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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 267 268 269 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES: 373
Smoothness, Regularity and Complete Intersection JAVIER MAJADAS ANTONIO G. RODICIO Universidad de Santiago de Compostela, Spain
c a m br i d g e u n ivers i ty p r e s s Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521125727 © J. Majadas and A. G. Rodicio 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-12572-7 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Introduction 1
2
page 1
Definition and first properties of (co-)homology modules 1.1 First definition 1.2 Differential graded algebras 1.3 Second definition 1.4 Main properties
4 4 5 11 17
Formally smooth homomorphisms 2.1 Infinitesimal extensions 2.2 Formally smooth algebras 2.3 Jacobian criteria 2.4 Field extensions 2.5 Geometric regularity 2.6 Formally smooth local homomorphisms of noetherian rings 2.7 Appendix: The Mac Lane separability criterion
22 23 26 29 34 39
3
Structure of complete noetherian local rings 3.1 Cohen rings 3.2 Cohen’s structure theorems
47 47 52
4
Complete intersections 4.1 Minimal DG resolutions 4.2 The main lemma 4.3 Complete intersections 4.4 Appendix: Kunz’s theorem on regular local rings in characteristic p
55 56 60 62
v
43 46
64
vi 5
6
Contents Regular homomorphisms: Popescu’s theorem 5.1 The Jacobian ideal 5.2 The main lemmas 5.3 Statement of the theorem 5.4 The separable case 5.5 Positive characteristic 5.6 The module of differentials of a regular homomorphism
Localization of formal smoothness 6.1 Preliminary reductions 6.2 Some results on vanishing of homology 6.3 Noetherian property of the relative Frobenius 6.4 End of the proof of localization of formal smoothness 6.5 Appendix: Power series Appendix: Some exact sequences Bibliography Index
67 68 74 83 87 91 108 109 109 115 117 120 121 126 130 134
Introduction
This book proves a number of important theorems that are commonly given in advanced books on Commutative Algebra without proof, owing to the difficulty of the existing proofs. In short, we give homological proofs of these results, but instead of the original ones involving simplicial methods, we modify these to use only lower dimensional homology modules, that we can introduce in an ad hoc way, thus avoiding simplicial theory. This allows us to give complete and comparatively short proofs of the important results we state below. We hope these notes can serve as a complement to the existing literature. These are some of the main results we prove in this book: Theorem (I) Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. Then the following conditions are equivalent: a) B is a formally smooth A-algebra for the n-adic topology b) B is a flat A-module and the K-algebra B ⊗A K is geometrically regular. This result is due to Grothendieck [EGA 0IV , (19.7.1)]. His proof is long, though it provides a lot of additional information. He uses this result in proving Cohen’s theorems on the structure of complete noetherian local rings. An alternative proof of (I) was given by M. Andr´e [An1], based on Andr´e–Quillen homology theory; it thus uses simplicial methods, that are not necessarily familiar to all commutative algebraists. A third proof was given by N. Radu [Ra2], making use of Cohen’s theorems on complete noetherian local rings. Theorem (II) Let A be a complete intersection ring and p a prime ideal of A. Then the localization Ap is a complete intersection. 1
2
Introduction
This result is due to L.L. Avramov [Av1]. Its proof uses differential graded algebras as well as Andr´e–Quillen homology modules in dimensions 3 and 4, the vanishing of which characterizes complete intersections. Our proofs of these two results follow Andr´e and Avramov’s arguments [An1], [Av1, Av2] respectively, but we make appropriate changes so as to involve Andr´e–Quillen homology modules only in dimensions ≤ 2: up to dimension 2 these homology modules are easy to construct following Lichtenbaum and Schlessinger [LS]. Theorem (III) A regular homomorphism is a direct limit of smooth homomorphisms of finite type (D. Popescu [Po1]–[Po3]). We give here Popescu’s proof [Po1]–[Po3], [Sw]. An alternative proof is due to Spivakovsky [Sp]. Theorem (IV) The module of differentials of a regular homomorphism is flat. This result follows immediately from (III). However, for many years up to the appearance of Popescu’s result, the only known proof was that by Andr´e, making essential use of Andr´e–Quillen homology modules in all dimensions. Theorem (V) If f : (A, m, K) → (B, n, L) is a local formally smooth homomorphism of noetherian local rings and A is quasiexcellent, then f is regular. This result is due to Andr´e [An2]; we give here a proof more in the style of the methods of this book, mainly following some papers of Andr´e, A. Brezuleanu and N. Radu. We now describe the contents of this book in brief. Chapter 1 introduces homology modules in dimensions 0, 1 and 2. First, in Section 1.1 we give the definition of Lichtenbaum and Schlessinger [LS], which is very concise, at least if we omit the proof that it is well defined. The reader willing to take this on trust and to accept its properties (1.4) can omit Sections (1.2–1.3) on first reading; there, instead of following [LS], we construct the homology modules using differential graded resolutions. This makes the definition somewhat longer, but simplifies the proof of some properties. Moreover, differential graded resolutions are used in an essential way in Chapter 4.
Introduction
3
Chapter 2 studies formally smooth homomorphisms, and in particular proves Theorem (I). We follow mainly [An1], making appropriate changes to avoid using homology modules in dimensions > 2. This part was already written (in Spanish) in 1988. Chapter 3 uses the results of Chapter 2 to deduce Cohen’s theorems on complete noetherian local rings. We follow mainly [EGA 0IV ] and Bourbaki [Bo, Chapter 9]. In Chapter 4, we prove Theorem (II). After giving Gulliksen’s result [GL] on the existence of minimal differential graded resolutions, we follow Avramov [Av1] and [Av2], taking care to avoid homology modules in dimension 3 and 4. As a by-product, we also give a proof of Kunz’s result characterizing regular local rings in positive characteristic in terms of the Frobenius homomorphism. Finally, Chapters 5 and 6 study regular homomorphisms, giving in particular proofs of Theorems (III), (IV) and (V). The prerequisites for reading this book are a basic course in commutative algebra (Matsumura [Mt, Chapters 1–9] should be more than sufficient) and the first definitions in homological algebra. Though in places we use certain exact sequences deduced from spectral sequences, we give direct proofs of these in the Appendix, thus avoiding the use of spectral sequences. Finally, we make the obvious remark that this book is not in any way intended as a substitute for Andr´e’s simplicial homological methods [An1] or the proofs given in [EGA 0IV ], since either of these treatments is more complete than ours. Rather, we hope that our book can serve as an introduction and motivation to study these sources. We would also like to mention that we have profited from reading the interesting book by Brezuleanu, Dumitrescu and Radu [BDR] on topics similar to ours, although they do not use homological methods. We are grateful to T. S´ anchet Giralda for interesting suggestions and to the editor for contributing to improve the presentation of these notes. Conventions. All rings are commutative, except that graded rings are sometimes (strictly) anticommutative; the context should make it clear in each case which is intended.
1 Definition and first properties of (co-)homology modules
In this chapter we define the Lichtenbaum–Schlessinger (co-)homology modules Hn (A, B, M ) and H n (A, B, M ), for n = 0, 1, 2, associated to a (commutative) algebra A → B and a B-module M , and we prove their main properties [LS]. In Section 1.1 we give a simple definition of Hn (A, B, M ) and H n (A, B, M ), but without justifying that they are in fact well defined. To justify this definition, in Section 1.3 we give another (now complete) definition, and prove that it agrees with that of 1.1. We use differential graded algebras, introduced in Section 1.2. In [LS] they are not used. However we prefer this (equivalent) approach, since we also use differential graded algebras later in studying complete intersections. More precisely, we use Gulliksen’s Theorem 4.1.7 on the existence of minimal differential graded algebra resolutions in order to prove Avramov’s Lemma 4.2.1. Section 1.4 establishes the main properties of these homology modules. Note that these (co-)homology modules (defined only for n = 0, 1, 2) agree with those defined by Andr´e and Quillen using simplicial methods [An1, 15.12, 15.13].
1.1 First definition Definition 1.1.1 Let A be a ring and B an A-algebra. Let e0 : R → B be a surjective homomorphism of A-algebras, where R is a polynomial A-algebra. Let I = ker e0 and j
0 → U → F −−→ I → 0
2 an exact sequence of R-modules with F free. Let φ : F → F be the R2 module homomorphism defined by φ(x ∧ y) = j(x)y − j(y)x, where F 4
1.2 Differential graded algebras
5
is the second exterior power of the R-module F . Let U0 = im(φ) ⊂ U . We have IU ⊂ U0 , and so U/U0 is a B-module. We have a complex of B-modules U/U0 → F/U0 ⊗R B = F/IF → ΩR|A ⊗R B (concentrated in degrees 2, 1 and 0), where the first homomorphism is induced by the injection U → F , and the second is the composite F/IF → I/I 2 → ΩR|A ⊗R B, where the first map is induced by j, and the second by the canonical derivation d : R → ΩR|A (here ΩR|A is the module of K¨ ahler differentials). We denote any such complex by LB|A , and define for a B-module M Hn (A, B, M ) = Hn (LB|A ⊗B M ) n
n
for n = 0, 1, 2,
H (A, B, M ) = H (HomB (LB|A , M ))
for n = 0, 1, 2.
In Section 1.3 we show that this definition does not depend on the choices of R and F .
1.2 Differential graded algebras Definition 1.2.1 Let A be a ring. A differential graded A-algebra (R, d) (DG A-algebra in what follows) is an (associative) graded A-algebra with unit R = n≥0 Rn , strictly anticommutative, i.e., satisfying xy = (−1)pq yx for x ∈ Rp , y ∈ Rq
and
x2 = 0 for x ∈ R2n+1 ,
and having a differential d = (dn : Rn → Rn−1 ) of degree −1; that is, d is R0 -linear, d2 = 0 and d(xy) = d(x)y + (−1)p xd(y) for x ∈ Rp , y ∈ R. Clearly, (R, d) is a DG R0 -algebra. We can view any A-algebra B as a DG A-algebra concentrated in degree 0. A homomorphism f : (R, dR ) → (S, dS ) of DG A-algebras is an Aalgebra homomorphism that preserves degrees (f (Rn ) ⊂ Sn ) such that dS f = f dR . If (R, dR ), (S, dS ) are DG A-algebras, we define their tensor product R ⊗A S to be the DG A-algebra having a) underlying A-module the usual tensor product R⊗A S of modules, with grading given by Rp ⊗A Sq R ⊗A S = n≥0
p+q=n
6
Definition and first properties of (co-)homology modules b) product induced by (x⊗y)(x ⊗y ) = (−1)pq (xx ⊗yy ) for y ∈ Sp , x ∈ Rq c) differential induced by d(x ⊗ y) = dR (x) ⊗ y + (−1)q x ⊗ dS (y) for x ∈ Rq , y ∈ S.
Let {(Ri , di )}i∈I be a family of DG A-algebras. For each finite subset J ⊂ I, we extend the above definition to A Ri ; for finite subsets i∈J J ⊂ J of I, we have a canonical homomorphism A Ri . A Ri → i∈J
i∈J
We thus have a direct system of homomorphisms of DG A-algebras. We say that the direct limit is the tensor product of the family of DG A algebras {(Ri , di )}i∈I . It is a DG A-algebra, that we denote by A Ri i∈I
(and is not to be confused with the tensor product of the underlying family of A-algebras Ri ). A DG ideal I of a DG A-algebra (R, d) is a homogeneous ideal of the graded A-algebra R that is stable under the differential, i.e., d(I) ⊂ I. Then R/I is canonically a DG A-algebra and the canonical map R → R/I is a homomorphism of DG A-algebras. An augmented DG A-algebra is a DG A-algebra together with a surjective (augmentation) homomorphism of DG A-algebras p : R → R , where R is a DG A-algebra concentrated in degree 0; its augmentation ideal is the DG ideal ker p of R. A DG subalgebra S of a DG A-algebra (R, d) is a graded A-subalgebra S of R such that d(S) ⊂ S. Let (R, d) be a DG A-algebra. Then Z(R) := ker d is a graded A-subalgebra of R with grading Z(R) = n≥0 Z(R) ∩ Rn , and B(R) := im(d) is a homogeneous ideal of Z(R). Therefore the homology of R H(R) = Z(R)/B(R) is a graded A-algebra. Example 1.2.2 Let R0 be an A-algebra and X a variable of degree n > 0. Let R = R0 X be the following graded A-algebra: a) If n is odd, R0 X is the exterior R0 -algebra on the variable X, i.e., R0 X = R0 1 ⊕ R0 X, concentrated in degrees 0 and n. b) If n is even, R0 X is the quotient of the polynomial R0 -algebra on variables X (1) , X (2) , . . . , by the ideal generated by the elements (i + j)! (i+j) X (i) X (j) − X for i, j ≥ 1. i!j!
1.2 Differential graded algebras
7
The grading is defined by deg X (m) = nm for m > 0. We set X (0) = 1, X = X (1) and say that X (i) is the ith divided power of X. Observe that i!X (i) = X i . Now let R be a DG A-algebra, x a homogeneous cycle of R of degree n − 1 ≥ 0, i.e., x ∈ Zn−1 (R). Let X be a variable of degree n, and R X = R ⊗R0 R0 X . We define a differential in R X as the unique differential d for which R → R X is a DG A-algebra homomorphism with d(X) = x for n odd, respectively d(X (m) ) = xX (m−1) for n even. We denote this DG A-algebra by R X; dX = x . Note that an augmentation p : R → R satisfying p(x) = 0 extends in a unique way to an augmentation p : R X; dX = x → R by setting p(X) = 0. Lemma 1.2.3 Let R be a DG A-algebra and c ∈ Hn−1 (R) for some n ≥ 1. Let x ∈ Zn−1 (R) be a cycle whose homology class is c. Set S = R X; dX = x and let f : R → S be the canonical homomorphism. Then: a) f induces isomorphisms Hq (R) = Hq (S) for all q < n − 1; b) f induces an isomorphism Hn−1 (R)/ c R0 = Hn−1 (S). Proof a) is clear, since Rq = Sq for q < n, b) Zn−1 (R) = Zn−1 (S) and Bn−1 (R) + xR0 = Bn−1 (S).
Definition 1.2.4 If {Xi }i∈I is a family of variables of degree > 0, we define R0 {Xi }i∈I := R0 R0 Xi as the tensor product of the DG i∈I
R0 -algebras R0 Xi for i ∈ I (as in Definition 1.2.1). If R is a DG A-algebra, we say that a DG A-algebra S is free over R if the underlying graded A-algebra is of the form S = R ⊗R0 S0 {Xi }i∈I where S0 is a polynomial R0 -algebra and {Xi }i∈I a family of variables of degree > 0, and the differential of S extends that of R. (Caution: it is not necessarily a free object in the category of DG A-algebras.) If R is a DG A-algebra and {xi }i∈I a set of homogeneous cycles of R, we define R {Xi }i∈I ; dXi = xi to be the DG A-algebra R ⊗R0 ( R0 R0 Xi ; dXi = xi ), i∈I
which is free over R. Lemma 1.2.5 Let R be a DG A-algebra, n − 1 ≥ 0, {ci }i∈I a set
8
Definition and first properties of (co-)homology modules
of elements of Hn−1 (R) and {xi }i∈I a set of homogeneous cycles with classes {ci }i∈I . Set S = R {Xi }i∈I ; dXi = xi , and let f : R → S be the canonical homomorphism. Then: a) f induces isomorphisms Hq (R) = Hq (S) for all q < n − 1; b) f induces an isomorphism Hn−1 (R)/ {ci }i∈I R0 = Hn−1 (S). Proof Similar to the proof of Lemma 1.2.3, bearing in mind that direct limits are exact. Theorem 1.2.6 Let p : R → R be an augmented DG A-algebra. Then there exists an augmented DG A-algebra pS : S → R , free over R with S0 = R0 , such that the augmentation pS extends p and gives an isomorphism in homology
R if n = 0, H(S) = H(R ) = 0 if n > 0. If R0 is a noetherian ring and Ri an R0 -module of finite type for all i, then we can choose S such that Si is an S0 -module of finite type for all i. Proof Let S 0 = R. Assume that we have constructed an augmented DG A-algebra S n−1 that is free over R, such that S0n−1 = R0 and the augmentation S n−1 → R induces isomorphisms Hq (S n−1 ) = Hq (R ) for q < n − 1. Let {ci }i∈I be a set of generators of the R0 -module ker Hn−1 (S n−1 ) → Hn−1 (R ) (equal to Hn−1 (S n−1 ) for n > 1), and {xi }i∈I a set of homogeneous cycles with classes {ci }i∈I . Let S n = S n−1 {Xi }i∈I ; dXi = xi . Then S n is a DG A-algebra free over R with S0n = R0 and such that the augmentation pS n : S n → R extending pS n−1 defined by pS n (Xi ) = 0 induces isomorphisms Hq (S n ) = Hq (R ) for q < n (Lemma 1.2.5). We define S := lim S n . −→ If R0 is a noetherian ring and Ri an R0 -module of finite type for all i, then by induction we can choose S n with Sin an S0n = R0 -module of finite type for all i, since if Sin−1 is an S0n−1 -module of finite type for all i, then Hi (S n−1 ) is an S0n−1 -module of finite type for all i. Definition 1.2.7 Let A → B be a ring homomorphism. Let R be a DG A-algebra that is free over A with a surjective homomorphism of DG
1.2 Differential graded algebras
9
A-algebras R → B inducing an isomorphism in homology. Then we say that R is a free DG resolution of the A-algebra B. Corollary 1.2.8 Let A → B be a ring homomorphism. Then a free DG resolution R of the A-algebra B exists. If A is noetherian and B an Aalgebra of finite type, then we can choose R such that R0 is a polynomial A-algebra of finite type and Ri an R0 -module of finite type for all i. Proof Let R0 be a polynomial A-algebra such that there exists a surjective homomorphism of A-algebras R0 → B. (If A is noetherian and B an A-algebra of finite type, then we can choose R0 a polynomial Aalgebra of finite type.) Now apply Theorem 1.2.6 to R0 → B. Definition 1.2.9 Let R be a DG A-algebra that is free over R0 , i.e., R = R0 {Xi }i∈I . For n ≥ 0, we define the n-skeleton of R to be the DG R0 -subalgebra generated by the variables Xi of degree ≤ n and their divided powers (for variables of even degree > 0). We denote it by R(n). Thus R(0) = R0 , and if A → B is a surjective ring homomorphism with kernel I and R a free DG resolution of the A-algebra B with R0 = A, then R(1) is the Koszul complex associated to a set of generators of I. Lemma 1.2.10 Let A be a ring and B an A-algebra. Let A → S R → B be a commutative diagram of DG A-algebra homomorphisms, where S is a free DG resolution of the S0 -algebra B and R is a DG A-algebra that is free over A. Then there exists a DG A-algebra homomorphism R → S that makes the whole diagram commute. Proof Let R(n) be the n-skeleton of R. Assume by induction that we have defined a homomorphism of DG A-algebras R(n − 1) → S so that the associated diagram commutes. We extend it to a DG A-algebra homomorphism R(n) → S keeping the commutativity of the diagram. a) If n = 0, R(0) = R0 and R0 → S0 exists because R0 is a polynomial A-algebra.
10
Definition and first properties of (co-)homology modules b) If n is odd, let R(n) = R(n−1) {Ti }i∈I . We have a commutative diagram R(n − 1)n−2 R(n − 1)n ⊕ i∈I R0 Ti R(n − 1)n−1 R(n)n −−−−→ R(n)n−1 −−−−→ R(n)n−2 R(n − 1)n @ R @ Sn −−−−→ Sn−2 −−−−→ Sn−1 and therefore a homomorphism R(n)n → ker(Sn−1 → Sn−2 ) = im(Sn → Sn−1 ), and so there exist an R0 -module homomorphism R(n)n → Sn extending R(n − 1)n → Sn . By multiplicativity using the map R(n)n → Sn , we extend R(n − 1) → S to a homomorphism of DG A-algebras R(n) → S. c) For even n ≥ 2, suppose that R(n) = R(n − 1) {Xi }i∈I . As above, we define R(n)n → Sn and then extend it to R(n) → S by multiplicativity using divided power rules based on the binomial and multinomial theorems. In more detail, suppose the map R(n)n → Sn is defined by Xi →
v
(rt,1 )
at Y1
· · · Ym(rt,m ) ∈ Sn ,
t=1
where the at are coefficients in S0 , the Yi are variables with (r ) deg Yi > 0, and the divided powers Yj t,j have integer exponents rt,j ≥ 0. (Of course, for deg Yj odd and r > 1, we understand (r) (l) Yj = 0.) Then for l > 0, the image of Xi is determined by the familiar divided power rules† (α ) (α ) (a) (Y1 + · · · + Yv )(l) = Y1 1 · · · Yv v ; and α1 +···+αv =l α1 ,...,αv ≥0 (l)
(b) (Y1 Y2 )(l) = Y1l Y2
(if deg Y1 and deg Y2 ≥ 2 are even).
Thus R(n) → Sn is given by v (rt,1 ) αt (r )
) · · · (Ym t,m )αt (l) αt (Y1 Xi → , at αt ! t=1 α1 +···+αv =l α1 ,...,αv ≥0
† Both are justified by observing that the two sides agree on multiplying by l!.
1.3 Second definition
11
where the monomial (rt,1 ) αt
(Y1
)
(r
)
· · · (Ym t,m )αt αt !
equals • 1 if αt = 0; • 0 if αt ≥ 2 and rt,j = 0 for every j with deg Yj even positive; (r ) (r α ) (r ) (rt,j αt )! × (Y1 t,1 )αt · · · Yj t,j t · · · (Ym t,m )αt if αt = 1, or • αt !(r α t,j !) t if for some j deg Yj is even and positive and rt,j αt ≥ 1; note that the coefficient (p)
Using the formula Yi
(q)
Yi
(rt,j αt )! αt !(rt,j !)αt
=
is an integer.
(p+q!) (p+q) , p!q! Yi
(rt,i ) αt
we see that (Yi
)
=
(rt,i αt )! (rt,i αt ) , (rt,i !)αt Yi
and so this definition does not depend on the chosen j. A straightforward computation (easier if we multiply “formally” by (p) (q) (p+q) p!q!), shows that under this map, Xi Xi and (p+q)! have the p!q! Xi same image. Remarks i) The assumption that S is free over S0 is only used to avoid defining divided powers structure. ii) For the definition of Hn (A, B, M ), for n = 0, 1, 2, we use free DG resolutions only up to degree 3, and so we could have used symmetric powers resolutions instead of divided powers resolutions (since they agree in degrees ≤3). However, in Chapter 4 we use minimal resolutions and there we need divided powers.
1.3 Second definition Definition 1.3.1 Let A → B be a ring homomorphism. Let e : R → B be a free DG resolution of the A-algebra B. Let J = ker(R ⊗A B → B, x ⊗ b → e(x)b). Let J (2) be the graded R0 ⊗A B-submodule of R ⊗A B generated by the products of the elements of J and the divided powers X (m) , m > 1 of variables of J of even degree ≥ 2. Note that J (2) is a subcomplex of R0 ⊗A B-modules of J. We define the complex ΩR|A ⊗R B := J/J (2) , which is in fact a complex of B-modules. In degree 0 it is isomorphic to ΩR0 |A ⊗R0 B, where ΩR0 |A is the usual
12
Definition and first properties of (co-)homology modules
R0 -module of differentials of the A-algebra R0 . For, we have an exact sequence of R0 -modules defined by the multiplication of R0 (considering R0 ⊗A R0 as an R0 -module multiplying in the right factor) 0 → I → R0 ⊗A R0 → R0 → 0, which splits, and so applying − ⊗R0 B we obtain an exact sequence 0 → I ⊗R0 B → R0 ⊗A B → B → 0, showing that I ⊗R0 B = J0 . On the other hand, the exact sequence of R0 -modules 0 → I 2 → I → ΩR0 |A → 0 gives an exact sequence I 2 ⊗R0 B = (I ⊗R0 B)2 = J02 → I ⊗R0 B = J0 → ΩR0 |A ⊗R0 B → 0, (2)
and therefore J0 /J0 = J0 /J02 = ΩR0 |A ⊗R0 B. In degree 1, (J/J (2) )1 = J1 /J0 J1 = (R1 ⊗A B)/J0 (R1 ⊗A B) = (R1 ⊗A B) ⊗R0 ⊗A B B = R1 ⊗R0 B is the free B-module obtained by base extension of the free R0 -module R1 . Similarly, in degree 2, (J/J (2) )2 = J2 /(J0 J2 + J12 ) = (R2 /R12 ) ⊗R0 B. In general, for n > 0, (ΩR|A ⊗R B)n = (R(n)/R(n − 1))n ⊗R0 B. Definition 1.3.2 We say that an A-algebra P has property (L) if for any A-algebra Q, any Q-module M , any Q-module homomorphism u : M → Q such that u(x)y = u(y)x for all x, y ∈ M , and for any pair of A-algebra homomorphisms f, g : P → Q such that im(f − g) ⊂ im(u), there exists a biderivation λ : P → M such that uλ = f − g P λ f g u M −−→ Q. Here we say that λ is a biderivation to mean that λ is A-linear and λ(xy) = f (x)λ(y) + g(y)λ(x). Lemma 1.3.3 Let A be a ring, P an A-algebra. a) If P is a polynomial A-algebra, then P has property (L). b) If P has property (L) and S is a multiplicative subset of P , then S −1 P has property (L).
1.3 Second definition
13
Proof a) Let Q, M, u, f, g be as in (1.3.2). Let P = A[{Xi }i∈I ]. Since im(f − g) ⊂ im(u), there exist elements Yi in M such that u(Yi ) = f (Xi ) − g(Xi ). Define λ on monomials Xi1 · · · Xin by λ(Xi1 · · · Xin ) =
n
f (Xi1 · · · Xij−1 )Yij g(Xij+1 · · · Xin ).
j=1
To see that λ is well defined, we show that the right-hand side of this formula remains invariant under transpositions (ip , ip+1 ) of the indices: λ(Xi1 · · · Xip+1 Xip · · · Xin ) − λ(Xi1 · · · Xip Xip+1 · · · Xin ) = f (Xi1 · · · Xip−1 )Yip+1 g(Xip Xip+2 · · · Xin ) + f (Xi1 · · · Xip−1 Xip+1 )Yip g(Xip+2 · · · Xin ) − f (Xi1 · · · Xip−1 )Yip g(Xip+1 · · · Xin ) − f (Xi1 · · · Xip )Yip+1 g(Xip+2 · · · Xin ) = f (Xi1 · · · Xip−1 )g(Xip+2 · · · Xin )(u(x)y − u(y)x) = 0, where x = Yip+1 and y = Yip . Extend λ to P by linearity. It follows easily that λ is a biderivation. Since f − g is also a biderivation and f (Xi ) − g(Xi ) = u(Yi ) = u(λ(Xi )), we have uλ = f − g. b) Let Q be an A-algebra and M a Q-module and u : M → Q a Q-module homomorphism such that u(x)y = u(y)x for all x, y ∈ M ; suppose that f , g : S −1 P → Q are two A-algebra homomorphisms such that im(f − g ) ⊂ im(u). Let f, g : P → Q be the respective composites of f , g with the canonical map P → S −1 P . Since P has property (L), there exists a biderivation λ : P → M such that uλ = f − g. P
λ S −1 P f g u
M −−→
Q.
Let p/s ∈ S −1 P with p ∈ P and s ∈ S. Solving the equation λ(p) = λ(s · p/s) = f (s)λ (p/s) + g (p/s)λ(s) for λ gives λ (p/s) = (g (s)λ(p) − g (p)λ(s))/(f (s)g (s)). Using the relation u(x)y = u(y)x for x, y ∈ M , one can show by a
14
Definition and first properties of (co-)homology modules
tedious but straightforward calculation that this formula actually defines a biderivation λ : S −1 P → M extending λ. Since f −g is a biderivation extending f − g, it is clear that uλ = f − g . Lemma 1.3.4 Let A → B be a ring homomorphism and p : R → B and q : S → B two free DG resolutions of the A-algebra B. Let f, g : R → S be two homomorphisms of augmented DG A-algebras, i.e., p = qf , p = qg. Then there exist B-module homomorphisms αi : (ΩR|A ⊗R B)i → (ΩS|A ⊗S B)i+1
for i = 0, 1, 2
such that dS1 α0 = f0 − g0 , and dSi+1 αi + αi−1 dR i = fi − gi , for i = 1, 2, where dR , dS are the differentials of R and S respectively, and denotes the induced map in the following diagram dR
dR
dR
dS
dS
dS
3 2 1 (ΩR|A ⊗R B)2 −−→ (ΩR|A ⊗R B)1 −−→ (ΩR|A ⊗R B)0 (ΩR|A ⊗R B)3 −−→ α2 α1 α0 f3 g3 f2 g2 f1 g1 f0 g0 3 2 1 (ΩS|A ⊗S B)2 −−→ (ΩS|A ⊗S B)1 −−→ (ΩS|A ⊗S B)0 . (ΩS|A ⊗S B)3 −−→
Proof Let I = ker(R ⊗A B → B), J = ker(S ⊗A B → B), so that ΩR|A ⊗R B = I/I (2) , ΩS|A ⊗S B = J/J (2) . We have fi (Ii ) ⊂ Ji , gi (Ii ) ⊂ Ji , for all i. We begin by defining maps α i R3 g3 f3 S3
dR
3 −−→
α 2
dS 3
−−→
R2 g2 f2 S2 /S12
dR
2 −−→
α 1
dS 2
−−→
R1 g1 f1 S1 /dS2 (S12 )
dR
1 −−→
α 0
dS
1 −−→
R0 g0 f0 S0
and the αi are the maps induced by the α i . Note that the lower row is still exact. 0 : R0 → Definition of α 0 : by Lemma 1.3.3, there exists a biderivation α S1 /dS2 (S12 ) such that dS1 α 0 = f0 − g0 . The A-linear map α 0 induces a map R0 ⊗A B → S1 /dS2 (S12 ) ⊗A B → S1 /dS2 (S12 ) ⊗S0 B = S1 ⊗S0 B = (ΩS|A ⊗S B)1 , and this composite factors through a map α0 : (ΩR|A ⊗R B)0 → (ΩS|A ⊗S B)1 , since f0 (I0 ) ⊂ J0 , g0 (I0 ) ⊂ J0 , α 0 is a biderivation and then α 0 ⊗A B takes I02 into the image of J0 J1 . 2 Definition of α 1 : let R1 = i R0 Ti . For each i, let yi ∈ S2 /S1 be
1.3 Second definition
15
0 dR such that dS2 (yi ) = f1 (Ti ) − g1 (Ti ) − α 1 (Ti ); such a yi exists, since 0 dR dS1 (f1 (Ti ) − g1 (Ti ) − α 1 (Ti )) = 0 )(dR (Ti )) = 0. f0 dR 0 dR 1 (Ti ) − (f0 − g 1 1 (Ti ) − g Define α 1 (Ti ) = yi and extend it to R1 by R0 -linearity (regarding S2 as an R0 -module via f0 ). It is clear that α 1 induces α1 : (ΩR|A ⊗R B)1 → (ΩS|A ⊗S B)2 , since (f0 ⊗A B)(I0 ) ⊂ J0 . By A-linearity, to see that 1 dS2 α 1 + α 0 dR 1 = f1 − g it is enough to see that 1 (aTi ) + α 0 dR 1 (aTi ) dS2 α 1 (aTi ) = f1 (aTi ) − g with a ∈ R0 . But
1 (aTi ) + α 0 dR 1 (aTi ) dS2 α 1 (aTi ) − f1 (aTi ) − g = f0 (a)dS2 α 1 (Ti ) + f0 (a) α0 dR 0 (dR α0 (a) 1 (Ti ) + g 1 (Ti )) − f0 (a)f1 (Ti ) − g0 (a) g1 (Ti ) 1 (Ti ) − g1 (Ti ) = f0 (a) dS2 α 1 (Ti ) + α 0 dR (T ) − f i 1 R + g0 (d1 (Ti )) α0 (a) − f0 (a) − g0 (a))( g1 (Ti ) 1 (Ti ) − g1 (Ti ) = f0 (a) dS2 α 1 (Ti ) + α 0 dR (T ) − f i 1 + dS1 ( α0 (a)) g1 (Ti ) g1 (Ti )) α0 (a)) − dS1 ( = f0 (a) dS2 α 1 (Ti ) + α 0 dR 1 (Ti ) 1 (Ti ) − f1 (Ti ) − g + dS2 ( g1 (Ti ) α0 (a)),
α0 (a) have the obvious meaning. Now the two where the products g1 (Ti ) summands are zero, the first by definition of α 1 and the second because S 2 it is in d2 (S1 ). Definition of α 2 : let R2 = ( k R0 Uk ) ⊕ ( i=j R0 Ti Tj ). By the 1 , there exist zk ∈ S3 such that dS (zk ) = equation dS2 α 0 dR 1 + α 1 = f1 − g 3 R f2 (Uk ) − g2 (Uk ) − α 1 d (Uk ). Define α 2 (Uk ) = zk . Define α 2 (Ti Tj ) ∈ S3 2
such that its image in (ΩS|A ⊗S B)3 is 0, e.g., define α 2 (Ti Tj ) = 0 or α1 (Tj ) − f1 (Tj ) α1 (Ti ) (this second definition has the α 2 (Ti Tj ) = f1 (Ti ) following advantage, here unnecessary since we do not need α 2 , but only α2 : (dS3 α 2 + α 1 dR )(T T ) = 0 = f (T T ) − g (T T )). Extend α 2 to i j 2 i j 2 i j 2 R2 by R0 -linearity (again regarding S3 as an R0 -module via f0 ). As for
16
Definition and first properties of (co-)homology modules
α 1 , it is clear that α 2 induces α2 : (ΩR|A ⊗R B)2 → (ΩS|A ⊗S B)3 . The equation dS3 α2 + α1 dR 2 = f2 − g2 1 dR follows from dS3 α 2 (Uk ) = dS3 (zk ) = f2 (Uk ) − g2 (Uk ) − α 2 (Uk ), and S R from the R0 -linearity of d3 , d2 , α 2 , α 1 and of f2 − g2 . Remark 1.3.5 Again, the assumption that S is free is not necessary. Definition 1.3.6 Let A → B be a ring homomorphism and M a Bmodule. Let R be a free DG resolution of the A-algebra B. For n = 0, 1, 2, define the (co-)homology B-modules by Hn (A, B, M ) = Hn (ΩR|A ⊗R M ) := Hn ((ΩR|A ⊗R B) ⊗B M ) H n (A, B, M ) = H n (HomB (ΩR|A ⊗R B, M )). In view of Lemma 1.2.10 and Lemma 1.3.4, the definition does not depend on the choice of R, and is natural in A, B and M . Proposition 1.3.7 The (co-)homology modules of Definition 1.3.6 agree with those of Definition 1.1.1, which are therefore well defined. Proof Let A be a ring and B an A-algebra. Let e0 : R → B be a surjective homomorphism of A-algebras, where R is a polynomial Aalgebra. Let I = ker(e0 ) and j
→I→0 0→U →F − 2 F → F be as an exact sequence of R-modules with F free. Let φ : in Definition 1.1.1, and U0 = im(φ). Let S be a free DG resolution of the A-algebra B with S0 = R (see the proof of corollary (1.2.8)) and with S1 = F as S0 = R-module (see the proof of Theorem 1.2.6). We prove that there is an isomorphism between the complex U/U0 → F/IF → ΩR|A ⊗R B of Definition 1.1.1, and (ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 → (ΩS|A ⊗S B)1 → (ΩS|A ⊗S B)0 .
1.4 Main properties
17
Since Hn (ΩS|A ⊗S B) for n = 0, 1, 2, do not depend on the choice of S, this proves the proposition (note also that (ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 ⊗B M = (ΩS|A ⊗S M )2 /d3 (ΩS|A ⊗S M )3 , HomB ((ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 , M ) = ker(HomB ((ΩS|A ⊗S B)2 , M ) → HomB ((ΩS|A ⊗S B)3 , M )), for any Bmodule M). It is clear that (ΩS|A ⊗S B)0 = ΩR|A ⊗R B, (ΩS|A ⊗S B)1 = S1 ⊗S0 B = F ⊗R B = F/IF . To see that U/U0 = (ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 , first note that U/U0 is the first Koszul homology module of the ideal I of R, that is, U/U0 = H1 (S(1)), where S(n) is the n-skeleton of S. Associated to the exact sequence 0 → S(1) → S(2) → S(2)/S(1) → 0 we have an exact sequence β2
δ
α
2 1 H1 (S(1)) −→ H1 (S(2)) = 0 H2 (S(2)) −→ H2 (S(2)/S(1)) −→
and so U/U0 = H1 (S(1)) = H2 (S(2)/S(1))/ im(β2 ). We have isomorphisms Hn (S(n)/S(n−1)) = (S(n)/S(n−1))n ⊗S0 B = (ΩS|A ⊗S B)n for n > 1, and with this identification, the diagram δ
3 H3 (S(3)/S(2)) −→
(ΩS|A ⊗S B)3
β2
H2 (S(2)) −→ H2 (S(2)/S(1)) d
3 −→
(ΩS|A ⊗S B)2
is commutative. On the other hand, H2 (S(3)) = 0, so that α2 = 0 and then δ3 is surjective. Thus (ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 = H2 (S(2)/S(1))/ im(β2 ) = U/U0 . So we have vertical isomorphisms in the diagram U/U0
−−−−→
F/IF
−−−−→ ΩR|A ⊗R B
(ΩS|A ⊗S B)2 /d3 (ΩS|A ⊗S B)3 → (ΩS|A ⊗S B)1 → (ΩS|A ⊗S B)0 and it is an easy computation to see that the diagram commutes.
1.4 Main properties Proposition 1.4.1 Let A be a ring, B an A-algebra, M a B-module. We have
18
Definition and first properties of (co-)homology modules a) H0 (B, B, M ) = 0 = H 0 (B, B, M ). b) H0 (A, B, M ) = ΩB|A ⊗B M, H 0 (A, B, M ) = HomB (ΩB|A , M ) = DerA (B, M ) (A-derivations from B to M ). c) If B = A/I where I is an ideal of A, H1 (A, B, M ) = I/I 2 ⊗B M, H 1 (A, B, M ) = HomB (I/I 2 , M ). d) If B is a polynomial A-algebra, Hn (A, B, M ) = 0 = H n (A, B, M ) for n = 1, 2.
Proof All the properties follow easily from Definition 1.1.1.
Proposition 1.4.2 Let A be a ring, B an A-algebra and 0 → M → M → M → 0 an exact sequence of B-modules. Then there exist exact sequences H2 (A, B, M ) → H2 (A, B, M ) → H2 (A, B, M ) → H1 (A, B, M ) → H1 (A, B, M ) → H1 (A, B, M ) → H0 (A, B, M ) → H0 (A, B, M ) → H0 (A, B, M ) → 0 and 0 → H 0 (A, B, M ) → H 0 (A, B, M ) → H 0 (A, B, M ) → H 1 (A, B, M ) → H 1 (A, B, M ) → H 1 (A, B, M ) → H 2 (A, B, M ) → H 2 (A, B, M ) → H 2 (A, B, M ). Proposition 1.4.3 (Base change) Let A → B and A → C be ring homomorphisms such that TorA i (B, C) = 0 for i = 1, 2. Let M be a B ⊗A C-module. Then Hn (A, B, M ) = Hn (C, B ⊗A C, M ) H n (A, B, M ) = H n (C, B ⊗A C, M )
for n = 0, 1, 2.
Proof Let R be a free DG resolution of the A-algebra B. Since R is a projective resolution of the A-module B, Hi (R ⊗A C) = TorA i (B, C) = 0 for i = 1, 2, and H0 (R ⊗A C) = B ⊗A C. The DG C-algebra R ⊗A C is free over C. By the proof of Theorem 1.2.6, we can extend R ⊗A C to a free DG resolution S of the C-algebra B ⊗A C adjoining variables only in degrees ≥ 4; in particular Sj = Rj ⊗A C for 0 ≤ j ≤ 3, and so Hn (C, B ⊗A C, M ) = Hn (ΩS|C ⊗S M ) = Hn (ΩR⊗A C|C ⊗R⊗A C M ) = Hn (ΩR|A ⊗R M ) = Hn (A, B, M ),
1.4 Main properties
19
for n = 0, 1, 2, and similarly for cohomology.
Proposition 1.4.4 Let A be a noetherian ring, B an A-algebra of finite type. Then we can choose a free DG resolution R of B so that ΩR|A ⊗R B is a complex of B-modules of finite type, and so if M is a B-module of finite type, Hn (A, B, M ) and H n (A, B, M ) are B-modules of finite type.
Proof Corollary 1.2.8.
Proposition 1.4.5 (Universal coefficient exact sequences) Let A → B → C be ring homomorphisms, M a C-module. a) If N is a flat C-module, Hn (A, B, M ⊗C N ) = Hn (A, B, M ) ⊗C N, n = 0, 1, 2. b) If N is a flat C-module, A noetherian and B an A-algebra of finite type, H n (A, B, M ⊗C N ) = H n (A, B, M ) ⊗C N, n = 0, 1, 2. c) If M is an injective C-module, H n (A, B, M ) = HomC (Hn (A, B, C), M ), n = 0, 1, 2. d) There exist exact sequences TorC 2 (ΩB|A ⊗B C, M ) → H1 (A, B, C) ⊗C M → H1 (A, B, M ) → TorC 1 (ΩB|A ⊗B C, M ) → 0 0 → Ext1C (ΩB|A ⊗B C, M ) → H 1 (A, B, M ) → HomC (H1 (A, B, C), M ) → Ext2C (ΩB|A ⊗B C, M ). Proof Suppose N is a flat C-module, A noetherian and B an A-algebra of finite type. Choose a free DG resolution R of B so that ΩR|A ⊗R B is a complex of B-modules of finite type. Then HomB (ΩR|A ⊗R B, M ⊗C N ) = HomB (ΩR|A ⊗R B, M ) ⊗C N , and so H n (A, B, M ⊗C N ) = H n (HomB (ΩR|A ⊗R B, M ⊗C N )) = H n (HomB (ΩR|A ⊗R B, M )⊗C N ) = H n (A, B, M ) ⊗C N . This proves b); c) and a) are similar and d) follows from the universal coefficient spectral sequences 2 Ep,q = TorC p (Hq (ΩR|A ⊗R C), M ) ⇒ Hp+q (ΩR|A ⊗R M )
E2p,q = ExtpC (Hq (ΩR|A ⊗R C), M ) ⇒ H p+q (HomC (ΩR|A ⊗R C, M )), where R is a free DG resolution of the A-algebra B (in the appendix we give a direct proof without using spectral sequences).
20
Definition and first properties of (co-)homology modules
Proposition 1.4.6 (Jacobi–Zariski exact sequences) Let A → B → C be ring homomorphisms and M a C-module. There exist exact sequences H2 (A, B, M ) → H2 (A, C, M ) → H2 (B, C, M ) → H1 (A, B, M ) → H1 (A, C, M ) → H1 (B, C, M ) → H0 (A, B, M ) → H0 (A, C, M ) → H0 (B, C, M ) → 0 0 → H 0 (B, C, M ) → H 0 (A, C, M ) → H 0 (A, B, M ) → H 1 (B, C, M ) → H 1 (A, C, M ) → H 1 (A, B, M ) → H 2 (B, C, M ) → H 2 (A, C, M ) → H 2 (A, B, M ). Proof Let R be a free DG resolution of the A-algebra B and S a DG A-algebra free over R (and in particular over A) such that it is a resolution of C (Theorem 1.2.6). Then B ⊗R S is a DG B-algebra free over B. Consider the maps S → B ⊗R S → C. The first map induces an isomorphism in homology, since R → B induces an isomorphism in homology and S is free over R, and also the composite S → C induces an isomorphism in homology by the definition of S. So the second map B ⊗R S → C also induces an isomorphism in homology, i.e., B ⊗R S is a free DG resolution of the B-algebra C. Now the exact sequence of complexes of free C-modules 0 → (ΩR|A ⊗R B)B C → ΩS|A ⊗S C → ΩB⊗R S|B ⊗B⊗R S C → 0 induces the required exact sequences.
Proposition 1.4.7 (Localization) a) Let A be a ring, T a multiplicative subset of A, B a T −1 A-algebra and M a B-module. Then Hn (A, B, M ) = Hn (T −1 A, B, M ), n = 0, 1, 2 H n (A, B, M ) = H n (T −1 A, B, M ), n = 0, 1, 2. b) Let A be a ring, B an A-algebra, T a multiplicative subset of B and M a T −1 B-module. Then Hn (A, B, M ) = Hn (A, T −1 B, M ), n = 0, 1, 2 H n (A, B, M ) = H n (A, T −1 B, M ), n = 0, 1, 2.
1.4 Main properties
21
c) Let A be a ring, B an A-algebra, T a multiplicative subset of B and M a B-module. Then T −1 Hn (A, B, M ) = Hn (A, B, T −1 M ), n = 0, 1, 2 and if A is noetherian and B an A-algebra of finite type, then also T −1 H n (A, B, M ) = H n (A, B, T −1 M ), n = 0, 1, 2. Proof a) follows from base change (1.4.3) by the homomorphism A → T −1 A, and c) follows from 1.4.5, a) and b). For b), note first that the case n = 0 is clear by (1.4.1.b). Let A → B be a surjective homomorphism of A-algebras where A is a polynomial A-algebra. By (1.4.6) and (1.4.1) we have H2 (A, B, M ) = H2 (A , B, M ) and an exact sequence 0 → H1 (A, B, M ) → H1 (A , B, M ) → H0 (A, A , M ) and the same isomorphism and exact sequence with B replaced by T −1 B. So it is enough to prove the result for A instead of A. Let then U be the inverse image of T in A . By a) and (1.4.3) we have Hn (A , T −1 B, M ) = Hn (U −1 A , T −1 B, M ) = Hn (U −1 A , B ⊗A U −1 A, M ) = Hn (A , B, M ). Proposition 1.4.8 (Direct limits) Let {Ai → Bi }i∈I be a (filtered) direct system of ring homomorphisms and {Mi }i∈I a direct system of {Bi }i∈I -modules. Then Hn (lim Ai , lim Bi , lim Mi ) = lim Hn (Ai , Bi , Mi ). −→ −→ −→ −→ Proof For each i, construct a free DG resolution of the Ai -algebra Bi as follows. Choose Ri such that (Ri )0 is the polynomial Ai -algebra over the elements of Bi , (Ri )0 = Ai [{Xb }b∈Bi ], (Ri )1 the free (Ri )0 -module over the elements of ker((Ri )0 → Bi ), and so on. Then there exists a direct system {Ri }i∈I with maps sending variables to variables such that R = lim Ri is a free DG resolution of the lim Ai -algebra lim Bi , −→ −→ −→ and then the result follows from the fact that the module of differentials of Definition 1.3.1 commutes with direct limits (details can be checked using Lemma 1.2.10 and its proof; alternatively, this same proof works with Definition 1.1.1 instead of Definition 1.3.6), and that direct limits are exact.
2 Formally smooth homomorphisms
In this chapter we prove some important results on smooth homomorphisms. Starting from basic definitions (Section 2.2), we interpret the first cohomology module in terms of infinitesimal extensions (Section 2.1) to characterize formal smoothness in terms of homology. The first key result is the Jacobian criterion of formal smoothness (2.3.5), a homological characterization of this property. This result and some other results on the homology of field extensions in Section 2.4 (in particular a homological characterization of separability) will allow us to prove the main theorems in this section: • Grothendieck’s Theorems 2.5.8 and 2.5.9, which assert that formal smoothness over a field is equivalent to geometric regularity [EGA, 0IV , 22.5.8]. Another ingenious proof by Faltings of this result can be seen in Matsumura’s book [Mt, Theorem 28.7]. • Corollary 2.6.5, which reduces formal smoothness over a noetherian local ring to formal smoothness (or geometric regularity) over a field. This result was proved also by Grothendieck [EGA, 0IV 19.7.1] and it is stated without proof in [Mt, Theorem 28.9]. The more difficult part is to prove that a formally smooth homomorphism is flat. Note that this is much easier if the homomorphism is of finite type (see, e.g., [Bo, Chap. X, §7.10, lemma 5], but the general case is needed. Our proofs follow those by Andr´e [An1, 7.27, 16.18], but we need a few changes in order to avoid the use of simplicial methods or homology modules in higher dimensions. For example, Lemma 2.5.4 would be immediate from Jacobi–Zariski exact sequence and Corollary 2.5.3 if we could make use of the characterization of complete intersection rings by the vanishing of some H3 [An1, 6.27]. Though these homological methods are spread along this book, it is 22
2.1 Infinitesimal extensions
23
perhaps in this section where they appear with more naturality, showing their great power in Commutative Algebra. Some suggestions for further reading on these methods could be [An1], [Qu], as well as many papers mainly by Andr´e himself, Avramov, Brezuleanu and Radu.
2.1 Infinitesimal extensions Definition 2.1.1 Let A be a ring, B an A-algebra and M a B-module. An infinitesimal extension of B over A by M is an exact sequence of A-module homomorphisms i
p
→E− → B → 0, 0→M − where E is also an A-algebra, p an A-algebra homomorphism and ei(x) = i(p(e)x) for all e ∈ E, x ∈ M . This last equality implies that M , as an ideal of E, is of square zero. Conversely, if p : E → B is a surjective homomorphism of A-algebras whose kernel is an ideal M of square zero, then we have an infinitesimal extension of B over A by M . We say that two infinitesimal extensions of B over A by M 0→M →E→B→0 0 → M → E → B → 0 are equivalent if there exists an A-algebra homomorphism f : E → E making the diagram 0
→
M 0 → M
→
E ↓f E
→
→ B → B
→
0
→ 0
commute. In such a case, f is an isomorphism. We denote by ExalcomA (B, M ) the set of equivalence classes of infinitesimal extensions of B over A by M . We say that an infinitesimal extension i
p
0→M − →E− →B→0 is trivial if there exists an A-algebra homomorphism q : B → E such that pq = 1. Lemma 2.1.2 Let A be a ring, B an A-algebra and M a B-module.
24
Formally smooth homomorphisms
Then there exists a trivial extension of B over A by M . Moreover, all trivial extensions of B over A by M are equivalent. Proof Let E = M ⊕B as A-module, with the A-algebra structure given by (x, b)(x , b ) = (bx + b x, bb ), x, x ∈ M, b, b ∈ B. The exact sequence p
i
0→M − →E− → B → 0, where i(x) = (x, 0), p(x, b) = b, is a trivial infinitesimal extension of B over A by M (q : B → E can be defined as q(b) = (0, b)). Now, let p
i
0→M − → E −→ B → 0 be a trivial infinitesimal extension of B over A by M . Let q : B → E be such that p q = 1. Then the A-algebra homomorphism f : E → E , f (x, b) = i (x) + q (b) makes the diagram 0
→ M
0 →
M
i
→ − i
− →
p
E ↓f
− →
B
→
0
E
−→ B
→
0
p
commute.
Definition 2.1.3 Let f : C → B be an A-algebra homomorphism, M a B-module. We define a map f ∗ = ExalcomA (f, M ) : ExalcomA (B, M ) → ExalcomA (C, M ) as follows: let E: 0 → M → E → B → 0 be an infinitesimal extension of B over A by M . The fibre product E ×B C gives a commutative diagram Ef : E:
0 → M 0 → M
→ E ×B C ↓ → E
→ →
C ↓f B
→
0
→
0
such that Ef is an extension of C over A by M . If E, E are two equivalent extensions of B over A by M , then Ef , Ef are also equivalent. So we define f ∗ ([E]) = [Ef ].
2.1 Infinitesimal extensions
25
Theorem 2.1.4 Let A be a ring, B an A-algebra and M a B-module. There exists a bijective map H 1 (A, B, M ) → ExalcomA (B, M ) satisfying: a) the element 0 ∈ H 1 (A, B, M ) goes to the class of trivial extensions b) if f : C → B is an A-algebra homomorphism, the canonical homomorphism H 1 (A, B, M ) → H 1 (A, C, M ) induced by f goes to f ∗ = ExalcomA (f, M ) : ExalcomA (B, M ) → ExalcomA (C, M ). Proof Let R be a polynomial ring over A and J an ideal of R such that B = R/J. We have an exact sequence (1.4.6), (1.4.1) α
DerA (R, M ) − → HomR (J, M ) → H 1 (A, B, M ) → 0, where for d ∈ DerA (R, M ), α(d) = d|J . So an element of H 1 (A, B, M ) is represented by an R-module homomorphism f : J → M and two homomorphisms f, g : J → M represent the same element of H 1 (A, B, M ) if and only if there exists an A-derivation d : R → M such that f −g = d|J . Let E: 0 → M → E → B → 0 be an infinitesimal extension of B over A by M . Since R is a polynomial ring over A, we have a commutative diagram 0 → 0 →
→
J ↓ f M
R ↓f E
→
→
B → B
→
0
→
0.
Moreover, if g : R → E, g : J → M are two other homomorphisms making the diagram commutative, then d : R → M defined by d(x) = f (x) − g(x) is an A-derivation and d|J = f − g . So we define a map u : ExalcomA (B, M ) → H 1 (A, B, M ) by u([E]) = [f ]. Conversely, let f : J → M be an R-module homomorphism. We have a commutative diagram 0 → E:
0 →
J ↓f M
→ →
R ↓ Ef
p
− → →
B B
→
0
→
0,
26
Formally smooth homomorphisms
where Ef = M ⊕J R is the quotient of the A-algebra M ⊕ R (with multiplication (x, r)(x , r ) = (rx + r x, rr ), x, x ∈ M, r, r ∈ R, as in the proof of Lemma 2.1.2, where we are considering M as an R-module via p) by the ideal generated by the elements {(f (s), −s) : s ∈ J}. We have then an extension E of B over A by M . If f, g : J → M are two R-module homomorphisms such that there is some d ∈ DerA (R, M ) such that f − g = d|J , then the associated extensions are equivalent. In effect, the A-algebra homomorphism h : Ef → Eg defined by h(x, r) = (x + d(r), r) is well defined and makes the diagram 0 0
→
M → M
→ Ef ↓ → Eg
→ →
B B
→ 0 → 0
commute. We then define v : H 1 (A, B, M ) → ExalcomA (B, M ) by v([f ]) = [E]. It is easy to check that u and v are inverse, and properties a) and b).
2.2 Formally smooth algebras Definition 2.2.1 Let A be a ring, B an A-algebra and J an ideal of B. We say that the A-algebra B is formally smooth in the J-adic topology (or J-smooth) if the following holds: for any surjective A-algebra homomorphism p : E → C with kernel an ideal of square zero, and for any A-algebra homomorphism f : B → C that is continuous for the J-adic topology in B and the discrete topology in C (i.e., such that f (J n ) = 0 for some n), there exists an A-algebra homomorphism g : B → E such that pg = f :
g p 0 → N → E −−→
B f C
→ 0.
Note that then g is continuous for the discrete topology in E, since f (J n ) = 0 =⇒ g(J 2n ) = 0. In the case J = 0, we say that B is a smooth A-algebra. So smooth implies J-smooth for any ideal J. Examples 2.2.2
2.2 Formally smooth algebras
27
a) A polynomial ring over A is a smooth A-algebra. b) If A is a ring and S a multiplicative subset of A, then S −1 A is a smooth A-algebra, since if N is a square zero ideal of a ring E and e ∈ E is such that e + N is a unit in E/N , then e is a unit in E (let x ∈ E be such that ex − 1 ∈ N . Since N 2 = 0, we have 0 = (ex − 1)2 = e2 x2 − 2ex + 1 and so 1 = e(2x − ex2 )). c) Let A → B, f : B → C be ring homomorphisms, J an ideal of B and T an ideal of C such that f (J) ⊂ T . If B is a J-smooth A-algebra and C is a T -smooth B-algebra, then C is a T -smooth A-algebra. Lemma 2.2.3 Let A → B be a ring homomorphism, J an ideal of B, C an A-algebra and T an ideal of C. Assume that B is a J-smooth A-algebra and C complete for the T -adic topology. Let f1 : B → C/T be a continuous A-algebra homomorphism (i.e., f1 (J n ) = 0 for some n). Then there exists a continuous A-algebra homomorphism f : B → C inducing f1 . Proof Since B is J-smooth, there exists a continuous A-algebra homomorphism f2 : B → C/T 2 making the diagram
f2 0
→ T /T 2
→
C/T 2
−→
B f1 C/T
→
0
commute. Similarly, if we have already constructed fn : B → C/T n , by the J-smoothness of B there exists fn+1 : B → C/T n+1 inducing fn . We define f = lim fn : B → C = lim C/T n . ←− ←− Lemma 2.2.4 Let A be a ring, B an A-algebra, J an ideal of B and 0 ≤ i ≤ 2 an integer. The following are equivalent: a) lim H i (A, B/J n , M ) = 0 for any B-module M annihilated by J. −→ b) lim H i (A, B/J n , M ) = 0 for any B-module M annihilated by −→ some power of J (here the limit is taken over the direct system for n large enough). Proof b) implies a) is trivial. To prove the converse, let M be a Bmodule such that J t M = 0 for some t. We will see by induction on
28
Formally smooth homomorphisms
t that lim H i (A, B/J n , M ) = 0. The case t = 1 holds by assumption. −→ Assume that the equality is valid for t − 1. The exact sequence 0 → JM → M → M/JM → 0 induces an exact sequence lim H i (A, B/J n , JM ) → lim H i (A, B/J n , M ) −→ −→ → lim H i (A, B/J n , M/JM ). −→ Since JM is annihilated by J t−1 , the term on the left is zero. The one on the right is also zero, since M/JM is annihilated by J. So lim H i (A, B/J n , M ) = 0. −→ Lemma 2.2.5 Let A be a ring, B an A-algebra, J an ideal of B. The following are equivalent: a) B is an A-algebra formally smooth for the J-adic topology b) lim H 1 (A, B/J n , M ) = 0 for any B/J-module M . −→ Proof b) =⇒ a) Let p : E → C be a surjective A-algebra homomorphism with kernel N satisfying N 2 = 0. Let E be the extension 0 → N → E → C → 0. Let f : B → C be an A-algebra homomorphism such that f (J t ) = 0. Then f induces ft : B/J t → C, and N is a B/J t -module. Consider the extension Eft (2.1.3). We have a commutative diagram 0 → 0
N
→
→ N
→ −
i
pt
E ft ↓g
−→
E
− →
B/J t ↓ ft
→
0
C
→
0
p
where E ft = E ×C B/J t . By (2.1.4) and (2.2.4), lim ExalcomA (B/J n , N ) = lim H 1 (A, B/J n , N ) = 0. −→ −→ Then there exists an integer s ≥ t such that the extension Efs = (Eft )qst is trivial, where qst : B/J s → B/J t is the canonical map: Efs :
0 → N
→
E fs ↓θ
−→
ps
B/J s ↓ qst
→
0
Eft :
0
→
→ −
i
E ft
−→
pt
B/J t
→
0,
N
2.3 Jacobian criteria
29
with E fs = E ft ×B/J t B/J s . That is, there exists an A-algebra homomorphism σ : B/J s → E fs such that ps σ = 1. Therefore pt θσ = qst . Let πs : B → B/J s be the canonical map B πs s B/J qst
θσ pt
Eft −→ g E
p
−−→
t B/J ft
C.
We have pgθσπs = ft pt θσπs = ft qst πs = f . So B is a J-smooth A-algebra. a) =⇒ b) Let ε
n B/J n → 0 0 → M → E −→
be an infinitesimal extension of B/J n over A by M . Let πn : B → B/J n be the canonical map. Since B is formally smooth over A, there exists ψn : B → E such that πn = εn ψn . Since M 2 = 0, we deduce that ψn (J 2n ) = 0 and so there exists φn : B/J 2n → E such that ψn = φn π2n . Let E q be as above: →
M
−→ E q
0 →
M
−→
0
E
−→ B/J 2n q φn −→ B/J n
→
0
→
0.
The homomorphism φn induces an A-algebra homomorphism B/J 2n → E q , which is a section of the map E q → B/J 2n . Therefore, 0 → M → E q → B/J 2n → 0 is a trivial extension. By (2.1.4) lim H 1 (A, B/J n , M ) = 0. −→
2.3 Jacobian criteria Theorem 2.3.1 (Jacobian criterion for smoothness) Let A be a ring and B an A-algebra. The following are equivalent: a) B is a smooth A-algebra
30
Formally smooth homomorphisms b) H 1 (A, B, M ) = 0 for any B-module M c) H1 (A, B, B) = 0 and ΩB|A is a projective B-module d) For any presentation B = R/I where R is a polynomial ring over A, the B-module homomorphism I/I 2 → ΩR|A ⊗R B has left inverse.
Proof a) ⇐⇒ b) is a consequence of (2.2.5) for J = 0, b) ⇐⇒ c) is a consequence of the universal coefficient exact sequence (1.4.5.d), and c) ⇐⇒ d) is a consequence of the exact sequence (1.4.1), (1.4.6) 0 → H1 (A, B, B) → I/I 2 → ΩR|A ⊗R B → ΩB|A → 0. Next we study the nondiscrete case. Lemma 2.3.2 Let A be a ring, I an ideal of A, B = A/I and M a B-module. We have a natural injective B-module homomorphism H 2 (A, B, M ) → Ext1A (I, M ). Proof In Definition 1.1.1, take R = A, F a free A-module with an exact sequence of A-modules 0 → U → F → I → 0. Let U0 be as in (1.1.1). We have H 2 (A, B, M ) = coker(HomB (F/IF, M ) = HomA (F, M ) → HomB (U/U0 , M ) = HomA (U/U0 , M )). The result follows from the commutative diagram of exact rows and columns
HomA (F, M ) → HomA (F, M ) →
0 ↓ HomA (U/U0 , M ) → ↓ HomA (U, M ) →
H 2 (A, B, M ) ↓ Ext1A (I, M )
→ 0 →
0.
2.3 Jacobian criteria
31
Lemma 2.3.3 Let A be a noetherian ring, I an ideal of A, n > 0 an integer. Then there exists an integer s ≥ n such that the canonical homomorphism A n s TorA 1 (I , A/I) → Tor1 (I , A/I)
is zero. A A n n n Proof We have TorA 1 (I , A/I) = Tor2 (A/I , A/I) = Tor1 (A/I , I). Let
0→U →F →I→0 be an exact sequence of A-modules with F free of finite type. From the exact sequence n n n 0 → TorA 1 (A/I , I) → U/I U → F/I F n n n we obtain TorA 1 (A/I , I) = (U ∩ I F )/I U . Artin–Rees lemma [Mt, Theorem 8.5] says that there exists an integer r > 0 such that I t F ∩ U = I t−r (I r F ∩ U ) for all t > r. Let s = n + r. Then I s F ∩ U ⊂ I n U , and so the map A s n TorA 1 (A/I , I) → Tor1 (A/I , I)
is zero.
Lemma 2.3.4 Let A be a ring, I an ideal of A, M an A/I-module. Then lim H 1 (A, A/I n , M ) = 0. −→ If A is noetherian, we also have lim H 2 (A, A/I n , M ) = 0. −→ Proof By (1.4.1), lim H 1 (A, A/I n , M ) = lim HomA/I n (I n /I 2n , M ) = −→ −→ lim HomA/I (I n /I n+1 , M ), and this last module is zero, since the map −→ I n+1 /I n+2 → I n /I n+1 is zero. This proves the first part. Now assume A is noetherian. Consider the exact sequence 0 → Ext1A/I (I n /I n+1 , M ) → Ext1A (I n , M ) n → HomA/I (TorA 1 (I , A/I), M ),
32
Formally smooth homomorphisms
which comes from the change-of-rings spectral sequence (see the Appendix for a direct proof without spectral sequences) p+q n n E2p,q = ExtpA/I (TorA q (I , A/I), M ) ⇒ ExtA (I , M ). n Since lim Ext1A/I (I n /I n+1 , M ) = 0 and lim HomA/I (TorA 1 (I , A/I), M ) = −→ −→ 1 n 0 by (2.3.3), we deduce that lim ExtA (I , M ) = 0. −→ On the other hand, by (2.3.2) we have an injective homomorphism
lim H 2 (A, A/I n , M ) → lim Ext1A (I n , M ) −→ −→ 2 n which shows that lim H (A, A/I , M ) = 0. −→
Theorem 2.3.5 (Jacobian criterion of formal smoothness) Let A be a ring, B a noetherian A-algebra and J an ideal of B. The following are equivalent: a) b) c) d)
B is a formally smooth A-algebra for the J-adic topology. H 1 (A, B, M ) = 0 for any B/J-module M . H1 (A, B, B/J) = 0 and ΩB|A ⊗B B/J is a projective B/J-module. For any presentation B = R/I where R is a smooth A-algebra, the B/J-module homomorphism I/I 2 ⊗B B/J → ΩR|A ⊗R B/J has left inverse.
(The implications a)⇐=b) ⇐⇒ c) ⇐⇒ d) hold without the noetherian assumption.) Proof For any B/J-module M we have the exact sequence (1.4.6) H 1 (B, B/J n , M ) → H 1 (A, B/J n , M ) → H 1 (A, B, M ) → H 2 (B, B/J n , M ), hence an exact sequence lim H 1 (B, B/J n , M ) → lim H 1 (A, B/J n , M ) → H 1 (A, B, M ) → −→ −→ lim H 2 (B, B/J n , M ). −→ a) =⇒ b) By (2.3.4) lim H 1 (A, B/J n , M ) = H 1 (A, B, M ) and the −→ result follows from (2.2.5). b) =⇒ a) By (2.3.4) lim H 1 (A, B/J n , M ) → H 1 (A, B, M ) is injective −→ and so the result follows again from (2.2.5). b) ⇐⇒ c) ⇐⇒ d) follows as in the proof of (2.3.1 b) ⇐⇒ c) ⇐⇒ d)).
2.3 Jacobian criteria
33
We now give an example which shows that the noetherian assumption is essential. This is a negative answer to a question of Brezuleanu [Br, Remark 1.3.(i)]. Lemma 2.3.6 Let A be a ring, N an ideal of A such that N 2 = N . Let I ⊂ N be another ideal of A such that I = IN . Let B = A/I, J = N/I. Then the A-algebra B is formally smooth for the J-adic topology and there exists a B/J-module M such that H 1 (A, B, M ) = 0. Proof Since J = J 2 , for any B/J = A/N -module M we have lim H 1 (A, B/J n , M ) = H 1 (A, B/J, M ) = H 1 (A, A/N, M ) −→ = HomA/N (N/N 2 , M ) = 0 since N 2 = N . So B is formally smooth over A by (2.2.5). On the other hand, H 1 (A, B, M ) = H 1 (A, A/I, M ) = HomA/I (I/I 2 , M ) = HomA/N (I/I 2 ⊗A/I A/N, M ) = HomA/N (I/IN, M ), which is not zero in general (take M = I/IN ), since I = IN .
Example 2.3.7 We show that there exists a ring A satisfying the assumptions of Lemma 2.3.6. Let A = C(R, R) be the ring of continuous functions from the real field R on itself. Let i ∈ C(R, R) be the identity map, and let I = (i) be the ideal of A generated by i. We know that i is not a unit in A since i(0) = 0, and is not a zero divisor since if φi = 0 for φ ∈ A, then φ(x) = 0 for all x ∈ R, x = 0, and so φ = 0 by continuity. The ideal I is not absolutely isolated [Bo, Chapitre II, §2, Ex. 15, c], and so rad(I) = I [Bo, loc. cit., a]. In particular, I is not a prime ideal. Now let N be a maximal ideal of A such that I ⊂ N . N is prime and so N 2 = N [Bo, loc. cit. d]. Finally, we see that I = IN . If I = IN , i ∈ IN = iN , so i = iψ for some ψ ∈ N . Then i(1 − ψ) = 0. Since i is not a zero divisor, ψ = 1, but this is impossible since ψ ∈ N . Remark 2.3.8 Let A be a ring, (B, n, L) a noetherian local A-algebra essentially of finite type, J ⊂ n an ideal of B. Then B is a smooth A-algebra if and only if it is J-smooth if and only if it is n-smooth. By definition, it is enough to show that n-smooth implies smooth. By (2.3.5), n-smooth implies H1 (A, B, L) = 0. By (1.4.5.d) we have then
34
Formally smooth homomorphisms
TorB 1 (ΩB|A , L) = 0, and since B is noetherian and ΩB|A a B-module of finite type, we deduce that ΩB|A is a projective B-module. Therefore, by (1.4.5.d) again, H1 (A, B, B) ⊗B L = 0, and so H1 (A, B, B) = 0 by Nakayama’s lemma and (1.4.4), (1.4.7). By (2.3.1) B is a smooth A-algebra.
2.4 Field extensions Let F |K be a field extension, M an F -module. By (1.4.5) we have Hn (K, F, M ) = Hn (K, F, F ) ⊗F M H n (K, F, M ) = HomF (Hn (K, F, F ), M ) for n = 0, 1, 2, so we concentrate on the F -vector spaces Hn (K, F, F ). Proposition 2.4.1 If F |K is a field extension, then H2 (K, F, F ) = 0. Proof By (1.4.8) we can assume that F |K is of finite type, and then by (1.4.6) we can assume F = K(t). If t is transcendental over K, then F is isomorphic to the field of fractions of the polynomial ring K[X] and so H2 (K, F, F ) = 0 by (1.4.1) and (1.4.7). If t is algebraic over K, then F = K[X]/(f ) with f ∈ K[X]. Since f is not a zero divisor, we can take R = K[X], I = (f ), F = K[X], U = 0 in Definition 1.1.1, and therefore H2 (K, F, F ) = 0. Corollary 2.4.2 Let F |K, E|F be field extensions. There exists an exact sequence 0 → H1 (K, F, E) → H1 (K, E, E) → H1 (F, E, E) → ΩF |K ⊗F E → ΩE|K → ΩE|F → 0. Proof This follows from (1.4.1), (1.4.6), (2.4.1).
Proposition 2.4.3 Let F |K be a field extension with F = K(t). Then: a) In the case that t is transcendental over K, H1 (K, F, F ) = 0 and dimF ΩF |K = 1. b) If t is separable algebraic over K, then H1 (K, F, F ) = 0 and ΩF |K = 0. c) If t is algebraic but inseparable over K, then dimF H1 (K, F, F ) = 1 and dimF ΩF |K = 1.
2.4 Field extensions
35
Proof a) F = K(t) is isomorphic to the field of fractions of K[X]. So H1 (K, F, F ) = 0 by (1.4.1) and (1.4.7). Also ΩF |K = ΩK[X]|K ⊗K[X] F , and so dimF ΩF |K = 1. b) and c) We have a surjective K-algebra homomorphism K[X] → F with kernel (f ), where f is the irreducible polynomial of t over K. Applying (1.4.6) to the sequence K → K[X] → F we obtain an exact sequence d
0 → H1 (K, F, F ) → (f )/(f 2 ) − → ΩK[X]|K ⊗K[X] F → ΩF |K → 0. Since dimF (f )/(f 2 ) = 1 and dimF ΩK[X]|K ⊗K[X] F = 1, we have dimF H1 (K, F, F ) = dimF ΩF |K ≤ 1. So we have to show that t is separable if and only if dimF H1 (K, F, F ) = 0. We have that t is separable over K if and only if f (t) = 0. We have isomorphisms ΩK[X]|K ⊗K[X] F
− →
≈
K[X] ⊗K[X] F
− →
≈
F
gdX ⊗ 1
→
g⊗1
→
g(t)
and so an exact sequence d
0 → H1 (K, F, F ) → (f )/(f 2 ) −−→ F with d(gf + (f 2 )) = g(t)f (t). Then t is separable if and only if f (t) = 0 and this is equivalent to d = 0. Since dimF (f )/(f 2 ) = 1, this is also equivalent to the injectivity of d, and this last condition is equivalent to H1 (K, F, F ) = 0. Theorem 2.4.4 (Cartier equality) Let F |K be a field extension of finite type. We have tr deg(F |K) = dimF ΩF |K − dimF H1 (K, F, F ). Proof By (2.4.2) we may assume that F = K(t), and so the result follows from (2.4.3). Theorem 2.4.5 A general field extension F |K is separable if and only if H1 (K, F, F ) = 0.
36
Formally smooth homomorphisms
Proof Assume F |K separable. By (1.4.8) we can assume that F |K is of finite type and separably generated. By (1.4.6), we can reduce the problem to the case F = K(t1 , . . . , tn ) where t1 , . . . , tn are algebraically independent over K, and to the case of a finite separable extension F |K. In either case, using (1.4.6) once more, we reduce the problem to the case F = K(t). Then (2.4.3) gives H1 (K, F, F ) = 0. Conversely, assume now that H1 (K, F, F ) = 0. We must show that any finitely generated subextension L|K of F |K is separably generated. Applying (2.4.2) to K → L → F we see that H1 (K, F, F ) = 0 implies H1 (K, L, L) = 0. So we may assume that F |K is finitely generated. By (2.4.4), tr deg F |K = dimF ΩF |K , that is, if n = tr deg(F |K), there exist n elements (x1 , . . . , xn ) such that {dx1 , . . . , dxn } is an F -basis of ΩF |K . Let E = K(x1 , . . . , xn ). In the exact sequence ΩE|K ⊗E F → ΩF |K → ΩF |E → 0 the map on the left is surjective, and so ΩF |E = 0. By (2.4.4) again, we see that F |E is algebraic and H1 (E, F, F ) = 0. Then tr deg E|K = tr deg F |K = n, so x1 , . . . , xn are algebraically independent over K, and so they form a transcendence basis of F |K. We must see that F |E is separable. Let t ∈ F . Applying (2.4.2) to E → E(t) → F and bearing in mind that H1 (E, F, F ) = 0, we see that H1 (E, E(t), E(t)) = 0. By (2.4.3) t is then separable over E. Corollary 2.4.6 A field extension F |K is separable if and only if F is a smooth K-algebra.
Proof (2.3.1).
Now we shall prove some lemmas on algebras over a field which we use in the next sections. Lemma 2.4.7 Let A be a ring containing a field of characteristic p > 0. Let B be a reduced A-algebra and M a B-module. Then the canonical homomorphism α
H1 (Ap , B p , M ) − → H1 (A, B, M ) is zero (here Ap is the subring of A of elements ap with a ∈ A, and similarly B p ).
2.4 Field extensions
37
Proof Let R be a polynomial A-algebra with a surjective A-algebra homomorphism R → B. We have a commutative diagram α
H1 (Ap , B p , M ) ↓
− →
H1 (Rp , B p , M )
− →
β
H1 (A, B, M ) ↓γ H1 (R, B, M ).
By (1.4.1) H1 (A, R, M ) = 0 and so by (1.4.6) γ is injective. So, α = 0 if β = 0. Thus we may assume B = A/I with I an ideal of A. Since B is reduced, B p = Ap /J, with J = {xp : x ∈ I}. Then α is the map α
J/J 2 ⊗B p M = H1 (Ap , B p , M ) − → H1 (A, B, M ) = I/I 2 ⊗B M, which is zero since J/J 2 → I/I 2 is zero.
Lemma 2.4.8 Let K be a field of characteristic p > 0, F |K a field extension such that K 1/p ⊂ F . Then the canonical homomorphism H1 (K, F, M ) → H1 (K 1/p , F, M ) is zero for any F -module M . Proof By (1.4.5) we may assume that M is an algebraically closed field L. In the commutative diagram H1 (K, F, L) ↓
− →
α
H1 (K 1/p , F, L) ↓γ
H1 (K, L, L)
− →
β
H1 (K 1/p , L, L)
β = 0 by (2.4.7) and γ is injective by (2.4.2). So α = 0.
Lemma 2.4.9 Let K be a field of characteristic p > 0, (B, n, L) a local K-algebra and F a field such that K ⊂ F ⊂ K 1/p . Then the ring B⊗K F is local with maximal ideal J = {z ∈ B ⊗K F : z p ∈ n ⊗K F }. Proof From the exact sequence 0 → n → B → L → 0 we obtain an exact sequence 0 → n ⊗K F → B ⊗K F → L ⊗K F → 0. Clearly J is a proper ideal of B ⊗K F . So it is enough to show that if x ∈ B ⊗K F − J, then x is a unit in B ⊗K F . The element xp belongs to the subring B ⊗K K of B ⊗K F , since F p ⊂ K. Moreover xp ∈ / n ⊗K F
38
Formally smooth homomorphisms
since x ∈ / J. So we can identify xp to an element of B − n. Therefore xp is a unit, and so is x. Lemma 2.4.10 Let K be a field of characteristic p > 0, B a local Kalgebra, F the residue field of the local ring B ⊗K K 1/p (Lemma 2.4.9) and M an F -module. There exists a natural isomorphism H2 (B ⊗K K 1/p , F, M ) = H1 (K, B, M ). Proof In the commutative diagram H1 (K, B, M ) α
−→
H1 (K, F, M ) γ
β
H1 (K 1/p , F, M )
H1 (K 1/p , B ⊗K K 1/p , M ) −−→
γ = 0 by (2.4.8) and α is an isomorphism by (1.4.3). So β = 0. By the Jacobi–Zariski exact sequence associated to K 1/p → B ⊗K K 1/p → F and (2.4.1) 0 → H2 (B ⊗K K 1/p , F, M ) β
δ
−−→ H1 (K 1/p , B ⊗K K 1/p , M ) −−→ H1 (K 1/p , F, M ) we deduce that δ is an isomorphism. The required isomorphism is then α−1 δ. Lemma 2.4.11 Let A → F → L be ring homomorphisms such that F and L are fields, and A contains a field. Then the canonical homomorphism h
→ H2 (A, L, L) H2 (A, F, L) − is an isomorphism. Proof By (1.4.6) and (2.4.1) h is surjective. Let K be a field contained in A. Applying (1.4.6) to the sequences K → A → L and K → A → F we obtain a commutative diagram with exact rows 0 = H2 (K, F, L) 0 = H2 (K, L, L)
→
H2 (A, F, L) → H1 (K, A, L) ↓h → H2 (A, L, L) → H1 (K, A, L)
(2.4.1), (1.4.5). So h is injective.
2.5 Geometric regularity
39
2.5 Geometric regularity Lemma 2.5.1 Let (A, m, K) be a noetherian local ring, I an ideal of A, a1 , . . . , an a set of generators of I, F = An and j : F → I the surjective homomorphism that sends the canonical basis of F to a1 , . . . , an . Let H1 (a1 , . . . , an ; A) be the first Koszul homology module associated to the set of generators a1 , . . . , an of I. We have an exact sequence 0 → H2 (A, A/I, M ) → H1 (a1 , . . . , an ; A) ⊗A/I M → j
F/IF ⊗A/I M − → I/I 2 ⊗A/I M → 0 for any A/I-module M , where j is the homomorphism induced by j. Proof This follows easily from Definition 1.1.1, bearing in mind that, with the notation of (1.1.1), U/U0 = H1 (a1 , . . . , an ; A). Theorem 2.5.2 Let (A, m, K) be a noetherian local ring and I an ideal of A. The following are equivalent: a) I is generated by a regular sequence b) H2 (A, A/I, K) = 0 c) H2 (A, A/I, −) = 0. Proof a) =⇒ c) Let a1 , . . . , an be a regular sequence that generates the ideal I and let F , j and H1 (a1 , . . . , an ; A) be as in (2.5.1). Then H1 (a1 , . . . , an ; A) = 0 and so H2 (A, A/I, −) = 0 by (2.5.1). b) =⇒ a) Let a1 , . . . , an be a minimal set of generators of the ideal I. Let F , j and H1 (a1 , . . . , an ; A) be as in (2.5.1). In the exact sequence of (2.5.1) 0 → H2 (A, A/I, K) → H1 (a1 , . . . , an ; A) ⊗A/I K → F/IF ⊗A/I K j
−−→ I/I 2 ⊗A/I K → 0 we have H2 (A, A/I, K) = 0 and j is an isomorphism, since it can be identified with the map F/mF → I/mI induced by j. Therefore H1 (a1 , . . . , an ; A) ⊗A/I K = 0, so H1 (a1 , . . . , an ; A) = 0 by Nakayama’s Lemma, and a1 , . . . , an is a regular sequence. Corollary 2.5.3 Let (A, m, K) be a noetherian local ring. The following are equivalent: a) A is a regular local ring
40
Formally smooth homomorphisms b) H2 (A, K, K) = 0.
Lemma 2.5.4 Let K → (B, n, L) be a local homomorphism of regular local rings. Then H2 (K, B, L) = 0. Proof Let b1 , . . . , bn be a regular sequence that generates the ideal n. Let f : K[X1 , . . . , Xn ] → B be the K-algebra homomorphism sending Xi to bi . In the Jacobi–Zariski exact sequence associated to K → f →B K[X1 , . . . , Xn ] − H2 (K, K[X1 , . . . , Xn ], L) → H2 (K, B, L) → H2 (K[X1 , . . . , Xn ], B, L) → H1 (K, K[X1 , . . . , Xn ], L) the first and last terms are zero by (1.4.1). Thus we have to show that H2 (K[X1 , . . . , Xn ], B, L) = 0. Let g : K[X1 , . . . , Xn ] → K be the K-algebra homomorphism sending Xi to 0. Since X1 , . . . , Xn is a regular sequence in K[X1 , . . . , Xn ], the Koszul complex K(X1 , . . . , Xn ; K[X1 , . . . , Xn ]) is a free resolution of the K[X1 , . . . , Xn ]-module K[X1 , . . . , Xn ]/(X1 , . . . , Xn ) = K. Then, considering K as a K[X1 , . . . , Xn ]-module via g, we have K[X1 ,...,Xn ]
Torj
(K, B) = Hj (X1 , . . . , Xn ; B) = Hj (b1 , . . . , bn ; B) = 0
for all j > 0. Then by (1.4.3), H2 (K[X1 , . . . , Xn ], B, L) = H2 (K, B ⊗K[X1 ,...,Xn ] K, L) = H2 (K, B/(b1 , . . . , bn ), L) = H2 (K, L, L) and this last term vanishes by (2.4.1), (2.5.3) and the Jacobi–Zariski exact sequence associated to K, its residue field, and L. Remark Using Andr´e–Quillen H3 we can show that if K is a regular local ring, the vanishing of H2 (K, B, L) is equivalent to B being a complete intersection. Lemma 2.5.5 Let K be a field, (B, n, L) a noetherian local K-algebra such that H1 (K, B, L) = 0. Then B is a regular local ring.
2.5 Geometric regularity
41
Proof Applying (1.4.6) and (2.4.1) to K → B → L 0 = H2 (K, L, L) → H2 (B, L, L) → H1 (K, B, L) = 0 we obtain H2 (B, L, L) = 0 and so the result follows from (2.5.3).
Definition 2.5.6 Let K be a field, B a noetherian K-algebra. We say that B is geometrically regular over K if for any finite field extension F |K, the noetherian ring B ⊗K F is regular. The next three theorems show in particular that a noetherian local algebra over a field is geometrically regular if and only if it is formally smooth (for the adic topology of its maximal ideal). Theorem 2.5.7 Let K be a field, (B, n, L) a noetherian local K-algebra. If B is formally smooth for the n-adic topology, then B is geometrically regular over K. Proof Let F |K be a finite field extension, C = B ⊗K F , N a maximal ideal of C. We have to show that CN is a regular local ring. C is a finite B-algebra and so integral over B. In particular nC ⊂ N , and so any C/N -module is an L-module. Let M be a C/N -module. By (1.4.3), H 1 (F, C, M ) = H 1 (K, B, M ), and this last term is zero by (2.3.5). Again by (2.3.5), C is formally smooth over F for the N -adic topology. Since CN is a smooth C-algebra (2.2.2.b) we obtain that CN is a formally smooth F -algebra (2.2.2.c). Then, by (2.3.5) and (2.5.5), CN is a regular local ring. Theorem 2.5.8 Let K be a field, (B, n, L) a noetherian local K-algebra such that the extension L|K is separable. The following are equivalent: a) B is a formally smooth K-algebra for the n-adic topology b) B is a geometrically regular K-algebra c) B is a regular local ring. Proof a) =⇒ b) is (2.5.7). b) =⇒ c) is clear. To see c) =⇒ a), consider the Jacobi–Zariski exact sequence H2 (B, L, L) → H1 (K, B, L) → H1 (K, L, L). The right term is zero by (2.4.5), and the left one is also zero by (2.5.3). So H1 (K, B, L) = 0 and then B is formally smooth by (2.3.5).
42
Formally smooth homomorphisms
Theorem 2.5.9 Let K be a field of characteristic p > 0, (B, n, L) a noetherian local K-algebra. The following are equivalent: a) B is a formally smooth K-algebra for the n-adic topology. b) B is a geometrically regular K-algebra. c) For any finite field extension F |K such that F ⊂ K 1/p , the noetherian local ring (2.4.9) B ⊗K F is regular. d) The local ring B ⊗K K 1/p is regular (and in particular noetherian). Proof a) =⇒ b) is (2.5.7). b) =⇒ c) is clear. To see c) =⇒ d), let K 1/p = lim Fi , with Fi |K finite extensions such that Fi ⊂ K 1/p . Let Ei −→ be the residue field of B ⊗K Fi , and E the residue field of B ⊗K K 1/p . By assumption, each local ring B ⊗K Fi is regular and so by (2.5.3), (1.4.5), H2 (B ⊗K Fi , Ei , E) = 0. By (2.4.11), H2 (B ⊗K Fi , Ei , E) = H2 (B ⊗K Fi , E, E), and so by (1.4.8), 0 = lim H2 (B ⊗K Fi , Ei , E) = lim H2 (B ⊗K Fi , E, E) −→ −→ = H2 (lim B ⊗K Fi , E, E) = H2 (B ⊗K K 1/p , E, E). −→ So by (2.5.3) it is enough to show that B ⊗K K 1/p is noetherian. Let Q be the field of fractions of B, and B 1/p = {x ∈ Q1/p : xp ∈ B}. We have a ring isomorphism B 1/p → B given by b → bp . Since B is regular, we deduce that B 1/p is regular. Let F |K be a finite field extension such that F ⊂ K 1/p . We have a canonical homomorphism αF : B ⊗K F → B 1/p , αF (b ⊗ x) = bx, which is local by (2.4.9). This homomorphism is integral, since if y ∈ B 1/p , y p = αF (y p ⊗1), and so dim(B ⊗K F/ ker(αF )) = dim(B 1/p ). The extension F |K is finite, so dim B = dim(B ⊗K F ), and so dim(B 1/p ) = dim(B ⊗K F ). Therefore αF is injective since B ⊗K F , being a regular local ring, is a domain. Let n = dim(B ⊗K F ) and p = (x1 , . . . , xn ) its maximal ideal. By base change, the homomorphism (B ⊗K F )/p → B 1/p /αF (p)B 1/p induced by αF is an integral extension and therefore dim(B 1/p /αF (p)B 1/p ) = dim((B ⊗K F )/p) = 0. Hence αF (p)B 1/p = αF (x1 ), . . . , αF (xn ) is an ideal of height n, and so αF (x1 ), . . . , αF (xn ) is a regular sequence in B 1/p [Mt, Theorem 17.4]. Thus grp (B ⊗K F ) = ((B ⊗K F )/p)[X1 , . . . , Xn ], and grp (B 1/p ) = grαF (p)B 1/p (B 1/p ) = (B 1/p /αF (p)B 1/p )[X1 , . . . , Xn ] [Mt, Theorem 16.2.i]. By the local flatness criterion [Mt, Theorem 22.3],
2.6 Formally smooth local homomorphisms of noetherian rings 43 αF is flat. Then lim αFi : lim B⊗K Fi = B⊗K K 1/p → B 1/p is flat and so −→ −→ faithfully flat, since it is local. Since B 1/p is noetherian, so is B ⊗K K 1/p . d) =⇒ a) We have 0 = H2 (B ⊗K K 1/p , E, E) = H1 (K, B, E) = H1 (K, B, L) ⊗L E by (2.5.3), (2.4.10), and (1.4.5) respectively. Then B is a formally smooth K-algebra for the n-adic topology by (2.3.5). 2.6 Formally smooth local homomorphisms of noetherian rings Proposition 2.6.1 Let A be a ring, B and C two A-algebras such that the canonical homomorphism A → B is surjective. Then for any B ⊗A C-module M there exists a natural exact sequence H2 (A, B, M ) → H2 (C, B ⊗A C, M ) → TorA 1 (B, C) ⊗C M → H1 (A, B, M ) → H1 (C, B ⊗A C, M ) → 0. Proof Let I = ker(A → B), and suppose that F is a free A-module such that there exists an exact sequence j
0 → U → F −−→ I → 0 of A-modules. Let φ, U0 be as in (1.1.1). We have a commutative diagram with exact rows A 0 → TorA 2 (B, M ) = Tor1 (I, M ) → U ⊗A M e p
0 → H2 (A, B, M )
→ F ⊗A M → I ⊗ A M
→ U/U0 ⊗A M → F ⊗A M → I ⊗A M
from which we deduce ker e = ker p. 2 The surjective homomorphism φ ⊗ 1 : A F ⊗A M → U0 ⊗A M and the exact sequence v
p
U0 ⊗A M −−→ U ⊗A M −−→ U/U0 ⊗A M give isomorphisms ker e = ker p = im v = im(v(φ ⊗ 1)). So we have an exact sequence 2 e F ⊗A M → TorA −→ H2 (A, B, M ). 2 (B, M ) − A
On the other hand, we have an exact sequence 0 → IC → C → B ⊗A C → 0
44
Formally smooth homomorphisms
and a surjective C-module homomorphism F ⊗A C → IC, where F ⊗A C is a free C-module, and so the above reasoning shows that there exists an exact sequence 2
C (F
2
⊗AC) ⊗C M
A
F ⊗A M
→
TorC 2 (B ⊗A C, M ) →
H2 (C, B ⊗A C, M ).
Consider the exact sequence C TorA 2 (B, M ) → Tor2 (B ⊗A C, M ) → A C TorA 1 (B, C) ⊗C M → Tor1 (B, M ) → Tor1 (B ⊗A C, M ) → 0
associated to the change-of-rings spectral sequence (see the Appendix for a direct proof without using spectral sequences) A A 2 Ep,q = TorC p (Torq (B, C), M ) ⇒ Torp+q (B, M ).
Then the required exact sequence follows from the diagram 2 A
F ⊗A M =
2 A
F ⊗A M
C A TorA 2 (B, M ) → Tor2 (B ⊗A C, M ) → Tor1 (B, C) ⊗C M →
H2 (A, B, M ) → H2 (C, B ⊗A C, M ) ↓
↓
0
0 C → TorA 1 (B, M ) → Tor1 (B ⊗A C, M ) → 0
and from the isomorphisms of 1.4.1.c, H1 (A, B, M ) = TorA 1 (B, M ) and H1 (C, B ⊗A C, M ) = TorC (B ⊗ C, M ). A 1 Lemma 2.6.2 Let A be a ring, B and C two A-algebras and M a B ⊗A C-module. Then the canonical homomorphism H1 (A, B, M ) → H1 (C, B ⊗A C, M ) is surjective.
2.6 Formally smooth local homomorphisms of noetherian rings 45 Proof Let R be a polynomial A-algebra such that B = R/I. We have an exact sequence p
→ B ⊗A C → 0. I ⊗A C → R ⊗A C − Let J = ker p. We have an epimorphism I ⊗A C → J. The Jacobi– Zariski exact sequences associated to A → R → B and C → RC → BC , where RC = R ⊗A C and BC = B ⊗A C, give a commutative diagram with exact rows 0 → H1 (A, B, M ) → I ⊗R M → ΩR|A ⊗R M
→
ΩB|A ⊗B M
0 → H1 (C, BC , M ) → J ⊗RC M → ΩRC |C ⊗RC M → ΩBC |C ⊗BC M. Then it is enough to show that I ⊗R M → J ⊗R⊗A C M is surjective. But I ⊗R M = (I ⊗A C) ⊗R⊗A C M , and the epimorphism I ⊗A C → J gives an epimorphism (I ⊗A C) ⊗R⊗A C M → J ⊗R⊗A C M . Lemma 2.6.3 Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings, such that H1 (A, B, L) = 0. Then a) H2 (A, K, L) → H2 (B, B ⊗A K, L) is surjective b) H1 (A, K, L) → H1 (B, B ⊗A K, L) is injective. Proof By (2.6.2), since H1 (A, B, L) = 0 we deduce H1 (K, B ⊗A K, L) = 0, and so H2 (K, B ⊗A K, L) = 0 by (2.5.5), (2.5.4). We have Jacobi– Zariski exact sequences associated to A → K → B ⊗A K and A → B → B ⊗A K α
n Hn (A, B ⊗A K, L) → Hn (K, B ⊗A K, L) → · · · · · · → Hn (A, K, L) −−→
βn
· · · → Hn (A, B, L) → Hn (A, B ⊗A K, L) −−→ Hn (B, B ⊗A K, L) → · · · The homomorphisms of a) and b) are β2 α2 and β1 α1 respectively. Since α1 and β1 are injective, and α2 and β2 are surjective, we obtain the desired result. Theorem 2.6.4 Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. The following are equivalent: a) H1 (A, B, L) = 0 b) H1 (K, B ⊗A K, L) = 0 and B is a flat A-module. Moreover, if these conditions hold, then H2 (A, B, L) = 0.
46
Formally smooth homomorphisms
Proof a) =⇒ b) Consider the exact sequence of (2.6.1) H2 (A, K, L) → H2 (B, B ⊗A K, L) → TorA 1 (B, K) ⊗B L → H1 (A, K, L) → H1 (B, B ⊗A K, L) → 0. A By (2.6.3) TorA 1 (B, K) ⊗B L = 0. Since Tor1 (B, K) is a B-module of finite type, by Nakayama’s lemma we have TorA 1 (B, K) = 0, and by the local flatness criterion [Mt, Theorem 22.3] B is a flat A-module. Finally, since H1 (A, B, L) = 0, by (1.4.3) we obtain H1 (K, B ⊗A K, L) = 0. b) =⇒ a) By (1.4.3), H1 (A, B, L) = H1 (K, B ⊗A K, L) = 0. Under these assumptions, H1 (K, B ⊗A K, L) = 0 implies that also H2 (K, B ⊗A K, L) = 0 by (2.5.5), (2.5.4) and then by (1.4.3),
H2 (A, B, L) = H2 (K, B ⊗A K, L) = 0.
Corollary 2.6.5 Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. The following are equivalent: a) B is a formally smooth A-algebra for the n-adic topology b) B is a flat A-module and the K-algebra B ⊗A K is geometrically regular. Proof (2.6.4), (2.3.5), (2.5.8) and (2.5.9).
2.7 Appendix: The Mac Lane separability criterion As a by-product of the theory so far, we now deduce Mac Lane’s criterion for separability of a field extension. Proposition 2.7.1 Let L|K be a field extension of characteristic p > 0. Then L|K is separable if and only if L ⊗K K 1/p is a field. Proof This follows from (2.5.9), (a) ⇐⇒ (d), (2.4.6) and (2.4.9).
3 Structure of complete noetherian local rings
In this chapter we prove Cohen’s 1946 structure theorems for complete noetherian local rings (Theorem 3.2.4). Our proof follows Grothendieck [EGA, 0IV §19.8] (and in some parts Bourbaki [Bo, Chapter IX]), thus using the results on formal smoothness proved in Section 2. We would like to point out that this way can be reversed. One first can prove Cohen’s theorems as in some standard books on Commutative Algebra (e.g., in Matsumura’s [Mt, sections 28 and 29], treating first the equicharacteristic case), and then Radu [Ra2] shows how to use them to prove that a formally smooth homomorphism is flat (the more difficult part of (2.6.5)).
3.1 Cohen rings Definition 3.1.1 Let f : (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. We say that B is a Cohen A-algebra if B is complete, A-flat and f (m)B = n. Proposition 3.1.2 Let (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings such that B is a Cohen A-algebra. If L|K is separable then B is a formally smooth A-algebra (for the n-adic topology). Proof By (2.6.5) it is enough to show that the K-algebra B ⊗A K = L is geometrically regular, and this is equivalent to the separability of L|K (2.4.6), (2.5.8), (2.5.9). 47
48
Structure of complete noetherian local rings
Lemma 3.1.3 Let A be a ring, I an ideal of A, and M an A-module which is Hausdorff for the I-adic topology. If M/IM = 0 then M = 0. Lemma 3.1.4 Let A be a ring, I an ideal of A, M an A-module which is complete for the I-adic topology and N an A-module which is Hausdorff for the I-adic topology. Let f : M → N be an A-module homomorphism. If the induced homomorphism f : M/IM → N/IN is surjective then f is surjective. If in addition N is A-flat and f is injective, then f is injective. Proof Let K = ker f , P = im(f ), C = coker(f ). Since M is complete, we have that P is complete. So from the commutative diagram with exact rows and columns
0 → 0 →
P ↓ P ↓ 0
0 ↓ → N ↓ → N
→ →
C ↓g C
→
0
we deduce that g is injective, i.e., C is Hausdorff. Since f is surjective, C/IC = 0 and so C = 0 by (3.1.3). Then we have an exact sequence f
0→K→M − → N → 0, that induces an exact sequence f
0 → K/IK → M/IM − → N/IN → 0 when N is A-flat. Thus K/IK = 0. Since K is Hausdorff, being a submodule of a Hausdorff module, again by (3.1.3) we deduce K = 0. Lemma 3.1.5 Let {(Ai , mi ), fji : Ai → Aj }i,j∈I be a filtered direct system of local rings and flat local homomorphisms. Assume that fji (mi )Aj = mj for all i ≤ j. Let A = lim Ai , m = lim mi , and for all i ∈ I, let −→ −→ fi : Ai → A be the canonical homomorphism. Then a) A is a local ring with maximal ideal m, and for all i ∈ I, the homomorphism fi : Ai → A is local, flat and satisfies fi (mi )A = m. b) If mi is an Ai -module of finite type for some i, then the (Haus of the local ring A is a noetherian ring. If dorff ) completion A moreover Ai is noetherian for all i, then A is a noetherian ring.
3.1 Cohen rings
49
Proof a) It is clear that m is a proper ideal of A. If x ∈ A − m, there exist i ∈ I and xi ∈ Ai such that fi (xi ) = x. We have xi ∈ / mi , and so xi is a unit in Ai , and then x is a unit in A. We deduce that A is a local ring with maximal ideal m. The equalities fi (mi )A = m are clear. Finally, fi is flat since lim is exact and commutes with tensor products. −→ b) Assume mi is an ideal of finite type of Ai for some i. Then [AM, = mA m = fi (mi )A is an ideal of finite type of A. Thus m is a Proposition 10.13] and it is an ideal of finite type of A. Then A 2 m , and so noetherian ring (grm generated by m/ (A) is an A/m-algebra noetherian, now apply [AM, corollary 10.25], or alternatively [ZS, VIII, §3, Theorem 7, Corollary 4]). Assume now that Ai is noetherian for all i. For all i ∈ I and all n > 0, m i (mn )A. n = A/f A ⊗Ai (Ai /mni ) = A/mn = A/ i i (mn )A is flat as Since A is flat as Ai -module, we deduce that A/f i n Ai /mi -module. Since Ai → A is a local homomorphism of noetherian rings (easy or see e.g. [Mt, Theorem 8.2] or [Bo, III, §2.13, Proposi is mi -adically ideal-Hausdorff, and by the tion 19]) the Ai -module A is a flat Ai -module. Taklocal flatness criterion [Mt, Theorem 22.3] A ing lim we see that A is a flat A-module, and therefore faithfully flat, −→ is local. Then, since A is noetherian, so is A. since A → A Remarks i) Note that if we do not assume fji (mi )Aj = mj for i ≤ j, but a) holds, then the proof shows that b) also holds (alternatively, note that a) implies fji (mi )Aj = mj for i ≤ j). ii) A similar result avoiding the flatness assumption can be seen in [Og1]. Let (A, m, K) be a local ring, f ∈ A[X] a monic polynomial of degree n > 0. Let B = A[X]/(f ), which as A-module is free of rank n, with basis {1, x, . . . , xn−1 }, where x is the image of X in B. Let B = B/mB = A[X]/(mA[X] + (f )), and f the image of f in K[X]. Then B = K[X]/(f ). Let f = i gi ei be the decomposition of f into irreducible factors in K[X], with distinct monic polynomials gi . For each i, let gi ∈ A[X] be a representant of gi . Lemma 3.1.6 The ideals mi = (m+gi )B of B are maximal and distinct. Moreover B/mi = K[X]/(gi ). There are no other maximal ideals of B.
50
Structure of complete noetherian local rings
Proof Let mi = (gi )B. The contraction of mi from B to B is mi . Since mi is maximal (B/mi = K[X]/(gi ) is a field) and B is a quotient of B, we see that mi is maximal. It is clear that the mi are distinct, and B/mi = K[X]/(gi ). Let n be a maximal ideal of B. Since B is an Amodule of finite type, by Nakayama’s lemma we have n + mB = B, and so mB ⊂ n. Then n is the contraction in B of some maximal ideal of B, i.e., some (gi )B = mi , and so n = mi . Lemma 3.1.7 Let (A, m, K) be a noetherian local ring, L = K(x) a field extension of K. Then there exist a noetherian local ring (B, n, L) and a flat local homomorphism f : (A, m, K) → (B, n, L) such that f (m)B = n. Proof First assume that x is algebraic over K. Let f ∈ K[X] be the irreducible polynomial of x over K, and let f ∈ A[X] be a monic polynomial representing f . Let B = A[X]/(f ). By (3.1.6) B is a noetherian local ring with maximal ideal n := mB, and B is a flat A-module (since it is free). The residue field of B is K[X]/(f ) = L. Now assume x transcendental over K. Let q = mA[X] = m[X] be the prime ideal extension of m in A[X]. Let B = A[X]q . Then B is a noetherian local ring with maximal ideal mB, and residue field K(X) = L. It is clear that B is flat as A-module. Lemma 3.1.8 Let (A, m, K) be a noetherian local ring, L|K a field extension. Then there exist a Cohen A-algebra (B, n, L). If L|K is separable then it is unique up to A-isomorphism. Proof Existence: let {Fi }i∈I be a family of subfields of L containing K, over a well-ordered indexing set I that possesses a greatest element ω, satisfying: a) i ≤ j =⇒ Fi ⊂ Fj ; b) the fields K and L belong to this family, K corresponds to the first element α of I, and Fω = L; c) if j ∈ I has a predecessor i, then the extension Fj |Fi is generated by one element; d) if j ∈ I has no predecessor, then Fj = i<j Fi . Such a family exists: for instance give L a well-ordering and define Fj as the K-subfield of L generated by the elements i ∈ L such that i < j. Then the family {Fi } ∪ {L} works.
3.1 Cohen rings
51
For each j ∈ I, we define a local ring (Cj , mj , Fj ) and a homomorphism fj : A → Cj by transfinite induction as follows: if j = α is the first element of I, then Cj = A. If j has a predecessor i, we define fji : (Ci , mi , Fi ) → (Cj , mj , Fj ) as in (3.1.7) and fj = fji fi . If j has no predecessor, Cj = lim Ci and fj = lim fi . −→ −→ i<j
i<j
Let I be the set of j ∈ I such that for all i < j the homomorphism fi : A → Ci is local flat and mCi = mi . If I = I, let β be the smallest element of I \ I . It is clear that the first element α of I is in I , and so ∅ = Sβ := {i ∈ I : i < β} ⊂ I . If Sβ does not have a maximum, then Cβ = lim Ci , and so β ∈ I by (3.1.5), which is a contradiction. −→ i∈Sβ
If Sβ has a maximum γ, we construct Cβ from Cγ as in (3.1.7), and so we have again the contradiction β ∈ I . Therefore I = I, and so the homomorphism fω : (A, m, K) → (Cω , mω , Fω = L) is local flat and mCω = mω . Similarly we see that Cω is noetherian: we consider the set I of elements j ∈ I such that Cj is noetherian. Using (3.1.5) and (3.1.7) as before we see that I = I and so Cω is noetherian. ω , m ω , L). Thus we define (B, n, L) := (C Uniqueness: now assume that L|K is separable. Let (B , n , L) be another Cohen A-algebra. By (3.1.2), B is n-smooth over A. By Lemma = 2.2.3, the canonical composite homomorphism B → B/n = L − → B /n = L is induced by a continuous (i.e., local) homomorphism f : B → B . By (3.1.4) f is an isomorphism. Definition 3.1.9 Let A be a local ring, and let p ≥ 0 be the characteristic of its residue field. Let P = Z(p) (localization of the ring of integers Z at the prime ideal (p) = pZ). We say that A is a Cohen ring if the canonical homomorphism P → A makes A a Cohen A-algebra. For example, the completion of Z(p) (i.e., the ring of p-adic integers) is a Cohen ring. Proposition 3.1.10 Let A be a local ring, and p ≥ 0 the characteristic of its residue field. Then a) If p = 0, then A is a Cohen ring if and only if it is a field (of characteristic zero).
52
Structure of complete noetherian local rings b) If p = 0, then A is a Cohen ring if and only if it is a complete discrete valuation domain with maximal ideal generated by p · 1.
Proof If A is a Cohen ring with p > 0, then by definition, the maximal ideal of A is generated by p · 1. Since A is P -flat, p · 1 is not a zero divisor in A and so A is a regular local ring of Krull dimension 1, i.e., a discrete valuation domain, and complete by definition. The case of p = 0 is clear.
3.2 Cohen’s structure theorems Theorem 3.2.1 a) Let C be a Cohen ring, B a complete noetherian local ring, and J an ideal of B. Then any local homomorphism C → B/J factors as a composite of local homomorphisms C → B → B/J. b) Let K be a field. There exist a Cohen ring C with residue field isomorphic to K. Any two such Cohen rings are isomorphic by an isomorphism inducing the identity map in the residue fields. Proof a) (3.1.2) and (2.2.3). b) (3.1.8).
Lemma 3.2.2 Let u : (C, m, K) → (B, n, L) be a local homomorphism of complete noetherian local rings, which induces an isomorphism between the residue fields K and L. Let {x1 , . . . , xm } be a set of generators of the maximal ideal m of C, and y1 , . . . , yn ∈ n. Let R = C[[Y1 , . . . , Yn ]]. Then a) There exists a unique ring homomorphism v : R → B extending u and such that v(Yi ) = yi for all i. b) v is surjective if and only if {u(x1 ), . . . , u(xm ), y1 , . . . , yn } generates the ideal n. c) v is finite if and only if {u(x1 ), . . . , u(xm ), y1 , . . . , yn } generates an ideal of definition of B. Proof a) is clear and b) follows from (3.1.4). c) The ideal generated by {u(x1 ), . . . , u(xm ), y1 , . . . , yn } is v(q)B, where q is the maximal ideal of R. We have that v(q)B is an ideal of definition of B ⇔ the length LB (B/v(q)B) < ∞ ⇔∗ LR/q (B/v(q)B) < ∞ ⇔ dimR/q (B/v(q)B) < ∞ ⇔∗∗ v is finite, where the equivalence marked with ∗ is as follows:
3.2 Cohen’s structure theorems
53
Assume LB (B/v(q)B) < ∞. Let B/v(q)B = Mn ⊃ Mn−1 ⊃ · · · ⊃ M0 ⊃ M−1 = 0 be a (maximal) composition series of the B-module B/v(q)B. Then the B-modules Mi /Mi−1 are simple, and so they are (simple) B/nmodules. Since R/q = B/n, the Mi /Mi−1 are simple R/q-modules, and so B/v(q)B = Mn ⊃ Mn−1 ⊃ · · · ⊃ M0 ⊃ M−1 = 0 is a (maximal) composition series for the R/q-module B/v(q)B. The converse is clear. Finally, ⇒∗∗ follows from (3.1.4) applied to a homomorphism Rt → B such that Rt /qRt → B/v(q)B is surjective, and ⇐∗∗ is clear. Lemma 3.2.3 Let u : (C, m, K) → (B, n, L) be a local homomorphism of complete noetherian local rings, which induces an isomorphism between the residue fields K and L. Assume that C is regular and let {x1 , . . . , xm } be a regular system of parameters of C. Let y1 , . . . , yn ∈ n, R = C[[Y1 , . . . , Yn ]], and v : R → B extending u as in (3.2.2). Then a) v : R → B is injective and finite if and only if u(x1 ), . . . , u(xm ), y1 , . . . , yn is a (not necessarily regular) system of parameters of B. In this case, dim B = m + n. b) If u(x1 ), . . . , u(xm ), y1 , . . . , yn is a part of a (not necessarily regular) system of parameters of B, then v is injective. Proof a) The set u(x1 ), . . . , u(xm ), y1 , . . . , yn is a system of parameters of B if and only if v(q) = (u(x1 ), . . . , u(xm ), y1 , . . . , yn ) is an ideal of definition of B and dim B = m + n, and by (3.2.2) this is equivalent to v finite and dim B = m + n. So we have to prove that if v is finite, then dim B = m + n ⇐⇒ v
is injective.
If B is a finite R-algebra, dim B = dim(R/ ker(v)). If ker(v) = 0, dim(R/ ker(v)) < dim R = m + n, since R is an integral domain. So if v is finite, dim B = m + n ⇐⇒ v is injective. b) If u(x1 ), . . . , u(xm ), y1 , . . . , yn is a part of a system of parameters of B, let yn+1 , . . . , yn+r ∈ n be such that u(x1 ), . . . , u(xm ), y1 , . . . , yn , yn+1 , . . . , yn+r is a system of parameters of B. There is a commutative diagram - C[[Y1 , . . . , Yn , Yn+1 , . . . , Yn+r ]]
R = C[[Y1 , . . . , Yn ]] Hv
j H
B
w
54
Structure of complete noetherian local rings
where the horizontal map is the inclusion, and w extends u by w(Yi ) = yi for i = 1, . . . , n + r. By a), w is injective, and so v is injective. Theorem 3.2.4 Let (B, n, K) be a complete noetherian local ring. a) There exist a Cohen ring C and a surjective ring homomorphism C[[Y1 , . . . , Yn ]] → B. If B contains a field, then there exists a surjective ring homomorphism K[[Y1 , . . . , Yn ]] → B. b) Suppose B is an integral domain or contains a field. Then there exists a finite injective local homomorphism C[[Y1 , . . . , Ym ]] → B, where C is a Cohen ring (if B does not contain a field) or C = K (if B contains a field). Proof a) Let C be a Cohen ring with residue field K (3.2.1), and C → B the homomorphism induced by C → K = B/n (3.2.1). By (3.2.2) there exists a surjective ring homomorphism C[[Y1 , . . . , Yn ]] → B. If B contains a field, it contains the field Fp (Fp = Z/pZ if p > 0,Fp = Q if p = 0), where p is the characteristic of K. Therefore K is a formally smooth Fp -algebra (2.4.6), and so by (2.2.3) there exists a section of the epimorphism B → K. So B contains a field isomorphic to its residue field, and the proof continues as in the previous case. b) If B contains a field, B contains a field isomorphic to K, as we have just seen. Then the result follows from (3.2.3). If B does not contain a field, then the characteristic of K is p > 0. Let C be a Cohen ring with residue field K, and C → B the homomorphism given by (3.2.1). The canonical homomorphism Z(p) → B is injective, since its kernel is a prime ideal of Z(p) (B is a domain), which is not (p)Z(p) (since B does not contain a field). So p · 1 ∈ B is a part of a system of parameters of B ([Mt, proof of Theorem 17.4.iii] or [ZS, VIII, §9, corollary 2 of Theorem 20]. By (3.2.3.b), C → B is injective. Now the result follows from (3.2.3.a). Corollary 3.2.5 Any complete noetherian local ring is a quotient of a complete regular local ring. Proof It follows from (3.2.4).
4 Complete intersections
The main purpose of this chapter is to prove the descent of the complete intersection property by flat local homomorphisms (4.3.8), which has as a consequence the localization theorem for complete intersections (4.3.9): if (A, m, K) is a complete intersection ring, p a prime ideal of A, then Ap is complete intersection. This is another important result which appears without proof in Matsumura’s book [Mt, end of Section 21]. The case when A is a quotient of a regular ring follows easily from the same localization property for regular rings (Serre’s theorem). The difficult part, solved by Avramov [Av1], is to reduce the problem to this case. We follow some papers by Avramov. We first need to present Gulliksen’s proof [GL] of the existence of minimal DG algebra resolutions (4.1.7). This result is used to prove Main Lemma 4.2.1 following [Av1]. We do not know any easier proof of this lemma (or, equivalently, of (4.2.2)). Finally, we characterize complete intersection rings in terms of homology modules in order to prove the main theorems (4.3.8), (4.3.9). In (higher) Andr´e–Quillen homology theory, complete intersections are characterized by the vanishing of an H3 module [An1, 6.27]. Since we want to avoid these higher homology modules, we characterize them by counting dimensions of the lower homology modules (4.3.5) as in [Av2, Section 3]. Avramov’s Lemma 4.2.1 is very powerful (as an example we give in (4.4.2) an alternative proof of Kunz’s characterization of regularity in characteristic p [Ku] using this lemma). For other applications and improvements of it see [Av2], [Av3].
55
56
Complete intersections 4.1 Minimal DG resolutions
Definition 4.1.1 Let A → B be a surjective homomorphism of noetherian local rings. Let (R, d) be a free DG resolution of the A-algebra B with R0 = A. Then (R, d) is said to be minimal if d(R) ⊂ mR, where m is the maximal ideal of R0 . Definition 4.1.2 Let (R, d) be a DG A-algebra. A derivation ϑ on R is an R0 -linear map ϑ : R → R of homogeneous degree w and such that a) dϑ = ϑd b) ϑ(xy) = (−1)wq ϑ(x)y + xϑ(y) where y ∈ Rq . Lemma 4.1.3 Let ϑ be a derivation on a DG A-algebra R, and x a cycle in R. Let R = R X; dX = x . Then ϑ can be extended to a derivation ϑ on R if and only if ϑ(x) ∈ B(R ). Proof If ϑ can be extended, ϑ(x) = ϑ(d(X)) = dϑ (X) ∈ B(R ). Conversely, if ϑ(x) ∈ B(R ), choose an element G ∈ R such that d(G) = ϑ(x). If deg X is odd, we have R = R ⊕ RX. For r, s ∈ R, we define ϑ (r + sX) = ϑ(r) + (−1)deg ϑ ϑ(s)X + sG. If deg X is even, we have R =
RX (i) .
i≥0
For r0 , . . . , rm ∈ R, we define m m m
(i) (i) ϑ = ri X ϑ(ri )X + ri X (i−1) G. i=0
i=0
i=1
The verification that ϑ is well defined and becomes a derivation on R is straightforward. Definition 4.1.4 If ϑ , X and G are as above, we call ϑ the canonical extension of ϑ satisfying ϑ (X) = G.
4.1 Minimal DG resolutions
57
Example 4.1.5 Suppose that R is a DG A-algebra of the form R = R X; dX = x . Then by (4.1.3) we may consider the canonical extension ϑ of the trivial derivation (the zero map) on X satisfying ϑ (X) = 1. Explicitly, • ϑ (r + sX) = s if deg X is odd; m m • ϑ ( i=0 ri X (i) ) = i=1 ri X (i−1) if deg X is even. This derivation ϑ is called the derivation associated with the extension R → R . Lemma 4.1.6 Let ϑ : R → R be a derivation on a DG A-algebra R. Let {xi }i∈I be a set of cycles in R. Assume that there exist elements Gi ∈ R such that d(Gi ) = ϑ(xi ) for all i ∈ I. Let R = R {Xi }i∈I ; dXi = xi . Then ϑ has an extension to a derivation ϑ : R → R satisfying • ϑ (Xi ) = Gi for all i ∈ I; (m) (m−1) • ϑ (Xi ) = Xi Gi for all i ∈ I and m > 0, if deg Xi is even. Such an extension ϑ is unique. Proof The uniqueness of ϑ is clear. The existence of ϑ follows from (4.1.3) in view of Definition 1.2.4 of R {Xi }i∈I ; dXi = xi . Theorem 4.1.7 Let (A, m, K) be a noetherian local ring. Then there exists a free DG resolution R of the A-algebra K with R0 = A which is minimal, i.e., such that d(R) ⊂ mR. Proof We use the construction of (1.2.6). We begin taking R0 = A. At each step n > 0, we choose a minimal set of generators {cn,i }i∈In of the R0 -module ker Hn−1 (Rn−1 ) → Hn−1 (K) (= Hn−1 (Rn−1 ) if n−1 > 0), and a set of homogeneous cycles {xn,i }i∈In representing {cn,i }i∈In , and take Rn = Rn−1 {Xn,i }i∈In ; dXn,i = xn,i . Finally, we define R = lim Rn . −→ For each n > 0 and i ∈ In , we define ϑn,i as the unique derivation on Rn satisfying ϑn,i (Rn−1 ) = 0, ϑn,i (Xn,i ) = 1 and ϑn,i (Xn,j ) = 0 if i = j, and (m)
(m−1)
ϑn,i (Xn,i ) = Xn,i
(m)
and ϑn,i (Xn,j ) = 0 for i = j
58
Complete intersections
if m > 0 and deg Xn,i is even. Such a derivation exists by (4.1.6). We will show that each of the derivations ϑn,i can be extended to a derivation θn,i on R. By a limit argument, it is enough to show the following: let t ≥ n and suppose that ϑn,i has been extended to a derivation ϑ on Rt . Then ϑ can be extended to a derivation ϑ on Rt+1 . We will prove this. If Ht (Rt ) = 0, then Rt+1 = Rt , and so we put ϑ = ϑ. Otherwise let {xt+1,j }j∈It+1 be the set of homogeneous cycles chosen above. The existence of ϑ will follow from (4.1.6) if we show that (∗) ϑ(xt+1,j ) ∈ B(Rt ) for all j ∈ It+1 . Since deg ϑ = − deg Xn,i = −n < 0, and deg xt+1,j = t, we have ϑ(xt+1,j ) ∈ Zt−n (Rt ). Thus if t > n, (∗) is satisfied, since Hq (Rt ) = 0 for 0 < q < t. Hence we may assume that t = n. In this case the chosen cycles can be expressed as follows (∗∗) xt+1,j =
k rj,k Xn,k
+ yj ,
where rj,k ∈ R0 = A, rj,k = 0 for only finitely many indices k, and yj ∈ Rn−1 . Differentiating (∗∗) yields 0 = k rj,k xn,k +dyj . Since {xn,i }i∈In represents a minimal set of generators of Hn−1 (Rn−1 ), it follows that rj,k ∈ m for all k ∈ In . From (∗∗) we obtain ϑ(xt+1,j ) = rj,i · 1 ∈ mR0n = B0 (Rt ) (the last equality follows since t = n > 0). Hence for each n > 0 and i ∈ In , we have a derivation θn,i : R → R. We now show that
ker(θn,i ⊗A K) = (R ⊗A K)0 = K.
n>0, i∈In
/ K, we can choose For this, let z ∈ n>0, i∈In ker(θn,i ⊗A K). If z ∈ n > 0 and i ∈ In , so that z ∈ Rn−1 {Xn,j }j∈In ; dXn,j = xn,j and z∈ / Rn−1 {Xn,j }j∈In , j=i ; dXn,j = xn,j . We have the following commutative diagram with exact top row, where
4.1 Minimal DG resolutions
59
we abbreviate dXn,j = xn,j by dX = x: 0 → Rn−1 {Xn,j }j∈In ; dX = x → R
j=i
{Xn,j }j∈In ; dX = x → Rn−1 {Xn,j }j∈In ; dX = x
n−1
R
θ n,i
−−→
R
where the vertical maps are inclusions, and the right-hand map is the derivation associated with the extension Rn−1 {Xn,j }j∈In ; dXn,j = xn,j → Rn−1 {Xn,j }j∈In ; dXn,j = xn,j , j=i
(see (4.1.5)). Since the top row is a split exact sequence of free modules over R0 , we obtain a commutative diagram 0 → Rn−1 {Xn,j }j∈In ; dX = x ⊗A K → j=i
Rn−1 {Xn,j }j∈In ; dX = x ⊗A K → Rn−1 {Xn,j }j∈In ; dX = x ⊗A K θ n,i ⊗ K
A −−−−− −→
R ⊗A K
R ⊗A K
where the top row is exact and similarly the vertical maps are injective. Since z ∈ ker(θn,i ⊗A K), we deduce that also z ∈ Rn−1 {Xn,j }j∈In ,j=i ; dXn,j = xn,j , which is a contradiction. Finally, we show by induction on q that Bq (R ⊗A K) = 0. For q = 0 this is clear since B0 (R) = m. Let p > 0, and assume that Bq (R⊗A K) = 0 for q < p. For every n > 0 and i ∈ In , θn,i ⊗A K is of negative degree and commutes with the differential on R ⊗A K. Hence (θn,i ⊗A K)(Bp (R ⊗A K)) ⊂ Bq (R ⊗A K) = 0. q
Therefore Bp (R ⊗A K) is contained in Bp (R ⊗A K) ∩ ker(θn,i ⊗A K) = Bp (R ⊗A K) ∩ K = 0, n>0 i∈In
since p > 0.
60
Complete intersections 4.2 The main lemma
We prove here the following key result of Avramov. Lemma 4.2.1 Let f : (A, m, K) → (B, n, L) be a flat local homomorphism of noetherian local rings. Let {u1 , . . . , un } be a minimal set of generators of the ideal m of A, and {u1 , . . . , un , t1 , . . . , tm } a set of generators of the ideal n of B (we denote an element of A and its image in B by the same symbol). Then the homomorphism induced by f between the first Koszul homology modules h : H1 (u1 , . . . , un ; A) ⊗K L → H1 (u1 , . . . , un , t1 , . . . , tm ; B) is injective. Proof Let Q be a minimal free DG resolution of the A-algebra K (4.1.7) having 1-skeleton Q(1) = A U1 , . . . , Un ; dUj = uj . Then P := Q ⊗A B is a free DG resolution of the B-algebra K ⊗A B (since f is flat) and minimal (since f is local). Let g be the homomorphism Z := P T1 , . . . , Tm ; dTi = 1 ⊗ ti → K ⊗A B T1 , . . . , Tm ; dTi = 1 ⊗ ti obtained by applying − ⊗B B T1 , . . . , Tm ; dTi = 1 ⊗ ti to the map P → K ⊗A B. Since this last map induces isomorphisms in homology, so does g. Let L = B U1 , . . . , Un , T1 , . . . , Tm ; dUj = uj , dTi = ti be the Koszul complex associated to the sequence u1 , . . . , un , t1 , . . . , tm in B. Let M = L V1 , . . . , Vr ; dVi = vi where V1 , . . . , Vr are the variables of degree 2 of the minimal resolution Q, and vi ∈ A U1 , . . . , Un ⊂ B U1 , . . . , Un , T1 , . . . , Tm their images by the differential of Q. Since the classes of the cycles vi are a minimal set of generators of H1 (Q(1)) = H1 (u1 , . . . , un ; A), to see that h is injective we have to see that the classes of v1 , . . . , vr are linearly independent in H1 (L) = H1 (u1 , . . . , un , t1 , . . . , tm ; B). So suppose that the classes of dV1 , . . . , dVr are linearly dependent in H1 (L). This means that there is a V = ri Vi with ri ∈ B, some rj ∈ / n, and an S ∈ L2 such that dV = dS. Let Wα = Uα for 1 ≤ α ≤ n, n+m Wα = Tα−n for n < α ≤ n + m. Let S = α,β=1 bαβ Wα Wβ with α<β
4.2 The main lemma bαβ ∈ B. Define (V − S)(k) =
γk ,εα,β
r
(γi )
riγi Vi
i=1
61
n+m α,β=1 α<β
(bαβ Wα Wβ )εαβ ,
where the sum runs over all γk ≥ 0, εα,β such that 0 ≤ εαβ ≤ 1
and
r
i=1
γi +
n+m
εαβ = k.
α,β=1 α<β
Observe that this definition extends divided powers on the variables of even degree to the element of even degree V −S, using, as in the proof of (1.2.10), the rules
(x + y)(k) = x(i) y (j) , and i+j=k
(k)
(xy)
=
xk y (k)
if deg x is even, and deg y is even and ≥ 2,
0
if deg x or deg y is odd and k ≥ 2.
We deduce from this that d((V −S)(k) ) = (V −S)(k−1) d(V −S) = 0, and so (V − S)(k) is a cycle for all k ≥ 1. The products of the elements Ti , Uj (s) and Vq for s > 0 form part of a basis of the free B-module Z, and in the decomposition of (V − S)(k) with respect to this basis, there is a (k) summand rjk Vj ∈ / mZ. Since dZ ⊂ mZ, we deduce that the homology class of (V − S)(k) in Z is not zero for all k > 0. But as we have seen, Hk (Z) = Hk (K ⊗A B T1 , . . . , Tm ; dTi = 1 ⊗ ti ), which is obviously zero for k > m. This is a contradiction. Corollary 4.2.2 Let (A, m, K) → (B, n, L) be a flat local homomorphism of noetherian local rings. Then the canonical homomorphism H2 (A, K, L) → H2 (B, L, L) is injective. Proof This follows from (4.2.1) and the exact sequences (2.5.1): 0 → H2 (A, K, L) → H1 (a1 , . . . , am ; A) ⊗K L → · · · 0 → H2 (B, L, L) →
H1 (b1 , . . . , bn ; B)
→ ···
62
Complete intersections 4.3 Complete intersections
Definition 4.3.1 We say that a noetherian local ring (A, m, K) is a complete intersection if its m-adic completion can be given as a quotient of a regular local ring by an ideal generated by a regular sequence. Definition 4.3.2 Let (A, m, K) be a noetherian local ring. We define the complete intersection defect of A as the integer (1.4.4) d(A) = δ2 (A) − δ1 (A) + dim(A), where δi (A) = dimK Hi (A, K, K) for i = 1, 2. its (m-adic) Lemma 4.3.3 Let (A, m, K) be a noetherian local ring, A completion. Then the canonical homomorphisms K, K) Hn (A, K, K) → Hn (A, K, K) → H n (A, K, K) H n (A, are isomorphisms for n = 0, 1, 2. Proof n = 0 is clear since all modules are zero. Let 1 ≤ n ≤ 2. For any positive integer s the ring homomorphisms A → A/ms → A/m = K induce an exact sequence H n−1 (A, A/ms , K) → H n (A/ms , K, K) → H n (A, K, K) → H n (A, A/ms , K). Taking direct limits in s > 0, by (2.3.4) we obtain that the canonical homomorphism lim H n (A/ms , K, K) → H n (A, K, K) −→ is an isomorphism. The same argument applied to the local noetherian m, K) shows that the canonical homomorphism ring (A, m K, K) s , K, K) → H n (A, lim H n (A/ −→ m s show is an isomorphism. The canonical isomorphisms A/ms = A/ K, K) → H n (A, K, K) is an isomorphism. that the map H n (A, K, K) thus follow from The isomorphisms Hn (A, K, K) → Hn (A, (1.4.5).
4.3 Complete intersections
63
Proposition 4.3.4 Let (A, m, K) be a noetherian local ring. We have: a) d(A) = d(A). b) If A = R/I where (R, n, K) is a regular local ring, then d(A) = dimK (I/nI) − (dim(R) − dim(A)). c) d(A) ≥ 0. Proof a) follows from (4.3.3). b) From the exact sequence → H2 (A, K, K)
0 = H2 (R, K, K)
→ H1 (R, A, K) = I/nI → H1 (R, K, K) = n/n → H1 (A, K, K) → 0 2
we obtain δ2 (A) − δ1 (A) = dimK (I/nI) − dimK (n/n2 ) = dimK (I/nI) − dim(R). ≥ 0. By (3.2.5), A = R/I c) By a), it is enough to show that d(A) where R is a regular local ring, and so by b), d(A) = dimK (I/nI)−ht I ≥ 0. Proposition 4.3.5 Let A be a noetherian local ring. Then A is complete intersection if and only if d(A) = 0. Proof It follows from (4.3.4) (see [Mt, Theorem 17.4]).
Proposition 4.3.6 Let A be a noetherian local ring, and R any regular local ring such that A = R/I. Then A is complete intersection if and only if I is generated by a regular sequence. Proof (4.3.4), and (4.3.5).
Proposition 4.3.7 Let (A, m, K) → (B, n, L) be a flat local homomorphism of noetherian local rings. Then d(B) = d(A) + d(B ⊗A K). Proof Bearing in mind that Hn (A, K, L) = Hn (B, B⊗A K, L) by (1.4.3) and using (4.2.2), the Jacobi–Zariski exact sequence associated to B → B ⊗A K → L 0 → H2 (A, K, L) → H2 (B, L, L) → H2 (B ⊗A K, L, L) → H1 (A, K, L) → H1 (B, L, L) → H1 (B ⊗A K, L, L) → 0
64
Complete intersections
and the equality dim(B) = dim(A) + dim(B ⊗A K), give the desired result. Theorem 4.3.8 Let (A, m, K) → (B, n, L) be a flat local homomorphism of noetherian local rings. Then B is complete intersection if and only if A and B ⊗A K are complete intersection. Proof (4.3.4.c), (4.3.5), (4.3.7).
Theorem 4.3.9 Let (A, m, K) be a noetherian local ring, p a prime ideal of A. If A is complete intersection, then Ap is complete intersection. Proof If A is a quotient of a regular local ring R, then the result follows from (4.3.6) and Serre’s theorem: if R is regular then Rq is regular for any prime ideal q of R. So by (3.2.5), the result is valid for A. such that Let p be a prime ideal of A, and let q be a prime ideal of A is faithfully flat). Then q ∩ A = p (such an ideal exists, since A → A q , we deduce applying (4.3.8) to the flat local homomorphism Ap → A that Ap is complete intersection.
4.4 Appendix: Kunz’s theorem on regular local rings in characteristic p Lemma 4.4.1 Let A be a ring that contains a field of characteristic p > 0 and B an A-algebra. Let φA : A → A, φ(a) = ap be the Frobenius homomorphism, φA the ring A considered as A-module via φA , and similarly φB , φ B. The map induced by φA , φB H1 (A, B, M ) → H1 (φA, φB, M ) is zero for any φB-module M . Proof As in the proof of (2.4.7) we may assume that the homomorphism A → B is surjective with kernel I. Since the Frobenius homomorphism induces the zero map I/I 2 → I/I 2 the result follows from (1.4.1.c). Theorem 4.4.2 Let (A, m, K) be a noetherian local ring containing a field of characteristic p > 0. Let φ : A → A be the Frobenius homomorphism, and φA as above. The following are equivalent: a) A is a regular local ring.
Appendix: Kunz’s theorem
65
b) φA is a flat A-module. Proof a) =⇒ b) Let x1 , . . . , xn be a regular sequence generating the ideal m. By (2.6.1) we have an exact sequence φ φ H2 (A, K, φK) → H2 (φA, K ⊗A φA, φK) → TorA 1 (K, A) ⊗φA K →
H1 (A, K, φK) → H1 (φA, K ⊗A φA, φK) → 0. We have that ker(φA → K ⊗A φA) is generated by the regular sequence [Mt, Theorem 16.1], and so H2 (φA, K ⊗A φA, φK) = 0 by (2.5.2). On the other hand, the homomorphism
xp1 , . . . , xpn
H1 (A, K, φK) → H1 (φA, K ⊗A φ A, φK) can be identified with the φK-module homomorphism f : m/m2 ⊗K φK → m[p] /m · m[p] , defined by f ((x + m2 ) ⊗ a) = axp + m · m[p] , where m[p] = xp : x ∈ m A . Since A is regular, grm (A) = K[X1 , . . . , Xn ]. Via this isomorphism, the map f composed with the canonical homomorphism m[p] /m · m[p] → mp /mp+1 identifies to g : K[X1 , . . . , Xn ]1 → K[X1 , . . . , Xn ]p , defined by g(q(X1 , . . . , Xn )) = q(X1p , . . . , Xnp ), where K[X1 , . . . , Xn ]i is the ith homogeneous component of the polynomial ring K[X1 , . . . , Xn ]. This map g is clearly injective, and so f is injective. φ φ Therefore from the exact sequence we obtain TorA 1 (K, A)⊗φA K = 0. A φ φ Since Tor1 (K, A) is a A-module of finite type, Nakayama’s lemma says φ φ that TorA 1 (K, A) = 0, and therefore A is a flat A-module by the local flatness criterion [Mt, Theorem 22.3]. b) =⇒ a) Let E = Fp be the prime subfield. By (1.4.6), (2.4.1), (2.4.5), we can identify α : H2 (A, K, φK) → H2 (φA, φK, φK) with the homomorphism H1 (E, A, φK) → H1 (φE, φA, φK) which is zero by (4.4.1). But α is injective by (4.2.2). So H2 (A, K, φK) = 0 and then A is regular by (2.5.3), (1.4.5). Remarks i) The maps Hn (A, B, M ) → Hn (φA, φB, M ) as in (4.4.1) are zero also for n = 0, 2. For n = 0 it is clear from (1.4.1.b), since dbp ⊗ m = pbp−1 db ⊗ m = 0 in ΩφB|φA ⊗φB M . For n = 2, note
66
Complete intersections that by (1.4.1.d), (1.4.6) we can assume that A → B is surjective. If I = ker(A → B) and H1 (I) is the first Koszul homology module associated to a set of generators of I, we have an injection H2 (A, B, M ) → H1 (I) ⊗B M (same proof as (2.5.1)). So it is enough to show that the Frobenius homomorphism induces the zero map in H1 (I). Let {ai } be a set of generators of I, α, β : A[{Xi }] → A the homomorphisms of A-algebras defined by α(Xi ) = 0, β(Xi ) = ai . As in the proof of (2.5.4), A[{X }] H(I) = Tor1 i (α A,β A) (the fact that the number of variables in (2.5.4) is finite is not necessary). Applying − ⊗A[{Xi }] β A to the exact sequence 0 → (Xi ) → A[{Xi }] →
α
A→0
we obtain an exact sequence A[{Xi }] α
0 → Tor1
( A,β A) →
A[{Xi }] (Xi ) → (Xi − ai ) (Xi )(Xi − ai )
i )∩(Xi −ai ) and so H1 (I) = (X (Xi )(Xi −ai ) . If z ∈ (Xi ) ∩ (Xi − ai ), then p z ∈ (Xi )(Xi − ai ) and so the result follows. ii) It is clear that the maps Hn (A, B, M ) → Hn (φA, φB, M ) are zero for all n ∈ N by using simplicial resolutions [An3, Lemma 53]. iii) a) ⇒ b) can be also easily deduced from [Mt, Theorem 23.1].
5 Regular homomorphisms: Popescu’s theorem
There are two important results in this chapter. The main one is that any regular homomorphism is a filtered inductive limit of finite type smooth homomorphisms. We will present here Popescu’s proof ([Po1][Po3]), taking into consideration some simplifications by Ogoma, Andr´e and Swan ([Og2], [An4], [Sw]). In fact, our exposition will follow closely Swan’s paper, which incorporates some ideas of all these papers. In order to be consistent with the flavour of this book and also to show examples of the usefulness of these homological tools in Commutative Algebra, in some places where we can choose homological versus non homological methods we have preferred the former. This choice is also justified by the fact that nonhomological proofs are already available in literature (compare, e.g., the proof of Proposition 5.5.8 with [Bo, Chapter IX, §2.2], or that of Lemma 5.4.3 with [Mt, Theorem 23.1]). The interested reader should notice that a different proof of Popescu’s theorem was given by Spivakovsky in [Sp]. The second important result in this chapter is Theorem 5.6.1: the module of differentials of a regular homomorphism is flat. This result is due to Andr´e in 1974 [An1, Suppl´ement, Th´eor`eme 30]. In homological terms it means that if A → B is a homomorphism of noetherian rings, the vanishing of H1 (A, B, Bq /qBq ) for all prime ideals q of B implies the vanishing of H1 (A, B, M ) for any B-module M (see the proof of Proposition 5.6.2). Though this statement involves only H1 , Andr´e’s proof uses Hn for all n in an essential way. Alternatively, since 1986, we can deduce this result from Popescu’s theorem. This is how we prove it in this book. However, the proof of Popescu’s theorem is long and difficult. It would be desirable to find a simpler proof of (5.6.1).
67
68
Regular homomorphisms: Popescu’s theorem 5.1 The Jacobian ideal
Definition 5.1.1 Let A be a ring, M an A-module of finite presentation. Let x = {x1 , . . . , xn } be a set of generators of M , F a free A-module with basis {e1 , . . . , en }, p : F → M the homomorphism with p(ei ) = xi . Let y = {y1 , . . . , yr } be a set of generators of N = ker p. Let yj =
n
aij ei
aij ∈ A,
j = 1, . . . , r.
i=1
Let s be an integer and let ∆x,s be the ideal of A generated by the (n − s) × (n − s) minors of the matrix a = (aij ) (if t ≤ 0, we define the determinant of a t × t matrix as 1). Lemma 5.1.2 The ideal ∆x,s does not depend on the choice of the set of generators y of N . Proof Let y = {y1 , . . . , yt } be another set of generators of N , yj =
n
aij ei
aij ∈ A,
j = 1, . . . , t.
i=1
For each j = 1, . . . , t, let yj =
r
bkj yk .
k=1
Then ab = a , where b = (bkj ), etc. Therefore any (n − s) × (n − s) minor of a is a linear combination of (n − s) × (n − s) minors of a (easy computation; see, e.g., Bourbaki, Alg`ebre, chapitre III, Ex.§8.6), and so ∆x,s ⊂ ∆x,s . By symmetry, ∆x,s = ∆x,s . Lemma 5.1.3 The ideal ∆x,s depends only on M (and s) and not on the set of generators x of M . Proof It is enough to show that if x = {x1 , . . . , xn , xn+1 } with xn+1 ∈ n M a superfluous element, then ∆x,s = ∆x ,s . Let xn+1 = i=1 bi xi , bi ∈ A. Let N, y, aij , be as in (5.1.1), and p
0 → N → F −−→ M → 0 an exact sequence with F a free A-module with basis {e1 , . . . , en , en+1 }, and p (ei ) = xi for i = 1, . . . , n + 1. Consider the set of generators n y = {y1 , . . . , yr , − i=1 bi ei + en+1 }, where yi is the image of yi ∈ N
5.1 The Jacobian ideal
69
in N via the inclusion F → F taking ei to ei . The matrix (aij ) of this presentation is a11 . . . a1r −b1 . .. .. .. . . . a n1 . . . anr −bn 0 ... 0 1 The result follows easily using (5.1.2) and the fact that ∆x,s ⊂ ∆x,s+1 for any s. So we shall denote ∆M,s instead of ∆x,s . Example 5.1.4 If M is a free A-module of rank n, ∆M,n = A, ∆M,s = 0 for s < n. Lemma 5.1.5 If B is an A-algebra, then ∆M,s B = ∆M ⊗A B,s .
Proof Clear. Lemma 5.1.6 Let M1 , M2 be A-modules and M = M1 ⊕ M2 . Then
∆M,s = ∆M1 ,s1 ∆M2 ,s2 . s1 +s2 =s
In particular, ∆M,s = ∆M ⊕Am ,s+m for any m. Proof Take a presentation direct sum of two presentations. The matrix is of the form # " (aij ) 0 0 (bkl ) and so the result follows easily.
Lemma 5.1.7 Let M, F, N, n, r, etc., be as in (5.1.1). Then ∆M,n−r = A if and only if N is a free direct summand of F with basis {y1 , . . . , yr }. Proof Assume first that N is a free direct summand of F with basis {y1 , . . . , yr }. Then M ⊕ N = F is free. By (5.1.6) and (5.1.4) ∆M,n−r = ∆M ⊕N,n = A. Conversely, if ∆M,n−r = A, the set {y1 , . . . , yr } is linearly independent, and so N is free with basis {y1 , . . . , yr }. By (5.1.5), for any A-algebra B = A/I, we have ∆M ⊗A B,n−r = B, and so, by the same argument, the B-submodule generated by {y1 ⊗ 1, . . . , yr ⊗ 1} is free,
70
Regular homomorphisms: Popescu’s theorem
hence the homomorphism N ⊗A B → F ⊗A B is injective. Therefore M is a flat A-module, and then, being of finite presentation, projective. Thus the exact sequence 0→N →F →M →0
is split.
Definition 5.1.8 Let A be a ring, B = A[X1 , . . . , Xn ]/I, where I is a finite type ideal of A[X1 , . . . , Xn ], C = S −1 B, where S is a multiplicative subset of B. Let T be a multiplicative subset of A[X1 , . . . , Xn ] such that T −1 A[X1 , . . . , Xn ]/T −1 I = S −1 (A[X1 , . . . , Xn ]/I) = C. For each finite type ideal J ⊂ I, let BJ = A[X1 , . . . , Xn ]/J, SJ the image of T in BJ . Let ΩBJ |A be the BJ -module of differentials, which is of finite presentation, and let ∆BJ |A = ∆ΩBJ |A ,n−µ(J) , where µ(J) is the minimum number of generators of J. We can compute ∆BJ |A using the presentation J/J 2 → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] BJ → ΩBJ |A → 0. That is, if {f1 , . . . , fµ(J) } is a minimal set of generators of J, ∆BJ |A is the ideal of BJ generated by the µ(J) × µ(J) minors of the Jacobian matrix (∂fi /∂Xj ). Finally we define
HC|A (B) = rad( ∆BJ |A (J : I)C), J
where (J : I) = {f ∈ A[X1 , . . . , Xn ] : f I ⊂ J}. If R is a multiplicative subset of C, then R−1 HC|A (B) = HR−1 C|A (B). Proposition 5.1.9 Let q be a prime ideal of C. The following are equivalent: i) Cq is a smooth A-algebra. ii) HC|A (B) ⊂ q. In particular, C is a smooth A-algebra if and only if HC|A (B) = C. Proof It is enough to show the last claim, and we can assume that C is local. Suppose HC|A (B) = C. Then, since C is local, there exists J such that ∆BJ |A (J : I)C = C. That means ∆BJ |A C = C and (J : I)C = C.
5.1 The Jacobian ideal
71
Let m be a maximal ideal of T −1 A[X1 , . . . , Xn ] such that its image in C is the maximal ideal of this local ring. Then (J : I)C = C implies (T −1 J : T −1 I) = T −1 (J : I) ⊂ m, and so ((T −1 J)m : (T −1 I)m ) = (T −1 J : T −1 I)m = T −1 A[X1 , . . . , Xn ]m that is, (T −1 J)m = (T −1 I)m , and therefore, replacing T by the inverse image in A[X1 , . . . , Xn ] of T −1 A[X1 , . . . , Xn ] − m, we can assume T −1 J = T −1 I. Thus SJ−1 BJ = T −1 A[X1 , . . . , Xn ]/T −1 J = C, where SJ is the image of T in BJ . Therefore, by the first equality SJ−1 ∆BJ |A = ∆BJ |A SJ−1 BJ = ∆BJ |A C = C and (5.1.7), we have a split exact sequence 0 → T −1 J/(T −1 J)2 = SJ−1 (J/J 2 ) → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] C → ΩC|A → 0. By (2.3.1), (1.4.7), C is a smooth A-algebra. Conversely, assume that C is a smooth A-algebra. By (2.3.1) we have a split exact sequence 0 → S −1 (I/I 2 ) →
ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] C → ΩC|A → 0,
and so, since C is local, S −1 (I/I 2 ) is a free summand of the module ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] C. Note that rank S −1 (I/I 2 ) = µ(T −1 I/(T −1 I)2 ) = µ(T −1 I) by Nakayama’s lemma. Let r = rank S −1 (I/I 2 ) and let y1 , . . . , yr ∈ I be elements such that their images in T −1 I generate this ideal. Let J = (y1 , . . . , yr ), so that T −1 J = T −1 I and so (J : I)C = C. The above exact sequence can be written as 0 → T −1 J/(T −1 J)2 → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] C → ΩC|A → 0 and by (5.1.7), ∆ΩBJ |A ,n−µ(J) C = ∆ΩBJ |A ,n−r C = ∆ΩC|A ,n−r = C. Therefore ∆BJ |A (J : I)C = C, and so HC|A (B) = C. Remark 5.1.10 From (5.1.9), we deduce that HC|A (B) does not depend on B, and so we will denote it simply by HC|A . Definition 5.1.11 Let C be an A-algebra essentially of finite presentation. Let a ∈ HC|A . We say that a is standard with respect to a finite presentation C = S −1 (A[X1 , . . . , Xn ]/I) if there exists a finite type ideal
72
Regular homomorphisms: Popescu’s theorem
J ⊂ I such that a ∈ rad(∆BJ |A (J : I)C). We say that a is standard if it is standard with respect to some finite presentation. We say that a is strictly standard if a ∈ ∆BJ |A (J : I)C. Proposition 5.1.12 In the notation as in (5.1.11), the following are equivalent: i) a is standard. ii) a is nilpotent, or Ca is a smooth A-algebra and ΩCa |A is a stably free Ca -module. Proof Assume that a is standard and not nilpotent. Let J ⊂ I be an ideal of finite type such that a ∈ rad(∆BJ |A (J : I)C). We have rad(∆BJ |A (J : I)C)a = Ca , and so ∆BJ |A (J : I)Ca = Ca . As in the proof of (5.1.9), using (5.1.7) we deduce that (SJ−1 J/J 2 )a is a free Ca module and the exact sequence 0 → (SJ−1 J/J 2 )a → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] Ca → ΩCa |A → 0 is split exact, and so ΩCa |A is a stably free Ca -module. Assume now that Ca is a smooth A-algebra and ΩCa |A is stably free. Let Ca = S −1 (A[X1 , . . . , Xn ]/I) for some S. We have a split short exact sequence (2.3.1) 0 → S −1 (I/I 2 ) → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] Ca → ΩCa |A → 0. Since ΩCa |A is a stably free Ca -module, so is S −1 (I/I 2 ), that is, S (I/I 2 ) ⊕ (Ca )r is a free Ca -module. Replacing the presentation A[X1 , . . . , Xn ]/I by A[X1 , . . . , Xn , Y1 , . . . , Yr ]/(I + (Y1 , . . . , Yr )), we obtain a similar exact sequence with the same term on the right and where the middle term is the direct sum of the former middle term and (Ca )r . So this exact sequence has S −1 (I/I 2 ) ⊕ (Ca )r as the left term. Thus we can assume that S −1 (I/I 2 ) is a free Ca -module. Now the proof concludes in the same way as that of (5.1.9). −1
Proposition 5.1.13 Let f : B → C be a homomorphism of A-algebras essentially of finite type. Then (HB|A C) ∩ HC|B ⊂ HC|A . Proof Let q be a prime ideal of C such that (HB|A C)∩HC|B ⊂ q. Then HB|A ⊂ f −1 q =: p, and HC|B ⊂ q, that is, Bp is a smooth A-algebra and Cq is a smooth B-algebra. So Cq is a smooth A-algebra and then HC|A ⊂ q.
5.1 The Jacobian ideal
73
Proposition 5.1.14 Let B be an A-algebra essentially of finite presentation, C an A-algebra and D = B ⊗A C. Then HB|A D ⊂ HD|C . Proof It follows from (5.1.9), (2.3.1), (1.4.3), (1.4.7).
Lemma 5.1.15 Let A be a ring, M an A-module. Let SA (M ) be the symmetric A-algebra on M . The following are equivalent: i) SA (M ) is a smooth A-algebra. ii) M is a projective A-module. Proof SA (M ) is smooth if and only if H1 (A, SA (M ), SA (M )) = 0 and ΩSA (M )|A (= M ⊗A SA (M )) is a projective SA (M )-module. ii) =⇒ i) follows easily from the definition of smoothness. Alternatively, using (2.3.1), if M is projective, there exists a free A-module F and a factorization of the identity map M → F → M , and so SA (M ) → SA (F ) → SA (M ). Therefore the identity map factors as H1 (A, SA (M ), SA (M )) → H1 (A, SA (F ), SA (M )) → H1 (A, SA (M ), SA (M )) with middle term zero by (1.4.1.d). So H1 (A, SA (M ), SA (M )) = 0 and ΩSA (M )|A = M ⊗A SA (M ) is a projective SA (M )-module by base change. i) =⇒ ii) If SA (M ) is a smooth A-algebra, ΩSA (M )|A = M ⊗A SA (M ) is a projective SA (M )-module, and so by base change, M = M ⊗A SA (M ) ⊗SA (M ) A is a projective A-module. Example 5.1.16 Suppose that C = R−1 (A[X1 , . . . , Xn ])/I is a smooth A-algebra essentially of finite presentation. Then we have a split exact sequence 0 → I/I 2 → ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] C → ΩC|A → 0. Let D = SC (I/I 2 ). By (5.1.15), D is a smooth C-algebra and so a smooth A-algebra. So we have a split exact sequence (1.4.6), (2.3.1) 0 → ΩC|A ⊗C D → ΩD|A → ΩD|C → 0 and then ΩD|A = (ΩC|A ⊗C D) ⊕ ΩD|C = (ΩC|A ⊗C D) ⊕ (I/I 2 ⊗C D) = ΩA[X1 ,...,Xn ]|A ⊗A[X1 ,...,Xn ] D is a free D-module. So let C be an A-algebra essentially of finite presentation. The above construction shows that there exists a C-algebra of finite presentation
74
Regular homomorphisms: Popescu’s theorem
D (since I/I 2 is a C-module of finite presentation) such that HC|A D ⊂ HD|A and ΩDa |A is a free Da -module for all a ∈ HC|A (since Ca is a smooth A-algebra and Da = SCa (Ia /Ia2 )). In particular D has a presentation such that the image in D of any element a ∈ HC|A is standard (5.1.12). Moreover, every A-algebra homomorphism C → E factors as C → D → E.
5.2 The main lemmas Lemma 5.2.1 Let f : A → B be a homomorphism of noetherian rings, a ∈ A an element such that (0 : a) = (0 : a2 ), (0 : f (a)) = (0 : f (a)2 ). = A/(a), B = B/a2 B, B = B/aB (denoting a Let A = A/(a2 ), A instead of f (a) when clear). Let A → C → B be a factorization with C an A-algebra of finite type. Then, there exists a factorization A → D → := B with D an A-algebra of finite type and a homomorphism C → D D/aD of A-algebras making the diagram C
→ B
D
→
B
commute, and such that π −1 (HC|A B) ⊂ rad(HD|A B), where π : B → B is the canonical map. Proof Let C = A[X1 , . . . , Xr ]/I. Let I be an ideal of A[X1 , . . . , Xr ] such that a2 ∈ I and I = I/a2 A[X1 , . . . , Xr ], so C = A[X1 , . . . , Xr ]/I. By (5.1.8) we can choose a finite number of elements hij ∈ I(i ∈ Γ, j ∈ Λi ), wi ∈ A[X1 , . . . , Xr ] (i ∈ Γ) such that, denoting hij , wi their images in I and A[X1 , . . . , Xr ], and denoting Ji = (a2 , {hij }j∈Λi ) for each i, J i = ({hij }j∈Λi ) its image in I, we have
HC|A = rad pi C , i∈Γ
where for each i, wi I ⊂ J i , pi := mi wi , pi := mi wi , where each mi is a |Λi | × |Λi | minor of the matrix (∂hij /∂Xt )j,t , and mi its image in A[X1 , . . . , Xr ]. Take a finite set of generators of I of the form {hij }i∈Γ ,j∈Λi , where
5.2 The main lemmas
75
Γ = Γ∪{∗ }. Let v : A[X1 , . . . , Xr ] → B be an A-algebra homomorphism making the diagram A[X1 , . . . , Xr ] →
B
→
B
C
commute. We have v(hij ) ∈ a2 B. Let ζij ∈ aB be such that v(hij ) = aζij . Let Z = {Zij }i∈Γ ,j∈Λi be a set of variables and let gij = hij − aZij ∈ A[X1 , . . . , Xr , Z]. For each k ∈ Γ, pk I ⊂ Jk = (a2 , {hkj }j∈Λk ). So for each i ∈ Γ , j ∈ Λi , let
pk hij = Hijkj0 hkj0 + a2 Gijk , j0 ∈Λk
where if i = k we choose Hijkj0 = pk δj,j0 and Gijk = 0. Let Fijk = pk Zij − j0 ∈Λk Hijkj0 Zkj0 − aGijk . So Fijk = 0 when i = k, and
aFijk = pk (hij − gij ) − Hijkj0 (hkj0 − gkj0 ) − a2 Gijk = −pk gij +
j0 ∈Λk
Hijkj0 gkj0 ∈ (g),
j0 ∈Λk
where (g) is the ideal generated by {gij }i∈Γ ,j∈Λi . Therefore Fijk ∈ ((g) : a). Also, it is clear that Fijk ∈ (Z) + (a). Let (F ) = ({Fijk }i,j,k ), T = (g) + (F ), D = A[X1 , . . . , Xr , Z]/T . The homomorphism v : A[X1 , . . . , Xr ] → B extends to A[X1 , . . . , Xr , Z] via Zij → ζij and that one induces a homomorphism D → B, since (g) goes to 0 in B, F ⊂ ((g) : a) ∩ ((Z) + (a)), and this intersection also goes to 0 in B, since if ϕ ∈ ((g) : a), we have that aϕ ∈ (g) and so av(ϕ) = v(aϕ) = 0; if moreover ϕ ∈ (Z) + (a), then v(ϕ) ∈ aB (we have v(Zij ) = ζij ∈ aB) and so v(ϕ) ∈ aB ∩ (0 : a) = 0 since (0 : a) = (0 : a2 ) in B by assumption. We also have I = ({hij }i∈Γ ,j∈Λi ) ⊂ (g) + (a) in A[X1 , . . . , Xr , Z], := D/aD. and so we have a homomorphism C = A[X1 , . . . , Xr ]/I → D Clearly, this homomorphism makes the required diagram commutative. Moreover, since Da = Aa [X1 , . . . , Xr , Z]/T = Aa [X1 , . . . , Xr ], we have a ∈ HD|A . Choose an ordering in the sets Γ , Λi (for all i ∈ Γ ), and let h be the
76
Regular homomorphisms: Popescu’s theorem
column vector {hij }i∈Γ ,j∈Λi with the order (i, j) ≤ (i , j ) if (i < i ) or (i = i , j ≤ j ). Let k ∈ Γ. We can write the equality
pk hij = Hijkj0 hkj0 + a2 Gijk j0 ∈Λk
as pk hij =
Hijki0 j0 hi0 j0 + a2 Gijk ,
Γ
i0 ∈ j0 ∈ Λi0
where
Hijki0 j0 =
Hijkj0
if i0 = k,
0
if i0 = k,
and so, if Hk is the square matrix (Hijki0 j0 )(i∈Γ ,j∈Λi ),(i0 ∈Γ ,j0 ∈Λi0 ) (here ij refers to the row and i0 j0 to the column), we have pk h ≡ Hk h
mod a.
Let Sk1 ,k2 = pk2 Hk1 − Hk2 Hk1 . We have Sk1 ,k2 h ≡ pk2 pk1 h − Hk2 pk1 h ≡ pk2 pk1 h − pk2 pk1 h = 0
mod a.
So, since a ∈ A, differentiation in each coordinate gives Sk1 ,k2 (∂h/∂Xt ) ∈ (a, h) = (a, g) for any t = 1, . . . r, the meaning of this relation being that all coordinates of Sk1 ,k2 (∂h/∂Xt ) are in the ideal of A[X1 , . . . , Xr , Z] generated by the elements a, gij . Since pk1 = mk1 wk1 with mk1 a |Λk1 | × |Λk1 | minor of the matrix (∂hk1 j /∂Xt )j,t , reordering Γ such that k1 is the first element, and reordering X1 , . . . , Xr if necessary, we can suppose that mk1 = det(M ), where M is the |Λk1 | × |Λk1 |-submatrix of the upper left corner of (∂hij /∂Xt )(i,j),t . Since Hijk1 i0 j0 = 0 if i0 = k1 , we have (Sk1 ,k2 )(ij)(i0 j0 ) = 0 for i0 = k1 , and so we can write (Sk1 ,k2 ) = (S|0), where S is the matrix formed by the first |Λk1 | columns. Therefore we can write the relation Sk1 ,k2 (∂h/∂Xt ) ∈ (a, g) for any t = 1, . . . r, as
" (S|0)
M ∗
# ∈ (a, g),
5.2 The main lemmas
77
and so SM ∈ (a, g). Multiplying by the adjoint matrix of M , then by wk1 , we have pk1 Sk1 ,k2 ∈ (a, g). Viewing Z as a column vector, F(k1 ) the ideal generated by {Fijk1 }ij , and Fk2 the column vector (Fijk2 )ij , we have Hk1 Z ≡ pk1 Z mod (a, F(k1 ) ), and so Sk1 ,k2 Z = pk2 Hk1 Z − Hk2 Hk1 Z ≡ pk2 pk1 Z − pk1 Hk2 Z = pk1 (pk2 Z − Hk2 Z) ≡ pk1 Fk2 Let
mod (a, F(k1 ) ).
Ek1 = A[X1 , . . . , Xr , Z]/ (g) + F(k1 )
. pk1
We will show that Ek1 is a smooth A-algebra. Since
Hijk1 j0 gk1 j0 , aFijk1 = −pk1 gij + j0 ∈Λk1
each gij is 0 in A[X1 , . . . , Xr , Z]/ ({gk1 j }j∈Λk1 ) + F(k1 ) , so that pk1 Ek1 = A[X1 , . . . , Xr , Z]/ ({gk1 j }j∈Λk1 ) + F(k1 ) . Also, Fijk1 = pk1 pk1 Zij − j0 ∈Λk Hijk1 j0 Zk1 j0 − aGijk1 , and so we can solve for the Zij 1 and conclude that . Ek1 = A[X1 , . . . , Xr , {Zk1 j }j∈Λk1 ]/ {gk1 j }j∈Λk1
pk1
Since gij = hij − aZij , we have ∂gk1 j /∂(Xt , Zk1 j0 ) j∈Λ , t=1,...r, j0 ∈Λ
k1
k1
= (∂hk1 j /∂Xt )| − aI|Λk1 | .
By definition of pk1 , some |Λk1 | × |Λk1 | minor of (∂hk1 j /∂Xt ) divides pk1 , and so Ek1 is a smooth A-algebra. In particular, Ek1 is a flat A-module, and so, applying − ⊗A Ek1 to a
a2
the exact sequences 0 → (0 : a) → A −→ A and 0 → (0 : a2 ) → A −→ A, we see that (0 : a) = (0 : a2 ) in Ek1 . Since pk1 Sk1 ,k2 ∈ (a, g), we see that (Sk1 ,k2 )ij goes to 0 in Ek1 /aEk1 and so Fijk2 goes to 0 in Ek1 /aEk1 , that is, Fijk2 ∈ aEk1 for all i, j. We have aFijk2 ∈ (g) and then aFijk2 = 0 in Ek1 . Since (0 : a) = (0 : a2 ) in Ek1 , Fijk2 = 0 in Ek1 . Therefore Ek1 = Dpk1 , showing that pk1 ∈ HD|A . Since π −1 (HC|A B) is contained in the radical of the ideal generated by a and by the pk , we deduce π −1 (HC|A B) ⊂ rad(HD|A B).
78
Regular homomorphisms: Popescu’s theorem
Lemma 5.2.2 Let f : A → B be a homomorphism of noetherian rings, = A/(a4 ), B = B/a4 B, a ∈ A such that (0 : f (a)) = (0 : f (a)2 ). Let A etc. Let A → C → B be a factorization of f with C an A-algebra of finite type. Assume that the image of a in C is strictly standard and that →A making the diagram there exists a retraction of A-algebras ρ: C −→ B C A 2 )A commute. Let a = 0 : (0:a . Then there exists a factorization A → (0:a)A A C → D → B with D an A-algebra of finite type such that aD ⊂ HD|A . Proof Let C = A[X1 , . . . , Xr ]/I be such that the image a of a in C is i ) ∈ strictly standard for this presentation, and for each i let yi = ρ(X A, where Xi is the image of Xi in C. Let yi be a representant in A of yi . Let J ⊂ I be an ideal such that a ∈ ∆CJ |A (J : I)C, where CJ = A[X1 , . . . , Xr ]/J, and let P (X) ∈ ∆CJ |A (J : I) (where X = (X1 , . . . , Xr ), etc.) be a representant of a in A[X], that is, P (X)−a ∈ I. and so P (y)−a ∈ a4 A. In particular, P (y) = as Then P ( y )− a = 0 in A, with s ∈ A satisfying s ≡ 1 mod a. Let P (X) = v lv mv where the mv are u×u minors of (∂f /∂X) where u = µ(J), and f = (f1 , . . . , fu ) is a set of generators of J. Reordering the fi and the Xt if necessary, suppose that ∂f1 /∂X1 . . . ∂f1 /∂Xr .. .. . H1 = . ∂fu /∂X1 . . . ∂fu /∂Xr 0
. . . Ir−u
is such that det H1 = m1 . For each v, we define Hv similarly, with the same first u rows, and changing the columns in the lower part of the matrix (in the r − u lower rows) such that det Hv = mv . Let Gv be the adjoint matrix of Hv and Gv (X) = lv Gv . Thus Gv Hv = Hv Gv = lv mv Ir . We have then P (X)Ir = v Gv Hv and, in particular, asIr = P (y)Ir = v Gv (y)Hv (y). Let ξi be the image of Xi in B and ηi the image of yi in B. So ξi ≡ ηi mod a4 B, that is, ξi = ηi + a3 εi with εi ∈ aB. Let ε be the column vector (ε1 , . . . , εr ) and similarly ξ and η. Let τ v = (τ1v , . . . , τrv ) be the
5.2 The main lemmas
79
column vector Hv (y)ε (τiv ∈ aB for all i). We have in B
Gv (y)τ v = P (y)ε = asε v
and so s(ξ − η) = sa3 ε = a2
Gv (y)τ v .
v
Since all the Hv have the same first u rows, the same is true for the v , . . . , τrv ). τ . So we can write τ v = (τ1 , . . . , τu , τu+1 v v Let W1 , . . . , Wr be variables, and T = (T1 , . . . , Tu , Tu+1 , . . . , Trv ) with v Tj , Tj , variables. Let
Gv (y)T v )j ∈ A[X, W , T ]. hj (X, W , T ) = s(Xj − yj ) − a3 Wj − a2 ( v
v
Let fi (X)−fi (y) =
∂fi (y)(Xj −yj )+higher degree terms in (Xj − yj ), ∂Xj j
Since s(Xj − yj ) ≡ a3 Wj + a2 ( v Gv (y)T v )j mod hj , we see that if m = max{deg fi }, then sm (fi (X) − fi (y)) ≡
sm−1
j
∂fi (y)(a3 Wj + a2 ( Gv (y)T v )j ) + a4 Qi ∂Xj v
modulo (h), where (h) = ({hj }), and Qi ∈ (W , T )2 A[W , T ]. So sm fi (X) − sm fi (y) ≡ sm−1 a2
∂fi
(y)( Gv (y)T v )j + a3 Qi mod(h) ∂X j v j
with Qi ∈ j AWj + a(W , T )2 A[W , T ]. ∂fi Now j ∂Xj (y)(Gv (y)T v )j = (Hv (y)Gv (y)T v )i = (lv (y)mv (y)T v )i = lv (y)mv (y)Ti (since i ≤ u), and so
∂fi (y)( Gv (y)T v )j = lv (y)mv (y)Ti = asTi . ∂Xj v v j Therefore sm fi (X) − sm fi (y) ≡ sm a3 Ti + a3 Qi
mod (h).
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Regular homomorphisms: Popescu’s theorem
Since fi (y) ∈ a4 A, let fi (y) = a3 ci with ci ∈ aA. Let gi = sm ci + sm Ti + Qi ∈ A[W , T ]. Then we have a3 gi = sm a3 ci + sm a3 Ti + a3 Qi = sm fi (y) + sm a3 Ti + a3 Qi ≡ sm fi (X)
mod (h).
Let D = A[X, W , T ]/(I + (g, h)) = C[W , T ]/(g, h), where (g) is the ideal generated by {gi } (in each of these rings). We have a canonical homomorphism of A-algebras ϕ : A[X, W , T ] → B, sending Xj to ξj , Wj to 0, Tj to τj , Tjv to τjv . This homomorphism takes h and I to 0. So the above congruence mod (h) shows that a3 ϕ(gi ) = sm ϕ(fi (X)) = 0. That is, ϕ(gi ) ∈ (0 : a3 )B = (0 : a)B . Since ϕ(gi ) ∈ aB (by definition of gi , since ci ∈ aA, τjv ∈ aB, Qi ∈ j AWj + a(W , T )2 A[W , T ] and ϕ(Wj ) = 0), we have ϕ(gi ) = ab with b ∈ B, and aϕ(gi ) = 0 implies b ∈ (0 : a2 )B = (0 : a)B . Therefore ϕ(gi ) = ab = 0, and so ϕ defines a homomorphism D → B. Next we see that a ∈ HD|A . Since a3 gi ≡ sm fi (X) mod (h), we have that gi is 0 in Ca [W , T ]/(h), and so Da = Ca [W , T ]/(h). We can solve in this ring the equations hi = 0 for the Wi , and then Da = Ca [T ]. So Da is a smooth Ca -algebra, and therefore a smooth A-algebra, that is, a ∈ HD|A . By definition of hj , Xj ≡ yj mod (a2 , hj )s in As [X, W , T ]. In particular, if q(X) ∈ I then q(X) ≡ q(y) ≡ 0 mod (a2 , h)s and then IAs [X, W , T ] ⊂ (a2 , h)As [X, W , T ]. Similarly, P (X) ≡ P (y) = as mod (a2 , h)s , that is, for some λ, sλ P (X) ≡ asλ+1 mod (a2 , h) in A[X, W , T ]. Since s ≡ 1 mod a, we have as ≡ a mod a2 . Therefore sλ P (X) ≡ a mod (a2 , h) in A[X, W , T ], and then sλ P (X) ≡ ta mod h in A[X, W , T ] for some t ∈ A[X, W , T ], t ≡ 1 mod a in A[X, W , T ]. Let E = A[X, W , T ]/(g, h). We have D = E/IE. From a3 gi ≡ m s fi (X) mod h, we deduce that fi = 0 in Es . Since P (X)I ⊂ J = (f1 , . . . , fu ), we have atIEs = 0, and so East = Dast . The A-algebra Da is smooth, and then so is Dast = East , that is, ast ∈ HE|A . We shall show that, in fact, st ∈ HE|A . In Es = As [X, W , T ]/(g, h) we can solve the equations hj = 0 for the Xj so that Es = As [W , T ]/(g). For each i, j = 1, . . . , u, we have ∂gi /∂Tj ≡ δij sm mod a, since Qi ∈ 2 mu + j AWj +a(W , T ) A[W , T ]. So, taking determinants, we see that s mu+1 ae ∈ HEs |As for some e ∈ Es . Thus s t + stae ∈ HE|A . Since ast ∈ HE|A , we obtain smu+1 t ∈ HE|A , and then st ∈ HE|A . Since IAs [X, W , T ] ⊂ (a2 , h)As [X, W , T ], we have IEst ⊂ aEst and then IEst = ab, where b is an ideal of Est . Since atIEs = 0, we have
5.2 The main lemmas
81
aIEst = 0 and so a2 b = 0. Hence b ⊂ (0 : a2 )Est ⊂ (0 : a(0 :
(0 : a2 ) ))Est = (0 : aa)Est (0 : a)
(denoting by a0 the image of a ∈ A in Est , we have b ⊂ (0 : a20 ) ⊂ (0 : a2 ) (0 : a20 ) )) = (0 : a(0 : ))Est = (0 : aa)Est , where in (0 : a0 (0 : (0 : a0 ) (0 : a) (0 : a2 )Est (0 : a20 ) = , . . . , we have used that Est , the middle equalities (0 : a0 ) (0 : a)Est being smooth over A, is a flat A-algebra). Therefore we deduce aab = 0, that is, aIEst = 0, and so for any α ∈ a, IEstα = 0, and then Dstα = Estα . Thus stα ∈ HD|A . Since a ∈ HD|A and s, t are ≡ 1 mod a, we see that 1 − st = γa ∈ HD|A for some γ, so α − stα ∈ HD|A , and then finally α ∈ HD|A . Lemma 5.2.3 Let A be a ring, B = A[X1 , . . . , Xn ]/I, where I is a finite type ideal of A[X1 , . . . , Xn ], C = S −1 B, where S is a multiplicative subset of B. Let A be an A-algebra, B = B ⊗A A , I = IA [X1 , . . . , Xn ], C = C ⊗A A . If a ∈ C is a strictly standard element for the presentation C = S −1 (A[X1 , . . . , Xn ]/I), then the image of a in C is strictly standard for the presentation C = S −1 (A [X1 , . . . , Xn ]/I ). Proof Let J ⊂ I be a finite type ideal such that a ∈ ∆BJ |A (J : I)C. Let J = JA [X1 , . . . , Xn ]. By (5.1.5) we have ∆BJ |A BJ = ∆ΩBJ |A ,n−µ(J) BJ = ∆ΩB
J
|A ,n−µ(J)
⊂ ∆ΩB
J
|A ,n−µ(J
)
= ∆B |A J
since µ(J ) ≤ µ(J). On the other hand, (J : I)A [X1 , . . . , Xn ] ⊂ (J : I ), so ∆BJ |A (J : I)C ⊂ ∆B |A (J : I )C . J
Lemma 5.2.4 Let f : A → B be a homomorphism of noetherian rings, a ∈ A such that (0 : a) = (0 : a2 ), (0 : f (a)) = (0 : f (a)2 ). Let = A/(a4 ), B = B/a4 B, etc. Let A → C → B, A → D → B be A factorizations of f where C and D are A-algebras of finite type. Assume →D making comthat there exists a homomorphism of A-algebras ϕ: C mutative the diagram C → D
B
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Regular homomorphisms: Popescu’s theorem
If the image of a in C is strictly standard, then there exists a factorization A → E → B of f with E an A-algebra of finite type, and A-algebra homomorphisms C → E, D → E, such that the diagram C
→
E ↓ B
←
D
is commutative and HD|A E ⊂ HE|A . = ϕ( Clearly, ρ( c ⊗ d) c)d. Proof Let F = C ⊗A D. Let ρ : F → D, composition with the canonical homomorphism D → F is the identity and the diagram of D, F → D
B
is commutative. Applying (5.2.2) to D → F → B and to the image of a in F (which is strictly standard in F by (5.2.3)), we see that there (0 : a2 )D )D exists D → F → E → B with aE ⊂ HE|D , where a = (0 : (0 : a)D (denoting also by a the image of a in D). Let x ∈ HD|A . Then Dx is smooth over A and in particular A-flat. Therefore (0 : a) = (0 : a2 ) implies (0 : a)Dx = (0 : a2 )Dx . So ax = Dx , and then Ex ⊂ (HE|D )x = HEx |Dx . Thus Ex is a smooth Dx -algebra and so a smooth A-algebra, that is x ∈ HE|A . Lemma 5.2.5 Let A → C → B be homomorphisms of noetherian rings with C an A-algebra of finite type. Let a ∈ A be such that (0 : a) = (0 : a2 ) in A and in B, and such that the image of a in C is strictly standard. Let c ≥ 8 be an integer and A = A/ac A, B = B/ac B, etc. Assume that we have a factorization A → C → F → B with F an A-algebra of finite type. Then, there exists a factorization A → C → E → B with E an A-algebra of finite type such that π −1 (HF |A B) ⊂ rad(HE|A B), where π : B → B is the canonical map. = B/a4 B, etc. By (5.2.1) (with = A/a4 A, B Proof Let c = 8. Let A 4 a instead of a), there exists a factorization A → D → B with D an A making a diagram algebra of finite type and a homomorphism F → D, −1 as in (5.2.1) commute, such that π (HF |A B) ⊂ rad(HD|A B). → F, and The homomorphism C → F induces a homomorphism C
5.3 Statement of the theorem
83
induces F → D. So we have a homothe homomorphism F → D → D. Applying (5.2.4) to the factorizations A → C → morphism C B, A → D → B, we obtain a factorization A → C → E → B with E an A-algebra of finite type such that HD|A E ⊂ HE|A .
This proves the result for c = 8. If c ≥ 8, by (5.1.14), HF |A F ⊂ H ,
F |A
where A = A/a8 A, F = F/a8 F , and so the result follows from the case c = 8.
5.3 Statement of the theorem Definition 5.3.1 A homomorphism A → B of noetherian rings is regular if it is flat and the fibers k(p) → B ⊗A k(p) are geometrically regular for all prime ideals p of A. The main result we want to prove in this chapter is: Theorem 5.3.2 Let f : A → B be a regular homomorphism of noetherian rings. Then f is a filtered inductive limit of smooth homomorphisms of finite type. Theorem 5.3.3 Let f : A → B be a regular homomorphism of noetheg rian rings. Assume that we have a factorization of f , A → C − → B, where C is an A-algebra of finite type. Then there exists a factorization C → D → B of g such that D is a smooth A-algebra of finite type. These two theorems are equivalent. For, if f = lim(fi : A → Di ) −→ g
i
with each fi smooth of finite type, and A → C − → B is a factorization of f with C = A[X1 , . . . , Xn ]/(h1 , . . . , hr ), then the homomorphism A[X1 , . . . , Xn ] → B factors through some Di , and we can take i large enough so that the images of the elements h1 , . . . , hr are 0. So g factors through some Di . Conversely, let f = lim(φi : A → Ci ) with each φi of finite type. For −→ i each i, we factor Ci → B as Ci → Di → B with D a smooth A-algebra of finite type. So we have a homomorphism lim Di → B which is surjective. −→ i To see that it is also injective, since Di is an A-algebra of finite type, reasoning as before, there exists some j ≥ i such that Di → B factors as Di → Cj → B. So if x ∈ lim Di goes to 0 in B, taking some i such −→ i
84
Regular homomorphisms: Popescu’s theorem
that x ∈ Di and some k ≥ j such that x goes to 0 in Ck , we see that x is 0 in Dk , and so in lim Di . −→ i
Lemma 5.3.4 Let B be an A-algebra of finite presentation, S a multiplicative subset of B. If S −1 B is a smooth A-algebra, then there exists some u ∈ S such that Bu is a smooth A-algebra. Proof As we saw in (5.1.8), S −1 HB|A = HS −1 B|A = S −1 B. Therefore there exists x ∈ HB|A , s ∈ S, such that x/s = 1 in S −1 B, that is, there exist s, t ∈ S such that (x − s)t = 1 in B. Thus (HB|A )st = Bst , i.e., HBst |A = Bst . Lemma 5.3.5 Let A → C → B be ring homomorphisms with C an Aalgebra of finite presentation. If there exists a factorization A → C → E → B with E a smooth A-algebra essentially of finite presentation, then there exists a factorization A → C → D → B with D a smooth A-algebra of finite presentation. Proof Let E = S −1 F , with F of finite presentation over A. By (5.3.4), there exists s ∈ S such that Fs is a smooth A-algebra. Let C = A[X1 , . . . , Xn ]/(f1 , . . . , fr ). Since n and r are finite, there exists t ∈ S such that C → E = S −1 F factors through C → Ft → E = S −1 F , and so C → E factors as C → D := Fst → E. Lemma 5.3.6 Let A → C → B be homomorphisms of noetherian rings, with C an A-algebra of finite type. i) Let a ∈ A be such that its image in B belongs to rad(HC|A B). Then there exists a factorization A → C → D → B with D an A-algebra of finite type such that HC|A D ⊂ HD|A and the image of a in D is standard with respect to a finite presentation of D. So there exists an integer s such that the image of as in D is strictly standard. ii) If HC|A B = B, then there exists a factorization C → D → B with D a smooth A-algebra of finite type. Proof i) Let r be an integer such that the image of ar in B belongs to HC|A B. If HC|A = (c1 , . . . , cn ), write ar = i bi ci with bi ∈ B. Let E = C[X1 , . . . , Xn ]/(ar − i ci Xi ), and let ϕ : E → B be the homomorphism of C-algebras that sends Xi to bi . Since Eci =
5.3 Statement of the theorem
85
Cci [X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ], which is smooth over A, we see that each ci ∈ HE|A and so HC|A E ⊂ HE|A . By definition of E, the image of ar in E belongs to HC|A E, and so to HE|A , and since this ideal is radical, the image of a in E belongs to HE|A . By (5.1.16) applied to A → E, we see that there exists an E-algebra of finite type D with HE|A D ⊂ HD|A , and the image of a in D is standard for some presentation of D. Moreover, the homomorphism E → B factors through D. ii) Take a = 1 in i). Theorem 5.3.7 Let f : A → B be a homomorphism of noetherian rings, A → C → B a factorization of f with C an A-algebra of finite type. Let q be a prime ideal of B minimal among those prime ideals containing HC|A B, and assume that p = f −1 (q) is also minimal over f −1 (HC|A B). Assume moreover that Ap → Bq is formally smooth. Then there exists A → C → D → B with D an A-algebra of finite type such that HC|A B ⊂ rad(HD|A B) ⊂ q. We now see that Theorem 5.3.7 implies Theorem 5.3.3. Assume that (5.3.7) holds. Let f : A → B be a regular homomorphism of noetherian rings, A → C → B a factorization with C an A-algebra of finite type. Since B is noetherian, we can choose a factorization A → C → C → B with rad(HC|A B) maximal. By (5.3.6.ii), in order to prove (5.3.3), we have to show that HC|A B = B. If not, let q be a prime ideal of B minimal among those containing HC|A B. It is enough to get a factorization A → C → D → B with D an A-algebra of finite type such that HC|A B ⊂ rad(HD|A B) ⊂ q. If a := f −1 (HC|A B) and p is a minimal prime ideal over a, then we can choose q minimal over HC|A B such that f −1 (q) = p, since if rad(HC|A B) = q1 ∩ · · · ∩ qn , then rad(a) = f −1 (q1 ) ∩ · · · ∩ f −1 (qn ) ⊂ p and therefore some f −1 (qi ) ⊂ p. Thus (5.3.7) implies (5.3.3). We prove Theorem 5.3.7 at the end of Sections 5.4 and 5.5. Definition 5.3.8 Let A → C → B be ring homomorphisms with C an A-algebra of finite type. Let q be a prime ideal of B. We say that A → C → B ⊃ q is resolvable if the conclusion of (5.3.7) holds, that is, if there exists a factorization A → C → D → B with D an A-algebra of finite type such that HC|A B ⊂ rad(HD|A B) ⊂ q. Proposition 5.3.9 Let A → C → B be homomorphisms of noetherian
86
Regular homomorphisms: Popescu’s theorem
rings, with C an A-algebra of finite type. Let q be a prime ideal of B, and a ∈ A such that its image in C is strictly standard. If HC|A B ⊂ q, then A → C → B ⊃ q is (trivially) resolvable. If HC|A B ⊂ q (and in particular the image of a in B belongs to q), let c ≥ 8 be an integer and A = A/ac A, B = B/ac B, q = q/ac B, etc. Assume that (0 : a) = (0 : a2 ) in A and in B. If A → C → B ⊃ q is resolvable, then A → C → B ⊃ q is resolvable. Proof Assume HC|A B ⊂ q. Let A → C → D → B be such that HC|A B ⊂ HD|A B ⊂ q. By (5.2.5), there exists A → C → E → B with π −1 (HD|A B) ⊂ rad(HE|A B), where π : B → B is the canonical map. By (5.1.14), HC|A C ⊂ HC|A , so HC|A B ⊂ HC|A B, and then HC|A B ⊂ π −1 (HC|A B). Therefore HC|A B ⊂ π −1 (HC|A B) ⊂ π −1 (HD|A B) ⊂ rad(HE|A B). On the other hand, rad(HE|A B) ⊂ q, since HD|A B ⊂ q and so π (HD|A B) ⊂ π −1 (q) = q. −1
Proposition 5.3.10 Let A → C → B be homomorphisms of noetherian rings, with C an A-algebra of finite type. Let q be a prime ideal of B containing HC|A B. Let e ≥ 1 be an integer and c = 8e. Let a1 , . . . , ar ∈ A be such that the image of each aei in C is strictly standard, and for each i, ((ac1 , . . . , aci−1 ) : a2i ) = ((ac1 , . . . , aci−1 ) : ai ) in A and in B (when i = 1 we understand (0 : a21 ) = (0 : a1 )). Let A = A/(ac1 , . . . , acr ), etc. If A → C → B ⊃ q is resolvable, then A → C → B ⊃ q is resolvable. Proof Replacing each ai by aei , we can assume that e = 1. If r = 1, the result follows from (5.3.9). Suppose that the result holds for r − 1, and let A = A/(ac1 , . . . , acr−1 ), etc. By (5.2.3) the image of ar in C is strictly standard and so the result follows by induction applying (5.3.9) to A → C → B ⊃ q . Proposition 5.3.11 If Theorem 5.3.7 holds whenever ht(p) = 0, then it holds in general. Proof With the notation as in (5.3.7), assume that ht(p) > 0. Since p is minimal over f −1 (HC|A B), there exists a ∈ f −1 (HC|A B) such that a∈ / n, for any prime ideal n ⊂ p of A with ht n = 0 [AM, Proposition 1.11]. That is, ht(a) ≥ 1, and so ht(p/(a)) < ht p. By (5.3.6), there
5.4 The separable case
87
exists A → C → D → B with HC|A D ⊂ HD|A and the image of a in D standard. Replacing a by as if necessary, we can suppose that the image of a in D is strictly standard and that (0 : a) = (0 : a2 ) in A and in B. If HD|A B ⊂ q, then the result holds. If HD|A B ⊂ q, then q is a minimal prime ideal over HD|A B (since it is minimal over HC|A B), and similarly p is minimal over f −1 (HD|A B). Then the result follows by induction on ht p, applying (5.3.9) to A → D → B ⊃ q.
5.4 The separable case Proposition 5.4.1 Let A → C → B be homomorphisms of noetherian rings, with C an A-algebra of finite type, and q a prime ideal of B minimal over HC|A B. If A → C → Bq ⊃ qBq is resolvable, then there exists A → C → D → B with HD|A B ⊂ q. Remark We do not claim that HC|A B ⊂ rad(HD|A B). See (5.4.2). Proof By assumption, there exists an A-algebra of finite type E and a factorization A → C → E → Bq such that HE|A Bq ⊂ qBq , that is, HE|A Bq = Bq . By (5.3.6.ii) we can assume that E is a smooth Aalgebra. Let E = C[X1 , . . . , Xn ]/(F1 , . . . , Fm ). Since n and m are finite, we can choose s ∈ B − q such that the homomorphism E → Bq factors through Bs . Also, replacing s by some power, we can assume that the image of each Xi in Bs is of the form zi /s with zi ∈ B. Consider the commutative diagram ϕ
C[X1 , . . . , Xn , Y ] −−→ λ B
−→
E[Y ] i Bs
defined by ϕ(Xi ) = Y xi (where xi ∈ E is the image of Xi ), ϕ(Y ) = Y , λ(Xi ) = zi , λ(Y ) = s, i(Y ) = s. Since the image of λ(ker ϕ) in Bs is 0, there exists an integer t such that st λ(ker ϕ) = 0 in B. Let D = C[X1 , . . . , Xn , Y ]/Y t (ker ϕ). It is clear that λ induces a C-algebra homomorphism D → B. If y is the class of Y in D, Dy = E[Y ]y = E[Y, 1/Y ] which is a smooth E-algebra, and so smooth over A. Thus s = λ(Y ) ∈ HD|A B and then HD|A B ⊂ q.
88
Regular homomorphisms: Popescu’s theorem
Proposition 5.4.2 Let A → C → B, q be as in (5.4.1). Assume moreover that ht q = 0. If there exists a factorization A → C → D → B with D an A-algebra of finite type and HD|A B ⊂ q, then there exists A → C → D → D → B with D an A-algebra of finite type and HC|A B ⊂ rad(HD |A B) ⊂ q. Proof If HC|A B is nilpotent, we can take D = D. Assume that HC|A B is not nilpotent, and in particular, q is not nilpotent and (0) is not a primary ideal. Let 0 = a ∩ b be a decomposition with a and b ideals of B such that Ass(B/a) = {q}, q ∈Ass(B/b) / [Mt, Theorems 6.6 and 6.8]. Since q is not nilpotent and rad a = q, we have b = 0. Suppose that a = (z1 , . . . , zr ) and b = (y1 , . . . , ys ). Then if D = C[X1 , . . . , Xn ]/(F1 , . . . , Fm ), let D = C[X1 , . . . , Xn , Y1 , . . . , Ys , Z1 , . . . , Zr )]/({Yi Fj }i,j , {Yk Zl }k,l ). The homomorphism C[X1 , . . . , Xn , Y1 , . . . , Ys , Z1 , . . . , Zr ] → B of algebras over C[X1 , . . . , Xn ] that sends Yi to yi and Zj to zj factors through a homomorphism ϕ : D → B, since the image of Yi Fj is 0, and the image of Yk Zl is yk zl ∈ ba ⊂ a ∩ b = 0. / q. Since b ⊂ q, reordering {y1 , . . . , ys } if necessary, we can assume y1 ∈ −1 Let p = ϕ (q). The image Y 1 of Y1 in D does not belong to p and DY 1 = (C[X1 , . . . , Xn , Y1 , . . . , Ys ]/(F1 , . . . , Fm ))Y 1 = D[Y1 , . . . , Ys ]Y 1 , which is a smooth D-algebra. Therefore, HD |D ⊂ p. By assumption, HD|A B ⊂ q, and so HD |A B ⊂ q by (5.1.13) and [AM, Proposition 1.11.ii]. Finally, we shall see that HC|A B ⊂ rad(HD |A B). Let n be a prime ideal of B such that q ⊂ n. We have a ⊂ n, and so, after reordering z1 , . . . , zr if necessary, we can assume z1 ∈ / n. The image Z 1 of Z1 in D −1 is isomorphic to a does not belong to m := ϕ (n) as above, and so DZ 1 localization of C[X1 , . . . , Xn , Z1 , . . . , Zr ] which is smooth over C. Then, q ⊂ n implies HD |C ⊂ m = ϕ−1 (n). Therefore, if n is a prime ideal of B such that HC|A B ⊂ n, we have q ⊂ n, and so HD |C ⊂ m. On the other hand, HC|A B ⊂ n implies HC|A D ⊂ m. By (5.1.13), HD |A ⊂ m, and so HD |A B ⊂ n. Thus HC|A B ⊂ rad(HD |A B). Lemma 5.4.3 Let f : (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings. Suppose that A, B and B ⊗A K are regular and dim B = dim A + dim B ⊗A K. Then f is flat. If moreover L|K is separable, then f is formally smooth.
5.4 The separable case
89
Proof By the local flatness criterion and (2.6.1), to prove the flatness of f we have to show that the homomorphism α : H2 (A, K, L) → H2 (B, B ⊗A K, L) is surjective, and the homomorphism β : H1 (A, K, L) → H1 (B, B ⊗A K, L) is injective. The surjectivity of α is clear, since H2 (B, B ⊗A K, L) = 0 by (2.5.4). On the other hand, H1 (A, K, L) = m/m2 ⊗K L, and then dimK H1 (A, K, L) = dim A, since A is regular. Similarly, since B and B ⊗A K are regular, dimK H1 (B, B ⊗A K, L) = dim B − dim B ⊗A K = dim A. Since β is surjective (2.6.2), it is an isomorphism. If L|K is separable, f is formally smooth by (2.6.5), (2.5.8). Remark The assumptions of (5.4.3), though enough for our purposes, are unnecessarily strong. See, e.g., [Mt, Theorem 23.1]. Lemma 5.4.4 Let A → B → C be local homomorphisms of noetherian local rings such that B and C are flat as A-modules. Let I be a proper ideal of A. If B/IB → C/IC is flat (resp. formally smooth), then B → C is flat (resp. formally smooth). Proof The flat case follows easily from the local flatness criterion [Mt, Theorem 22.3, (4) =⇒ (1)]. The formally smooth case follows from the flat case by (2.3.5), bearing in mind that if L is the residue field of C, then H1 (B, C, L) = H1 (B/IB, C/IC, L) by (1.4.3).
Proof of Theorem 5.3.7 for separable residue field extension k(q)|k(p) By (5.3.11) we can assume ht p = 0. Note that this reduction does not affect the separability assumption. Induction on ht q. If ht q = 0, by (5.4.1), (5.4.2) we can assume that A and B are artinian local rings, and the homomorphism is local. In this case, the fibre B/pB is regular (p is the maximal ideal of A) by (2.5.8), and so is a field, that is, pB is the maximal ideal of B. Therefore B is a Cohen A-algebra (3.1.1). Since the residue field extension is separable, by (3.1.8) and its proof, we see that B is a filtered inductive limit of local subrings that are A-algebras essentially of finite type, and that are formally smooth
90
Regular homomorphisms: Popescu’s theorem
by (3.1.2). So, as we saw in the proof that Theorems 5.3.2 and 5.3.3 are equivalent, we see now that for any A → C → B with C an Aalgebra of finite type, there exists A → C → E → B with E a smooth algebra essentially of finite type (2.3.8), and so by (5.3.5) there exists A → C → D → B with D a smooth algebra of finite type, that is HD|A = D. If ht q > 0, since q is minimal over HC|A B, we have rad(HC|A B)Bq = qBq . Since Bq /pBq is a regular local ring (2.5.8), there exists an element ξ ∈ rad(HC|A B) whose image in Bq /pBq is part of a regular system of parameters. Let A = A[X], f : A → B the homomorphism of Aalgebras defined by f (X) = ξ. Let p = (f )−1 (q) = pA + (X). We have a local homomorphism Ap /pAp = k(p)[X](X) → Bq /pBq . The fibre Bq /p Bq is a regular ring since ξ is a regular parameter, and so by (5.4.3) Ap /pAp → Bq /pBq is formally smooth. Then applying (5.4.4) to Ap → Ap → Bq , we deduce that Ap → Bq is formally smooth. Consider the factorization A = A[X] → C[X] → B. By (5.1.14), ξ ∈ rad(HC[X]|A[X] B). By (5.3.6) there exists a factorization A[X] → C[X] → E → B such that HC[X]|A[X] E ⊂ HE|A[X] and the image of X in E is standard. Let a = X s with s large enough such that (0 : a2 ) = (0 : a) in A[X] and in B, and that the image of a in E is strictly standard. Let A = A /a8 A , etc. Since a is not a zero divisor in A , and Bq is a flat A -module, then a is not a zero divisor in Bq . Therefore ht q < ht(q). If HE|A B ⊂ q we have resolved A → E → B ⊃ q. If HE|A B ⊂ q, since HC[X]|A E ⊂ HE|A , and by (5.1.14) we have HE|A E ⊂ HE|A , we have that q is a minimal prime ideal of HE|A B. Applying then the induction assumption to A → E → B ⊃ q (note that ht p = 0 since ht p = 0, that A p → B q is formally smooth by base change and that the residue field extension remains separable, in fact, it is the same), we deduce that A → E → B ⊃ q is resolvable. Then, by (5.3.9), A → E → B ⊃ q is resolvable, that is, there exists A → E → D → B with HE|A B ⊂ rad(HD|A B) ⊂ q. Then A → C → D → B ⊃ q resolves A → C → B ⊃ q, since HD|A ⊂ HD|A by (5.1.13) (note that A → A is smooth) and so rad(HC|A B) ⊂ rad(HC[X]|A B) ⊂ rad(HE|A B) ⊂ rad(HD|A B) ⊂ rad(HD|A B), and rad(HD|A B) ⊂ q.
5.5 Positive characteristic
91
5.5 Positive characteristic Let A be an artinian local ring with residue field K of characteristic p > 0. By (3.2.1) there exists a Cohen ring A0 and a local homomorphism h : A0 → A inducing the identity map on the residue fields. If C = A0 / ker(h), then C is an artinian local subring of A with maximal ideal generated by p and residue field K. The construction of A0 and so that of C is made in (3.1.7), (3.1.8). Following [An4], now we will need, in this particular case, a construction of C (of Nagata and Narita) in some sense more canonical (for each choice of a p-basis of K). This construction also works in the non-noetherian case. See also [Bo, Chapter IX, §2.2]. Definition 5.5.1 Let A be a ring, p a prime number, I an ideal of A. We define I = {ap + pb : a, b ∈ I}. If I = A, we have that A is a subring of A. In general, I is an ideal of A . We define inductively for each integer n ≥ 0, A0 = A, An = (An−1 ) for n > 0, and similarly In . Lemma 5.5.2 If p ∈ I, then In ⊂ I n+1 for all n ≥ 0. Proof I0 = I ⊂ I 1 . Assume In−1 ⊂ I n . Then In = (In−1 ) ⊂ (I n ) ⊂ (I n )p + pI n ⊂ I n+1 . Lemma 5.5.3 Let (A, m, K) be a local ring such that the characteristic of K is p > 0 and mr+1 = 0 for some r. Let D be a subring of A such that D + m = A. Then Dn = An for all n ≥ r. Proof We have D + m = A , since if x ∈ A, x = d + u with d ∈ D, u ∈ m, then xp = dp + (up + pv) with v ∈ m, and so xp ∈ D + m . Similarly, px = pd + pu ∈ D + m . Analogously, Dn + mn = An for all n ≥ 0, and since mr ⊂ mr+1 = 0 by (5.5.2), we have Dr = Ar . Lemma 5.5.4 Let (A, m, K) be a local ring such that the characteristic of K is p > 0 and mr+1 = 0 for some r. Let {αi } be a set of elements of A such that if {αi } is its image in K, then K p ({αi }) = K. Then An [{αi }] = An+1 [{αi }] for all n ≥ r. Proof The image of A1 by the canonical map A → K is K p , and so the n+1 image of An+1 is K p . Therefore the image of An+1 [{αi }] by this map n+1 2 is K p ({αi }) = K (note that K = K p ({αi }), so K p = K p ({αpi }), 2 2 and then K = K p ({αi }) = K p ({αpi })({αi }) = K p ({αi }); in general, n K = K p ({αi })). Therefore A = An+1 [{αi }] + m, and by (5.5.3), for
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Regular homomorphisms: Popescu’s theorem
all n ≥ r we have An = (An+1 [{αi }])n ⊂ An+1 [{αi }]. We have seen an inclusion An [{αi }] ⊂ An+1 [{αi }]. The reverse inclusion is obvious. Definition 5.5.5 Let K|F be a field extension of characteristic p > 0. We say that a finite set {α1 , . . . , αn } of elements of K is p-independent over F if the set {α1r1 · · · αnrn : 0 ≤ ri < p for all i} is linearly independent over K p (F ), in other words, if [K p (F )(α1 , . . . , αn ) : K p (F )] = pn . We say that a subset B of K is p-independent over F if any finite subset of B is p-independent. We say that a subset B of K is a p-basis of K over F if it is p-independent over F and K p (F )(B) = K. In this case we have ⊗ K p (F ) K p (F )(b) = K. If F = Fp is the field of p elements, b∈B
we simply say a p-basis of K. For later use in Chapter 6, we point out now that if B is a ring, and A a subring containing a field of characteristic p, we also say that a finite set {β1 , . . . , βn } of elements of B is p-independent over A if the set {β1r1 · · · βnrn : 0 ≤ ri < p for all i} is linearly independent over A[B p ]. Lemma 5.5.6 Let K|F be a field extension of characteristic p > 0, and D a subset of K such that K p (F )(D) = K. Then there exists a p-basis B ⊂ D of K over F . Proof For each G ⊂ D, let EG = K p (F )(G). Let B be a maximal subset of p-independent elements of D. It is enough to show that EB = K, and so it is enough to see that D ⊂ EB . Let d ∈ D. If d ∈ / EB , then for any finite subset H ⊂ B we have d ∈ / EH , and dp ∈ K p ⊂ EH . Then [EH (d) : EH ] = p, and so [EH∪{d} : K p (F )] = [EH∪{d} : EH ][EH : K p (F )] = pp|H| = p|H∪{d}| . Thus H ∪ {d} is p-independent for any H, and then B ∪ {d} is p-independent, contradicting the maximality of B. Proposition 5.5.7 A subset B of K is a p-basis of K over F if and only if {db : b ∈ B} is a basis of the K-vector space ΩK|F . Proof In the exact sequence α
→ ΩK|F → ΩK|K p (F ) → 0 ΩK p (F )|F ⊗K p (F ) K − we have α = 0 since in ΩK|F we have dap = pap−1 da = 0 and dλ = 0, for a ∈ K, λ ∈ F . Thus ΩK|F = ΩK|K p (F ) .
5.5 Positive characteristic
93
Let B be a p-basis of K over F . Assume first that B = {b} consists in one element. Then K = K p (F )(b) and so ΩK|K p (F ) is generated by db. Since K|K p (F ) is a non-trivial inseparable algebraic extension, by (2.4.3) we have ΩK|K p (F ) = 0 and so {db} is a basis of this vector space. p In general, if B is a p-basis, K p (F ) K (F )(b) = K, then b∈B
ΩK|K p (F ) =
ΩK p (F )(b)|K p (F ) ⊗K p (F )(b) K
b∈B
and so {db}b∈B is a basis of ΩK|K p (F ) by the particular case above. Conversely, suppose that {db}b∈B is a basis of ΩK|K p (F ) . Consider the homomorphism p C := K p (F ) K (F )(b) → K b∈B
and the induced exact sequence ΩC|K p (F ) ⊗C K =
ΩK p (F )(b)|K p (F ) ⊗K p (F )(b) K
b∈B γ
−→ ΩK|K p (F ) → ΩK|C → 0. We have that γ is an isomorphism and so ΩK|C = 0. If L is the image of C in K (which is a field, since L ⊂ K is an integral extension of domains and K is a field), we have ΩK|L = ΩK|C = 0, and since K p ⊂ L, from (2.4.3) we deduce L = K. Then, by (5.5.6) there exists a p-basis D ⊂ B. We have already seen that in this case {db}b∈D is a basis of ΩK|K p (F ) and so D = B. Proposition 5.5.8 Let (A, m, K) be a local ring such that the characteristic of K is p > 0 and mr+1 = 0 for some r. Let {αi } be a set of elements of A whose image {αi } in K is a p-basis. Let C = Ar [{αi }]. Then C is an artinian local subring of A with maximal ideal generated by p and residue field K. If moreover A is artinian, then A is a C-module of finite type. Proof A ⊂ A is an integral extension and so C ⊂ A is also integral. Therefore C is a local ring with maximal ideal n := m ∩ C. From the
94
Regular homomorphisms: Popescu’s theorem
commutative diagram C
→
A
C/n
→
K
we deduce that the residue field C/n of C is the image of C by the map r A → K, and this image is K p [{αi }] = K. Since p ∈ m ∩ C = n, we have a commutative diagram Z(p) ↓ Fp
→ →
C ↓ K
(a) We have H1 (Z(p) , K, K) = (p)/(p2 ) ⊗Fp K. This follows from the exact sequence 0 = H2 (Fp , K, K) → H1 (Z(p) , Fp , K) → H1 (Z(p) , K, K) → H1 (Fp , K, K) = 0 associated to Z(p) → Fp → K, and from (2.4.1), (2.4.5) and (1.4.1). (b) ΩC|Z(p) ⊗C K is generated as K-vector space by {dαi ⊗ 1}. For, consider the commutative diagram with exact row ΩAr+1 |Z(p) ⊗Ar+1 K ε
β
−−→
γ
ΩC|Z(p) ⊗C K − → ΩC|Ar+1 ⊗C K → 0
ΩAr |Z(p) ⊗Ar K The elements dap , d(pb), with a, b ∈ Ar , generate ΩAr+1 |Z(p) and so ε(dap ⊗ 1) = dap ⊗ 1 = pap−1 da ⊗ 1 = 0, ε(d(pb) ⊗ 1) = d(pb) ⊗ 1 = pdb ⊗ 1 = 0 in ΩAr |Z(p) ⊗Ar K. Thus ε = 0, then β = 0, and so γ is an isomorphism. Now the result follows, since C = Ar+1 [{αi }], and so ΩC|Ar+1 is generated by {dαi }. λ
→ ΩK|Z(p) is injective (in fact, (c) The homomorphism ΩC|Z(p) ⊗C K − an isomorphism). This homomorphism takes the set of generators {dαi ⊗ 1} into {dαi }, and this last is a basis of ΩK|Z(p) = ΩK|Fp by (5.5.7), so the claim follows. (d) n/n2 is generated by the image of p. This follows from the exact sequence associated to Z(p) → C → K η
λ
H1 (Z(p) , K, K) − → H1 (C, K, K) → ΩC|Z(p) ⊗C K − → ΩK|Z(p) → 0,
5.5 Positive characteristic
95
where H1 (Z(p) , K, K) = (p)/(p2 ) ⊗Fp K by (a), H1 (C, K, K) = n/n2 , λ is injective by (c), and so η is surjective. (e) C is a noetherian ring. Since the maximal ideal n of C is nilpotent, it is enough to show that C/ni is noetherian for all i. Any complete local ring with maximal ideal of finite type is noetherian [AM, 10.25]. So by (d), C/n2 is noetherian. Assume that C/ni is noetherian and we shall see that C/ni+1 is noetherian. Since dimK H1 (Z(p) , K, K) < ∞ by (a), from the exact sequence H1 (Z(p) , K, K) → H1 (Z/(pi ), K, K) → H0 (Z(p) , Z/(pi ), K) = 0 we deduce dimK H1 (Z/(pi ), K, K) < ∞. Since C/ni is noetherian, dimK H2 (C/ni , K, K) < ∞ by (1.4.4), and so from the exact sequence H2 (C/ni , K, K) → H1 (Z/(pi ), C/ni , K) → H1 (Z/(pi ), K, K) we deduce dimK H1 (Z/(pi ), C/ni , K) < ∞. Now using the exact sequence H1 (Z(p) , Z/(pi ), K) → H1 (Z(p) , C/ni , K) → H1 (Z/(pi ), C/ni , K) we obtain dimK H1 (Z(p) , C/ni , K) < ∞. Consider now the exact sequence ηi
H1 (Z(p) , C/ni , K) −→ H1 (C, C/ni , K) → ΩC|Z(p) ⊗C K λ
i −→ ΩC/ni |Z(p) ⊗C/ni K → 0.
We have that λi is injective, since the composite homomorphism λ λ
i ΩC/ni |Z(p) ⊗C/ni K → ΩK|Z(p) ΩC|Z(p) ⊗C K −→
is injective by (c). So ηi is surjective and then dimK H1 (C, C/ni , K) < ∞, that is dimK ni /ni+1 < ∞. That means that ni /ni+1 is a C/ni+1 module of finite type. Since C/ni is noetherian, n/ni is a C/ni -module of finite type, and so a C/ni+1 -module of finite type. Then from the exact sequence 0 → ni /ni+1 → n/ni+1 → n/ni → 0 we deduce that the maximal ideal n/ni+1 of C/ni+1 is of finite type and so C/ni+1 is noetherian. (f) n = (p). This follows from (d) and (e) using Nakayama’s lemma. If moreover A is artinian, then each mi /mi+1 is a K-vector space of finite dimension, and so a C-module of finite type, since K is also the residue field of C. Since m is nilpotent, we deduce that m is a C-module
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Regular homomorphisms: Popescu’s theorem
of finite type. The C-module A/m = K is also of finite type and then so is A. This could be also deduced from (3.2.2). Theorem 5.5.9 Suppose that (A, m, K) → (B, n, L) is a local inclusion of artinian local rings with residue fields of characteristic p > 0. If dimL H1 (K, L, L) < ∞, then there exists a filtered family D of local subrings of B and local inclusions satisfying: (i) Each D ∈ D contains A, and is an A-algebra essentially of finite type. (ii) B = lim D. −→ (iii) If D1 ⊂ D2 in D have maximal ideals m1 , m2 , resp., then m1 D2 = m2 , m1 B = n. (iv) The homomorphisms D → B are flat for all D ∈ D, and so D1 → D2 are flat for all D1 ⊂ D2 in D. (v) Let D ∈ D, E its residue field, x ∈ B, x its image in L. Then for any large enough power q of p, there exists D ∈ D, D ⊂ D , with residue field E = E(xq ), such that xq ∈ D . Proof Consider the exact sequence H1 (K, L, L) → ΩK|F ⊗K L → ΩL|F → ΩL|K → 0, where F = Fp is the prime subfield of K. Let {αi }i∈I ⊂ A be a set such that its image {αi }i∈I in K is a p-basis of K, that is, {dαi }i∈I is a basis of the K-vector space ΩK|F . Choose a (finite) basis of H1 (K, L, L), represent the images in ΩK|F ⊗K L of the elements of this basis as linear combination of the dαi ⊗ 1 and remove from {dαi }i∈I the elements dαi appearing in this representation. We obtain a set {dαj }j∈J0 , with J0 ⊂ I and I − J0 finite, such that its image in ΩL|F is linearly independent. We can choose a basis of ΩL|F of the form {dαj }j∈J0 ∪ {dβ j }j∈J1 , with {βj }j∈J1 ⊂ B. Let Ba = Br [{αj }j∈J0 ∪ {βj }j∈J1 ], where nr+1 = 0, that by (5.5.8) is an artinian local subring of B with maximal ideal (p), residue field L, and B is a Ba -module of finite type. Analogously we define Aa = Ar [{αi }i∈I ]. Let E be a field, K ⊂ E ⊂ L, with E|K a field extension of finite type. Consider the exact sequence ΩK|F ⊗K E → ΩE|F → ΩE|K → 0. The set {dαj ⊗ 1}j∈J0 generates a subspace of finite codimension in ΩK|F ⊗K E. Therefore, since dimE ΩE|K < ∞, the image of {dαj ⊗ 1}j∈J0 in ΩE|F generates a subspace of finite codimension and is linearly
5.5 Positive characteristic
97
independent (since so is in ΩL|F ). Let {γh }h∈HE ⊂ Ba be such that HE is finite and {dαj }j∈J0 ∪ {dγ h }h∈HE is a basis of ΩE|F . We consider all possible choices of {γh }h∈HE . Let π : B → L be the canonical homomorphism. Then π −1 (E) is a local subring of B containing A and with maximal ideal n. Let BHE = π −1 (E)r [{αj }j∈J0 ∪ {γh }h∈HE ]. By (5.5.8), BHE is an artinian local subring of π −1 (E), with maximal ideal generated by p and residue field E. Moreover, since {γh }h∈HE ⊂ Ba , we have BHE ⊂ Ba . Since HE is finite, BHE is a π −1 (E)r [{αj }j∈J0 ]-algebra of finite type. Now, for each E, let ME = {j ∈ J1 : β j ∈ E}. The set {dαj }j∈J0 ∪ {dβ j }j∈ME is linearly independent in ΩE|F and so ME is finite. Let CE = π −1 (E)r [{αj }j∈J0 ∪{βj }j∈ME ]. Clearly {CE }E is a filtered system and lim CE = Ba . Then, since π −1 (E)r [{αj }j∈J0 ] ⊂ CE , and BHE is a −→ E
π −1 (E)r [{αj }j∈J0 ]-algebra of finite type, there exists E ⊃ E such that BHE ⊂ CE . Conversely, given E , some HE satisfies ME = {j ∈ J1 : β j ∈ E } ⊂ {γh }h∈HE , and so CE ⊂ BHE . So {BHE } is also a filtered system and its limit is Ba . Since BHE ⊂ Ba and the maximal ideals of these two rings are generated by p, by the local flatness criterion [Mt, theorem 22.3], Ba is a flat BHE -module, and so it is faithfully flat. By (5.5.8) or (3.2.2), B is a Ba -algebra of finite type of the form B = Ba [Y1 , . . . , Yn ]/(F1 , . . . , Fs ), where the image of each Yi in B belongs to n. Since nr+1 = 0, we can assume that Yir+1 ∈ {F1 , . . . , Fs } for all i. Let H = {HE : BHE contains all the coefficients of the polynomials F1 , . . . , Fs }. We have that {BHE }HE ∈H is a filtered subsystem of {BHE } with the same limit Ba . For each HE ∈ H, let DHE = BHE [Y1 , . . . , Yn ]/(F1 , . . . , Fs ), and let D = {DHE : HE ∈ H}. The rings DHE are local (with maximal ideal (p, y1 , . . . , yn ), where yi is the image of Yi in DHE , since Yir+1 ∈ {F1 , . . . , Fs } and so yir+1 = 0), and the homomorphisms DHE → DHE are also local. Since BHE → Ba is faithfully flat, by base change, so is DHE → Ba ⊗BHE DHE = B. In particular, DHE ⊂ B. The homomorphisms DHE ⊂ DHE are then faithfully flat [Mt, p. 46]. We have already proved (ii), (iii) and (iv) for D . By (5.5.8), A is an Aa -module of finite type, and so an Ar [{αj }j∈J0 ]algebra of finite type. Therefore, since Ar [{αj }j∈J0 ] ⊂ DHE for any HE , and {DHE } is a filtered system whose limit contains A, we have A ⊂ DHE for some HE ∈ H and this inclusion is local since A ⊂ B is
98
Regular homomorphisms: Popescu’s theorem
local. We define D = {DHE : HE ∈ H and A ⊂ DHE }. Since A → DHE is a local homomorphism with residue field extension E|K of finite type, we deduce that E is an A-algebra essentially of finite type and so DHE is an A-algebra essentially of finite type, since it is artinian. Clearly we still have (ii), (iii) and (iv) for D, and now also (i). We shall prove (v). Let DHE ∈ D, x ∈ B, x its image in L. Assume first that x is algebraic over E. Replacing x by xq , where q is a large enough power of p, we can assume that x is separable over E and x ∈ Br ⊂ Ba . Let E = E(x). By (2.4.3), (2.4.5), ΩE|F ⊗E E = ΩE |F and so {dαj }j∈J0 ∪ {dγ h }h∈HE also is a basis of ΩE |F . Thus, we take HE = HE , BHE = π −1 (E )r [{αj }j∈J0 ∪ {γh }h∈HE ]. Since x ∈ E , we have x ∈ π −1 (E ), and so xq ∈ π −1 (E )r ⊂ BHE ⊂ DHE with q = pr . Moreover, since x is separable over E, we have E = E(x) = E(xq ). Assume now that x is transcendental over E. Replacing as above x by xq , we can assume that x ∈ Ba . Let E = E(x). By (2.4.3), (2.4.4), {dαj }j∈J0 ∪{dγ h }h∈HE ∪{dx} is a basis of ΩE |F . Taking HE = HE ∪ {w} with γw = x, we have x ∈ BHE = π −1 (E )r [{αj }j∈J0 ∪ {γh }h∈HE ∪ {x}] ⊂ DHE . Lemma 5.5.10 Let (A, m, K) → (B, n, L) be a formally smooth homomorphism of noetherian local rings. Then dimL H1 (K, L, L) < ∞. Proof From the Jacobi–Zariski exact sequence 0 = H1 (A, B, L) → H1 (A, L, L) → H1 (B, L, L) = n/n2 we deduce dimL H1 (A, L, L) < ∞, and then from the exact sequence H1 (A, L, L) → H1 (K, L, L) → H0 (A, K, L) = 0 we obtain dimL H1 (K, L, L) < ∞.
Proposition 5.5.11 Let f : A → B be a homomorphism of noetherian rings. Let q be a prime ideal of B such that Ap → Bq is formally smooth, where p = f −1 (q). Let K = k(p), L = k(q) be their residue fields. Assume that the characteristic of K is p > 0. Let n > 0 be an = Bq /qn Bq , A the image of Ap in B. Let G be a finite subset integer, B of B. Then there exists a homomorphism f : A := A[Y1 , . . . , Ym ] → B
5.5 Positive characteristic
99
of local subrings that extends f , and a (not necessarily filtered) family D satisfying: of B (i) p Bq = qBq where p = (f )−1 (q). (ii) Ap → Bq is flat. = A /(p )n A . For any D ∈D we have (iii) Let A p p ⊂ D; (a) A (b) G ⊂ D; → D is smooth essentially of finite type; (c) A (d) D → B is flat; is the maximal ideal of D. (e) p D ∈D and x then there exists a power q of p and D ∈ D, (iv) If D ∈ B, q ⊂D and x . such that D ∈D Proof Let b1 , . . . , br be a set of generators of q. Enlarging G if necessary, By (5.5.10), we can assume that G contains the images of b1 , . . . , br in B. satisfies dimL H1 (K, L, L) < ∞. Since the residue field of A is K, A → B the assumption of (5.5.9), and so there exists a family D of local subrings that satisfy the conditions of (5.5.9). Since G is finite, there exists of B some D ∈ D such that G ⊂ D. Let E be the residue field of D and let e1 , . . . , et ∈ D be such that {de1 , . . . , det } is a basis of the E-vector B, etc., we denote by e its image space ΩE|K (if e is an element of D, A, in the residue field). of) We are going to see that we can assume that (the images in B e1 , . . . , et are images of elements y1 , . . . , yt ∈ B. Let y1 , . . . , yt ∈ B, s ∈ B − q be such that each ei is the image of yi /s ∈ Bq . If s ∈ L is algebraic over E, replacing s by sq , yi /s by sq−1 yi /sq with q some power of p, we can assume that s is separable over E. By (5.5.9.(v)), replacing again s by some sq , there exists D ∈ D, D ⊂ D , and the residue field of such that s ∈ D ( s is the image of s in B) D is E = E(s). Replacing once again, if necessary, s by sp , we can assume that ds = 0 in ΩE |K (notice that E(s) = E(sp )). From the exact sequence 0 = H1 (E, E , E ) → ΩE|K ⊗E E → ΩE |K → ΩE |E = 0 (2.4.3), (2.4.5), we deduce that {de1 , . . . , det } is also a basis of ΩE |K , and then, since y i = sei ∈ E , we have dy i = sdei + ei ds = sdei in ΩE |K . Thus {dy 1 , . . . , dy t } is a basis of ΩE |K . So replacing ei by sei , D by D , we have that e1 , . . . , et are images of elements y1 , . . . , yt ∈ B. If s ∈ L is transcendental over E, replacing again s by sq , let D ∈ D,
100
Regular homomorphisms: Popescu’s theorem
D ⊂ D , be such that s ∈ D and the residue field of D is E = E(s) as in (5.5.9.(v)). We have an exact sequence 0 = H1 (E, E , E ) → ΩE|K ⊗E E → ΩE |K → ΩE |E → 0 showing that {de1 , . . . , det , ds} is a basis of the E -vector space ΩE |K (2.4.3), (2.4.5). Similarly to the algebraic case, we have dy i = sdei +ei ds, and so we see that {dy 1 , . . . , dy t , ds} also is a basis of ΩE |K . Therefore, replacing D by D , {e1 , . . . , et } by {se1 , . . . , set , s} we get the result. Let ϕ : A[Y1 , . . . , Yt ] → B be the A-algebra homomorphism taking Yi into yi . We show that the induced homomorphism A[Y1 , . . . , Yt ]p → Bq , where p = ϕ−1 (q), is formally smooth. By (5.4.4), it is enough to show that H1 (K[Y1 , . . . , Yt ], B ⊗A K, L) = 0. Let K[Y1 , . . . , Yt ] → E be the homomorphism of K-algebras sending Yi into ei . In the Jacobi– Zariski exact sequence 0 = H1 (K, K[Y1 , . . . , Yt ], L) → H1 (K, E, L) → H1 (K[Y1 , . . . , Yt ], E, L) → ΩK[Y1 ,...,Yt ]|K ⊗K[Y1 ,...,Yt ] L → ΩE|K ⊗E L associated to K → K[Y1 , . . . , Yt ] → E, the final homomorphism is injective, because {de1 , . . . , det } is a basis of ΩE|K , and so H1 (K, E, L) = H1 (K[Y1 , . . . , Yt ], E, L). We thus have a commutative diagram with exact rows and columns H2 (E, L, L) = 0
0 = H1 (Ap , Bq , L) = H1 (K, B ⊗A K, L)
→ H1 (K, E, L) = H1 (K[Y1 , . . . , Yt ], E, L) → H1 (B ⊗A K, E, L) 0 = H2 (B ⊗A K, L, L) → H1 (K[Y1 , . . . , Yt ], B ⊗A K, L) →
?
H1 (K[Y1 , . . . , Yt ], L, L)
H1 (E, L, L)
? → H1 (B ⊗A K, L, L) =
H1 (E, L, L)
showing that H1 (K[Y1 , . . . , Yt ], B ⊗A K, L) = 0. Let {y1 , . . . , ys } ⊂ {b1 , . . . , br } be a set whose image in Bq /p Bq is a regular system of parameters. Let A = A[Y1 , . . . , Yt , Y1 , . . . , Ys ] and let A → B be the homomorphism extending ϕ by sending Yi into yi .
5.5 Positive characteristic
101
Let p be the inverse image of q in A . The homomorphism Ap → Bq is flat (by (5.4.4) applied to Ap → Ap → Bq with I = pAp , we can reduce the flatness to the case where Ap and Bq are regular; then (5.4.3) applies). Since p Bq = qBq , by base change, the homomorphism = Bq /qn Bq = Ap /(p )n Ap → B A is flat, and since it is local, faithfully flat. In particular, it is injective. are contained in D, we Since the images of yi and yi ∈ {b1 , . . . , br } in B have a homomorphism A → D which is injective. From the inclusion ⊂ mD B = qBq /qn Bq . p D ⊂ mD (maximal ideal of D) we deduce p B n Now, from the faithful But p B = qBq /q Bq , and then p B = mD B. → B and we deduce p D = mD . Using again that A flatness of D → B, D → B are faithfully flat, we deduce that A → D is faithfully flat. be the family of those D ∈ D such that D ⊂ D and the residue Let D is separable over the residue field E of D. Properties (i), field E of D (ii), (iii)(a), (iii)(b), (iii)(d) are now clear. Condition (iii)(e) follows from (5.5.9.(iii)). . We have an exact sequence Let K be the residue field of A ΩK |K ⊗K E → ΩE|K → ΩE|K → 0 and, since ΩE|K is generated by dy 1 , . . . , dy t , we see that the left homomorphism is surjective and then ΩE|K = 0. Therefore E|K is separable (2.4.4), (2.4.5), and thus E |K is separable. We have seen that the local → D is flat and its fibre K → E homomorphism of noetherian rings A is → D, is a separable field extension (since p D = mD ). By (2.6.5), A formally smooth, proving (iii)(c). By (5.5.9.(v)), ∈D with residue field E , and x ∈ B. To see (iv), let D there exists D with D ⊂ D and residue field E = E (xq ) such that . Taking q large enough so that xq is separable over E , we have x q ∈ D ∈ D. E |E separable and so D Lemma 5.5.12 Let A be a noetherian local ring.
(i) Let 0 → M → M → M → 0 be an exact sequence of A-modules of finite type. Let a1 , . . . , as ∈ A be a regular sequence on M and M . Then it is a regular sequence in M . (ii) Let a1 , . . . , as , b1 , . . . , bs ∈ A be such that c1 , . . . , cs is a regular sequence for any choice of each ci ∈ {ai , bi }. Then, for any j, a1 b1 , . . . , aj−1 bj−1 , cj , . . . , cs is a regular sequence for any choice of cj , . . . , cs as before.
102
Regular homomorphisms: Popescu’s theorem
Proof (i) Applying the snake lemma to the diagram 0
−−−−→
0
−−−−→
M ↓. a1 M
−−−−→
M ↓. a1 M
−−−−→
−−−−→ −−−−→
M ↓. a1 M
−−−−→ 0 −−−−→ 0
we deduce that a1 is M -regular, and we have an exact sequence
0 → M /a1 M → M/a1 M → M /a1 M → 0, so we repeat the process until as . (ii) Applying (i) to the exact sequence .a
1 A/(a1 b1 ) → A/(a1 ) → 0 0 → A/(b1 ) −−→
we deduce that a1 b1 , c2 , . . . , cs is a regular sequence on A for any choice of each ci ∈ {ai , bi }. We repeat with the exact sequence .a
2 A/(a1 b1 , a2 b2 ) → A/(a1 b1 , a2 ) → 0 0 → A/(a1 b1 , b2 ) −−→
(the first map is injective since a1 b1 , a2 , . . . is a regular sequence and so a2 is regular on A/(a1 b1 )), and we deduce that a1 b1 , a2 b2 , c3 , . . . , cs is a regular sequence, etc. Lemma 5.5.13 Let A be a noetherian ring, M an A-module of finite type, S ⊂ A a multiplicative subset. Let a ∈ A be such that (0 : a)S −1 M = (0 : a2 )S −1 M . Then there exists t ∈ S such that for all n > 0, if s = tn , (0 : as)M = (0 : (as)2 )M . Proof We have S
−1
"
(0 : a2 )M (0 : a)M
# =
(0 : a2 )S −1 M = 0. (0 : a)S −1 M
So there exists u ∈ S such that u(0 : a2 )M ⊂ (0 : a)M . Let N be large enough such that (0 : um )M = (0 : um+1 )M for all m ≥ N and let t = uN , s = tn . Then s(0 : a2 )M ⊂ (0 : a)M and (0 : s)M = (0 : s2 )M . If x ∈ (0 : (as)2 )M , xa2 s2 = 0 and then xa2 ∈ (0 : s2 )M = (0 : s)M . Therefore xs ∈ (0 : a2 )M and then xs2 ∈ (0 : a)M . This gives ax ∈ (0 : s2 )M = (0 : s)M , and so x ∈ (0 : as)M . Lemma 5.5.14 Let A be a ring, p a prime ideal of A, d1 , . . . , dr ∈ p elements which are a regular sequence in Ap , and c > 0 an integer. Then
5.5 Positive characteristic
103
there exist s1 , . . . , sr ∈ A − p such that, for all i, (((d1 s1 T1 )c , . . . , (di−1 si−1 Ti−1 )c ) : di si Ti ) = (((d1 s1 T1 )c , . . . , (di−1 si−1 Ti−1 )c ) : (di si Ti )2 ) in the polynomial ring A[T ] = A[T1 , . . . , Tr ]. Proof By (5.5.12), for any i, dc1 , . . . , dci−1 , di is a regular sequence in Ap , c and again by (5.5.12), dc1 T1c , . . . , dci−1 Ti−1 , di Ti is a regular sequence in Ap [T ]. Then, if S = A − p, and we have already s1 , . . . , si−1 , setting Mi = A[T ]/((d1 s1 T1 )c , . . . , (di−1 si−1 Ti−1 )c ), we have S −1 Mi = Ap [T ]/((d1 T1 )c , . . . , (di−1 Ti−1 )c ) and then (0 : di Ti )S −1 Mi = (0 : (di Ti )2 )S −1 Mi . Then the result follows from (5.5.13). Proof of (5.3.7) for residue field k(q) of positive characteristic Let r = ht q and let N be large enough so that (ηBq )N = 0, where ηBq is the nilradical of Bq , and so that qN Bq ⊂ HC|A Bq (such an N exists, since q is minimal over HC|A B). Let c1 , . . . , cM be a set of generators of HC|A , and let
cj Zij }i=1,...,r ). C = C[X1 , . . . , Xr , {Zij }i=1,...,r;j=1,...,M ]/({Xi2N − j
Cc j
is isomorphic to Ccj [X1 , . . . , Xr , {Zih }h=j ], it follows that Since Cc j is a smooth Ccj [X1 , . . . , Xr ]-algebra, and so a smooth A[X1 , . . . , Xr ]algebra, since cj ∈ HC|A . That is, the image of each cj in C belongs to HC |A[X1 ,...,Xr ] and then HC|A C ⊂ HC |A[X1 ,...,Xr ] . Since Xi2N ∈ HC|A C for any i, we deduce that Xi2N ∈ HC |A[X1 ,...,Xr ] and so Xi ∈ HC |A[X1 ,...,Xr ] . Let C = SC (I/I 2 ), where I is the kernel of a finite free presentation of the A[X1 , . . . , Xr ]-algebra C (see (5.1.16)), and ρ : C → C the aug mentation homomorphism. As we saw in (5.1.16), HC |A[X1 ,...,Xr ] C ⊂ HC |A[X1 ,...,Xr ] and so HC|A C ⊂ HC |A[X1 ,...,Xr ] . Moreover, C has a presentation such that the image in C of any element of HC |A[X1 ,...,Xr ] is standard. Let e be such that the image of any Xie in C is strictly standard over A[X1 , . . . , Xr ], and let c = 8e. We will apply (5.5.11) with n = N + rc and G containing the images of a set of generators of C over is contained in the rings D. A, so that the image of C in B
104
Regular homomorphisms: Popescu’s theorem
Lemma 5.5.15 Let d1 , . . . , dr ∈ p (notation of (5.5.11)), and denote ∈D and a by the same symbols their images in B. Then there exists D homomorphism C → B sending Xi into di εi with εi ∈ B − q, and such is contained in D. that the image of C in B have been fixed, Remark We need this result after C , d1 , . . . , dr and D so this lemma does not follow immediately from (5.5.11). ∈ D. Since qN Bq ⊂ HC|A Bq , we have that dN ∈ HC|A Bq Proof Let D i belongs to HC|A B. Since D →B for any i, and then its image dN in B i is faithfully flat, dN i ∈ HC|A B ∩ D = HC|A D [Mt, Theorem 7.5]. So, for each i, there exist elements zij ∈ D (j = 1, . . . , M ) such that dN i = M ij . j=1 cj z M n Let zij ∈ Bq representing zij . Then dN i − j=1 cj zij ∈ q Bq ⊂ M n−N N n−N q HC|A Bq and then di = Bq . j=1 cj (zij + zij ) with zij ∈ q M 2N N N Therefore, di = j=1 cj zij , where the image zij of zij = di zij +di zij belongs to D, since the image of dN z belongs to D (dN , z ∈ D) in B i ij i ij is 0. and dN z ∈ qN qn−N Bq = qn Bq , and so the image of dN z in B i
ij
i
ij
Let zij = wij /si with wij ∈ B and si ∈ B − q. After replacing si by a by (5.5.11.iv), we can assume sufficiently high power, and enlarging D and in particular w Let ti ∈ B − q be that si ∈ D, ij = zij si ∈ D. M 2N −1 2N wij ) = 0. Again, replacing ti by a such that ti ((si di ) − j=1 cj si we can assume that sufficiently high power and enlarging D, ti ∈ D. Define εi = si ti , and so the homomorphism C → B sending Xi into di εi and Zij into si2N −1 t2N i wij is well defined and its image in B is contained in D. By (5.3.11), we can assume ht p = 0. Since the homomorphism Ap → Ap is formally smooth, it is flat with regular fibre. Therefore, since Ap is Cohen–Macaulay (it is artinian local), we deduce that Ap is Cohen– Macaulay [Mt, corollary to Theorem 23.3]. Choose d1 , . . . , ds ∈ p such that its classes in Ap ⊗Ap k(p) = Ap /pAp form a regular system of parameters. Since pAp is nilpotent, d1 , . . . , ds form a system of parameters of Ap and so a regular sequence. The images of d1 , . . . , ds in Bq also form a regular sequence, since Ap → Bq is flat, and in particular, dim Bq ≥ s = dim Ap . Since d1 , . . . , ds is a system of parameters in Ap , (p )t Ap ⊂ (d1 , . . . , ds )Ap for some t, and so by (5.5.11.(i)), qt Bq ⊂ (d1 , . . . , ds )Bq and then r = dim Bq = s. So from now on, we denote them by d1 , . . . , dr as before. We
5.5 Positive characteristic
105
replace now d1 , . . . , dr by elements d1 s1 , . . . , dr sr with s1 , . . . , sr ∈ A −p satisfying (5.5.14), that we continue to denote d1 , . . . , dr ; that is, we have ((d1 T1 )c , . . . , (di−1 Ti−1 )c : di Ti ) = ((d1 T1 )c , . . . , (di−1 Ti−1 )c : (di Ti )2 ). ∈D such that there exist elements Lemma 5.5.16 We can enlarge D δi ∈ B − q satisfying: (i) δi ∈ D.
(ii) There exists a homomorphism C → B sending Xi into di δi such is contained in D. that the image of C in B (iii) The elements di δi satisfy ((d1 δ1 )c , . . . , (di−1 δi−1 )c : di δi ) = ((d1 δ1 )c , . . . , (di−1 δi−1 )c : (di δi )2 ) in B, for all i. Proof Let εi ∈ B − q be the elements of (5.5.15). Assume that we have already found elements η1 , . . . , ηi−1 ∈ B − q such that δj := εj ηj (j < i) so satisfy condition (iii) and that by (5.5.11.(iv)) we have enlarged D that the ηj lie in D. Since d1 ε1 , . . . , dr εr is a regular sequence in Bq , by (5.5.13), there exists ηi ∈ B − q such that di εi ηiq satisfies (iii) for all for so that ηq lies in D q > 0. Again by (5.5.11.(iv)) we can enlarge D i q some q. Replacing ηi by ηi gives the result. Note that then condition (ii) follows: (5.5.15) (see its proof) gives a map C → B sending Xi into di εi , Zij into some ζij . If we send Xi into di εi ηi and Zij into ηi2N ζij we also have a map C → B. Lemma 5.5.17 qn Bq ⊂ (dc1 , . . . , dcr )Bq and (p )n Ap ⊂ (dc1 , . . . , dcr )Ap . Proof We have chosen d1 , . . . , dr so that (d1 , . . . , dr ) + pAp = p Ap , where pAp is nilpotent. Then qBq = p Bq = (d1 , . . . , dr )Bq +ηBq , where ηBq is the nilradical of Bq . Therefore qn Bq =
N −1
(d1 , . . . , dr )n−i (ηBq )i ⊂ (d1 , . . . , dr )n−(N −1) Bq
i=0
= (d1 , . . . , dr )rc+1 Bq ⊂ (dc1 , . . . , dcr )Bq (recall that (ηBq )N = 0). The second inclusion follows from this one applying − ⊗Ap Bq , since Ap → Bq is faithfully flat. Let A = A [T1 , . . . , Tr ], and let A[X1 , . . . , Xr ] → A be the map
106
Regular homomorphisms: Popescu’s theorem
sending Xi into di Ti . Let f : A → B extending the map f : A → B by sending Ti into δi . Let Γ = A ⊗A[X1 ,...,Xr ] C and Γ → B the homomor phism which over A is f , and over C is the composite C → C with the homomorphism C → B of (5.5.16). We have HC |A[X1 ,...,Xr ] Γ ⊂ HΓ|A (5.1.14), and we already saw that HC|A C ⊂ HC |A[X1 ,...,Xr ] . Then HC|A Γ ⊂ HΓ|A .
Assume that A → Γ → B ⊃ q can be resolved, i.e., there exists A → Γ → E → B ⊃ q such that HΓ|A B ⊂ rad(HE|A B) ⊂ q. Since A is a smooth A-algebra HE|A ⊂ HE|A and then HC|A B ⊂ HΓ|A B ⊂ rad(HE|A B) ⊂ rad(HE|A B) ⊂ q, and so A → C → E → B resolves A → C → B ⊃ q. Therefore it is enough to resolve A → Γ → B ⊃ q. We shall apply (5.3.10) to ai := di Ti . We have already seen that the assumption over the quotient ideals holds in A , and by (5.5.16), it holds also in B. On the other hand, the image of Xie in C is strictly standard over A[X1 , . . . , Xr ] (by the choice of e) and then, by (5.2.3), its image aei in Γ is strictly standard over A . So (5.3.10) says that it is enough to show that
A /I → Γ/IΓ → B/IB ⊃ q/IB is resolvable, where I = ((d1 T1 )c , . . . , (dr Tr )c ). We have ht(q/IB) = 0 by (5.5.17) (notice that the images δi of Ti in Bq are units). So by (5.4.1) and (5.4.2), it is enough to see that
A /I → Γ/IΓ → Bq /IBq ⊃ qBq /IBq is resolvable. It is enough to find
A /I → Γ/IΓ → E → Bq /IBq
with E a smooth A /I-algebra of finite type, and by (5.3.5) it is enough to prove that there exists
R−1 (A /I) → R−1 (Γ/IΓ) → E → Bq /IBq ,
(∗)
where R = A − p , with p := (f )−1 (q) and where E is a smooth R−1 (A /I)-algebra of finite type. Let J = (dc1 , . . . , dcr ) ⊂ A . Since the elements Ti belong to R, R−1 I = IR−1 A = JR−1 A , and so (*) can be written as
R−1 (A /JA ) → R−1 (Γ/JΓ) → E → Bq /JBq . Let S = A − p ⊂ R. By flat base change, it is enough to show that
5.5 Positive characteristic
107
there exists
S −1 (A /JA ) → S −1 (Γ/JΓ) → F → Bq /JBq
with F a smooth S −1 (A /JA )-algebra of finite type, that is, Ap [T1 , . . . , Tr ]/JAp [T1 , . . . , Tr ] → Ap [T1 , . . . , Tr ]/JAp [T1 , . . . , Tr ] ⊗A [T1 ,...,Tr ] Γ → F → Bq /JBq and again by base change, since (p )n Ap [T1 , . . . , Tr ] ⊂ JAp [T1 , . . . , Tr ] by (5.5.17), it is enough to see that there exists Ap [T1 , . . . , Tr ]/(p )n Ap [T1 , . . . , Tr ] → Ap [T1 , . . . , Tr ]/(p )n Ap [T1 , . . . , Tr ] ⊗A [T1 ,...,Tr ] Γ → F → Bq /(p )n Bq , that is, [T1 , . . . , Tr ] → A [T1 , . . . , Tr ] ⊗A [T ,...,T ] Γ → F → B A 1 r or equivalently [T1 , . . . , Tr ] → A [T1 , . . . , Tr ] ⊗A[X ,...,X ] C → F → B. A 1 r are contained in D by [T1 , . . . , Tr ] and C in B The images of A [T1 , . . . , Tr ] ⊗A[X ,...,X ] C → D (5.5.16), so it is enough to see that A 1 r factors through the homomorphism D[T1 , . . . , Tr ] → D that sends Ti [T1 , . . . , Tr ] → D[T 1 , . . . , Tr ] is a smooth homomorphism to δi , since A essentially of finite type (5.5.11.(iii)) and then we apply (5.3.5). Let C = C[X1 , . . . , Xr , {Zij }]/({Xi2N − j cj Zij }i ) → B be the homomorphism of (5.5.16), sending Xi into di δi , and let ζij be the image are contained in D of Zij in B. The images ζij of these elements in B
by (5.5.16). The homomorphism A[X1 , . . . , Xr ] → A = A [T1 , . . . , Tr ] sends Xi into di Ti and then we have a homomorphism of A[X1 , . . . , Xr ] 1 , . . . , Tr ] sending Xi to di Ti . So we define algebras C[X1 , . . . , Xr ] → D[T a homomorphism C → D[T1 , . . . , Tr ] of A[X1 , . . . , Xr ]-algebras sending since it is a unit in Zij into (Ti /di )2N ζij (notice that di is a unit in D, B, and the extension of the maximal ideal of D is the maximal ideal of which makes the diagram B), → D D[T1 , . . . , Tr ] C
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Regular homomorphisms: Popescu’s theorem
commute. Composing with the augmentation C → C , we have a 1 , . . . , Tr ] which induces the required homohomomorphism C → D[T morphism [T1 , . . . , Tr ] ⊗A[X ,...,X ] C → D[T 1 , . . . , Tr ]. A 1 r
5.6 The module of differentials of a regular homomorphism Theorem 5.6.1 Let f : A → B be a regular homomorphism. Then ΩB|A is a flat B-module. Proof By (5.3.2), (2.3.1) and (1.4.8), ΩB|A is the direct limit of a filtered system {ΩBi |A } of flat Bi -modules. Since filtered direct limits commute with ⊗ and are exact, they preserve flatness and then we deduce the result. Proposition 5.6.2 Let f : A → B be a homomorphism of noetherian rings. The following are equivalent: (i) f is regular. (ii) H1 (A, B, k(q)) = 0 for any prime ideal q of B, where k(q) = Bq /qBq . (iii) H1 (A, B, B) = 0 and ΩB|A is a flat B-module. (iv) H1 (A, B, −) = 0. Proof (i) ⇐⇒ (ii) By (2.6.5), f is regular if and only if for any prime ideal q of B, the homomorphism Ap → Bq is formally smooth, where p = f −1 (q). By (2.3.5) this last condition is equivalent to H1 (Ap , Bq , k(q)) = 0, and by (1.4.7), H1 (Ap , Bq , k(q)) = H1 (A, B, k(q)). (i) =⇒ (iii) follows from (5.3.2), (5.6.1) and (1.4.8), and (iii) =⇒ (iv) follows from (1.4.5.d). Finally, (iv) =⇒ (ii) is trivial.
6 Localization of formal smoothness
We will prove in this chapter the theorem by Andr´e [An2] of localization of formal smoothness (6.4.2). This result is also stated without proof in Matsumura’s book [Mt, end of Section 32]. In the first section we prove some lemmas which will allow us to reduce the problem to simpler cases. Sections 2 and 3 are needed for the proof of the characteristic p case (6.4.1), but they also contain some results of independent interest, for instance proposition (6.3.1) which is due to Grothendieck [EGA, 0IV 22.2.6]. We will finish the proof in Section 4. We follow essentially [An3], [BR1], [Ra1], but we also use ideas from some papers in the nineties ([An5], [An6], [BM], [Ra3]) using the relative Frobenius homomorphism, in order to switch, in some sense, from H1 to H2 , so that we can use results like (6.2.2), (6.2.4).
6.1 Preliminary reductions Proposition 6.1.1 i) Let f : (A, m, K) → (B, n, L) be a flat local homomorphism of noetherian local rings. If B is regular, then A is regular. If A and B ⊗A Kare regular, then so is B. f
g
→ B − → C be homomorphisms of noetherian rings such ii) Let A − that g is faithfully flat and gf regular. Then f is regular. f
g
→B− → C be regular homomorphisms of noetherian rings. iii) Let A − Then gf is regular. 109
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Localization of formal smoothness
Proof i) This follows from the exact sequence in the proof of Proposition 4.3.7: 0 → H2 (A, K, L) → H2 (B, L, L) → H2 (B ⊗A K, L, L). ii) Let p be a prime ideal of A, F |k(p) a finite field extension. The ring C ⊗A F is regular, since gf is regular. The homomorphism B ⊗A F → C ⊗A F is faithfully flat, and so any prime ideal q of B ⊗A F is the contraction of a prime ideal q of C ⊗A F . Since the homomorphism (B ⊗A F )q → (C ⊗A F )q is local and flat, we deduce from i) that (B ⊗A F )q is regular. iii) It follows from (5.6.2) and (1.4.6). Lemma 6.1.2 Let f : (A, m, K) → (B, n, L) be a formally smooth hoˆ is formally momorphism of noetherian local rings. Then fˆ: Aˆ → B smooth. Proof Consider the commutative diagram f
A −−→ B β α Aˆ
fˆ
−→
ˆ B
where the vertical maps are the canonical completion homomorphisms. We have a commutative diagram with exact rows (1.4.6) H2 (A, L, L) →H2 (B, L, L)→ H1 (A, B, L)→ H1 (A, L, L) → H1 (B, L, L) = = = = ˆ L, L) →H2 (B, ˆ L, L)→ H1 (A, ˆ B, ˆ L)→ H1 (A, ˆ L, L) → H1 (B, ˆ L, L) H2 (A, ˆ B, ˆ L), and so the result (see (4.3.3)). Therefore H1 (A, B, L) = H1 (A, follows from (2.3.5). Lemma 6.1.3 Let f : (A, m, K) → (B, n, L) be a local homomorphism of noetherian local rings such that the canonical homomorphism A → Aˆ ˆ is regular then f is regular. is regular. If fˆ: Aˆ → B Proof Let f
A −−→ B β α Aˆ
fˆ
−→
ˆ B
6.1 Preliminary reductions
111
be as above. If fˆ is regular then βf = fˆα is regular by (6.1.1) and so, since β is faithfully flat, f is regular by (6.1.1). Lemma 6.1.4 Let A be a ring, B an A-algebra, n an ideal of B. Assume that B is formally smooth for the n-adic topology. Let E, E be A-algebras, M , M nilpotent ideals of E, E respectively, f : E → E a surjective homomorphism of A-algebras such that f (M ) ⊂ M . Let f : E/M → E /M be the induced map, p : E → E/M , p : E → E /M the canonical maps. Let u : B → E/M , v : B → E be continuous homomorphisms of A-algebras (for the discrete topologies in E and E ) such that p v = f u. Then there exists a continuous homomorphism of A-algebras v : B → E such that v = f v, u = pv. Proof Let J = ker f . We have a commutative diagram of exact rows and columns 0 ↓ M ∩J ↓
0
→
0
→
→
E ↓p
0
→ J/(M ∩ J) → ↓ 0
E/M ↓ 0
J ↓
→
0 ↓ M ↓
→
0 ↓ M = (J + M )/J ↓
f
− → f
E = E/J ↓ p
− → E /M = E/(J + M ) ↓ 0
→
0
→
0
For any b ∈ B, we have p v (b) = f u(b). Let x, y ∈ E be such that u(b) = x + M , v (b) = y + J (we are identifying E with E/J). Then x − y = n + m, n ∈ J, m ∈ M . Therefore we can define v0 : B → E/(M ∩ J) by sending b to the class of x−m = y+n, and so the composite canonical v0 v0 homomorphisms B −→ E/(M ∩ J) → E/M, B −→ E/(M ∩ J) → E/J are u, v resp. Since ker(u) ∩ ker(v ) ⊂ ker(v0 ), v0 is continuous (for the discrete topology in E/(M ∩ J)). Since M ∩ J is nilpotent, by (2.2.3) there exists a continuous homomorphism of A-algebras v : B → E such v that v0 is the composite B − → E → E/(M ∩ J). Lemma 6.1.5 Let f : (A, m, K) → (B, n, L) be a formally smooth homomorphism of noetherian local rings. Let C be an A-algebra, I and J two ideals of C. Assume that i) C is complete for the J-adic topology.
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ii) The sequence {I n }n≥0 converges to 0, that is, each (some) power of J contains some power of I. iii) I is closed in C. Then, for any continuous homomorphism of A-algebras u : B → C/I, there exists a continuous homomorphism of A-algebras v : B → C such v that u is the composite B − → C → C/I. Proof Some power of I is contained in J, and hence (I + J)/J is a nilpotent ideal of C/J. Applying (2.2.3) to the diagram B u v1 0 →
(I + J)/J
→
C/J
→
C/I C/(I + J)
→
0,
we deduce the existence of a continuous homomorphism of A-algebras v1 : B → C/J making the diagram commute. By recurrence on n, using (6.1.4), we have a continuous homomorphism of A-algebras vn+1 : B → C/J n+1 making the diagram
u C/I
B H H vn HH vn+1 HH ? j → C/J n C/J n+1
→ C/(I + J n+1 ) →
C/(I + J n )
→
0
commute. Take v = lim vn : B → lim C/J n = C. Since I is closed in ←− ←− C, C/I is Hausdorff and so complete. Thus C/I = lim C/(I + J n ), and ←− we obtain u as the composite of v with the canonical homomorphism C → C/I. Lemma 6.1.6 Let g
A −−→ B β α f
A −−→ B
6.1 Preliminary reductions
113
be a commutative diagram of local homomorphisms of noetherian local rings, with B complete, f formally smooth, α finite and β surjective. If g is regular, then f is regular. Proof Let B = B ⊗A A. Let m be the maximal ideal of B . Consider B with the mB -adic topology and B with the topology of its maximal ideal. With this topology, B is complete (since B is complete) and the canonical homomorphism p : B → B is surjective (since β is surjective) and continuous. Thus its kernel I is a closed ideal, and some power of I is contained in mB (since B /mB is a finite B /m-algebra and so artinian). We can apply (6.1.5): there exists an A-algebra homomorphism j : B → B such that pj = 1. Therefore, for any B-module M , the identity map of H1 (A, B, M ) factors through H1 (A, B ⊗A A, M ) = H1 (A , B , M ) = 0 (the first equality by (1.4.3), and the second by (5.6.2)). So H1 (A, B, M ) = 0, and again by (5.6.2) A → B is regular. Proposition 6.1.7 Let (A, (p), K) be a Cohen ring of positive characteristic p (3.1.10) and (C, n, L) a formally smooth complete local noetherian K-algebra. Then there exists a formally smooth complete local noetherian A-algebra B such that B/pB is isomorphic to C as K-algebra. Proof i) Assume first that L|K is separable. Then C contains a field isomorphic to L as K-algebra, since the canonical K-algebra epimorphism C → L has a section (2.4.6), (2.2.3). Therefore C = L[[X1 , . . . , Xn ]] with n = dim C as K-algebras by (3.2.2), (3.2.3), (2.5.7). Let A be a Cohen A-algebra with residue field L (3.1.8). Then we can take B = A [[X1 , . . . , Xn ]]. ii) If L|K is not separable, let A0 be a Cohen ring with residue field Fp = Z/pZ (3.2.1). Applying i) to (A0 , (p), Fp ), (C, n, L), we see that there exists a formally smooth complete local noetherian A0 -algebra B such that B/pB is isomorphic to C. Let B → C be the canonical homomorphism. Since A0 → A is formally smooth (3.2.1), (3.1.2), there exists a local A0 -algebra homomorphism f : A → B making the diagram A −→ B K −→ C commute (6.1.5). By construction, B is a local regular ring and so p is
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Localization of formal smoothness
not a zero-divisor in B. Thus, since B/pB = C is flat as K-module, f is flat by the local flatness criterion [Mt, theorem 22.3] (or by (5.4.4) applied to A0 → A → B). Then by (2.6.5) f is formally smooth. Proposition 6.1.8 Let f : A → B be a formally smooth homomorphism of complete noetherian local rings. There exists a commutative diagram g
A −−→ B β α f
A −−→ B where A , B are complete regular local rings, dim A ≤ dim A + 1, g is formally smooth, α finite, β surjective. If A is an integral domain, we can choose A with dim A = dim A. If A contains a field, so does A . Proof i) Assume first that A contains a field. Then A contains a field isomorphic to its residue field K (3.2.4). Since B ⊗A K is a formally smooth K-algebra, B⊗A K is a regular local ring (2.5.7), and there exists a K-algebra homomorphism β0 : B ⊗A K → B such that the composite B ⊗A K → B → B ⊗A K is the identity map ((2.2.3) or (6.1.5)). Let α : A := K[[X1 , . . . , Xn ]] → A be a finite injective local homomorphism with n = dim A (3.2.4). Let β1 : B ⊗A K[[Y1 , . . . , Ym ]] → B be a surjective homomorphism extending β0 (3.2.2) and β : B := B ⊗A K[[Y1 , . . . , Ym , X1 , . . . , Xn ]] → B the homomorphism extending β1 by β(Xi ) = f (α(Xi )). Finally, let g : A → B be the composite of the canonical homomorphisms A = K[[X1 , . . . , Xn ]] → B ⊗A K[[X1 , . . . , Xn ]] → B = B ⊗A K[[Y1 , . . . , Ym , X1 , . . . , Xn ]], which is formally smooth because both the homomorphisms are formally smooth (the left one by base change (2.6.5), (1.4.3), (2.3.5) and the right one by (2.2.2.a), (6.1.2)). ii) Assume now that A does not contain a field. Let A0 → A be a local homomorphism where A0 is a Cohen ring with residue field K (3.2.1). By (6.1.7), there exists a formally smooth complete local noetherian A0 algebra C such that C ⊗A0 K = B ⊗A K. Let β : C → B be a local β
homomorphism of A0 -algebras such that C −→ B → C ⊗A0 K is the canonical homomorphism ((2.2.3) or (6.1.5)). Let A := A0 [[X1 , . . . , Xn ]] → A be a finite homomorphism with n = dim A − 1 or n = dim A (depending on whether the characteristic p of
6.2 Some results on vanishing of homology
115
K is part of a system of parameters of A or not; see (3.2.2), (3.2.3). In particular, n = dim A − 1 if A is an integral domain, as we saw in the proof of (3.2.4.b)). Let B := C[[Y1 , . . . , Ym , X1 , . . . , Xn ]] be as in i). The proof concludes in the same way as that of i).
6.2 Some results on vanishing of homology Lemma 6.2.1 Let S be a noetherian ring, n an integer and P∗ a complex of S-modules such that for any S-module of finite type N , Hn (P∗ ⊗S N ) is an S-module of finite type (e.g., if P∗ is a complex of S-modules of finite type). If Hn (P∗ ⊗S S/m) = 0 for any maximal ideal m of S, then Hn (P∗ ⊗S M ) = 0 for any S-module M . Proof We can assume that S is local with maximal ideal m. Using direct limits we can assume that M is of finite type, and by the homology long exact sequence we can assume that M is cyclic. Let X be the set of ideals I of S such that Hn (P∗ ⊗S S/I) = 0. Since S is noetherian, if X = ∅, then X has a maximal element J (with the inclusion order). Let a ∈ S − J. Consider the exact sequence .
a
0 → S/(J : a) −→ S/J → S/(J + (a)) → 0 and so the exact sequence .
a
Hn (P∗ ⊗S S/(J : a)) −→ Hn (P∗ ⊗S S/J) → Hn (P∗ ⊗S S/(J + (a))). Since Hn (P∗ ⊗S S/J) = 0 and Hn (P∗ ⊗S S/(J + (a))) = 0 by the maximality of J, we have Hn (P∗ ⊗S S/(J : a)) = 0, and so, again by the maximality of J, J = (J : a). So we have a surjective homomorphism .
a
Hn (P∗ ⊗S S/J) −→ Hn (P∗ ⊗S S/J), that is, aHn (P∗ ⊗S S/J) = Hn (P∗ ⊗S S/J). Since Hn (P∗ ⊗S S/J) is an S-module of finite type, from Nakayama’s lemma we deduce that a is a unit in S. Therefore the elements a of S − J are units and so J = m, contradicting the assumption. Lemma 6.2.2 Let R → S be a local homomorphism of regular local rings containing a field. Then H2 (R, S, −) = 0.
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Localization of formal smoothness
Proof Let K be the prime subfield of R. Let M be an S-module, and consider the Jacobi–Zariski exact sequence H2 (K, S, M ) → H2 (R, S, M ) → H1 (K, R, M ). We will see that H2 (K, S, M ) = 0. By (2.5.8), the homomorphism K → S is regular, and so by (5.3.2), S = lim Si , where each Si is a −→ smooth K-algebra of finite type. Let K i be the residue field of Si . By (2.5.4) we have H2 (K, Si ,K i ) = 0, and so from (1.4.4), (6.2.1) we deduce that H2 (K, Si , −) = 0. Finally, by (1.4.8), H2 (K, S, −) = 0. Using (2.5.8) again, the homomorphism K → R is regular, and by (5.6.2) we see that H1 (K, R, M ) = 0. From the exact sequence we deduce H2 (R, S, M ) = 0. Remarks i) In the proof of (6.2.2), we used (6.2.1) to prove the following: Let R be a noetherian ring, S an R-algebra of finite type such that H2 (R, S, S/m) = 0 for any maximal ideal m of S. Then H2 (R, S, −) = 0.
The reader will notice that this fact could be proved without using (6.2.1) as follows: by (1.4.1.d), (1.4.7), (1.4.6) we can assume R → S surjective and local, and so the result would follow from (2.5.2). However (6.2.1) will be used in (6.2.4). ii) The assumption that the rings contain a field is not necessary, as can be seen in [An1, Suppl´ement, Proposition 32]. Lemma 6.2.3 Let f : A → B be a local homomorphism of complete noetherian local rings containing a field. Then for any B-module of finite type M , H2 (A, B, M ) is a B-module of finite type. The same result holds replacing the assumption A complete by A regular. Proof Let g
A −−→ B β α f
A −−→ B be a commutative diagram of local homomorphisms of complete noetherian local rings containing a field with α and β surjective, A and B
6.3 Noetherian property of the relative Frobenius
117
regular; indeed, by (3.2.4) there exist surjective homomorphisms A := K[[X1 , . . . , Xn ]] − →A α
and B := L[[Y1 , . . . , Ym ]] → B,
where K and L are the residue fields of A and B respectively. Take B := B [[X1 , . . . , Xn ]] and the obvious homomorphisms g and β. In the exact sequence H2 (A , B , M ) → H2 (A , B, M ) → H2 (B , B, M ) we have H2 (A , B , M ) = 0 by (6.2.2), and H2 (B , B, M ) is a B-module of finite type by (1.4.4). Therefore H2 (A , B, M ) is a B-module of finite type. By (1.4.4) again, H1 (A , A, M ) is a B-module of finite type, and so from the exact sequence H2 (A , B, M ) → H2 (A, B, M ) → H1 (A , A, M ) we deduce the result. In the case that A is regular (not necessarily complete), consider the exact sequence associated to A → Aˆ → B ˆ M ) → H2 (A, B, M ) → H2 (A, ˆ B, M ). H2 (A, A, ˆ B, M ) is a B-module of ˆ M ) = 0 by (6.2.2), and H2 (A, Since H2 (A, A, finite type by the previous case, we deduce that H2 (A, B, M ) is of finite type. Lemma 6.2.4 Let (A, m, K) → (B, n, L) be a local homomorphism of complete noetherian local rings containing a field. If H2 (A, B, L) = 0, then H2 (A, B, −) = 0. The same result holds replacing the assumption A complete by A regular. Proof (6.2.1) and (6.2.3).
6.3 Noetherian property of the relative Frobenius Proposition 6.3.1 Let L|K be a field extension. The following are equivalent: i) There exists a formally smooth noetherian local K-algebra A with residue field K-isomorphic to L. ii) dimL H1 (K, L, L) < ∞.
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Localization of formal smoothness
Proof i) =⇒ ii) This is a particular case of (5.5.10). ii) =⇒ i) Let V = H1 (K, L, L), A0 = SL (V ) the symmetric Lalgebra on the L-module V and n = ker(SL (V ) → L) the augmentation ideal. By (1.4.5.c), (2.1.4), there exists an infinitesimal extension of L over K by H1 (K, L, L) associated to the identity map id ∈ HomL (H1 (K, L, L), H1 (K, L, L)) = H 1 (K, L, H1 (K, L, L)). This extension is of the form 0 → H1 (K, L, L) → A0 /n2 → L → 0 for some K-algebra structure f : K → A0 /n2 on A0 /n2 , in general different from the structure given by the composite of the canonical homomorphisms K → L → SL (V ) → SL (V )/n2 (note that the ring structure in A0 /n2 is the canonical one, since any infinitesimal extension of L over K by H1 (K, L, L) is trivial regarded as an extension over the prime subfield F of K: H 1 (K, L, H1 (K, L, L)) → H 1 (F, L, H1 (K, L, L)) = 0 by (2.4.5), (1.4.5.c)). Let A = S L (V ) be the n-adic completion of A0 , m = nA its maximal ideal. We have a commutative diagram of F -algebras (where F is the prime subfield of K)
) A −→ A/m2
K f
g =
A0 /n2
where g exists by (2.2.3). Consider A as K-algebra via g. We must show that A is formally smooth over K. Since A is a regular local ring (A = L[[X1 , . . . , Xn ]] with n = dim V ), we have H2 (A, L, L) = 0 by (2.5.3). On the other, inspecting the isomorphisms (1.4.5.c) and (2.1.4), we see that the canonical map H1 (K, L, L) → H1 (A, L, L) corresponds with the canonical isomorphism ≈ H1 (K, L, L) − → m/m2 deduced from the extension 0 → H1 (K, L, L) → A0 /n2 = A/m2 → L → 0. Thus the Jacobi–Zariski exact sequence ≈
→ H1 (A, L, L) 0 = H2 (A, L, L) → H1 (K, A, L) → H1 (K, L, L) − shows that H1 (K, A, L) = 0 and so A is formally smooth over K.
6.3 Noetherian property of the relative Frobenius
119
Corollary 6.3.2 Let K be a field of characteristic p > 0, (B, n, L) a noetherian local K-algebra such that dimL H1 (K, L, L) < ∞. Then the ring B ⊗K K 1/p is noetherian. Proof Let A be a complete formally smooth noetherian local K-algebra with residue field K-isomorphic to L (6.3.1). By (2.2.3), the homoˆ where B ˆ is morphism A → L induces a local homomorphism A → B, the n-adic completion of B. By (3.2.2), replacing A by A[[X1 , . . . , Xn ]] ˆ is surjective. By (2.5.9), if necessary, we can assume that A → B ˆ ⊗K K 1/p is surA ⊗K K 1/p is noetherian, and since A ⊗K K 1/p → B 1/p ˆ ˆ jective, B ⊗K K is noetherian. Since B → B is faithfully flat, so is ˆ ⊗K K 1/p , and then B ⊗K K 1/p is noetherian. B ⊗K K 1/p → B Lemma 6.3.3 Let f : (A, m, K) → (B, n, L) be a formally smooth homomorphism of complete regular local rings containing a field of characteristic p > 0. Let φ : A → A be the Frobenius homomorphism and φA as in (4.4.1). Then φA ⊗A B is a noetherian local ring. Proof By (5.5.10), dimL H1 (K, L, L) < ∞. So, identifying K with a subfield of A (3.2.4), φK⊗K B is a noetherian ring (6.3.2), and so φK⊗A B is noetherian. Let φm be the maximal ideal of φA. From the exact sequence 0 → φm → φA → φK → 0 we obtain an exact sequence 0 → φm ⊗A B → φA ⊗A B → φK ⊗A B → 0. Note that Spec(φA ⊗A B) = Spec(B), since we have homomorphisms α : B → φA ⊗A B, α(b) = 1 ⊗ b, β : φA ⊗A B → φB, β(a ⊗ b) = f (a)bp , such that αβ and βα are the Frobenius homomorphisms of φA ⊗A B and B. In particular φA ⊗A B is local. The ideal φm ⊗A B of φA ⊗A B is of finite type, and the maximal ideal of φK ⊗A B is also of finite type (since φK ⊗A B is noetherian). Thus the maximal ideal of φA ⊗A B is of finite type. Let φA = lim Ai , where each homomorphism Ai → Aj is local and flat, −→ and the A-algebras Ai are of finite type (6.5.6). We have φA ⊗A B = lim(Ai ⊗A B). The homomorphism Ai → φA is local and flat, so faithfully −→ flat, and then the homomorphism Ai ⊗A B → φA ⊗A B is also faithfully flat. Since φA ⊗A B is local, so is Ai ⊗A B. The maximal ideal of φ A ⊗A B is of finite type, so taking i large enough, we can assume that
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Localization of formal smoothness
the image of the maximal ideal of Ai ⊗A B by the local homomorphism Ai ⊗A B → φA ⊗A B generates the maximal ideal of φA ⊗A B. So we have a filtered direct system {Ai ⊗A B}i of local flat homomorphisms of noetherian local rings, such that for each i, the image of the maximal ideal of Ai ⊗A B in the limit φA ⊗A B generates the maximal ideal of this ring. As we saw in (3.1.5) and its remark, φA ⊗A B is noetherian.
6.4 End of the proof of localization of formal smoothness Proposition 6.4.1 Let f : A → B be a formally smooth homomorphism of complete regular local rings containing a field of characteristic p > 0. Then f is regular. Proof Let φ : A → A, φ : B → B be the Frobenius homomorphisms, and φA, φB as in (4.4.1). Let β : φA ⊗A B → φB, β(x ⊗ y) = f (x)y p . Step 1. We shall see that φA ⊗A B is regular. Let L be the residue field of B, and E the residue field of φA ⊗A B. This ring is noetherian and local by (6.3.3). The Jacobi–Zariski exact sequence associated to φ A → φA ⊗A B → φB (6.2.2), (1.4.3), (2.3.5) 0 = H2 (φA, φB, φL) → H2 (φA ⊗A B, φB, φL) → H1 (φA, φA ⊗A B, φL) = H1 (A, B, φL) = 0 shows that H2 (φA ⊗A B, φB, φL) = 0. So from the Jacobi–Zariski exact sequence associated to φA ⊗A B → φB → φL (2.5.3) 0 = H2 (φA ⊗A B, φB, φL) → H2 (φA ⊗A B, φL, φL) → H2 (φB, φL, φL) = 0 we deduce that H2 (φA ⊗A B, φL, φL) = 0. Then by (2.4.11) we have H2 (φA ⊗A B, E, φL) = 0 and so φA ⊗A B is regular by (1.4.5.a) and (2.5.3). Step 2. We have H2 (φA ⊗A B, φB, −) = 0. In fact, in the proof of step 1 we saw that H2 (φA ⊗A B, φB, φL) = 0, and so the result follows from (6.2.4). Step 3. Finally we shall show that f is regular. Let M be a B-module
6.5 Appendix: Power series
121
(φM the B-module via φ : B → B), and consider the Jacobi–Zariski exact sequence α
→ H1 (φA, φB, φM ). 0 = H2 (φA ⊗A B, φB, φM ) → H1 (φA, φA ⊗A B, φM ) − Then the composite homomorphism (1.4.3) ≈
α
→ H1 (φA, φB, φM ) → H1 (φA, φA ⊗A B, φM ) − H1 (A, B, φM ) − is injective. But this homomorphism is zero by (2.4.7) or (4.4.1), and so H1 (A, B, φM ) = 0. In particular, if q ∈ Spec B, 0 = H1 (A, B, φk(q)) = H1 (A, B, k(q)) ⊗k(q) φk(q) (1.4.5.a), and so H1 (A, B, k(q)) = 0 for all q. Therefore f is regular by (5.6.2). Theorem 6.4.2 Let f : A → B be a formally smooth homomorphism of is regular), noetherian local rings. If A is quasi-excellent (that is, A → A then f is regular. Proof First, we prove the following result for any n ≥0: (∗n) Let f : A → B be a formally smooth homomorphism of noetherian local rings. If A is a quasi-excellent domain of dimension n, then f is regular. The case n = 0 is (2.5.8), (2.5.9). Assume we have proved (∗n) for n < r. By (6.1.2), (6.1.3), (6.1.6), (6.1.8), we can assume A and B regular and complete (note that the dimension of A has not changed). We have to prove that for any prime ideal p of A, the fiber in 0 of the homomorphism A/p → B/pB is geometrically regular. The homomorphism A/p → B/pB is formally smooth by (2.3.5), (1.4.3), (2.6.5). So by (∗n) for n < r, it is enough to consider the case p = 0. If the field of fractions Q of A is of characteristic zero, since B is regular, and so B ⊗A Q = S −1 B with S = A − {0} is also regular, the result follows from (2.5.8). If the characteristic of Q is p > 0, then A contains the field Z/pZ and this case was proved in (6.4.1). So we have proved the result whenever A is a domain. Since we can assume A regular local by (6.1.2), (6.1.3), (6.1.6), (6.1.8), the proof is complete.
6.5 Appendix: Power series We prove here a theorem of Andr´e [An3, th´eor`eme 23] used in (6.3.3).
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Localization of formal smoothness
Definition 6.5.1 Let m ≥ 1 be an integer. We define an order relation in I = {r = (r1 , . . . , rm ) : r1 , . . . , rm are integers ≥ 0} by: r < s if and only if one of the following conditions holds: m m ri < i=1 si . i) m i=1 m ii) i=1 si and there exists an integer 1 ≤ j ≤ m such i=1 ri = that ri = si for all i > j and rj < sj . It is a total order. Lemma 6.5.2 Let {r(n)}n ≥ 0 be an infinite sequence in I. Then there exists 0 ≤ v < u such that r(v)i ≤ r(u)i for all i = 1, . . . , m (we will say that r(v) is below r(u)). Proof We omit the proof of this lemma since it is easier to think by oneself than to read (hint: think first the case m = 2). Let L|K be a field extension of characteristic p with Lp ⊂ K. Let T = (T1 , . . . , Tm ) be variables, A := K[[T ]] ⊂ B := L[[T ]]. We have B p ⊂ A. If t ∈ I, we denote τt ( Σ αr T r ) = Σ αr T r , where if r = (r1 , . . . , rm ) rm . then T r = T1r1 · · · Tm
r∈I
r≤t
Definition 6.5.3 Let b1 , . . . , bn ∈ B, let bi (0) be the degree 0 term of bi . We say that b1 , . . . , bn are perfectly p-independent over A if the elements b1 (0), . . . , bn (0) ∈ L are p-independent over K (5.5.5). If R is a ring containing a field of characteristic p, let R{n} be the submodule of the R-module R[X] = R[X1 , . . . , Xn ] consisting in the polynomials of the form
αv1 ,...,vn X1v1 · · · Xnvn . P (X) = 0≤vi
In the case R = A, each coefficient αv1 ,...,vn is of the form αv1 ,...,vn = Σ λv1 ,...,vn ,r T r with λv1 ,...,vn ,r ∈ K. r∈I
For each r ∈ I, let fr (X) = Then we can write
0≤vi
λv1 ,...,vn ,r X1v1 · · · Xnvn ∈ K{n}.
P (X) = Σ fr (X)T r ∈ K{n}[[T ]] r∈I
since the number of monomials X1v1 · · · Xnvn is finite.
6.5 Appendix: Power series
123
Remark 6.5.4 If b1 , . . . , bn ∈ B are perfectly p-independent over A, then they are p-independent over A in the sense of (5.5.5). In fact, if b1 , . . . , bn ∈ B are not p-independent over A, there exists 0 = P (X) ∈ A{n} as before such that P (b1 , . . . , bn ) = 0. Let r be the least index such that fr (X) = 0 in K{n}. Then fr (b1 (0), . . . , bn (0)) = 0 showing that b1 (0), . . . , bn (0) are not p-independent over K. Proposition 6.5.5 Let u be an element of B and b1 , . . . , bα a finite number of elements of B perfectly p-independent over A. Then there exists a finite number of elements c1 , . . . , cβ of B such that b1 , . . . , bα , c1 , . . . , cβ are perfectly p-independent over A and u belongs to the A-subalgebra generated by them. Proof Let bi = bi (0), cj = cj (0). Let t be an element of I and let ∫t be the following situation (in which β depends on t) i) the elements b1 , . . . , bα , c1 , . . . , cβ of L are p-independent over K. ii) there exists a polynomial P (X1 , . . . , Xα , Y1 , . . . , Yβ ) =
fr (X1 , . . . , Xα , Y1 , . . . , Yβ )T r
r∈I
of K{α + β}[[T ]] such that τt (P (b1 , . . . , bα , c1 , . . . , cβ )) = τt (u). iii) there exist elements r1 < · · · < rβ ≤ t of I, none of them below another, such that for all i = 1, . . . , β, the polynomial τri (P (X1 , . . . , Xα , Y1 , . . . , Yβ ) − T ri Yi ) is in fact a polynomial in the variables X1 , . . . , Xα , Y1 , . . . , Yi−1 . Let s be the next element to t. We are going to see how to obtain ∫s from ∫t . Let γ ∈ L be such that τs (P (b1 , . . . , bα , c1 , . . . , cβ ) − γT s ) = τs (u). Case a. γ ∈ K(b1 , . . . , bα , c1 , . . . , cβ ). Then there exists f (X1 , . . . , Xα , Y1 , . . . , Yβ ) ∈ K{α + β} with f (b1 , . . . , bα ,c1 , . . . , cβ ) = γ. Consider the elements c1 , . . . , cβ as in ∫t , replace P (X1 , . . . , Xα , Y1 , . . . , Yβ ) by P (X1 , . . . , Xα , Y1 , . . . , Yβ ) + T s f (X1 , . . . , Xα , Y1 , . . . , Yβ ), and r1 < · · · < rβ as in ∫t . We obtain ∫s .
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Localization of formal smoothness
Case b. γ ∈ / K(b1 , . . . , bα , c1 , . . . , cβ ) and at least one ri is below s. Let rλ be the first of the ri that is below s. We have s − rλ ∈ I. We keep the same polynomial P , the ri and the cj with j = λ, and we replace cλ by cλ + γT s−rλ . Since rλ ≤ t < s, we have s − rλ > 0 and so cλ = cλ (0) does not change, so the condition i) of ∫s holds. Condition iii) holds trivially, and ii) follows since by iii), P (X1 , . . . , Xα , Y1 , . . . , Yλ + W, . . . , Yβ ) − P (X1 , . . . , Xα , Y1 , . . . , Yβ )
= T rλ W + gr (X1 , . . . , Xα , Y1 , . . . , Yβ , W )W T r r>rλ
with gr ∈ K{α + β + 1}. Therefore τs (P (b1 , . . . , bα , c1 , . . . , cλ + γT s−rλ , . . . , cβ )) = τs (P (b1 , . . . , bα , c1 , . . . , cβ ) + γT s ) = τs (u). Case c. γ ∈ / K(b1 , . . . , bα , c1 , . . . , cβ ) and there is no ri below s. In this case we replace β by β + 1, we keep c1 , . . . , cβ but adding cβ+1 = γ, we replace P (X1 , . . . , Xα , Y1 , . . . , Yβ ) by P (X1 , . . . , Xα , Y1 , . . . , Yβ , Yβ+1 ) = P (X1 , . . . , Xα , Y1 , . . . , Yβ ) + T s Yβ+1 , and we keep r1 < · · · < rβ but adding rβ+1 = s. We obtain ∫s . Notice that in cases a and b we change neither β nor the elements ri . In case c, we replace β by β + 1 and we add an element rβ+1 = s. So by (6.5.2), this last case occurs only a finite number of times. On the other hand, the polynomial P in the step from ∫t to ∫s changes only in degree s and the ci change only in degree s − rλ . In particular, once we have done a finite number of steps large enough so that β does not change, the elements ci from ∫t to ∫s do not change in degrees < s − rβ , and in particular this does not affect their property of being perfectly p-independent. We can then prove the proposition by induction. We start with β = 0, P (X1 , . . . , Xα ) = 0, and we take γ = u = u(0). We take limits, obtaining a polynomial P (X1 , . . . , Xα , Y1 , . . . , Yβ ) ∈ A{α + β} and elements c1 , . . . , cβ such that b1 , . . . , bα , c1 , . . . , cβ are perfectly p-independent over A and such that P (b1 , . . . , bα , c1 , . . . , cβ ) = u. Theorem 6.5.6 Let B be a complete noetherian local ring containing a field of characteristic p > 0. Then we have an isomorphism of Balgebras φB = lim Bi where each Bi is a local B-algebra of finite type −→ and the homomorphisms Bi → Bj are local and flat.
6.5 Appendix: Power series
125
Proof First we will see that if a ring B has the property of the theorem and N is an ideal of B, then C = B/N has the same property. Let N = (x1 , . . . , xn ). Let φB = lim Bi be as in the statement, and let j0 −→ be such that there exist elements y1 , . . . , yn ∈ Bj0 whose images in φB are x1 , . . . , xn . Let J = {j : j0 ≤ j}. We have φB = lim Bj . For each −→ j∈J
j ∈ J, let Cj = Bj /(y1 , . . . , yn )Bj . The homomorphisms Cj → Ct are obtained from Bj → Bt by base change, and so they are flat. On the other hand, taking limits in the exact sequences 0 → (y1 , . . . , yn )Bj → Bj → Cj → 0 we see that φC = lim Cj . −→ j∈J
Therefore, by (3.2.4) we can suppose B = L[[T ]] where L is a field. Since B is reduced, we can identify the homomorphism φ : B → φB to the inclusion B p ⊂ B. Set K = Lp and we apply (6.5.5). We take as {Bi } the A = K[[T ]]-subalgebras of B generated by a finite set {bi1 , . . . , biαi } of elements of B perfectly p-independent over A, and when {bi1 , . . . , biαi } ⊂ {bj1 , . . . , bjαj } we define a homomorphism Bi → Bj as the inclusion map, which is local and flat (by (6.5.4)). Since each Bi is an A-algebra of finite type, it is also a B p = K[[T p ]]-algebra of finite type. So the result follows.
Appendix: Some exact sequences
In the text we have used some exact sequences, which we have deduced from spectral sequences. In this appendix we give an elementary proof of those exact sequences, avoiding spectral sequences, so that the text could be read without the use of spectral sequences. Proposition 1.4.5.d. Let A → B → C be ring homomorphisms and M a C-module. Then there exist exact sequences TorC 2 (ΩB|A ⊗B C, M ) → H1 (A, B, C) ⊗C M → H1 (A, B, M ) → TorC 1 (ΩB|A ⊗B C, M ) → 0 and 0 → Ext1C (ΩB|A ⊗B C, M ) → H 1 (A, B, M ) → HomC (H1 (A, B, C), M ) → Ext2C (ΩB|A ⊗B C, M ). Proof Let p : R → B be a surjective A-algebra homomorphism, with R a polynomial ring over A. We can write the Jacobi–Zariski exact sequence associated to A → R → B H1 (A, R, M ) → H1 (A, B, M ) → H1 (R, B, M ) → H0 (A, R, M ) → H0 (A, B, M ) → H0 (R, B, M ) as (∗): 0 → H1 (A, B, M ) → I/I 2 ⊗B M → ΩR|A ⊗R M → ΩB|A ⊗B M → 0, where I = ker p (1.4.1). 126
Appendix: Some exact sequences
127
Taking M = C, we obtain exact sequences (defining W ) 0 → H1 (A, B, C) → I/I 2 ⊗B C → W → 0 0 → W → ΩR|A ⊗R C → ΩB|A ⊗B C → 0 and hence exact sequences (∗∗) C TorC 2 (ΩB|A ⊗B C, M ) = Tor1 (W, M ) → H1 (A, B, C) ⊗C M
→ I/I 2 ⊗B M → W ⊗C M → 0 and (∗ ∗ ∗) 0 → TorC 1 (ΩB|A ⊗B C, M ) → W ⊗C M → ΩR|A ⊗R M → ΩB|A ⊗B M → 0. We can put (∗), (∗∗) and (∗ ∗ ∗) together into a commutative diagram with exact rows and columns: TorC 2 (ΩB|A ⊗B C, M )
0→
H1 (A, B, M )
H1 (A, B, C) ⊗C M → I/I 2 ⊗B M → ΩR|A ⊗R M → ΩB|A ⊗B M → 0
0 → TorC 1 (ΩB|A ⊗B C, M ) → W ⊗C M → ΩR|A ⊗R M → ΩB|A ⊗B M → 0 0
and so we deduce an exact sequence TorC 2 (ΩB|A ⊗B C, M ) → H1 (A, B, C) ⊗C M → H1 (A, B, M ) → TorC 1 (ΩB|A ⊗B C, M ) → 0 The proof for cohomology is similar.
Lemma (Exact sequence for proof of (2.6.1)) Let A be a ring, B an A-module, C an A-algebra and M a C-module. We have an exact sequence C TorA 2 (B, M ) → Tor2 (B ⊗A C, M ) → A C TorA 1 (B, C) ⊗C M → Tor1 (B, M ) → Tor1 (B ⊗A C, M ) → 0.
128
Appendix: Some exact sequences λ
µ
→P − → B → 0 be an exact sequence of A-modules Proof Let 0 → K − with P projective. We have an exact sequence µ∗
λ
∗ 0 → TorA 1 (B, C) → K ⊗A C −→ P ⊗A C −→ B ⊗A C → 0.
Let D = ker(µ∗ ) = im(λ∗ ). Applying − ⊗C M to the exact sequences 0 → TorA 1 (B, C) → K ⊗A C → D → 0 0 → D → P ⊗A C → B ⊗A C → 0 and − ⊗A M to the exact sequence 0→K→P →B→0 we obtain a commutative diagram with exact rows and columns 0
0
↓ TorC 1 (K
⊗A C, M ) →
TorC 1 (D, M )
→
*
TorA 1 (B, M )
TorA 1 (B, C)
↓ →
↓
⊗C M → K ⊗A M
→
D ⊗C M → 0 ↓
= (P ⊗A C) ⊗C M
↓ B ⊗A M
⊗A C, M ) ↓
↓ P ⊗A M
TorC 1 (B
↓ = (B ⊗A C) ⊗C M
Moreover, since 0 → D → P ⊗A C → B ⊗A C → 0 is a projective C presentation over C, we have TorC 1 (D, M ) = Tor2 (B ⊗A C, M ). So from the above diagram we obtain an exact sequence C TorC 1 (K ⊗A C, M ) → Tor2 (B ⊗A C, M ) → A C TorA 1 (B, C) ⊗C M → Tor1 (B, M ) → Tor1 (B ⊗A C, M ) → 0.
So it is enough to show an epimorphism C TorA 2 (B, M ) → Tor1 (K ⊗A C, M ). A Since TorA 2 (B, M ) = Tor1 (K, M ), the same reasoning for the Amodule K instead of B gives an epimorphism C TorA 1 (K, M ) → Tor1 (K ⊗A C, M ).
Appendix: Some exact sequences
129
A similar proof in cohomology gives an exact sequence 0 → Ext1A/I (J/IJ, M ) → Ext1A (J, M ) → HomA/I (TorA 1 (J, A/I), M ) for ideals J ⊂ I of A and an A/I-module M , which in the case J = I n was used in the proof of (2.3.4).
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Index
φ
A (4.4.1), 64 A , Ar (5.5.1), 91 ∆M,s (5.1.1)–(5.1.3), 68 HC|A (5.1.8), (5.1.10), 70–1 Cohen algebra (3.1.1), 47 Cohen ring (3.1.9), 51 complete intersection defect (4.3.2), 62 complete intersection ring (4.3.1), 62 derivation on a DG algebra (4.1.2), 56 differential graded algebra (1.2.1), 5 formally smooth (2.2.1), 26 free DG resolution (1.2.8), 9 geometrically regular (2.5.6), 41 homology module (1.1.1), (1.3.6), 4, 16 infinitesimal extension (2.1.1), 23 minimal DG resolution (4.1.1), 56 n-skeleton (1.2.9), 9 p-independent, p-basis (5.5.5), 92 perfectly p-independent (6.5.3), 122 regular homomorphism (5.3.1), 83 resolvable (5.3.8), 85 smooth (2.2.1), 26 standard, strictly standard element (5.1.11), 71
134
