊗ R +DX ⊗ R OX

OX

and ∂(1 ⊗ 1R − 1) ∈ DX ⊗ R + OX where ri ∈ R, ∂ ∈ DX , 1R is the unit of R. This construction is compatible with localization. The functor J: Sch(X) → SchD (X) is Spec R 7→ Spec JR for affine schemes; for arbitrary schemes use gluing. Note that for any Z ∈ Sch(X) the canonical projection JZ → Z induces a bijection ∼ (JZ)D (X) −→ Z(X) where (JZ)D (X) is the set of all horizontal sections X → Z.

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81

2.3.3. Here is a different description of JR. Denote by I the kernel of the ⊗2 n product map O⊗2 X = OX ⊗ OX → OX ; for n ≥ 1 set OX (n) := OX /I . These OX (n) k

form a projective system of algebras; there are two obvious morphisms OX ⇒ OX (n) . Therefore any OX -algebra B yields two OX (n) -algebras: B ⊗ OX (n) and OX

OX (n) ⊗ B. OX

Lemma. For any R, B ∈ Comu(X) one has   (2.3.3.1) Hom(JR, B) = lim ←− Hom OX (n) ⊗ R, B ⊗ OX (n) OX

OX

where the left and right Hom mean, respectively, the morphisms of OX - and OX (n) algebras. Proof. Our statement is X-local, so we can assume that X is affine, X = Spec C. Thus OX -algebras are the same as C-algebras. For a C-algebra B denote by JB n the kernel of the map B ⊗ C → B, b ⊗ c 7→ bc. Set TB := lim ←− B ⊗ C/JB . Let us k

consider TB as a C-algebra via the morphism C → TB, c 7→ 1 ⊗ c. The projection TB → C, b ⊗ c 7→ bc, is a morphism of C-algebras. Note that TB is a DX -algebra in the obvious way (the elments b ⊗ 1 ∈ TB are horizontal). It is easy to see that the functor Comu(X) → ComuD (X), B 7→ TB, is right adjoint to the functor o: ComuD (X) → Comu(X). Therefore we have Hom(oJR, B) = Hom(JR, TB) = Hom(R, oTB). The latter set coincides with ← lim − Hom in the statement of the lemma; we are done.  So for any X-scheme Y the k-points of JY are the same as pairs (x, γ) where x ∈ X and γ is a section of Y on the formal neighborhood of x. We call JY the jet scheme of Y . Remark. The above constructions are compatible with ´etale localization, so we may replace schemes by algebraic DX -spaces. 2.3.4. For A` ∈ ComuD (X) we denote by A` [DX ] the ring of differential operators with coefficients in A` . By definiton, this is a sheaf of associative algebras equipped with a morphism of algebras A` → A` [DX ] and that of Lie algebras ΘX → A` [DX ] which satisfy the relations τ · a − a · τ = τ (a), f · τ = f τ for a ∈ A` , f ∈ OX ⊂ A` , τ ∈ ΘX , and universal with respect to these data in the obvious sense. The embedding of ΘX extends to the morphism of associative algebras DX → A` [DX ]. One checks easily that the morphism of sheaves A` ⊗ DX → A` [DX ], a ⊗ ∂ 7→ a · ∂ is an isomorphism. OX

For any algebraic DX -space Y we have the corresponding sheaf OY [DX ] on Y´et .

2.3.5. An A` -module in the tensor category M` (X) is the same as a left A [DX ]-module quasi-coherent as an OX -module. Similarly, an A-module in M! (X) is the same as a right A` [DX ]-module quasi-coherent as an OX -module. We denote by M` (X, A` ), M(X, A` ) = Mr (X, A` ) the categories of left, resp. right, A` [DX ]modules. These are tensor categories (canonically identified) with unit object equal to A` , resp· A. Every morphism of commutative DX -algebras f : A` → B ` yields an adjoint pair of functors f ∗ : M(X, A` ) → M(X, B ` ), f· : M(X, B ` ) → M(X, A` ) where `

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f ∗ M := B ` ⊗ M , f· N is N considered as an A` [DX ]-module. Notice that f ∗ is a A`

tensor functor. The faithfully flat descent works in any abelian tensor category, so A` [DX ]modules are local objects with respect to the flat topology.41 So for any algebraic DX -space Y we have the corresponding categories M` (Y), Mr (Y) = M(Y). Their objects are sheaves of left, resp. right, OY [DX ]-modules which are quasi-coherent as OY -modules. The categories M` (U), M(U) for U ∈ Y´et form sheaves of categories M` (Y´et ) on Y´et . One has M` (Spec A` ) = M` (X, A), M(Spec A` ) = M(X, A` ). Most of the facts mentioned in 2.1 render easily to the OY [DX ]-modules setting. E.g., if G, L are left OY [DX ]-modules and M is right one, then G ⊗ L is a left OY

OY [DX ]-module in a natural way, and L ⊗ M is a right one. We have the standard OY

equivalence M` (Y) → Mr (Y), L 7→ LωX . Any N ∈ M(Y) admits a canonical finite left resolution (DR(N ))D whose terms are induced OY [DX ]-modules. 2.3.6. Theorem. Assume that Y is a DX -scheme which is affine (as a kscheme). Then projectivity of an OY [DX ]-module is a local property for the Zariski or ´etale topology. Proof. It is found in 2.3.7–2.3.9. 2.3.7. Let us first prove the theorem for a finitely generated OY [DX ]-module P (which is sufficient for the most of applications). Suppose that P is projective over some ´etale covering of Y. We need to show that P is projective. Take N ∈ M(Y). Consider the sheaf Hom(P, N ) on Y´et . It depends on N in an exact way, so projectivity of P follows if we prove that H i (Y´et , Hom(P, N )) = 0 for i > 0. This is true if N is an induced module, N = F ⊗ DX . Indeed, in this situOY

ation Hom(P, N ) admits a structure of a quasi-coherent OY -module. Namely, the OY -module structure on Hom(P, N ) comes from an OY -action on N inherited from F . Quasi-coherence is an ´etale local property, so it suffices to check it under the assumption that P is projective. Then Hom(P, N ) is isomorphic to a direct summand of a sum of several copies of N , which is obviously quasi-coherent. To prove acyclicity of Hom(P, N ) for arbitrary N , use the fact that N admits a finite left resolution by induced modules (see 2.1.9 and 2.3.5). We are done. 2.3.8. Without the finite generatedness assumption the theorem is nontrivial even if X is a point. In this case it was conjectured by Grothendieck (see Remark 9.5.8 from [Gr1]) and proved by Raynaud and Gruson (even for the fpqc topology); see [RG] 3.1.4. We will use the method of [RG]. It is based on the following fact: a flat module M over a ring R is projective if and only if M has the following properties: (i) M is a direct sum of countably generated modules; (ii) M is a Mittag-Leffler module (the implication projectivity ⇒(i) is due to Kaplansky [Ka]; the fact that (i) and (ii) imply projectivity is proved in [RG], p. 74). A flat module M is said to be Mittag-Leffler if for some (or for every) representation of M as a direct limit of finitely generated projective modules Pi (i belongs to a directed ordered set I) the 41 f is faithfully flat as a morphism of commutative algebras in the tensor category M` (X) if and only if it is faithfully flat as a morphism of plain OX -algebras.

2.3. DX -SCHEMES

83

projective system (Pi∗ ) satisfies the Mittag-Leffler condition (i.e., for every i ∈ I there exists j ≥ i such that Im(Pj∗ → Pi∗ ) = Im(Pk∗ → Pi∗ ) for all k ≥ j). Here Pi∗ := HomR (Pi , R). Now the theorem is easily reduced to the following lemma. 2.3.9. Lemma. Assume that X is affine and Y is an affine DX -scheme. Then a flat OY [DX ]-module M is globally flat; i.e., Γ(Y, M ) is flat over Γ(Y, OY [DX ]). Proof. M has a finite left resolution consisting of globally flat OY [DX ]-modules (indeed, M is OY -flat, so the canonical resolution (DR(M ))D has the required property). Therefore it suffices to show that if there is an exact sequence 0 → F1 → F0 → M → 0 with F0 , F1 globally flat and M flat, then M is globally flat; i.e., for every left OY [DX ]-module L the morphism Γ(Y, F1 ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) → Γ(Y, F0 ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) is injective. The map Γ(Y, Fi ) ⊗Γ(Y,OY [DX ]) Γ(Y, L) → Γ(Y, Fi ⊗OY [DX ] L) is an isomorphism (represent Fi as a direct limit of finitely generated free OY [DX ]-modules). Finally, the morphism F1 ⊗OY [DX ] L → F0 ⊗OY [DX ] L is injective because M is flat.  2.3.10. We see that for any algebraic DX -space Y there is a notion of Y-locally projective OY [DX ]-modules. Notice that for OY [DX ]-modules on an affine Y the property of being finitely generated is obviously ´etale local. So we know what Ylocally finitely generated OY [DX ]-modules on any algebraic DX -space Y are. We call a Y-locally finitely generated and projective OY [DX ]-module a vector DX -bundle on Y. 2.3.11. According to 1.4.6 (and 2.2) for a commutative DX -algebra A` the category M(X, A` ) carries a canonical augmented compound tensor structure. As in 1.4.6 for Li , M ∈ M(X, A` ) we denote the corresponding space of A` -polylinear ∗ operations by PA∗ I ({Li }, M ). In fact, this structure comes from an augmented compound strict A` [DX ]∗! operad B∗! A . We leave the explicit construction of BA to the reader (repeat 2.2.12 ` replacing DX by A [DX ] everywhere). The discussion in 2.2.13–2.2.18 remains literally valid for A` [DX ]-modules if we replace the words “coherent DX -module” by “finitely presented A` [DX ]-module”. The lemma from 2.2.19 also remains valid. According to Remark (i) of 1.4.6 for every morphism of commutative DX algebras f : B ` → A` the corresponding tensor functor f ∗ from 2.3.5 extends to a compound tensor functor (base change) (2.3.11.1)

f ∗ : M(X, A` )∗! → M(X, B ` )∗! .

We also have a forgetful pseudo-tensor functor (2.3.11.2)

f· : M(X, B ` )∗ → M(X, A` )∗

right adjoint to f ∗ considered as a ∗ pseudo-tensor functor (see Remark (ii) of 1.4.6). Lemma. M(X, A` )∗! has a local nature with respect to the flat topology of Spec A` . Proof. We already discussed the flat descent of A` [DX ]-modules in 2.3.5, so it remains to check that the A-polylinear ∗ operations satisfy the flat descent property. For Li , M ∈ M(X, A` ) the above adjunction identifies the descent data

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with operations ψ ∈ PA∗ I ({Li }, B ` ⊗ M ) whose composition with the two arrows A` B ` ⊗ M −→ −→B ` ⊗ B ` ⊗ M , b ⊗ m 7→ b ⊗ 1 ⊗ m, 1 ⊗ b ⊗ m, coincide. Now the exact A`

A`

A`

sequence 0 → M → B ` ⊗ M → B ` ⊗ B ` ⊗ M (the latter arrow is the difference of A`

A`

A`

the above two standard arrows) together with the left exactness of PA∗ implies the desired descent property.  2.3.12. For Li , M as above a ∗ operation ϕ ∈ PI∗ ({Li }, M ) is A-polydifferential if for every finitely generated A` [DX ]-submodules L0i ⊂ Li the restriction ϕ|{L0i } ∈ PI∗ ({L0i }, M ) is an A-polydifferential ∗ operation of finite order (see 1.4.8). The composition of A-polydifferential operations is A-polydifferential, so they define another pseudo-tensor structure on M(X, A` ). Lemma. A-polydifferential ∗ operations have a local nature with respect to the ´etale topology of Spec A` . Proof. The case of arbitrary A-polydifferential operations reduces immediately to that of A-polydifferential operations of finite order, and then to that of A(n) (n) polylinear ones (replace Li by Li as in 1.4.8 and notice that the Li have an ´etale local nature). Then use the previous lemma.  Therefore for any algebraic DX -space Y we have the compound tensor category M(Y)∗! and the sheaf of compound tensor categories M(Y´et )∗! on Y´et , the pull-back functors f ∗ are compound tensor functors, etc. The polydifferential ∗ operations also make sense in this setting. Remarks. (i) One has the following immediate analog of 2.2.13. Let {Li } be a finite collection of finitely presented A` [DX ]-modules, M any A` [DX ]-module. Then the A` [DX ]-module P∗A I ({Li }, M ) is well defined. The functor P∗A I is compatible with flat pull-backs. So the functor P∗I makes sense on any algebraic DX -space Y. (ii) In particular, for every finitely presented OY [DX ]-module L its dual L◦ := Hom∗ (L, OY ωX ) is well defined. In particular, if the OY [DX ]-module of differentials ΩY = ΩY/X is finitely presented,42 then ΘY := Ω◦Y is well defined. (iii) If L is a vector DX -bundle on Y, then L◦ is also a vector DX -bundle on Y, and the canonical pairing is non-degenerate (see 1.4.2). 2.3.13. We are going to discuss the notion of smoothness for algebraic DX spaces. First let us recall the definition of formally smooth algebras. Let M be any abelian tensor k-category with unit object (so ⊗ is right exact). A commutative algebra A in M is formally smooth if for any (commutative) algebra R and an ideal I ⊂ R such that I 2 = 0, any morphism f : A → R/I may be lifted to a morphism f˜: A → R. It suffices to check this in the case when f is an e := A × R, isomorphism; i.e., R/I = A, f = id (replace arbitrary R, I, f by R R/I

e → A) = I, f = id). Note that the symmetric algebra Sym V is formally Ie := Ker(R smooth if and only if V is a projective object of M. Assume that M has many projective objects. We may represent A as a quotient B/J, B = Sym V for a projective V . Then A is formally smooth iff the projection B/J 2 → B/J = A has a section σ: A → B/J 2 (if σ exists, then for any R, I, 42 This

happens if Y is (locally) finitely DX -presented or if ΩY is a vector DX -bundle.

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85

f as above f lifts to fˆ: B/J 2 → R and one may set f˜ = f σ). Note that the sections σ correspond bijectively to derivations ∂: B → J/J 2 such that ∂(x) = xmod J 2 for x ∈ J (namely, σ(bmod J) = b − ∂(b)). We may consider instead of ∂ the corresponding morphism of B-modules ϕ: ΩB /J · ΩB → J/J 2 ; the condition on ∂ means that ϕν = idJ/J 2 where ν: J/J 2 → ΩB /J · ΩB is the usual map ν(x mod J 2 ) = dx mod J · ΩB . Therefore A = B/J is formally smooth iff ν is a split (−1) injection. Set ΩA := Ker ν. Since Coker ν = ΩA and ΩB /J · ΩB is a projective A-module, we see (as in [Gr1] 9.5.7) that (−1) A is formally smooth iff ΩA = 0 and ΩA is a projective A-module. (−1)

Remark. The above modules ΩA , ΩA are cohomology of the cotangent complex L ΩA (see [Gr1], [Il], or [H] §7) which is the left derived functor of the functor A 7→ ΩA . To define it, one extends the category of commutative algebras to that of commutative unital DG algebras where the standard homotopy theory formalism is available (see, e.g., [H]). Now for a cofibrant DG algebra R its cotangent complex coincides with the R-module ΩR considered as the object of the derived category D(R) of R-modules. For arbitrary DG algebra R one considers its cofi˜ → R and defines L ΩR as the image of Ω ˜ under the equivalence brant resolution R R ∼ ˜ ˜ If A is a plain algebra, then D(R) −→ D(R); it does not depend on the choice of R. (−1) L ΩA is acyclic in degrees > 0 and H 0 (L ΩA ) = ΩA , H −1 (L ΩA ) = ΩA . 2.3.14. Let us return to our tensor category M` (X) assuming for the moment that X is affine. Let A` be a commutative DX -algebra. One check immediately that ΩA considered as a mere A` -module (we forget about the DX -action) coincides with the module of relative differentials ΩA` /OX . The same is true for the cotangent complex. Proposition. (i) A` is formally smooth iff it is formally smooth as an OX algebra and ΩA` is a projective A` [DX ]-module. (ii) A` is formally smooth as an OX -algebra iff it is formally smooth as a kalgebra. Proof. (i) Use the remark above, together with the fact that every projective A` [DX ]-module is a projective A` -module. (ii) Follows from the next lemma (the morphism ΩA` → ΩX ⊗ A` left inverse OX

to µ corresponds to the canonical derivation ∇: A` → ΩX ⊗ A` ). OX

Lemma. If A is an OX -algebra, then A is a formally smooth OX -algebra iff A is formally smooth as a k-algebra and the morphism of A-modules µ: ΩX ⊗ A → ΩA OX

is a split injection. Proof of Lemma. Since X is smooth over k, the standard long exact sequence (−1) (−1) reduces to 0 → ΩA → ΩA/OX → ΩX ⊗ A → ΩA → ΩA/OX → 0; this implies the OX

lemma. Here is another ad hoc proof. Let us represent A as B/J, B = Sym V , V a free OX -module. The formal smoothness of A over OX means that the morphism of A-modules ν: J/J 2 → ΩB/O /J ·ΩB/O admits a left inverse. The formal smoothness over k means that the morphism ν˜: J/J 2 → ΩB/k /J · ΩB/k admits a left inverse. It remains to apply the following easy sublemma to P = J/J 2 , Q = ΩB/k /J · ΩB/k , R = ΩX ⊗ A. OX

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g

Easy Sublemma. Suppose we have morphisms P −→Q←−R where g is a split injection. Then P → Q/Im g is a split injection iff both f : P → Q and R → Q/Imf are such. Proof of Easy Sublemma. Write Q as R⊕S, so g(r) = (r, 0), f (p) = (ϕ(p), ψ(p)). The morphism R → Q/f (P ) is a split injection iff there is a γ: S → R such that ϕ = γψ, i.e., when ϕ ∈ Hom(S, R)ψ; the morphism f is a split injection iff idP ∈ Hom(R, P )ϕ + Hom(S, P )ψ. We need to show that idP ∈ Hom(S, P )ψ if and only if idP ∈ Hom(R, P )ϕ + Hom(S, P )ψ and ϕ ∈ Hom(S, R)ψ, which is clear.  2.3.15. From now on we do not assume that X is affine. (−1) Let A` be a DX -algebra. Back in 2.3.13 we defined ΩA , ΩA assuming that X is affine. The construction is compatible with the localization of X, and we define these A` [DX ]-modules for arbitrary X by gluing. More generally, we have all H a (L ΩA ) ∈ M(X, A` ).43 Definition. We say that A` is – formally smooth if ΩA is a projective A` [DX ]-module locally on X (see 2.3.6) (−1) and ΩR vanishes; – smooth if, in addition, A` is finitely generated as a DX -algebra; – very smooth if it is smooth and H 6=0 (L ΩA ) = 0. Notice that the H a (L ΩA ) are compatible with the ´etale localization of Spec A` , so the above notions make sense for an arbitrary algebraic DX -space Y. If Y is smooth, then ΩY is a vector DX -bundle on Y. Remarks. (i) Y is formally smooth if and only if for every affine scheme S equipped with a DX -scheme structure and every closed DX -subscheme S0 ⊂ S defined by an ideal I with I 2 = 0, the map MorDX (S, Y) → MorDX (S0 , Y) is surjective.44 (ii) We do not know if there exists a smooth A` which is not very smooth.45 We also do not know if a formally smooth A` is necessary OX -flat and reduced. (iii) Suppose that A` is smooth and, as a mere OX -algebra, it can be represented locally on X as an inductive limit of smooth OX -algebras. Then A` is very smooth (see the remark that opens 2.3.14). In particular, the jet DX -scheme JZ for a smooth X-scheme Z is very smooth. 2.3.16. A morphism ϕ : Y → Z of algebraic DX -spaces is formally smooth if for every (S, S0 ) as in Remark (i) from 2.3.15, any pair of ϕ-compatible morphisms f : S0 → Y, g : S → Z lifts to a morphism f˜ : S → Y. The reformulation of this property in terms of relative differentials is left to the reader. One says that ϕ is formaly ´etale if, in addition, f˜ is unique. 43 In

fact, at least when X is quasi-projective, one has a well-defined object L ΩA of the derived category DM(X, A` ) defined by means of a global semi-free resolution of A` ; see 4.6.1 and 4.6.4. 44 To prove the “if” statement, notice that locally a D -morphism f : S → Y can be 0 X extended to a morphism S → Y and these extensions form a torsor over Hom(f ∗ ΩY , I). Such a torsor is the same as an extension of f ∗ ΩY by I. Now f ∗ ΩY is locally projective and therefore projective (see 2.3.6), so Ext(f ∗ ΩY , I) = 0 and our torsor is trivial. 45 There seems to be no known example of a formally smooth algebra over a field k of characteristic 0 such that H 6=0 (L ΩA ) does not vanish. For char k 6= 0 such an example was communicated to us by O. Gabber.

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2.3.17. Coordinate systems. For a DX -scheme Y a coordinate system on Y is an ´etale morphism of DX -schemes ν: Y → Spec Sym· (DnX ), n ≥ 0. Any Y that admits locally a coordinate system is very smooth.46 The jet scheme JZ for Z/X smooth admits a local coordinate system (for any ´etale ϕ: Z → AnX the morphism Jϕ: JZ → JAnX = Spec Sym· (DnX ) is a coordinate system on JZ). This is not true (even at a generic point!) for arbitrary smooth DX -schemes. Here is the reason. Suppose that our Y = Spec A` is a smooth reduced DX scheme such that ΩA` ' A` [DX ]. Notice that any invertible element of A` [DX ] lies ∼ in A` ⊂ A` [DX ]. Denote by L the image of A` by any isomorhism A` [DX ] −→ ΩA` . This is a canonically defined line subbundle of ΩA . If Y admits locally a coordinate system, then L is an integrable subbundle; i.e., d(L) ⊂ L ∧ ΩA` ⊂ Ω2A` . 2.3.18. Example. Let us construct a smooth A` with non-integrable L. Take X = Spec k[t], B ` = k[t, u, v, u0 , v 0 , . . . , v 0−1 ] (here u0 := ∂t u, etc., so B ` is the localization of Sym· (D2X ) = k[t, u, v, u0 , . . . ] by v 0 ). Let I ⊂ B ` be the DX -ideal generated by ϕ = v + u0 v 0 ; set A` := B ` /I. Consider the morphism of B ` [DX ]modules χ: B ` [DX ] ⊕ B ` [DX ] → ΩB ` , (∂1 , ∂2 ) 7→ ∂1 dϕ + ∂2 ξ, where ξ := du + (u0 /v 0 )dv ∈ ΩB ` . Localizing B ` (and A` ) further by v 0 − u00 v 0 + u0 v 00 , we see that χ becomes an isomorphism. Now A` is smooth, and the map A` [DX ] → ΩA` , ∂ 7→ ∂ξA` , is an isomorphism. Here ξA` is the restriction of ξ to Spec A` ; this is a generator of L. One has ξA` ∧ dξA` = ξA` ∧ d(u0 /v 0 ) ∧ dv = 2 −ξA` ∧ d(v/v 0 ) ∧ dv = αξA` ∧ dv ∧ dv 0 , α 6= 0, where u and v are considered as ` elements of A . It is easy to check that dv = β∂t ξA` , β 6= 0, so ξA` ∧ dξA` 6= 0 and L is non-integrable. 2.3.19. Vector DX -schemes. By definition, these are vector space objects in the category Af f SchD (X) of DX -schemes affine over X. For a left D-module N ` and R` ∈ ComuD (X) morphisms of DX -algebras Sym(N ` ) → R` are the same as morphisms of D-modules N ` → R` , so they form a vector space. Thus V(N ` ) := Spec Sym(N ` ) is a vector DX -scheme.47 It is easy to see that the functor (2.3.19.1)

V : M` (X)◦ → {vector DX -schemes}

is an equivalence of categories. The inverse functor assigns to a vector DX -scheme V the pull-back of ΩV by the zero section morphism X → V . Consider the category Φ = Φ(X) of all functors F : ComuD (X) → Sh(X);48 this is a tensor category in the obvious way. A vector DX -scheme V can be considered as an object of Φ: for an open j : U → X one has V (R` )(U ) := V |U (R` |U ) = V (j∗ j ∗ R` ). It is clear that the category of vector DX -schemes is a full subcategory of Φ, so we have a fully faithful embedding V : M` (X)◦ ,→ Φ. On the other hand, ˜ : M(X) → Φ. according to 2.2.8, we have a fully faithful embedding h Lemma. The intersection of these two subcategories is the category of vector DX -bundles. 46 Use

the remark that opens 2.3.14. a closed point x ∈ X the fiber V(N ` )x is the profinite-dimensional vector space Nx`∗ dual to Nx` . 48 Recall that Sh(X) is the category of sheaves of k-vector spaces on the ´ etale topology of X. 47 For

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Proof. Let M be a vector DX -bundle; M ◦ := HomDX (M ` , DX ) is its dual (see ∼ ◦ ˜ 2.2.16). One has h(M )(R` ) = h(R` ⊗ M ◦ ) = M ◦ ⊗ R` −→ HomDX (M ` , R` ) = DX

◦ ˜ V(M ` ); hence h(M ) = V(M ` ). It remains to show that if for some N ` ∈ M(X)` there exists M ∈ M(X) such ˜ that h(M ) = V(N ` ), then N is a vector DX -bundle. We can assume that X is affine. We need some notation. For L` ∈ M` (X) the multiplicative group k ∗ acts on ` L by homotheties, so for any F ∈ Φ it acts on the vector space F (SymL` ). Denote by F 1 (L` ) ⊂ F (SymL` ) the subsheaf on which k ∗ acts by the standard character. Therefore F yields a functor F 1 : M` (X) → Sh(X). ˜ )1 (L` ) = M ⊗ L` = h(M ⊗ One has V(N ` )1 (L` ) = HomDX (N ` , L` ) and h(M DX

L` ). The latter functor commutes with direct limits and is right exact. If the functors are equal, then N ` is a finitely presented and projective D-module, i.e., a vector DX -bundle. We are done.  2.3.20. The calculus of variations. To end the section let us explain, as a mere illustration, the first notions of the calculus of variations in the present context. Suppose we have a DX -scheme Y = Spec R` such that ΘY is well defined (see Remark (ii) of 2.3.12), e.g., Y is a smooth DX -scheme. Any ν ∈ h(Ω1Y/X ) defines a morphism of R` [DX ]-modules iν : ΘY → R; its image is the Euler-Lagrange ideal Iν . One gets the Euler-Lagrange DX -scheme Yν := Spec R` /Iν` . One usually considers the situation when ν = df¯ for certain f ∈ R called the lagrangian; f¯ ∈ h(R) is called the action. Remark. A more natural object is the DG DX -algebra SymR` (Θ`Y [1]) with the differential dν which equals iν on the degree −1 component Θ`Y [1]. If ν is closed (in particular, if it comes from a lagrangian), then it is an odd coisson DG algebra (see Exercise in 1.4.18). Example. Suppose that Y = JZ for a smooth X-scheme Z. Let π : Y → Z be the canonical morphism. Then dπ : π ∗ Ω1Z/X → Ω1Y/X identifies Ω1Y/X with the left R` [DX ]-module induced from π ∗ Ω1Z/X . Thus h(Ω1Y/X ) = π ∗ Ω1Z/X ⊗ ωX = R ⊗ Ω1Z/X . The de Rham differential h(d) : h(R) → h(Ω1Y/X ) = R ⊗ Ω1Z/X is OZ

OZ

called the variational derivative. By (2.2.16.1) there is a canonical isomorphism (2.3.20.1)

∼

(π ∗ ΘZ/X )D −→ ΘY

of right R` [DX ]-modules.49 Thus for a local coordinate system xi on Z the EulerLagrange ideal is generated, as a DX -ideal, by the df¯-images of ∂xi ∈ ΘZ/X ⊂ ΘY in R. We leave it to the reader to check that these are indeed the usual EulerLagrange equations. Remark. Here is another way to define (2.3.20.1). Consider ΘZ/X as a Lie algebra acting on Z/X. It acts on Y = JZ by transport of structure. Since ΘZ/X is a Lie OZ -algebroid, this action permits us to define a Lie OY -algebroid structure on 49 Explicitly,

it is the composition of the identifications ΘY := HomR` [DX ] (Ω1Y/X , R` [DX ]) =

HomR` (π ∗ Ω1Z/X , R` [DX ]) = HomR` (π ∗ Ω1Z/X , R` ) ⊗ DX = (π ∗ ΘZ/X )D .

2.4. THE SPACES OF HORIZONTAL SECTIONS

89

π ∗ ΘZ/X . Thus (π ∗ ΘZ/X )D is naturally a Lie∗ algebra acting on Y, and (2.3.20.1) is the action morphism. Suppose we have a Lie∗ algebra L acting on Y (see 1.4.9, 2.5.3, 2.5.6(c)). We say ˜ ˜ that our action is L-invariant if the h(L)-action on h(R) leaves f¯ invariant. In other ¯ words, the Lie algebra h(L) fixes f ∈ h(R), and this happens also after any base change, i.e., replacing L by L ⊗ F ` , R` by R` ⊗ F ` where F ` is any commutative DX -algebra. Since h is reliable (see 2.2.8) this happens if and only if the map h(R) → HomDX (L, R) defined by the action map (see (2.2.7.1)) L  R → ∆∗ R sends f¯ to 0. This is essentially Noether’s theorem. d

Remark. The above map is equal to the composition h(R` ) − → h(ΩY/X ) → Hom(ΘY , R) → Hom(L, R) where the last arrow comes from the action morphism L → ΘY . So the property of being L-invariant actually depends on df¯. 2.4. The spaces of horizontal sections In this section we assume that X 6= ∅ is connected of dimension n. We consider the space of global horizontal sections of a DX -scheme Spec, R` affine over X, which is the same as the space of global solutions of the corresponding system of differential equations. This is an ind-affine ind-scheme (see 2.4.1) which is a true affine scheme if X is proper (see 2.4.2); the corresponding algebra of functions is denoted by hRi. The non-proper situation can be reduced to the proper situation by considering all possible extensions of R` to a compactification (see 2.4.3). In 2.4.4–2.4.6 we give some explicit descriptions of hRi which will be of use in Chapter 4. In 2.4.7 we explain how one checks the smoothness of hRi assuming that R` is smooth. In the case of dim X = 1 we consider the ind-scheme of horizontal sections of Spec R` over the formal punctured disc at a point x ∈ X, describe its (topological) algebra of functions Rxas in 2.4.8–2.4.11, and relate the local and global pictures in 2.4.12. For the (more manageable) derived version of the functor R` 7→ hRi, see 4.6. 2.4.1. We will play with ind-schemes in the strict sense calling them simply ind-schemes. So an ind-scheme Y is a functor on the category of commutative algebras, R 7→ Y (R), which can be represented as the inductive limit of a directed family of quasi-compact schemes and closed embeddings. Some basic material about ind-schemes can be found in [BD] 7.11–7.12. When discussing a topological commutative algebra Q, we always tacitly assume that the topology is complete and has a base formed bySopen ideals Iα , so Q=← lim − Q/Iα . For more details, see 3.6.1. We write Spf Q := Spec Q/Iα ; this is an ind-affine ind-scheme. Denote by Comu(k) the category of commutative unital k-algebras. For any F ∈ Comu(k) the constant left D-module F ⊗ OX is a DX -algebra in the obvious way. This functor Comu(k) → ComuD (X) identifies Comu(k) with the 50 full subcategory Comu∇ It has a right adjoint D (X) of constant DX -algebras. ∇ ∇ ` ∇ functor Γ : ComuD (X) → Comu(k), Γ (R ) = Γ (X, R` ) = Hom(OX , R` ) = 0 HDR (X, R[−n]). Let us see if our embedding admits a left adjoint. So for R` ∈ ComuD (X) consider the functor F 7→ Hom(R` , F ⊗ OX ) on Comu(k). 50 I.e.,

those DX -algebras which are constant as a DX -module; see 2.1.12.

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Lemma. (i) This functor is representable by a topological commutative algebra hRi = hRi(X). For any x ∈ X(k), hRi is naturally a completion of Rx` . ˆ α i. (ii) R 7→ hRi is a tensor functor: one has h⊗Rα i = ⊗hR Proof. (i) Consider the set CR of DX -ideals I ` ⊂ R` such that R` /I ` is a constant D-algebra. Since any subquotient of a constant D-module is constant, we see that CR is a subfilter in the ordered set of all ideals in R` . For every I ` ∈ CR we have a commutative algebra hRiI := Γ∇ (X, R` /I ` ), S and for I 0 ⊃ I a surjective ` map hRiI  hRiI 0 . Therefore Hom(R , F ⊗ OX ) = Hom(R` /I ` , F ⊗ OX ) = S Hom(hRiI , F ) = Homcont (hRi, F ) where hRi is the CR -projective limit of the hRiI ’s. For any I ` ∈ CR one has hRiI = Rx` /Ix` , so we have a natural morphism ` Rx → hRi with dense image; q.e.d. (ii) Evident.  S By construction, the ind-affine ind-scheme Spf hRi := SpechRiI is the space of horizontal sections X → Spec R` . It is naturally an ind-subscheme of any fiber Spec Rx` . We denote it also by hSpec R` i. Exercise. If R is formally smooth and X is affine, then hRi is formally smooth. Consider the category hRimod of discrete hRi-modules. Since hRi ⊗ OX is a completion of R` , we have a fully faithful embedding (2.4.1.1)

hRimod → M(X, R` ),

V 7→ V ⊗ ωX .

Exercise. Show that its essential image is equal to the subcategory of R` [DX ]modules which are constant as DX -modules. Notice that hRi(X) is covariantly functorial with respect to the ´etale localization of X. Precisely, for any morphism U 0 → U in Xe´t , where U 0 , U are connected, there is a natural morphism hRi(U 0 )  hRi(U ). It is immediate that U 7→ SpfhRi(U ) is a sheaf of ind-schemes on Xe´t ; we denote it by SpfhRiX = hSpec R` iX . 2.4.2. Suppose that X is proper. Proposition. (i) hRi is a discrete algebra. (ii) If R is finitely generated as a DX -algebra, then hRi is finitely generated. Proof. (i) We want to show that R` admits the maximal constant D-algebra quotient R` /I0` . By 2.1.12, R` admits the maximal constant D-module quotient ` σR : R`  Rconst (see 2.1.12). Then I0` is the ideal generated by the kernel of σR . ∼ (ii) Clear, since the equivalence Comu∇ → Comu(k) identifies constant D (X) − DX -algebras which are finitely generated as DX -algebras with finitely generated algebras.  2.4.3. Let j : U ,→ X be a non-empty Zariski open subset. For R` ∈ ComuD (U ) the left D-module j· R` := H 0 j∗ R` is a DX -algebra in the obvious way. Let Ξc (R) denotes the set of DX -subalgebras Rξ` ⊂ j· R` such that j ∗ Rξ` = R` . The intersection of two subalgebras from Ξc (R) belongs to Ξc (R), so this is a subfilter in the ordered set of all DX -subalgebras of j· R` . We have a Ξc (R)-projective system of topological algebras51 hRξ i(X) connected by epimorphisms. The canonical maps 51 If

X is proper, then the hRξ i(X) are discrete; see 2.4.2.

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91

hRi(U ) → hRξ i(X) yield a morphism (the limit is taken over Ξc (R)) hRi(U ) → lim ←−hRξ i(X).

(2.4.3.1)

Proposition. This is an isomorphism. Proof. It suffices to show that any morphism52 ϕ: R` → F ⊗OU is the restriction to U of some morphism ϕX : Rξ` → F ⊗ OX , Rξ` ∈ Ξc (R). Note that ϕ defines a morphism of DX -algebras j· (ϕ): j· R` → j· (F ⊗OU ), and F ⊗OX is a DX -subalgebra  of j· (F ⊗ OU ). Set Rξ` := j· (ϕ)−1 (F ⊗ OX ), ϕX := j· (ϕ) ` . We are done. Rξ

Corollary. If R is a finitely generated DX -algebra, then SpfhRiX is a sheaf of ind-schemes of ind-finite type.  2.4.4. In 2.4.4–2.4.7 we assume that X is proper. Here are some “explicit” descriptions of hRi. ` We saw in 2.1.12 that Rconst = H0DR (X, R) ⊗ OX , so hRi is a quotient of H0DR (X, R). Let I` ⊂ R` ⊗ R` be the kernel of the multiplication map R` ⊗ R` → R` . ⊗2 Consider the morphism σR : R` ⊗ R` → H0DR (X, R)⊗2 ⊗ OX . Restricting it to I, ∼ passing to the de Rham cohomology, and using tr : H0DR (X, ωX ) −→ k, we get a morphism (2.4.4.1)

ν : H0DR (X, I) → H0DR (X, R)⊗2 .

The projection H0DR (X, R)  hRi together with the product on hRi identifies hRi with a quotient of H0DR (X, R)⊗2 . Lemma. One has hRi = Coker ν. Proof. Consider a commutative diagram R` x  

hRi ⊗ OX x  

−−−−→

R` ⊗ R` −−−−→ H0DR (X, R)⊗2 ⊗ OX According to the proof of 2.4.2(i), hRi ⊗ OX is the quotient of R` modulo the ideal R` · Ker σR , or, equivalently, the quotient of R` ⊗ R` modulo I + Ker σR ⊗ R` + R` ⊗ Ker σR . This is Cokerν ⊗ OX ; q.e.d.  2.4.5. Here is another description of hRi. Let ∆: X ,→ X × X be the diagonal embedding, j: U := X × X r ∆(X) ,→ X × X its complement. Consider the morphism Rj∗ j ∗ R  R → ∆∗ R[−n + 1] in DM(X × X) defined as the composi∼ tion of the canonical “residue” morphism Rj∗ j ∗ (R  R) → ∆∗ ∆! (R  R)[1] −→ L

L

∆∗ (R ⊗ ! R)[−n + 1] and the product R ⊗ ! R → R ⊗! R → R. Passing to de Rham cohomology,53 we get a morphism of vector spaces (2.4.5.1) 52 Here 53 One

0 µ : HDR (U, R  R[2n − 1]) → H0DR (X, R).

F is a commutative algebra. · · · · has HDR (X × X, j∗ j ∗ B) = HDR (U, B), HDR (X × X, ∆∗ C) = HDR (X, C).

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Lemma. One has hRi = Coker µ. Proof. Denote by C the cone of our morphism j∗ j ∗ R  R[−1] → ∆∗ R[−n]; this 2n is an object of DM(X × X).54 One has55 Coker µ = HDR (X × X, C). Consider the obvious morphism R  R → C. Denote by D its cone, so we have the exact triangle R  R → C → D. The complex D is supported on the diagonal; 2n−1 its top non-zero cohomology is H n−1 D = ∆∗ I. Therefore HDR (X × X, D) = n HDR (X, I) and all the higher de Rham cohomologies vanish. So the long exact 2n sequence of de Rham cohomology identifies HDR (X × X, C) with the cokernel of 2n−1 2n the arrow HDR (X × X, D) → HDR (X × X, R  R) which may be rewritten as n n ν 0 : HDR (X, I) → HDR (X, R)⊗2 . So we see that Coker µ = Coker ν 0 . We leave it to the reader to check that our 0 ν equals the map ν from (2.4.4.1). Now use 2.4.4.  2.4.6. Now let {xs } ⊂ X, s ∈ S, be a finite non-empty subset. Q ` We have the n−1 canonical residue map ResS = (Resxs ): HDR (X r S, R) → Rxs (see 2.1.12); denote its image by VS . Let ⊗Rx` s  hRi be the ring morphism that combines the surjections Rx` s  hRi, s ∈ S (see 2.4.1 and 2.4.2). As follows from the definition Q of hRi (and 2.1.12), its kernel contains f (VS ) where the map f : Rx` s → ⊗Rx` s Q ` sends (as ) ∈ Rxs to Σas ⊗ 1⊗Sr{s} . Hence it contains the ideal IS generated by f (VS ). We have defined αS : (⊗Rx` s )/IS  hRi. ∼

Lemma. One has αS : (⊗Rx` s )/IS −→ hRi. ` )xs = H0DR (X, R). Proof. Consider the natural projections πxs : Rx` s  (Rconst Q ` DR According to 2.1.12, the kernel of πS = Σπxs : Rxs  H0 (X, R) equals VS . (a) Assume that |S| = 1, so we have a single point xs = x ∈ X. As follows from the proof of 2.4.2(i), hRi ⊗ OX is the quotient of R` modulo the ideal generated by ` the kernel of the projection π : R`  Rconst . Therefore, passing to the x-fibers, we ` see that hRi is the quotient of Rx modulo the ideal Ix generated by Ker πx = Vx , and we are done. (b) Assume that |S| ≥ 2. Take any 0 ∈ S. Then Rx` 0 ∩ VS = Vx0 (indeed, both subspaces are equal to the kernel of πx0 ). Therefore the morphism Rx` 0 → ⊗ Rx` s Q ` S yields a morphism ζ : Rx` 0 /Ix0 → (⊗ Rx` s )/IS . Since Rx` 0 + VS = Rxs (which S

follows from the surjectivity of πx0 ), we see that ζ is surjective. The composition of ζ with αS is the map αx0 : Rx` 0 /Ix0 → hRi for the 1-point set {x0 } which we know to be an isomorphism by (a). Thus αS is an isomorphism.  Remark. One may also deduce 2.4.6 in the case |S| = 1 (i.e., (a) above) from 2.4.5. Namely, set Ux := X r {x}; denote by ix , ˜ix the embeddings {x} ,→ X, X = {x} × X ,→ X × X. Then ˜i!x (R  R) = (i!s R) ⊗ R, so we have a commutative diagram µ

0 HDR (U, R  R[2n − 1]) −−−−→ H0DR (X, R) x x    

(2.4.6.1)

µx

0 Rx` ⊗ HDR (Ux , R[n − 1]) −−−−→ 54 In

Rx`

the most important case of n = 1 our C is actually a D-module, not merely a complex. 2n (U, B) = 0 for every B ∈ M(U ). U is non-compact, HDR

55 Since

2.4. THE SPACES OF HORIZONTAL SECTIONS

93

the vertical arrows come from canonical morphisms ˜ix∗˜i!x (RR) → RR, ix∗ i!x R → R, and µx sends a ⊗ b to a Resx b. The standard long exact sequences show that the vertical arrows are surjective; the kernel of the right one is Im (Resx ). Since Resx (b) = µx (1⊗b), one has Im (Resx ) ⊂ Im(µx ). Therefore, Coker µ = Rx` /Im(µx ) ∼ = Rx` /Ix , so, by 2.4.5, we have Rx` /Ix −→ hRi. One checks immediately that this is the morphism of 2.4.6. 2.4.7. For M ∈ M(X, R` ) the quotient MhRi := M/I0` M is an hRi ⊗ DX module; therefore H0DR (X, MhRi ) is an hRi-module. It follows from 2.1.12 that the functor M 7→ H0DR (X, MhRi ) is left adjoint to the above embedding. ∼

Proposition. (i) One has a canonical isomorphism ΩhRi −→ H0DR (X, (ΩR )hRi ). (ii) Suppose R` is smooth, H1DR (X, (ΩR )hRi ) = 0, and H0DR (X, (ΩR )hRi ) is a projective hRi-module. Then hRi is smooth. The same is true for “smooth” replaced by “formally smooth”. Proof. (i) Follows from the adjunction. (ii) By 2.4.2 it suffices to consider formal smoothness. We want to show that every extension P of hRi by an ideal N of square 0 (which is an hRi-module) ˜ ` be the pull-back of P ⊗ OX by the morphism R` → hRi ⊗ OX ; splits. Let R this is a DX -algebra extension of R` by the ideal N ⊗ OX of square 0. By adjunction, it suffices to show that this extension splits. It happens locally on X since R` is formally smooth. The obstruction to the existence of global splitting lies in Ext1R` [DX ] (ΩR , N ⊗ ωX ) = Ext1hRi⊗DX ((ΩR )hRi , N ⊗ ωX ) (the latter equality comes since ΩR is a locally projective R` [DX ]-module). By duality, it equals HomhRi (RΓDR (X, ΩR )hRi [n − 1], N ) which vanishes by our conditions.  Remark. Suppose dim X = 1. For smooth R the conditions of (ii) amount to the smoothness of hRi = H0ch (X, R) and the vanishing of H1ch (X, R) (see 4.6). 2.4.8. For the rest of the section, X is a curve. Let x ∈ X be a (closed) point, jx : Ux ,→ X the complement to x. For R` ∈ ComuD (Ux ) consider the as topology Ξas x = Ξx (jx∗ R) on jx∗ R at x (see 2.1.13 for terminology) formed by all DX -subalgebras Rξ` such that jx∗ Rξ = R. We have (2.4.8.1)

` Rξx = i!x ((jx∗ R)/Rξ ) = h((jx∗ R)/Rξ )x .

` We see that for Rξ` 0 ⊂ Rξ` the morphism of fibers Rξ` 0 x → Rξx is surjective. Denote as as ` 56 . As a plain topoby Rx the Ξx -projective limit of the commutative algebras Rξx logical vector space it is equal to the Ξas -completion of the vector space h(jx∗ R)x x (see 2.1.13).

Remark. The functor R` 7→ Rxas is not compatible with the localization of R` , so it cannot be extended to non-affine DUx -schemes. Let Q be any topological commutative algebra. Following [BD] 7.11.1, we call an ideal I ⊂ Q (and the quotient Q/I) reasonable if I is open and for every open ideal J ⊂ I the ideal I/J ⊂ Q/J is finitely generated. We say that Q (and the ind-scheme Spf Q) is reasonable if the reasonable ideals form a base of the topology. 56 The

in 3.6.

superscript “as” means “associated” or “associative”; this notation will be clearified

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GEOMETRY OF D-SCHEMES

Example. The topological algebras k[t0 , t1 , · · · ]] := lim ←− k[t0 , t1 , · · · , tn ] and k[· · · , t−1 , t0 , t1 , · · · ]] := ← lim k[· · · , t , t , t , · · · , t ] are reasonable, while the alge−1 0 1