TRUBATCH 303 Number theory and algebraic geometry, M.. STALLARD eds 349 Model theory with applications to algebra and analysis I, Z.. WILKIE eds 350 Model theory with applications to alg
Trang 3Managing Editor: Professor M Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
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222 Advances in linear logic, J.-Y GIRARD, Y LAFONT & L REGNIER (eds)
223 Analytic semigroups and semilinear initial boundary value problems, K TAIRA
224 Computability, enumerability, unsolvability, S.B COOPER, T.A SLAMAN & S.S WAINER (eds)
225 A mathematical introduction to string theory, S ALBEVERIO et al
226 Novikov conjectures, index theorems and rigidity I, S.C FERRY, A RANICKI & J ROSENBERG (eds)
227 Novikov conjectures, index theorems and rigidity II, S.C FERRY, A RANICKI & J ROSENBERG (eds)
228 Ergodic theory of Zd-actions, M POLLICOTT & K SCHMIDT (eds)
229 Ergodicity for infinite dimensional systems, G DA PRATO & J ZABCZYK
230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S CASSELS & E.V FLYNN
231 Semigroup theory and its applications, K.H HOFMANN & M.W MISLOVE (eds)
232 The descriptive set theory of Polish group actions, H BECKER & A.S KECHRIS
233 Finite fields and applications, S COHEN & H NIEDERREITER (eds)
234 Introduction to subfactors, V JONES & V.S SUNDER
235 Number theory: Séminaire de théorie des nombres de Paris 1993–94, S DAVID (ed)
236 The James forest, H FETTER & B GAMBOA DE BUEN
237 Sieve methods, exponential sums, and their applications in number theory, G.R.H GREAVES et al (eds)
238 Representation theory and algebraic geometry, A MARTSINKOVSKY & G TODOROV (eds)
240 Stable groups, F.O WAGNER
241 Surveys in combinatorics, 1997, R.A BAILEY (ed)
242 Geometric Galois actions I, L SCHNEPS & P LOCHAK (eds)
243 Geometric Galois actions II, L SCHNEPS & P LOCHAK (eds)
244 Model theory of groups and automorphism groups, D.M EVANS (ed)
245 Geometry, combinatorial designs and related structures, J.W.P HIRSCHFELD et al (eds)
246 p-Automorphisms of finite p-groups, E.I KHUKHRO
247 Analytic number theory, Y MOTOHASHI (ed)
248 Tame topology and O-minimal structures, L VAN DEN DRIES
249 The atlas of finite groups - Ten years on, R.T CURTIS & R.A WILSON (eds)
250 Characters and blocks of finite groups, G NAVARRO
251 Gröbner bases and applications, B BUCHBERGER & F WINKLER (eds)
252 Geometry and cohomology in group theory, P.H KROPHOLLER, G.A NIBLO & R STÖHR (eds)
253 Theq-Schur algebra, S DONKIN
254 Galois representations in arithmetic algebraic geometry, A.J SCHOLL & R.L TAYLOR (eds)
255 Symmetries and integrability of difference equations, P.A CLARKSON & F.W NIJHOFF (eds)
256 Aspects of Galois theory, H VÖLKLEIN, J.G THOMPSON, D HARBATER & P MÜLLER (eds)
257 An introduction to noncommutative differential geometry and its physical applications (2nd edition), J MADORE
258 Sets and proofs, S.B COOPER & J.K TRUSS (eds)
259 Models and computability, S.B COOPER & J TRUSS (eds)
260 Groups St Andrews 1997 in Bath I, C.M CAMPBELL et al (eds)
261 Groups St Andrews 1997 in Bath II, C.M CAMPBELL et al (eds)
262 Analysis and logic, C.W HENSON, J IOVINO, A.S KECHRIS & E ODELL
263 Singularity theory, W BRUCE & D MOND (eds)
264 New trends in algebraic geometry, K HULEK, F CATANESE, C PETERS & M REID (eds)
265 Elliptic curves in cryptography, I BLAKE, G SEROUSSI & N SMART
267 Surveys in combinatorics, 1999, J.D LAMB & D.A PREECE (eds)
268 Spectral asymptotics in the semi-classical limit, M DIMASSI & J SJÖSTRAND
269 Ergodic theory and topological dynamics of group actions on homogeneous spaces, M.B BEKKA & M MAYER
271 Singular perturbations of differential operators, S ALBEVERIO & P KURASOV
272 Character theory for the odd order theorem, T PETERFALVI Translated by R SANDLING
273 Spectral theory and geometry, E.B DAVIES & Y SAFAROV (eds)
274 The Mandelbrot set, theme and variations, T LEI (ed)
275 Descriptive set theory and dynamical systems, M FOREMAN, A.S KECHRIS, A LOUVEAU & B WEISS (eds)
276 Singularities of plane curves, E CASAS-ALVERO
277 Computational and geometric aspects of modern algebra, M ATKINSON et al (eds)
278 Global attractors in abstract parabolic problems, J.W CHOLEWA & T DLOTKO
279 Topics in symbolic dynamics and applications, F BLANCHARD, A MAASS & A NOGUEIRA (eds)
280 Characters and automorphism groups of compact Riemann surfaces, T BREUER
281 Explicit birational geometry of 3-folds, A CORTI & M REID (eds)
282 Auslander-Buchweitz approximations of equivariant modules, M HASHIMOTO
283 Nonlinear elasticity, Y.B FU & R.W OGDEN (eds)
284 Foundations of computational mathematics, R DEVORE, A ISERLES & E SÜLI (eds)
285 Rational points on curves over finite fields, H NIEDERREITER & C XING
286 Clifford algebras and spinors (2nd Edition), P LOUNESTO
287 Topics on Riemann surfaces and Fuchsian groups, E BUJALANCE, A.F COSTA & E MARTÍNEZ (eds)
288 Surveys in combinatorics, 2001, J.W.P HIRSCHFELD (ed)
289 Aspects of Sobolev-type inequalities, L SALOFF-COSTE
290 Quantum groups and Lie theory, A PRESSLEY (ed)
291 Tits buildings and the model theory of groups, K TENT (ed)
292 A quantum groups primer, S MAJID
293 Second order partial differential equations in Hilbert spaces, G DA PRATO & J ZABCZYK
294 Introduction to operator space theory, G PISIER
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299 Kleinian groups and hyperbolic 3-manifolds, Y KOMORI, V MARKOVIC & C SERIES (eds)
300 Introduction to Möbius differential geometry, U HERTRICH-JEROMIN
301 Stable modules and the D(2)-problem, F.E.A JOHNSON
302 Discrete and continuous nonlinear Schrödinger systems, M.J ABLOWITZ, B PRINARI & A.D TRUBATCH
303 Number theory and algebraic geometry, M REID & A SKOROBOGATOV (eds)
304 Groups St Andrews 2001 in Oxford I, C.M CAMPBELL, E.F ROBERTSON & G.C SMITH (eds)
305 Groups St Andrews 2001 in Oxford II, C.M CAMPBELL, E.F ROBERTSON & G.C SMITH (eds)
306 Geometric mechanics and symmetry, J MONTALDI & T RATIU (eds)
307 Surveys in combinatorics 2003, C.D WENSLEY (ed.)
308 Topology, geometry and quantum field theory, U.L TILLMANN (ed)
309 Corings and comodules, T BRZEZINSKI & R WISBAUER
310 Topics in dynamics and ergodic theory, S BEZUGLYI & S KOLYADA (eds)
311 Groups: topological, combinatorial and arithmetic aspects, T.W MÜLLER (ed)
312 Foundations of computational mathematics, Minneapolis 2002, F CUCKER et al (eds)
313 Transcendental aspects of algebraic cycles, S MÜLLER-STACH & C PETERS (eds)
314 Spectral generalizations of line graphs, D CVETKOVI ´ C, P ROWLINSON & S SIMI ´ C
315 Structured ring spectra, A BAKER & B RICHTER (eds)
316 Linear logic in computer science, T EHRHARD, P RUET, J.-Y GIRARD & P SCOTT (eds)
317 Advances in elliptic curve cryptography, I.F BLAKE, G SEROUSSI & N.P SMART (eds)
318 Perturbation of the boundary in boundary-value problems of partial differential equations, D HENRY
319 Double affine Hecke algebras, I CHEREDNIK
320 L-functions and Galois representations, D BURNS, K BUZZARD & J NEKOVÁ ˇ R (eds)
321 Surveys in modern mathematics, V PRASOLOV & Y ILYASHENKO (eds)
322 Recent perspectives in random matrix theory and number theory, F MEZZADRI & N.C SNAITH (eds)
323 Poisson geometry, deformation quantisation and group representations, S GUTT et al (eds)
324 Singularities and computer algebra, C LOSSEN & G PFISTER (eds)
325 Lectures on the Ricci flow, P TOPPING
326 Modular representations of finite groups of Lie type, J.E HUMPHREYS
327 Surveys in combinatorics 2005, B.S WEBB (ed)
328 Fundamentals of hyperbolic manifolds, R CANARY, D EPSTEIN & A MARDEN (eds)
329 Spaces of Kleinian groups, Y MINSKY, M SAKUMA & C SERIES (eds)
330 Noncommutative localization in algebra and topology, A RANICKI (ed)
331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A PINKUS, E SÜLI & M.J TODD (eds)
332 Handbook of tilting theory, L ANGELERI HÜGEL, D HAPPEL & H KRAUSE (eds)
333 Synthetic differential geometry (2nd Edition), A KOCK
334 The Navier-Stokes equations, N RILEY & P DRAZIN
335 Lectures on the combinatorics of free probability, A NICA & R SPEICHER
336 Integral closure of ideals, rings, and modules, I SWANSON & C HUNEKE
337 Methods in Banach space theory, J.M.F CASTILLO & W.B JOHNSON (eds)
338 Surveys in geometry and number theory, N YOUNG (ed)
339 Groups St Andrews 2005 I, C.M CAMPBELL, M.R QUICK, E.F ROBERTSON & G.C SMITH (eds)
340 Groups St Andrews 2005 II, C.M CAMPBELL, M.R QUICK, E.F ROBERTSON & G.C SMITH (eds)
341 Ranks of elliptic curves and random matrix theory, J.B CONREY, D.W FARMER, F MEZZADRI & N.C SNAITH (eds)
342 Elliptic cohomology, H.R MILLER & D.C RAVENEL (eds)
343 Algebraic cycles and motives I, J NAGEL & C PETERS (eds)
344 Algebraic cycles and motives II, J NAGEL & C PETERS (eds)
345 Algebraic and analytic geometry, A NEEMAN
346 Surveys in combinatorics 2007, A HILTON & J TALBOT (eds)
347 Surveys in contemporary mathematics, N YOUNG & Y CHOI (eds)
348 Transcendental dynamics and complex analysis, P.J RIPPON & G.M STALLARD (eds)
349 Model theory with applications to algebra and analysis I, Z CHATZIDAKIS, D MACPHERSON, A PILLAY & A WILKIE (eds)
350 Model theory with applications to algebra and analysis II, Z CHATZIDAKIS, D MACPHERSON, A PILLAY & A WILKIE (eds)
351 Finite von Neumann algebras and masas, A.M SINCLAIR & R.R SMITH
352 Number theory and polynomials, J MCKEE & C SMYTH (eds)
353 Trends in stochastic analysis, J BLATH, P MÖRTERS & M SCHEUTZOW (eds)
354 Groups and analysis, K TENT (ed)
355 Non-equilibrium statistical mechanics and turbulence, J CARDY, G FALKOVICH & K GAWEDZKI
356 Elliptic curves and big Galois representations, D DELBOURGO
357 Algebraic theory of differential equations, M.A.H MACCALLUM & A.V MIKHAILOV (eds)
358 Geometric and cohomological methods in group theory, M.R BRIDSON, P.H KROPHOLLER & I.J LEARY (eds)
359 Moduli spaces and vector bundles, L BRAMBILA-PAZ, S.B BRADLOW, O GARCÍA-PRADA & S RAMANAN (eds)
360 Zariski geometries, B ZILBER
361 Words: Notes on verbal width in groups, D SEGAL
362 Differential tensor algebras and their module categories, R BAUTISTA, L SALMERÓN & R ZUAZUA
363 Foundations of computational mathematics, Hong Kong 2008, F CUCKER, A PINKUS & M.J TODD (eds)
364 Partial differential equations and fluid mechanics, J.C ROBINSON & J.L RODRIGO (eds)
365 Surveys in combinatorics 2009, S HUCZYNSKA, J.D MITCHELL & C.M RONEY-DOUGAL (eds)
366 Highly oscillatory problems, B ENGQUIST, A FOKAS, E HAIRER & A ISERLES (eds)
367 Random matrices: High dimensional phenomena, G BLOWER
368 Geometry of Riemann surfaces, F.P GARDINER, G GONZÁLEZ-DIEZ & C KOUROUNIOTIS (eds)
369 Epidemics and rumours in complex networks, M DRAIEF & L MASSOULIÉ
370 Theory of p-adic distributions, S ALBEVERIO, A.YU KHRENNIKOV & V.M SHELKOVICH
371 Conformal fractals, F PRZYTYCKI & M URBA ´ NSKI
372 Moonshine: The first quarter century and beyond, J LEPOWSKY, J MCKAY & M.P TUITE (eds)
373 Smoothness, regularity and complete intersection, J MAJADAS & A.G RODICIO
Trang 5Smoothness, Regularity and Complete Intersection
J AV I E R M A J A D A S
A N TO N I O G R O D I C I O
Universidad de Santiago
de Compostela, Spain
Trang 6Cambridge, 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
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First published 2010 Printed in the United Kingdom at the University Press, Cambridge
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Trang 72.7 Appendix: The Mac Lane separability criterion 46
3 Structure of complete noetherian local rings 47
Trang 85 Regular homomorphisms: Popescu’s theorem 67
6.3 Noetherian property of the relative Frobenius 117
6.4 End of the proof of localization of formal
Trang 9These 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 islong, though it provides a lot of additional information He uses thisresult in proving Cohen’s theorems on the structure of complete noethe-rian 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 meth-ods, that are not necessarily familiar to all commutative algebraists Athird 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
Trang 10This result is due to L.L Avramov [Av1] Its proof uses differentialgraded algebras as well as Andr´e–Quillen homology modules in dimen-sions 3 and 4, the vanishing of which characterizes complete intersec-tions.
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 followingLichtenbaum 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 thestyle 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 duces homology modules in dimensions 0, 1 and 2 First, in Section 1.1
intro-we give the definition of Lichtenbaum and Schlessinger [LS], which isvery concise, at least if we omit the proof that it is well defined Thereader willing to take this on trust and to accept its properties (1.4) canomit 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 ofsome properties Moreover, differential graded resolutions are used in
an essential way inChapter 4
Trang 11Introduction 3
Chapter 2 studies formally smooth homomorphisms, and in ular proves Theorem (I) We follow mainly [An1], making appropriate
partic-changes to avoid using homology modules in dimensions > 2 This part
was already written (in Spanish) in 1988
Chapter 3 uses the results ofChapter 2 to deduce Cohen’s theorems
on complete noetherian local rings We follow mainly [EGA 0IV] andBourbaki [Bo, Chapter 9]
InChapter 4, we prove Theorem (II) After giving Gulliksen’s result[GL] on the existence of minimal differential graded resolutions, we fol-low 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 sult characterizing regular local rings in positive characteristic in terms
re-of the Frobenius homomorphism
Finally, Chapters 5 and 6 study regular homomorphisms, giving inparticular proofs of Theorems (III), (IV) and (V)
The prerequisites for reading this book are a basic course in mutative algebra (Matsumura [Mt, Chapters 1–9] should be more thansufficient) and the first definitions in homological algebra Though inplaces we use certain exact sequences deduced from spectral sequences,
com-we give direct proofs of these in the Appendix, thus avoiding the use ofspectral sequences
Finally, we make the obvious remark that this book is not in anyway 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 alsolike 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
Trang 12Definition and first properties of
(co-)homology modules
In this chapter we define the Lichtenbaum–Schlessinger (co-)homology
modules H n (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 H n (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 giveanother (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 completeintersections More precisely, we use Gulliksen’s Theorem 4.1.7 on theexistence of minimal differential graded algebra resolutions in order toprove Avramov’s Lemma 4.2.1 Section 1.4 establishes the main prop-erties 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 e0and
0→ U → F −−→ I → 0 j
an exact sequence of R-modules with F free Let φ : 2
F → F be the module homomorphism defined by φ(x ∧y) = j(x)y −j(y)x, where2
R-F
4
Trang 131.2 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/I2 → 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 |Ais the
module of K¨ahler differentials) We denote any such complex by LB |A,and define for a B-module M
H n (A, B, M ) = H n(LB |A ⊗B M ) for n = 0, 1, 2,
H n (A, B, M ) = H n(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 R n, strictly anticommutative, i.e., satisfying
xy = ( −1) pq yx for x ∈ Rp , y ∈ Rq and x2= 0 for x ∈ R 2n+1,
and having a differential d = (d n : R n → 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, d R) → (S, dS ) of DG algebras is an algebra homomorphism that preserves degrees (f (R n)⊂ Sn) such that
A-dS f = f dR
If (R, d R ), (S, d S ) 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
Trang 14b) 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
i ∈J A
R i; for finite subsets
J ⊂ J of I, we have a canonical homomorphism
An augmented DG A-algebra is a DG A-algebra together with a jective (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.
sur-A DG subalgebra S of a DG sur-A-algebra (R, d) is a graded sur-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) =
Example 1.2.2 Let R0 be an A-algebra and X a variable of degree
n > 0 Let R = R0X be the following graded A-algebra:
a) If n is odd, R0X is the exterior R0-algebra on the variable X, i.e., R0X = R01⊕ R0X, concentrated in degrees 0 and n b) If n is even, R0X is the quotient of the polynomial R0-algebra
on variables X(1), X(2), , by the ideal generated by the
ele-ments
X (i) X (j) − (i + j)!
i!j! X
(i+j) for i, j ≥ 1.
Trang 151.2 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 ⊗R0R0X 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 settingp(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) = H q (S) for all q < n − 1; b) f induces an isomorphism H n −1 (R)/ c R0 = H n −1 (S).
Proof a) is clear, since R q = S q for q < n,
b) Z n −1 (R) = Z n −1 (S) and B n −1 (R) + xR0= B n −1 (S).
Definition 1.2.4 If {Xi}i ∈I is a family of variables of degree > 0, we define R0{Xi}i ∈I :=
i ∈I R0
R0Xi as the tensor product of the DG
R0-algebras R0Xi 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 ; dX i = x i to be the DG A-algebra
R ⊗R0(
i ∈I
R0R0Xi ; dX i = x 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
Trang 16of 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 ; dX i = x i , and let f : R → S be the canonical homomorphism Then:
a) f induces isomorphisms Hq (R) = H q (S) for all q < n − 1; b) f induces an isomorphism Hn −1 (R)/ {ci}i ∈I R0 = H n −1 (S).
Proof Similar to the proof of Lemma 1.2.3, bearing in mind that direct
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 p S extends p and gives an morphism in homology
iso-H(S) = H(R ) = R if n = 0,
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 S i is an S0-module of finite type for all i.
Proof Let S0 = R Assume that we have constructed an augmented
DG A-algebra S n −1 that is free over R, such that S n −1
0 = R0 and the
augmentation S n −1 → R induces isomorphisms H q (S n −1 ) = H q (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 H n −1 (S n −1 ) for n > 1), and {xi}i ∈I a set of homogeneouscycles with classes {ci}i ∈I Let S n = S n −1 {Xi}i ∈I ; dX i = x i Then
S n is a DG A-algebra free over R with S n
0 = R0 and such that the
augmentation p S n : S n → R extending p
S n−1 defined by p S n (X i) = 0
induces isomorphisms H q (S n ) = H q (R ) for q < n (Lemma 1.2.5).
We define S := lim −→ S n.
If R0 is a noetherian ring and R i an R0-module of finite type for all
i, then by induction we can choose S n with S n
0 -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
Trang 171.2 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 A- algebra of finite type, then we can choose R such that R0is a polynomial A-algebra of finite type and R i an R0-module of finite type for all i.
Proof Let R0 be a polynomial A-algebra such that there exists a jective homomorphism of A-algebras R0 → B (If A is noetherian and
sur-B an A-algebra of finite type, then we can choose R0 a polynomial algebra of finite type.) Now apply Theorem 1.2.6 to R0→ B.
A-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 X iof 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
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) = R0and R0→ S0 exists because R0 is a polyno-
mial A-algebra.
Trang 18b) If n is odd, let R(n) = R(n −1) {Ti}i ∈I We have a commutative
and therefore a homomorphism R(n) n → ker(Sn −1 → Sn −2) =
im(S n → Sn −1 ), and so there exist an R0-module homomorphism
t=1 atY1(r t,1)· · · Y (r t,m)
where the a t are coefficients in S0, the Y i are variables with
deg Y i > 0, and the divided powers Y j (r t,j)have integer exponents
rt,j ≥ 0 (Of course, for deg Yj odd and r > 1, we understand
Y j (r) = 0.) Then for l > 0, the image of X i (l)is determined by thefamiliar divided power rules†
Trang 19if for some j deg Y j is even and positive and r t,j α t ≥ 1;
note that the coefficient α (r t,j α t)!
t !(r t,j!)αt is an integer
Using the formula Y i (p) Y i (q) = (p+q!) p!q! Y i (p+q) , we see that (Y i (r t,i))α t =
(r t,i α t)!
(r t,i!)αt Y i (r t,i α t), and so this definition does not depend on the chosen j.
A straightforward computation (easier if we multiply “formally” by
p!q!), shows that under this map, X i (p) X i (q) and (p+q)! p!q! X i (p+q) have the
Remarks
i) The assumption that S is free over S0is only used to avoid ing divided powers structure
defin-ii) For the definition of H n (A, B, M ), for n = 0, 1, 2, we use free DG
resolutions only up to degree 3, and so we could have used metric powers resolutions instead of divided powers resolutions(since they agree in degrees≤3) However, in Chapter 4we useminimal resolutions and there we need divided powers
sym-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,
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 ΩR |A ⊗R B, where Ω R |Ais the usual
Trang 20R0-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− ⊗R0B we obtain an exact sequence
0→ I ⊗R0B → R0⊗A B → B → 0, showing that I ⊗R0B = J0 On the other hand, the exact sequence of
R0-modules
0→ I2→ I → ΩR0|A → 0
gives an exact sequence
I2⊗R0B = (I ⊗R0B)2= J02→ I ⊗R0B = J0→ ΩR0|A ⊗R0B → 0, and therefore J0/J0(2) = J0/J2= ΩR0|A ⊗R0B.
In degree 1, (J/J(2))1= J1/J0J1= (R1⊗A B)/J0(R1⊗A B) = (R1⊗A B) ⊗R0⊗ A B B = R1⊗R0B is the free B-module obtained by base extension of the free R0-module R1
Similarly, in degree 2, (J/J(2))2= J2/(J0J2+ J12) = (R2/R21)⊗R0B.
In general, for n > 0, (Ω R |A ⊗R B)n = (R(n)/R(n − 1))n ⊗R0B.
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
g
M −−→ Q u 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).
Trang 211.3 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(Y i) =
f (Xi)− g(Xi ) Define λ on monomials X i1 · · · Xi n by
f (Xi)− g(Xi ) = u(Y i ) = u(λ(X i )), 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
Trang 22tedious but straightforward calculation that this formula actually defines
a biderivation λ : S −1 P → M extending λ Since f −g is a biderivationextending 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 d S
1α0= f0− g0, and d S
i+1 α i + α i −1 d R i = f i − gi , for i = 1, 2, where d R , d S are the differentials of R and S respectively, and denotes the induced map in the following diagram
α1
g1
α0
g0
α1
g1
α0
f0
... ⊗R B, M )).
In view of Lemma 1.2.10 and Lemma 1.3.4, the definition does not
de-pend on the choice of R, and is natural in A, B and M
Proposition 1.3.7 The... homomorphisms and M a C-module There exist exact quences
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... 2.5.4 would beimmediate from Jacobi–Zariski exact sequence and Corollary 2.5.3 if wecould make use of the characterization of complete intersection rings by
the vanishing of some H3