17 using simplicial methods We shall get more elementary proof based on the Lichtenbaum Schlessinger cohomology theory 3 and counterexample showing that this result is not true for arbitrary First we recall the definition of the LichtenbaumSchlessing ID: 27355 Download Pdf

173K - views

Published bytawny-fly

17 using simplicial methods We shall get more elementary proof based on the Lichtenbaum Schlessinger cohomology theory 3 and counterexample showing that this result is not true for arbitrary First we recall the definition of the LichtenbaumSchlessing

Download Pdf

Download Pdf - The PPT/PDF document "Publicacions Matemtiques Vol Abstract..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

Publicacions Matemátiques, Vol 33 (1989), 339-343 Abstract ON THE JACOBIAN CRITERION OF FORMAL SMOOTHNESS ANTONIO RODICIO We give short proof of the jacobian criterion of formal smoothness using the Lichtenbaum-Schlessinger cotangent complex The aim of this note is to get proof of the following jacobian criterion of formal smoothness Theorem Let be ring, noetherian A-algebra, an ideal of B, and B/J Let us consider over and the discret and J-adic topology, respectively Then, the following statements are equivalent 1) The A-algebra is formally smooth 2) For every representation

R/I, where is smooth A-algebra, the canonical homomorphism is left invertible I/I ®B -> QRI ®R This theorem has been obtained by André [1, prop 16 .17] using simplicial methods We shall get more elementary proof, based on the Lichtenbaum- Schlessinger (co-)homology theory [3], and counter-example showing that this result is not true for arbitrary First we recall the definition of the Lichtenbaum-Schlessinger (co-)homology functors Let be ring, an A-algebra and B-module Choose polynomial algebra over such that R/I, and free R-module such that there exists an exact sequence of R-modules 0---+ >F

Let Uo be the image of the homomorphism ®R -> F, O(x y) j(x)y j(y)x and consider the complex of B-modules LBIA --> U/U -> F/IF --> ®R ---,

Page 2

34 RODICIO Then, (BIA,M) H¡(LBIA ®B M), Ti(BIA,M) H'(HOMB(LBIA,M)), i=0,1,2 There exist isomorphisms (BIA,M) QBI ®B M, T°(BIA,M) DerA(B, M) HOM (Q BIA M), T'(BI A, M) ExalcomA(B, M) the set of equivalente classes of infinitesimal extensions of over by M, and (B¡A, M) HOMB(I/I M) if A/I Proposition Let be ring, an A-algebra, an ideal of and B/J Assume that R/I, where is smooth A-algebra Then, the following conditions are equivalent 1) The

canonical homomorphism I/I ®B -> SZRI ®R is left invertible 2) (B ¡A, C) and QBI ®B is projective C-module 3) (B1 A, M) for all -module Proof Since is smooth A-algebra, we have (RIA, -) (RI A, -) [3, prop .1 .3] Hence there exist exact sequences [3, .3 .5] -> (B ¡A, C) ---> I/I2 ®B -> RIA ®R --> BIA ®B --, Hom (Q BIA (ZB C, M) -> Homc(Q RIA (DR C, M) HornC(I/I ®B C, M) -> (B ¡A, M) -) 0, where is C-module The result follows from this sequences having into account that QRIA ®R is projective C-module [3, prop .1 .3] For every C-module M, the homomorphisms -> --> B/ .I induce an exact sequence

(Bn1 B, M) (B ¡A, M) ---, TI(BIA M) -+ (Bn1 B, M) and, therefore, an exact sequence limT (B,IB, M) limT (B,jA, M) -> (BIA, M) i) limT (A,,~A,M) ii) limT (A ~A, M) --> limT2(Bn1B,M) The formal smoothness of over is equivalent, by [2, prop 19 .4 .4], to the vanishing of lim 1A,M) for all C-module Then, Theorem is --, consequence of Proposition and Proposition Let be noetherian ring, an ideal of A, A/ In, and an A/I- module Then

Page 3

Part i) is easy limT (A .JA,M) lim HOM (h`/I %M) lim HOMA/l(In/In+1, M) To prove par¡ ü) we need the following lemmms JACOBIAN CRITERION 34 Lemma Le¡ be

ring, an ideal of A, A/I, and B-module Then, there exists natural monomorphism of B-modules (BIA, M) Ext'A(I, M) Proof Let be free A-module such that there exists an exact sequence of A-modules -- -~ -i -> Let Uo be the image of the homo- morphism ®A F, O(x y) j(x)y j(y)x Then, (BIA,M) Coker(HOMA(F, M) -+ HOMA(U/UO, M)) The result follows from the diagram of exact sequences HOMA(F, M) --~ HOMA(U/U M) (BIA, M) >0 HomA(F, M) -> HomA(U, M) -> ExtÁ(I, M) --> Lemma Let be noetherian ring, an ideal of and an integer number Then, there exists such that the canonical homomorphism is trivial or (I A/I)

--> Tori (In, A/I) Proof We have Tori(In, A/I) Tor (A/I A/I) Tor (A/I I) Let -> -> -> -> be an exact sequence of A-modules where is free and of finite type Then Tor (A/I I) -- (U InF)/InU By Artin-Rees lemma [4, th 15] there exists such that PF t-r (I U) for Taking we obtain PF C_ PU and therefore Tori(I A/I) --> Tori (In, A/I) is trivial We now prove part ii) of Proposition By lemma there exists mono- morhism lim (An ¡A, M) --> lim ExtÁ(I M)

Page 4

34 RODICIO On the other hand, for each we have an exact sequence -> Ext'A/ (I /I n+ ', M) -> Ext'(I M) -+ HOM (TorA(I A/I), M) which is

deduced, for instance, from the change-rings spectral sequence Ez'9 ExtÁ lI (Torq (In, A/I), M) =* ExtA+9(In, M) Since 1im ExtA /I (In/In+', M) and lim HOM /I(TorA(In, A/I), M) 0, by lemma 2, we obtain lim Ext'A(In, M) Therefore, lim (A ¡A, M) Theorem is not true for arbitraiy To exhibit counter-example, we need the following result Lemma Let be ring and C_ two ideals of such that T2 and IT qÉ Let A/I, T/I and B/J Then, the A-algebra is formally smooth for ¡he J-adic topology, but there exists C-module such that Tl(BIA, M) :~ Proof We have JZ (T2 I)/I (T I)/I T/I Hence, for each A/T-module lim

T'(BnIA, M) T'(CI A, M) HOM (T/T M) 0, where B/Jn Therefore, is formally smooth On the other hand (BI A, M) HOMB(I/I M) HOM IT(I/IT, M) Since I/IT qÉ 0, we obtain T'(BIA, I/IT) :~ Counter-example C(R, R) the ring of all real-valued continous functions on R, principal ideal of generated by the identity function, and maximal ideal of containing (see (5], Ch 2, 2, Ex 15) This counter-example solves negatively question of Brezuleanu (6, Re- marle .3 (i)J References ANDRE, "Homologie des Algebres Commutatives," Springer, 1974 GROTHENDIECK, EGA OIL, Publ Math IHES 20, Paris, 1964 LICHTENBAUM,

SCHLESSINGER, The cotangent complex of mor- phism, Trans AMS 128 (1967), 41-70 MATSUMURA, "Commutative Algebra," Benjamin/Cummings, 1980

Page 5

JACOBIAN CRITERION 34 BREZULEANU, Smoothness and regularity, Compositio Math 24 (1972), 1-10 Departamento de Algebra Facultad de Matemáticas 15771 Santiago de Compostela SPAIN Rebut el 12 de Setembre de 1988

Â© 2020 docslides.com Inc.

All rights reserved.