Cohomology Operators Defined by a Deformation

204, 684]710 Ž1998. JA977317

JOURNAL OF ALGEBRA ARTICLE NO.

Cohomology Operators Defined by a Deformation Luchezar L. AvramovU and Li-Chuan Sun† Department of Mathematics, Purdue Uni¨ ersity, West Lafayette, Indiana 47907 Communicated by D. A. Buchsbaum Received April 16, 1996

INTRODUCTION A lot of Žco.homological information on modules over a commutative ring R is encoded in terms of composition products of various Ext and Tor modules. Two main difficulties in using this information are that the resulting algebra and module structures are seldom finite, and the products are almost never commutative. One significant exception occurs when R s QrŽx. for a Koszul-regular set x s  x 1 , . . . , x c 4 in a commutative ring Q; we think of Q as a deformation of R over a regular base. Indeed, Gulliksen w8x then constructs a set of commuting operators  X 1 , . . . , X c 4 acting on ExtUR Ž M, N . by increasing degrees by 2 and on RŽ Tor# M, N . by decreasing degrees by 2, making Ext and Tor graded modules over a polynomial ring S s Rw X 1 , . . . , X c x with variables of cohomological degree 2. He proves that if Ext Qn Ž M, N . is noetherian over Q for each n and vanishes for n c 0, then the graded S-module ExtUR Ž M, N . is noetherian. This partly overcomes the first obstacle referred to above. Under more restrictive conditions on Q and x, Mehta w9x interprets Gulliksen’s operators as composition products: this makes the second difficulty manageable, as the actions of ExtUR Ž M, M . and ExtUR Ž N, N . on ExtUR Ž M, N . are ‘‘essentially central.’’ * L.L.A. was partially supported by a grant from the National Science Foundation. E-mail: [email protected]. † E-mail: [email protected]. 684 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

COHOMOLOGY OPERATORS

685

These results have provided the basis for extensive studies of the Žco.homological properties of modules over local complete intersections by Eisenbud w7x, and of more general classes of modules of infinite projective dimension by Avramov w1x. Each of the papers quoted above also uses alternative constructionŽs. of cohomology operators. It has been repeatedly stated in the literature that they yield the same result, but a close inspection of the arguments supporting such claims has revealed serious defects. On the other hand, no single approach seems to give all the essential properties of the cohomology operators which have been extensively used in w4x. In an attempt to provide complete proofs for the coincidence of the various operators, we were led to two new constructions. Both of them come from viewing R-modules as DG Žs differential graded. modules over the Koszul complex K resolving R over Q, and our arguments make a full-fledged use of techniques of DG homological algebra, summarized in Section 1. As a bonus, we construct the operators and establish their main properties directly for complexes of R-modules. We first introduce operators a ` la Gulliksen, as connecting homomorphisms. Unlike previous approaches, both module arguments are involved from the start. This is important in proving that ‘‘left’’ and ‘‘right’’ versions of earlier operators agree}at least up to sign}which suffices for the applications in w7, 1x. On our second approach we produce cohomology operators from chainlevel maps, as did Eisenbud. However, our chain endomorphisms arise not from lifting the differential of a complex of R-modules, but from descending the differential of a DG K-module. These constructions are presented in Section 2. They are used in Section 3 to provide direct proofs of many formal properties of the cohomology operators, in particular, of their centrality. Section 4 compares our approach to earlier ones, filling in gaps and correcting misconceptions concerning the relations between the various operators. Trying to avoid further inaccuracies, we include sufficient detail for most of the essential computations. In Section 5, we note that some of the existing proofs of the noetherian nature of ExtUR Ž M, N . over the ring of cohomology operators carry over to complexes, and establish a new property of these operators}their primiti¨ ity. When k is the residue field of R, for certain finite R-modules M we prove that ExtUR Ž M, k . is a finite graded module over the subalgebra of ExtUR Ž k, k . generated by the central and primitive elements of degree 2, even though R itself may not have a deformation. This result plays a key role in the investigation of finite CI-dimension in the recent paper w4x.

686

AVRAMOV AND SUN

1. BACKGROUND This section contains a synopsis of DG homological algebra, taken from w3x. All modules in sight are defined over a fixed commutative ring I- , which is usually suppressed from the notation. In particular, m stands for mI and Hom for Hom I . -

-

1.1. Complexes and DG Modules. Differentials have degree y1 and are ubiquitously denoted ­ . The notation m g M means that m is a homogeneous element of M; we denote its degree by < m <. If m is a cycle, then w m x is its homology class. The functor forgetful of differentials is denoted Ž ] . h. Let M and N be complexes. The complex HomŽ M, N . has dth component Ł i g Z HomŽ Mi , Niqd ., and differential ­ Žg . s ­ (g y Žy1. < g
687

COHOMOLOGY OPERATORS

A bounded below Žrespectively, above. A-module M such that M h is projective Žrespectively, injective. over Ah is DG-projective Žrespectively, DG-injective.. Let X be an arbitrary A-module. There exists a quasi-isomorphism ,

P ª X from a DG-projective module P; any such morphism is called a DG-projecti¨ e resolution of X. Dually, X has a DG-injecti¨ e resolution, that ,

is, a quasi-isomorphism X ª I with I DG-injective. For each X we fix a ,

DG-projecti¨ e resolution « XA : PXA ª X and a DG-injecti¨ e resolution hAX : ,

X ª IAX . Let w : AX ª A be a homomorphism of DG algebras, let M and N be A-modules, and let M X and N X be AX-modules. A w-contra¨ ariant homomorphism n : N ª N X is a I- -linear map such that n Ž w Ž aX . n. s X X Žy1. < n < < a < a n Ž n. for all aX g AX and all n g N. A w-co¨ ariant homomorphism m : M X ª M is a I- -linear map such that m Ž aX mX . s X < a < < m < Žy1. w Ž aX . m Ž mX . for all aX g AX and all mX g M X . In other words, n and m are homomorphisms of AX-modules for the structures induced on N and M through w . For w , m , n , as above, and a w-covariant chain map l: LX ª L, by Subsection 1.2 there exist unique up to homotopy w-equivariant chain maps l ? , m ? , n? , such that the squares

6

6

M

6

L

,

,

IAN

M

n

NX

n?

,

IANX

6

m

X

N 6

,

PMA

6

,

m?

6

PMAX

6

6 L

l

6

X

X

PLA

6

,

l?

6

X

PLAX

X

commute up to homotopy. They induce a w-contravariant chain map X

X

Hom w Ž m ?, n? . : Hom A Ž PMA , IAN . ª Hom AX Ž PMAX , IANX . ,

g ¬ Ž y1 .

< m <Ž < g
n?gm ?

and a w-covariant chain map

l ?mw m ?: PLAX

X

mA

pX m qX ¬ Ž y1 .

X

X

PMAX ª PLA

< m < < pX <

mA

PMA ,

l ? Ž pX . m m ? Ž qX .

both of which are unique up to homotopy. 1.3. Deri¨ ed Functors. With the resolutions chosen in Subsection 1.2, AŽ set ExtUA Ž M, N . s H Hom AŽ PMA , IAN . and Tor# L, M . s HŽ PLA mA PMA .. The morphisms from Subsection 1.2 uniquely define homomorphisms of

688

AVRAMOV AND SUN

graded I- -modules ExtUw Ž m , n . : ExtUA Ž M, N . ª ExtUAX Ž M X , N X . and X

w Tor# Ž l , m . : Tor#A Ž LX , M X . ª Tor#A Ž L, M .

which make ExtU and Tor# functors of three arguments. If w , l, m , n , are wŽ quasi-isomorphisms, then ExtUw Ž m , n . and Tor# l, m . are isomorphisms. When A s A 0 is a ring, the construction yields the usual hyper Ž co . homology of complexes. By further specialization to modules M s M0 and N s N0 , it produces the classical deri¨ ed functors of Hom A and mA . Let g g ExtUA Ž M, N . and d g ExtUA Ž L, M . be represented by chain maps g ˜ : PMA ª IAN and d˜: PLA ª IAM . By Subsection 1.2, choose unique up to homotopy chain maps g ?: PMA ª PNA and d ?: PLA ª PMA such that g˜ s hAN ( « NA (g ? and d˜ s hAM ( « MA ( d ?. The composition product is the degree 0 homomorphism of graded I- -modules ExtUA Ž M, N . m ExtUA Ž L, M . ª ExtUA Ž L, N . ,

g m d ¬ g ? d s w g˜ ( d ? x . If t is the homology class of a cycle Ý k e k

mA

f k g PLA

mA

PMA , then

A ExtUA Ž M, N . m Tor# Ž L, M . ª Tor#A Ž L, N . ,

gmt¬g?ts

< < < ek <

Ý Ž y1. g

ek

mA g ? Ž f k .

k

defines a degree 0 homomorphism of graded I- -modules. Similarly, if u is the homology class of Ý k e k mA g k g PLA

mA

PNA , then

ExtUA Ž L, M . m Tor#A Ž L, N . ª Tor#A Ž M, N . ,

dmu¬d?us

Ý d ? Ž ek . mA

gk

k

is a degree 0 homomorphism of graded I- -modules. These pairings are associative, and so define a structure of graded I- -algebra on ExtUA Ž M, M . with the unit given by the class of the map hAM ( « MA : PMA ª IAM ; they make ExtUA Ž M, N . into a graded left ExtUA Ž N, N .-, right ExtUA Ž M, M .-bimodule, while Tor#A Ž L, M . becomes a graded left ExtUA Ž L, L.-, left ExtUA Ž M, M .-bimodule.

689

COHOMOLOGY OPERATORS

2. CONSTRUCTION We start by describing the setup for much of the discussion in this paper. 2.1. Koszul Regularity. Let x s  x 1 , . . . , x c 4 be a set of elements in Q, a s Žx. the ideal it generates, and R s Qra the quotient ring. We denote by K the Koszul complex K Žx, Q .; thus, K h is the exterior algebra on a free Q-module with basis j 1 , . . . , j c with < j j < s 1, and ­ Ž j j . s x j for 1 F j F c. We denote A the kernel of the canonical augmentation k : K ª H 0 Ž K . s R. We assume that x is Koszul-regular, that is, that H i Ž K . s 0 for i / 0; equivalently, k is a quasi-isomorphism. The cotangent module C s ara 2 is then free over R on x, where x j s x j q a 2 , and for each R-module M and each i g Z there are isomorphisms Ext Qi Ž R , M . , Hyi Hom Q Ž K , M . s Hom Q Ž K i , M . i

(

H Hom R Ž C, R . mR

M

and ToriQ Ž R, M . , H i K mQ M s K i mQ M (

ž

/

i

H C mR

M.

kŽ The canonical isomorphisms k U s ExtUk Ž M, N . and k# s Tor# L, M . are computed as follows. For each R-module X, factor its DG resolutions over K as

X

Žh KX . ?

Ž 0 : A . IKX

Žid X . ?

IKX .

6

Ž « XK . ?

6

PXKrA PXK

6

Žid X . ?

6

PXK

Note that the intermediate complexes of R-modules are Žrespectively. DG-projective and DG-injective, and that the induced maps Ž « XK . ? and ŽhKX . ? are quasi-isomorphisms. Then choose « XR s Ž « XK . ? and hRX s ŽhKX . ? as the DG resolutions of X over R. 2.2. Fundamental Sequences. We consider for each j the DG subalgebra K w j x ; K generated over K 0 s Q by j 1 , . . . , jˆj , . . . , j c , and the multiplication map

b j : K mK w j x K ª K

with b j Ž b m c . s bc.

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AVRAMOV AND SUN

PROPOSITION.

Let L, M, N, be K-modules.

M Ža. The canonical inclusion b j < M N and the degree one map a j < N defined

by
g ¬ m ¬ Ž y1 . g Ž j j m . y j jg Ž m .

ž

/

yield an exact sequence of chain maps Hom K w j x Ž M, N .

aj< M N

Hom K Ž M, N .

6

bj < M N

6

0 ª Hom K Ž M, N .

h h h in which a j < M N is surjecti¨ e if M is projecti¨ e or N is injecti¨ e o¨ er K . Žb. The canonical projection b j < L M and the degree one map a j < L M defined by l mK m ¬ Ž y1 . l mK w j x Ž j j m . y Ž j j l .
mK w j x

m

yield an exact sequence of chain maps L mK w j x M

b j < LM

L mK M ª 0

6

a j < LM

6

L mK M

in which a j < L M is injecti¨ e if Lh or M h is flat o¨ er K h. Proof. To simplify notation, we set A s K w j x and j s j j . A direct computation yields equalities Ker b j s Ž K mA K .Ž 1 mA j y j mA 1 . s 0: K mAK Ž 1 mA j y j mA 1 . .

ž

/

Thus, left multiplication by 1 mA j y j mA 1 on K mA K defines a degree 1 homomorphism of graded K mA K-modules

a j : K ª K mA K

given by

a ¬ 1 mA Ž j a . y j mA a

which fits into a fundamental exact sequence aj

bj

0 ª K ª K mA K ª K ª 0. More verifications show that the action ŽŽ b mA c .g .Ž m . s Žy1. < c < < g < bg Ž cm. gives Hom AŽ M, N . a structure of K mA K-module, and that the action Ž b mA c .Ž l mA m. s Žy1. < c < < l < bl mA cm produces such a structure for L mA M. Furthermore, if X is a K mA K-module and K acts on X mK M by aŽ x m m. s Ž ax . m m, then there is a canonical

691

COHOMOLOGY OPERATORS

isomorphism Hom Ž K mAK . Ž X , Hom A Ž M, N . . ( Hom K Ž X mK M, N . given by g ¬ ŽŽ x mK m. ¬ g Ž x .Ž m.., and a canonical isomorphism

Ž L mA

M.

mŽ K m K . A

X ( L mK Ž X mK M .

given by Ž l m m. m x ¬ Žy1. < m < < x < l m Ž x m m.. To obtain the exact sequence in Ža., apply the functor Hom Ž K mAK . Ž ], Hom AŽ M, N .. to the fundamental exact sequence, and set a j < M N s HomŽ a j , HomŽ M, N ... For the sequence in Žb., apply Ž L mA M . mŽ K m K . y to the fundamental seA quence, and set a j < L M s Ž L m M . m a j . Finally, use the isomorphisms above with X s K to identify the outer terms. The first isomorphism shows that the sequence in Ža. can be obtained in two stages. The first is an application of ymK M to the fundamental exact sequence, with the K-module structure on K mA K given by aŽ b m c . s Žy1. < a< < b < b m Ž ac .. The map b jh is then split by a ¬ 1 m a, so we get an exact sequence of K-modules K mA M

bj< KM

M ª 0.

6

aj< KM

6

0ªM

h h h Thus, a j < M N is surjective if N is injective or M is projective over K . h h < A similar argument proves the injectivity of a j L M if L or M is flat over K h. 2.3. Cohomology Operators. Let x s  x 1 , . . . , x c 4 ; Q be a Koszul-regular set. Let L, M, N, be complexes of R-modules. We view them as K-modules via k and introduce shorthand notation for their resolutions over K U s PLK ,

V s PMK ,

Y s IKN .

Then we choose as in Subsection 2.1 resolutions over R E s UrAU s PLR ,

F s VrAV s PMR ,

J s Ž 0 : A . Y s IRN ,

and let « U : U ª E, « V : V ª F, and h Y : J ª Y be the canonical maps. By Subsection 2.2 there are exact sequences of K-modules Hom K w j x Ž V , Y .

a j < VY

Hom K Ž V , Y . ª 0

6

b j < VY

6

0 ª Hom K Ž V , Y . and

U mK w j x V

b j
U mK V ª 0.

6

a j
6

0 ª U mK V

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AVRAMOV AND SUN

The connecting homomorphisms ­] j of the associated long Žco.homology exact sequences define operators x j by commutativity of the squares

xj

­] j

(

ToriR Ž L, M . xj

­] j

6

6

Ž M, N . Ext iq2 K

K Ž Toriy2 L, M .

(

R Ž Toriy2 L, M .

6

(

6

6

Ž M, N . Ext iq2 R

ToriK Ž L, M .

6

Ext iK Ž M, N .

6

(

6

Ext iR Ž M, N .

kŽ where the isomorphisms are the maps ExtUk Ž M, N . and Tor# L, M . induced by

,

Hom R Ž F , J . ª Hom K Ž V , Y .

with g ¬ h Y (g ( « V

and ,

U mK V ª E mR F

with u mK ¨ ¬ « U Ž u .

mR

«V Ž ¨ . .

We call x s Ž x j .1c the family of cohomology operators defined by x. For another construction of cohomology operators we study 2.4. DG-Projecti¨ es. Let P be a DG-projective K-module. The projectivity of Ph over K h, cf. Subsection 1.2, implies that of h Ph over Q, hence the epimorphism Ph ª P is split by a P s PhrKq homomorphism s of graded Q-modules. The degree zero homomorphism of K h-modules p : K h mQ ImŽ s . ª Ph given by p Ž a m p . s ap induces an isomorphism p mK h Q, so CokerŽp . s 0. Thus, we have an exact sequence of graded K h-modules 0 ª Ker p ª K h mQ Im Ž s . ª Ph ª 0 p

which splits because Ph is projective. In the exact sequence obtained by applying ymK h Q the map p mK h Q is bijective, hence KerŽp . mK h Q s 0. h s Ž j , . . . , j . K h, we have KerŽp . s Due to the nilpotency of the ideal Kq 1 c 0, hence p is bijective. For  h1 , . . . , h i 4 s H : Z s  1, . . . , c4 with h1 - ??? - h i , set j H s j h1 n ??? nj h i g K. Each ¨ g P can be written in the form ¨ s Ý H : Z j H ²¨:H with uniquely defined ² ¨ :H g ImŽ s ., so for each H : Z we have a Q-linear endomorphism

­ H : Im Ž s . ª Im Ž s .

with ­ H Ž p . s Ž y1 .


² ­ p :H

693

COHOMOLOGY OPERATORS

of degree Žy< H < y 1.. The equation ­ 2 Ž p . s 0 then yields c

­B2 Ž p . s

Ý x j ­j Ž p . ;

Ž 2.4.1.

js1

­B­ j Ž p . ' ­ j ­BŽ p .

mod a Im Ž s .

for j s 1, . . . , c. Ž 2.4.2.

Note that D s R mK P , ImŽ s .ra ImŽ s . s PrA P is a complex of projective R-modules with differential ­ Ž1 m p . s 1 m ­BŽ p ., that the maps

t jD : D ª D

given by 1 m p ¬ 1 m ­ j Ž p . for j s 1, . . . , c

are degree y2 chain endomorphisms of D, and that the family t D s Žt jD .1c is uniquely determined by x. It allows for chain level computations of the operators. PROPOSITION. Let L, M, N, be complexes of R-modules, and let « LR : E ª L, « MR : F ª M, hRM : M ª I, be their DG resolutions o¨ er R described in Subsection 2.3. If t 1E, . . . , tcE : E ª E and t 1F, . . . , tcF: F ª F are families of chain endomorphisms determined by x, then for j s 1, . . . , c there are equalities

x j s H Hom R Ž t jF , J . : ExtUR Ž M, N . ª ExtUR Ž M, N . ; x j s H Ž t jE

mR

R F . : Tor# Ž L, M . ª Tor#R Ž L, M . ;

R x j s yH Ž E mR t jF . : Tor# Ž L, M . ª Tor#R Ž L, M . .

Proof. Along with the chosen in Subsection 2.3 DG-projective resolutions V and F of M over K and R, we consider the complex of R-modules F w j x s R mK w j x V. Identifying F w j x h with F h m j j F h, we see that its differential is given by

­ Ž f X q j j f Y . s ­ Ž f X . y j j Ž t jF Ž f X . q ­ Ž f Y . .

for f X , f Y g F.

Due to Subsection 2.2 we have a commutative diagram of chain maps with exact rows bj< KV

0

,

F

6

6

Fw jx

b¨j < R F

6

,

V

6

6 F

a¨ j < R F

6

6

0

K mK w j x V

6

,

aj< KV

6

V

6

6

0

0

Ž 2.4.3.

694

AVRAMOV AND SUN

where the middle vertical arrow is k mK w j x V and the maps in the bottom row are a ¨j < R F Ž f . s j j f and b¨j < R F Ž f X q j j f Y . s f X . Apply Hom K Ž ], Y . to Ž2.4.3.. As Y is DG-injective over K, this functor is exact. Furthermore, on complexes of R-modules it reduces to Hom R Ž ], J ., so we get a commutative diagram with exact rows Hom K Ž V , Y . 6

6 0

,

a¨ j < FJ

Hom R Ž F , J .

6

Hom R Ž F w j x , J .

a j < VY

6

b¨j < FJ

,

6

6

Hom R Ž F , J .

Hom K w j xŽ V , Y . 6

6

,

0

6

6

0

b j < VY

Hom K Ž V , Y . 6

0

and chain maps b¨j < FJ Ž g˜ .Ž f X q j j f Y . s g˜ Ž f X . and a ¨ j < FJ Ž g˜ .Ž f . s < g < Žy1. g ˜ Ž j j f .. Its commutativity shows that the operator x j on ExtUR Ž M, N . can be computed by the connecting homomorphism ­¨] j of the lower row. A direct computation shows that if g is the class of a chain map g ˜ : F ª J, then ­¨] j Žg . s w g ˜ (t jF x. In other words, x j Žg . s H Hom R Žt jF, J .Žg .. Similar considerations applied to the DG-projective resolutions U and E of L over K and R produce a commutative diagram of chain maps with exact rows complex

(

0

Ž 2.4.4.

(

U

6

b j < UK

6

U mK w j x K

E 6

6

b˙j < ER

6

6

U

0

a j < UK

Ew j x 6

6

(

a˙ j < ER

6

6

E 6

0

0

with Ew j x s E mK w j x R. In it the middle quasi-isomorphism is U mK w j x k , and the maps in the top row are a ˙j < E R Ž e . s yj j e and b˙j < E R Ž eX q j j eY . s eX . Applying ŽymK V . to Ž2.4.4. and ŽU mK ] . to Ž2.4.2., we arrive at the diagram

U mR V

6

,

0

,

6

b¨j < EF

0

E mR F

6

6

E mR F w j x

bj
6

a¨ j < EF

,

Em 6R F

6

U mK w j x V

6

6

E mR F

6

0

,

b˙j < EF

6

,

a j
6

U mR V

6

0

Ew j x mR F 6

6

,

a ˙ j < EF

6

Em 6R F

6

0

0

which is commutative with exact rows. By the bottom portion, x j RŽ on Tor# L, M . is the connecting homomorphism ­˙] j of a short exact sequence of complexes in which a ¨j < EF Ž e m f . s Žy1. < e < e m Ž j j f . and b¨j < EF Ž e m Ž f X q j i f Y .. s e m f X . An easy calculation yields ­˙] j ŽwÝ k e k m f k x. s ywÝ k e k m t jF Ž f k .x, hence x j s yHŽ F mR t jF ..

695

COHOMOLOGY OPERATORS

RŽ The top part shows that the action of x j on Tor# L, M . is also given by the connecting homomorphism ­] j of an exact sequence a ˙j < EF Ž e m f . s X Y X ˙ Ž . < ŽŽ . . y j j e m f, and b j EF e q j j e m f s e m f. This time, ­] j ŽwÝ k e k m f k x. s wÝ kt jE Ž e k . m f k x.

COROLLARY. If hRM : M ª I is a DG-injecti¨ e resolution o¨ er R and 1 M s whRM ( « MR x g Ext 0R Ž M, M ., then x j Ž1 M . s whRM ( « MT (t jF x g Ext 2R Ž M, M . for j s 1, . . . , c. An alternative approach to operators in cohomology is obtained by using 2.5. DG-injecti¨ es. In addition to the assumptions and notation of Subsections 2.1 and 2.3, let  j HU 4H : Z be the Q-basis of Hom Q Ž K h, Q . dual to the basis  j H 4H : Z of K h. The action of K h on Hom Q Ž K h, Q . is described by j j j HU s 0 if j f H, and j j j HU s Žy1. ry1j HU _ j if j s h r , where H s  h1 , . . . , h i 4 and h1 - ??? - h i . Let I be a DG-injective K-module. h . h is then injective over Q, due to the The graded module I s Ž0 : Kq I h h injectivity of I over K , cf. Subsection 1.2. Let r be a Q-linear splitting of I : I h, and set i Ž y .Ž a. s Žy1. < y < < a
ž

/

The sequence splits, due to the injectivity of I h. Applying to it Hom K hŽ Q, ] . we get an exact sequence whose first map is bijective, so Hom K hŽ Q, Coker i . s 0. We remark that if X is a K h-module such that h implies Hom K hŽ Q, X . s 0, then the nilpotency of the graded ideal Kq that X s 0. Thus, Coker i s 0 and i is bijective. The map û : K h mQ I ª Hom Q Ž K h, Q . mQ I with a m w ¬ Žy1. c < a< a j ZU m w is a bijective degree Žyc . homomorphism of graded K h-modules. Thus, y g I has a unique expression of the form y s Ý H : Z j H ² y :H with ² y :H g W s iy1û Ž1 mQ I .. In particular, for each y g I there is a unique w g W such that y s j Z w. For H : Z we define a Q-linear degree Žy< H < y 1. endomorphism

­H : I ª I

with ­ H Ž y . s Ž y1 .

cy < H <

j Z ² ­ w :H .

696

AVRAMOV AND SUN

By a direct computation, the equation ­ 2 Ž w . s 0 now yields a congruence c

² ­ ² ­ w :B:B ' y

c

Ý x j ² ­ w :j q Ý j j Ž ² ­ ² ­ w :j :B y ² ­ ² ­ w :B:j . js1

mod

ž

Ý

js1

j j a ² ­ w :h , j4 q

 h , j 4:Z

Ý

< H
jH W .

/

Multiplication with j Z on the left with j Z gives j Z ² ­ ² ­ w :B:B s yj Z Ýcjs1 x j ² ­ w :j . Thus,

­B2 Ž y . s Ž y1 .

c

c

Ý x j ­j Ž y . js1

and the congruence above simplifies to c

Ý j j Ž ² ­ ² ­ w :B:j y ² ­ ² ­ w :j :B. ' 0 js1

mod

ž

j j a ² ­ w :h , j4 q

Ý

 h , j 4:Z

Ý

< H
jH W .

/

Multiplying the new congruence on the left with j Z _ j for j s 1, . . . , c, we get c

­ j ­BŽ y . ' ­B­ j Ž y .

mod

ž

Ý a ² ­ w :h , j4 hs1

/

.

Remark that y is in B s Ž0 : a .I s Ž0 : A .I if and only if w is in Ž0 : a .W , and that the restriction of ­B to B is equal to the differential ­ of B. Thus, setting ¨ j s Žy1. c­ j and denoting ¨ jB the restriction of ¨ j to B, we obtain a family © B s Ž ¨ jB .1c of chain endomorphisms of degree y2 of the complex of R-modules B. PROPOSITION. and J s Ž0 : A . Y ¨ 1J , . . . , ¨ cJ : J ª J j s 1, . . . , c there

Let M and N be complexes of R-modules, let F s VrAV be their DG resolutions as in Subsection 2.3, and let be the family of chain endomorphisms described abo¨ e. For are equalities

x j s H Hom R Ž F , ¨ jJ . : ExtUR Ž M, N . ª ExtUR Ž M, N . . Proof. Consider the complex of R-modules J w j x s Hom K w j xŽ R, Y ., and h m j Ž0 : a . h . The differential of J w j x is identify J w j x h with j Z Ž0 : a .W Z_ j W then given by

­ Ž j Z wX q j Z _ j wY . s ­ J Ž j Z wX . q Ž y1 . ¨ j Ž j Z wY . q Ž y1 . j

cy1

j Z _ j ² ­ wY :B

697

COHOMOLOGY OPERATORS

h . Thus, from Subsection 2.2 we get a commutative for wX , wY g Ž0 : a .W diagram of chain maps 0

,

Y

6

6

Hom K w j xŽ K , Y .

a j < YK

6

6 Y

J

6

,

6

6

0

b j < YK

6

,

a˙ j < R J

J w jx

6

J

6

b˙j < R J

6

0

0

in which the rows are exact, the middle vertical arrow is induced by Hom K w j xŽ k , Y . and is a quasi-isomorphism because the external two arrows are, and where

b˙j < RJ Ž j Z w . s j Z w

a ¨j < RJ Ž j Z wX q j Z _ j wY . s Ž y1. j Z wY . j

and

The rest of the argument is similar to that for Proposition 2.4.

3. PROPERTIES We derive the main formal properties of cohomology operators. 3.1. Naturality. Given a commutative diagram of ring homomorphisms Q

X

6

R

w

r

6

X

6

r

c

6

QX

R

with aX s KerŽ r X . generated by a Koszul-regular set x X s xX1 , . . . , xXcX ; QX , let x X s  x 1X , . . . , xcXX 4 be the family of cohomology operators defined by x X , and let

c Ž xXi . s

c

Ý qi j x j

with qi j g Q for 1 F i F cX

js1

be equalities resulting from the inclusion c Žx X . ; KerŽ r .. We can now describe the naturality of the cohomology operators. THEOREM. Let m : M X ª M be a w-co¨ ariant chain map. If l: LX ª L is a w-co¨ ariant chain map, then X

w

x j (Tor# Ž l , m . s

c

Ý qi j Tor#w Ž l , m . ( xiX is1

for j s 1, . . . , c.

698

AVRAMOV AND SUN

If n : N ª N X is a w-contra¨ ariant chain map and qi j s c Ž qXi j ., then X

c

Ý qXi j xiX (ExtUw Ž m , n . s ExtUw Ž m , n . ( x j

for j s 1, . . . , c.

is1

Proof. This is immediate from Subsection 1.2, Proposition 2.4, and the following lemma. LEMMA. Let K X be the Koszul complex K Ž xX , QX ., and let C: K X ª K be the homomorphism of DG algebras defined by C Ž j iX . s Ýcjs1 qi j j j for 1F X i F cX . Let P X be a DG-projecti¨ e module o¨ er K X , and let tX s Žt iX .1c be the family of chain endomorphisms of the complex of RX-modules DX s P XrA X P X defined by x X . Gi¨ en a C-co¨ ariant chain map g : P X ª P, let 1 m g : DX ª D be the induced co¨ ariant chain map of complexes o¨ er the ring homomorphism w : RX ª R. X X The chain maps t jD (Ž1 m g . and Ýcis1 qi j Ž1 m g .(t iD are homotopic. Remark. In view of Proposition 4.2 the lemma is equivalent to w7, Proposition 1.7x. Proof. In the notation of Subsection 2.4, for each H : Z s  1, . . . , c4 consider the degree Ž< g < y < H <. homomorphisms of graded QX-modules

g H : Im Ž s X . ª Im Ž s .

with g H Ž yX . s Ž y1 .


² g Ž y X . :H .

By definition, we have

­g Ž yX . ' ­BgBŽ yX . y

c

Ý x jg j Ž yX . js1

c

q

Ý jj js1

ž

­Bg j Ž yX . y ­ jgBŽ yX . q

" x h gh , j4Ž yX .

Ý

 h , j 4:Z

/

modulo ŽÝ < H < G 2 j H P .. Due to the C-covariance of g , we get X

X

g­ Ž y . '

gB­BX

X

Žy.y

c

Ý

X

j jg j ­BX

X

Ž y . y Ž y1.

js1

s gB­BX Ž yX . q


c

Ý C Ž j iX . gB­ iX Ž yX . is1

X

c

c

< < Ý j j Ž y1. g y1 Ý qi jgB­ iX Ž yX . y g j ­BX Ž yX .

js1

ž

is1

/

modulo ŽÝ < H < G 2 j H P .. Noting that ­g s Žy1. < g
699

COHOMOLOGY OPERATORS

of j j that X

c

t j (Ž 1 m g . y D

<

X

j<

Ý qi j Ž 1 m g . (t iD s ­g j y Ž y1. g g j ­ X is1

for j s 1, . . . , c. X

X

Thus, g j is a homotopy from t jD (Ž1 m g . to Ýcis1 qi j Ž1 m g .(t iD . 3.2. Centrality. Cohomology operators Žanti.commute with products: THEOREM.

For j s 1, . . . , c there are equalities

x j Ž g . ? d s x j Ž g ? d . s g ? x j Ž d . g ExtUR Ž L, N . R x j Ž g . ? t s yx j Ž g ? t . s yg ? x j Ž t . g Tor# Ž L, N . R x j Ž d . ? u s x j Ž d ? u . s d ? x j Ž u . g Tor# Ž M, N .

RŽ whene¨ er g g ExtUR Ž M, N ., d g ExtUR Ž L, M ., t g Tor# L, M ., u g RŽ . Tor# L, N .

Proof. In addition to the resolutions F, E, J, described in Subsection ,

,

2.3, we consider DG resolutions M ª IRM s I and G s PNR ª N. Choose chain maps g ˜ : F ª J and d˜: E ª I such that g s w g˜ x and d s w d˜x, cf. Subsection 1.2. Extend g ˜ to g˜?g Hom R Ž I, J . and lift d˜ to d˜ ?g Hom R Ž E, F .. Lemma 3.1 applied to id K shows that t jF ( d˜ ? and d˜ ?(t jE are homotopic, so Proposition 2.4 gives

xj Ž g . ? d s g ˜ (t jF ? d s g˜ ( t jF ( d˜ ?

ž

s

Ž g˜ ( d˜ ? . (t jE

/

s g ˜ ( d˜ ?(t jE

ž

/

s xj Ž g ? d . .

On the other hand, by w6, Sect. 7, No. 2x and Proposition 2.4 we have

xj Ž g ? d . s xj g ˜ ( d˜ ?

ž

?

/ s Ž g˜ ( d˜ . (t

s g ? d˜ ?(t jE s g ? x j Ž d . .

E j

s

Ž g˜?( d˜ . (t jE

700

AVRAMOV AND SUN

With Ý k e k m f k representing t and g ˜ ?: F ª G lifting g˜ , Proposition 2.4 yields

x j Ž g . ? t s g ( t jF ?

Ý ek m fk k

s

Ý Ž y1.

Ž < g
ek m g ˜ ? Ž t jF Ž f k . . ;

k

xj Ž g ? t . s xj

žÝ

< < < ek <

Ž y1. g

ek m g ˜ ? Ž fk .

k

sy

< < < ek <

Ý Ž y1. g

/

e k m t jG Ž g ˜ ? Ž fk . . ;

k

g ? x j Ž t . s g ? y Ý e k m Ž t jF Ž f k . . k

sy

< < < ek <

Ý Ž y1. g

ek m g ˜ ? Ž t jF Ž e k . . .

k

As t jG (g ˜ ? and g˜ ?( t jF are homotopic by Lemma 3.1, we get the second statement. Finally, if Ý k e k m g k represents u, then by Proposition 2.4 we have

x j Ž d . ? u s d˜(t jE ?

Ý ek m g k

s

k

xj Ž d ? u. s xj

d ? Ž ek . . m g k

k

Ý Ž t jE Ž e k . . m g k k

;

k

ž ÝŽ˜

d ? xj Ž u. s d ?

Ý ž d˜ ?(t jE Ž ek . / m g k

/

s

s

Ý ž t jF ( d˜ ? Ž ek . / m g k

;

k

Ý ž d˜ ?(t jE Ž e k . / m g k

.

k

As t jF ( d˜ ? and d˜ ?(t jE are homotopic, we have the last statement. 2.7. Characteristic Homomorphism. Let CU s Hom R Ž C, R . be the dual module of C s ara 2 , and let xU1 , . . . , xUc denote the basis of CU s Ž ara 2 .U dual to the basis x s  x 1 q a 2 , . . . , x c q a 2 4 of C. THEOREM. The unique homomorphism of R-modules CU ª Ext 2R Ž M, M .

with xUj ¬ x j Ž 1 M . for j s 1, . . . , c

is independent of the choice of the Koszul-regular set x. Its image lies in the center of ExtUR Ž M, M ., and so it extends canonically to a characteristic

COHOMOLOGY OPERATORS

701

homomorphism of graded R-algebras from the symmetric algebra S s Sym R Ž CU .:

z M : S ª ExtUR Ž M, M . . The maps z M and z N induce on ExtUR Ž M, N . the same S-module structure,

z N Ž xUj . ? g s x j Ž g . s g ? z M Ž xUj .

for g g ExtUR Ž M, N . .

RŽ The maps z L and z M induce on Tor# L, M . opposite S-module structures,

z LŽ xUj . ? t s x j Ž t . s yz M Ž xUj . ? t

R for t g Tor# Ž L, M . .

RŽ COROLLARY. On ExtUR Ž M, N . and Tor# L, M . there are equalities

xi x j s x j xi

for 1 F i , j F c.

Remark. When M is an R-module the characteristic homomorphism z M coincides with one constructed by Mehta, cf. Subsection 4.4; his definition does not seem to generalize to complexes of R-modules. When c s 1, he proves in w9, Proposition 2.3x that the image of z M is central and in w9, Proposition 2.4x that over complete local rings the actions on Tor coincide. The theorem extends Žand corrects some signs of. w9, Proposition 2.3x. Proof of the Theorem. Theorem 3.2 yields the formulas for the action of z M Ž xUj ., and the first one of them implies that the image of z M is in the center of ExtUR Ž M, M .. Along with x, consider a Koszul-regular set x X ; Q such that Žx X . s Žx.. As both x and x X are bases of C, the set x X also consists of c elements, say xX1 , . . . , xXc . Let t 1X F , . . . , tcX F be the chain endomorphisms of Subsection 2.4 defined by x X on the chosen DG projective resolution F of M over R. Writing xXi s Ýcjs1 qi j x j , and applying Lemma 3.1 to the identity maps of K and of P s PMK , we get c

t jF s

Ý qi jt iX F

for j s 1, . . . , c.

is1

It follows that the degree zero R-linear homomorphism Ž ara 2 .U ª Hom R Ž F, F . with xUj ¬ t jF for j s 1, . . . , c does not depend on the choice of the Koszul-regular set x. In view of the definition of z M and Corollary 2.4 we have

z M Ž xUj . s hRM ( « MR (t jF s x j Ž 1 M . . This establishes the independence of z M from the choice of x.

702

AVRAMOV AND SUN

4. COMPARISON This section presents the cohomology operators constructed by Gulliksen w8x, Eisenbud w7x, Mehta w9x, Avramov w1x, and compares them to the operators x j from Subsection 2.3. Throughout this section, r : Q ª R is a surjective homomorphism of commutative rings with KerŽ r . s a generated by a Koszul-regular set x s x 1 , . . . , x c . We consider R-modules L, M, N; as a further adjustment, we assume that DG-resolutions U, V, Y of L, M, N over K have been chosen with Ui s 0 and Vi s 0 for i - 0, respectively Yi s 0 for i ) 0. This can always be done, cf. w3x, and has the effect that the chosen in Subsection 2.3 complexes F s UrAU and E s VrAV are classical projective resolutions of the R-modules L and M, while the complex J s Ž0 : A . Y is a classical injective resolution of the R-module N. Finally, recall the identifications ExtUR Ž M, N . s H Hom R Ž F , J .

R Tor# Ž L, M . s H Ž E mR F . .

and

4.1. Gulliksen’s Construction w8, p. 176x. This starts from short exact sequences a Xj

b jX

a Yj

b jY

0 ª E ª Ew j x ª E ª 0 and 0 ª F ª F w j x ª F ª 0, where Ew j x and F w j x are as in Subsection 2.4; the maps in the first sequence are given by a Xj Ž e . s Žy1. < e
Hom R Ž b j , N .

Hom R Ž F w j x , N .

6

0ªHom R Ž F , N .

Y

Hom R Ž F , N . ª 0

6

Hom R Ž a j , N .

and 6

Ew j x

mR

M

b jXmR M

E mR M ª 0

6

a XjmR M

0 ª E mR M

X

whose connecting homomorphisms we denote ­]j . X

X

Y

Y

1 The relations a j ­ s ­a j and a j ­ s ­a j are opposite to those in Subsection 1.1 for degree 1 chain maps.

703

COHOMOLOGY OPERATORS

Gulliksen defines X j for j s 1, . . . , c by commutativity of the squares (

6

Ext iR Ž M, N .

Hyi Hom R Ž F , N . X

Žy1 . ­i]j

Xj

6

6

6

H i Ž E mR M .

6

Ž M, N . Ext iq2 R

(

Hyiy2 Hom R Ž F , N . (

ToriR Ž L, M .

X

Žy1 . ­i]j

Xj

6

6

6

H iy2 Ž E mR M .

(

R Ž Toriy2 L,

M.

with isomorphisms H Hom R Ž F, hRN . and HŽ E mR « MR .. PROPOSITION.

There are equalities x j s X j for j s 1, . . . , c.

Proof. Set v Ž f . s Žy1. < f < f, and paste Ž2.4.3. to the commutative diagram

6 6

b jY

F

6

6

6

Fw jx

6

a Yj

F

b¨j < R F

6

v

0

Fw jx

6

6

a¨ j < R F

F

0

F

0 0

along their common row. The diagram induced by applying Hom R Ž ], N . implies the desired assertion for ExtUR Ž M, N .. The argument for RŽ Tor# L, N . is similar. 4.2. Eisenbud’s Construction w7, Sect. 1x. This begins with a lifting of a complex of free R-modules Ž F X , ­ . to a pair Ž F˜X , ­˜. of a graded free Q-module F˜X and its endomorphism ­˜ of degree y1, such that Ž F X mQ R, ­˜ mQ R . , Ž F X , ­ .. It is easy to see that liftings always exist. Fixing one, note that ­˜2 mQ R s 0, hence ­˜2 s Ýcjs1 x j ˜t j Ž F˜X , ­˜. for some family Ž˜t j Ž F˜X , ­˜..1c of degree y2 endomorphisms of the graded Q-module F˜X . As x is a basis of the cotangent module, X ˜t j Ž F˜X , ­˜. mQ R is a chain endomorphism of F˜X mQ R. We denote by t jF the corresponding degree y2 chain endomorphism of F X . For j s 1, . . . , c Eisenbud defined e jleft by commutativity of the square 6

Ext iR Ž M, N .

(

Hyi Hom R Ž F X , N . X

e jleft

H Hom R Ž t jF , N .

6

(

6

6

Ž M, N . Ext iq2 R

Hyiy2 Hom R Ž F X , N .

704

AVRAMOV AND SUN

and e jright by that of the square 6

H i Ž L mR F X .

(

ToriR Ž L, M .

X

e jright

H Ž LmR t jF . (

6

6

R Ž Toriy2 L, M .

6

H iy2 Ž L mR F X . ,

where F X ª M is a free resolution of the R-module M, q : F ª F X is a comparison of resolutions, and the isomorphisms are H Hom R Žq , hRN . and HŽ « LR mR q .. Operators e jleft are similarly defined on ToriR Ž L, M ., by ,

means of a free resolution EX ª L.

PROPOSITION. On ExtUR Ž M, N . there are equalities x j s e jright for j s 1, . . . , c. RŽ On Tor# L, M . there are equalities x j s e jleft s ye jright for j s 1, . . . , c. Remark. This corrects a sign error in w7, Proposition 1.6x, which asserts that e jleft s e jright . The argument given there may not be used to compare the two maps: it assumes incorrectly that the canonical projections EX mR M ¤ EX mR F X ª L mR F X commute with the chain endomorphisms t jE

X

mR

M, t jE

X

mR

F X q EX mR t jF , EX m t jF . X

,

X

Proof. For an R-free resolution F ª M, pick a homogenous basis  flX 4l g L of the graded R-module F Xh, and let Ph be a graded K h-module on a basis  g l < < g l < s < flX <4l g L . Set v - 0 s 0, and assume by induction that a X k-covariant morphism v - h : P- h s @ < g l < - h K h g l ª Fh has been conX structed such that v - hŽ g l . s fl when < g l < - h, and v - h is a surjective quasi-isomorphism. If < flX < s h, then ­ Ž flX . is a cycle in ImŽ v - h ., hence ­ Ž flX . s v - hŽ zl . for some cycle zl g P- h . Extend the differential from P- h to PF h by setting X X Ž . ­ Ž g l . s zl , and extend v - h to v F h : PF h ª FF h by v F h g l s fl for all l with < g l < F h. By the choices made v F h is a morphism and by the Five-Lemma it is a quasi-homomorphism onto its image. The induction step is complete. Now set P s lim h P- h and v s lim h v - h : P ª F X . Clearly, v is a k-covariant surjective morphism, and the induced map R mK P ª F X is an isomorphism of complexes of R-modules. Using the h Ph, ­ . is a lifting of notation of Subsection 2.4, we see that the pair Ž PhrKq B X the complex F . In view of Ž2.4.1. we can choose ­ 1 , . . . , ­c to be the chain endomorphisms associated with this lifting. This choice leads to equalities of Eisenbud’s endomorphism with those from Subsection 2.4:

Ž 4.2.1.

X

t jF s t jF

X

for j s 1, . . . , c.

705

COHOMOLOGY OPERATORS X

Lemma 2.5 with w s id R and g s q shows that qt jF and t jFq are homotopic, hence X

H Hom R Ž q , hRN . (H Hom R Ž t jF , N . s H Hom R Ž t jF , J . (H Hom R Ž q , hRN . . As H Hom R Žt jF , J . s x j , we see that x j s e jright . The argument for Tor is similar. 4.3. A¨ ramo¨ ’s Construction w1, Sect. 1x. This dualizes that of Eisenbud described in Subsection 4.2. Recall that J is the chosen injective resolution of the R-module N. Choose a graded Q-module J˜ such that the Q-module J˜i is an injective envelope of the Q-module Ji for each i. Clearly, Hom Q Ž R, J˜. ( J. Since J˜ is a graded injective Q-module, there exists a endomorphism ­˜ of degree y1 on J˜ such that ­˜< J s ­ . By w1, Proposition 1.2x there are degree y2 endomorphisms u ˜ j Ž J,˜ x.: J˜ª J,˜ which satisfy ­˜2 s Ýcjs1 x j u˜ j Ž J,˜ x., and which restrict to degree y2 chain endomorphisms u jJ : J ª J. The operators e jright studied in w1x are defined by commutativity of the square 6

Ext iR Ž M, N .

(

Hyi Hom R Ž M, J .

e jright

H Hom R Ž M , u jJ .

6

6

6

Ž M, N . Ext iq2 R

(

Hyiy2 Hom R Ž M, J .

with isomorphisms H Hom R Ž « MR , J .. PROPOSITION. 1, . . . , c.

On ExtUR Ž M, N . there are equalities x j s e right for j s

Remark. Comparison with Proposition 4.2 shows that e jleft s e jright . This is stated in w1, Ž1.5.x, as a consequence of w1, Proposition 1.4x. However, that proposition is not correct: the morphisms Hom R Ž F, hRN . and Hom R Ž « MR , J . do not commute with the actions induced by the chain endomorphisms t jF and u jJ.

h . and Proof. With the notation of Subsection 2.5, we set Y s Ž0 : Kq Y consider the endomorphisms ­ H : Y ª Y constructed in Subsection 2.5. Because of the equalities J s Ž0 : a . Y s Ž0 : A . Y and ­ J s ­B< J s ­ Y < J , there is an injective homomorphism of graded Q-modules ˜i : J˜ª Y with ˜i < J s id J and ˜i ( ­˜y ­B(˜i s Ýcjs1 x j h j for appropriate degree y1 homomorphisms of graded Q-modules h j : H˜ ª Y. A direct computation gives c

c

Ý x j ž ˜i ( u˜jJ Ž J , x . y ¨ jJ (˜i / s Ý x j ž h j ( ­˜q ­B( h j / .

js1

js1

706

AVRAMOV AND SUN

As the Koszul complex is exact, u jJ is homotopic to ¨ jJ, yielding the middle equality

e jright s H Hom R Ž F , u jJ . s H Hom R Ž F , ¨ jJ . s x j ; the last one comes from Proposition 2.5 4.4. Mehta’s Construction w9, Sect. 1x. This proceeds from a presentation «

0 ª B ª G0 ª M ª 0 with a projective Q-module G 0 . In view of the isomorphism Tor1Q Ž R, G 0 . s 0 and Tor1Q Ž R, M . ( C m M, where C s ara 2 is the cotangent module, it induces an exact sequence of R-modules EM

«X

i

0 ª C m M ª BX ª GX0 ª M ª 0,

where ] X s R m ], and C s ara 2 is the cotangent module, cf. Subsection 2.1. In the notation introduced there, the homomorphism i ŽÝ k x k m m k . s 1 m Ý k x k mXk , where mXk g G 0 satisfy « Ž mXk . s m k . For each j we get an exact sequence Ej M

«X

0 ª M ª M j ª GX0 ª M ª 0,

as the pushout of E M along xUj m M: C mR M ª R mR M s M, where xU1 , . . . , xUc is the basis of CU dual to x 1 , . . . , x c . The iterated connecting homomorphisms induced by Ej M yield operaRŽ tors d jright on ExtUR Ž M, N . and d jleft on Tor# L, M . studied by Mehta. N Furthermore, the exact sequences Ej and Ej L yield operators d jleft on RŽ ExtUR Ž M, N . and d jright on Tor# L, M .. PROPOSITION. On ExtUR Ž M, N . there are equalities x j s d jleft s yd jright for 1 F j F c. RŽ On Tor# L, M . there are equalities x j s yd jleft s d jright for 1 F j F c. Remarks. Ž1. The last statement corrects a sign error in w9, Proposition RŽ 2.4x, which asserts that d jeft s d jright on Tor# L, M .. Ž2. Mehta also considers an analogous exact sequence obtained as the pullback of an injective corepresentation of M; when c s 1 he proves that these sequences are congruent with Ej M , and hence the same cohomology operators. His argument in w9, Proposition 2.5x can be extended to handle the case of arbitrary codimension. Proof. Let Ž G, ­ . be a free resolution of M over Q. As x j M s 0, for each j there is a homotopy s j from x j id G to 0 G . In the diagram of

707

COHOMOLOGY OPERATORS

R-modules 0

GX0

0x

«X

M

0

6

GX0

6

6

6

Mj

6

6

6

M

6 0

M

0

6

x Uj mM

«X

M

6

GX0

6

6

BX

6

i

«X

6

­ 1X

6

GX1 p

CmM

0

­ X2 x

6

w Cm «

X

ws0

6

Ž C m G 0 . [ GX2 6

­ 3X

6

Ž C m G1 . [ GX3

s1

6

Cm ­ 1

0

the middle and lower rows are the exact sequences E and Ej above, p comes from the canonical factorization of ­ 1X and si ŽÝ k x k m g k . s 1 m Ý k si k Ž g k . for i s 0, 1. The commutativity of the upper left hand square is clear. The one to its right commutes because ­ 1 s 0j s x j id G . Thus, the diagram is commutative. It is checked directly that the first row is exact.2 We extend it to a free resolution F X of M over R, and identify ExtUR Ž M, M . with H Hom R Ž F X , I . via the canonical isomorphism. Let m jM : F X ª I be the chain map which on F2X is the composition of U x j m « with the embedding h : M ª I0 , and is trivial elsewhere. The commutativity of the diagram implies that yw m jM x is the extension class associated with the exact sequence Ej M , cf. w6, Sect. 7, No. 3, Defintion 1x. Now Bourbaki w6, Sect. 7, No. 6, Corollary 3x yields M

d jleft Ž g . s yg ? ym jM

M

d jright Ž g . s ym jN ? g

and

for g g ExtUR Ž M, N . , while w6, Sect. 7, No. 8, Corollaries 2 and 4x show that

d jright Ž t . s ym jM ? t

d jleft Ž t . s ym jL ? t

and

R for t g Tor# Ž L, M . .

On the other hand, lifting F2X ª F1X ª F0X to homomorphisms of free Q-modules ­˜2

­˜1

G 0c [ G 2 ª G1 ª G 0 with ­˜1 s ­ 1 and ­˜2 ŽŽ g 1 , . . . , g c . q g . s Ý k s 0k Ž g k . q ­ 2 Ž g ., we see that X Eisenbud’s operator t jF from Subsection 4.2 acts on F2X by the formula X X t jF ŽÝ k x k m g k q 1 m g . s 1 m g j . This implies w m jM x s whRM ( « MR ( t jF x. As X X t jF s t jF by Ž3.2.1., Corollary 2.4 and Theorem 2.7 yield x j s d jleft s RŽ yd jright on ExtUR Ž M, N . and x j s d jright s yd jleft on Tor# L, M .. 2 In fact, it represents the beginning of a free resolution of M over R, constructed from G by Eisenbud w7x, but we shall not need the rest of it.

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5. APPLICATION We extend the important finiteness result discussed in the Introduction and its recent converse to Ext’s of complexes of R-modules. We also establish a new property of the cohomology operators}their primitivity. For the first two subsections we keep the notation and hypothesis of Subsection 2.1, and as in Proposition 3.3 denote by S the symmetric algebra of the R-module Hom R Ž ara 2 , R ., graded by assigning to its generators cohomological degree 2. 5.1. Finiteness. The R-module ExtUQ Ž M, N . is noetherian when Ext Qn Ž M, N . is noetherian over R Žor, equivalently, over Q . for each n, and vanishes for n c 0. THEOREM. If M and N are complexes of R-modules, then the R-module ExtUQ Ž M, N . is noetherian if and only if the S-module ExtUR Ž M, N . is noetherian. Comments in Place of Proof. Eisenbud’s constructions of cohomology operators e jleft , recalled in Subsection 4.2, can easily be extended to complexes, and the proof that they coincide with the cohomology operators x j carries over. Thus, by Theorem 3.3, we may consider ExtUR Ž M, N . as a module over the ring S of Eisenbud operators. If ExtUQ Ž M, N . is noetherian over Q, then the spectral sequence argument given in w2x for finiteness over S shows that ExtUR Ž M, N . is noetherian over S . ŽIf HŽ M . is bounded below and HŽ N . is bounded above, then Gulliksen’s original proof of w8, Theorem 3.1x can also be applied, in view of Proposition 3.1.. If M and N are Q-modules and ExtUR Ž M, N . is noetherian over R, then it is proved in w4, Theorem 4.2x that ExtUQ Ž M, N . is noetherian over Q. That argument only depends on the formal properties of the spectral sequence of w4, Theorem 4.4x, whose construction works equally well for complexes. Let ZRU Ž M . denote the subalgebra of ExtUR Ž M, M . generated over RrannŽ M . : Hom R Ž M, M . by the central elements of degree 2. From Theorems 4.1 and 2.7 we get COROLLARY.

If the Q-module ExtUQ Ž M, N . is noetherian, then the ExtUR Ž M, N . is noetherian o¨ er each ring.

ZRU Ž N .-bimodule ZRU Ž M .-Z

5.2. Primiti¨ ity. A DG algebra with divided powers Žor DG G-algebra. over R is a graded commutative DG R-algebra A with A i s 0 for i F 0, equipped with maps A 2 n 2 a ¬ aŽ i. g A 2 ni , defined for all n G 1 and i G 0, and satisfying some standard conditions, cf. e.g., w5, Ž1.3.x. A Gderi¨ ation is an R-linear map u : A ª A such that

u Ž aX aY . s u Ž aX . aY q Ž y1 .

< u < < aX < X

au Ž aY .

and

u Ž aŽ i. . s u Ž a . aŽ iy1. .

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COHOMOLOGY OPERATORS

Let S and T be DG G algebras over R. An extension of Tate’s procedure w10x of adjunction of G-variables to R factors the structure homomorphism R ª S as the composition of R ª E, where Eh s R ² X : is a free , G-algebra over R, with a surjective quasi-isomorphism E ª S. By w5, RŽ Ž2.16.x, Tor# S, T . , H#Ž E mR T . inherits a structure of G-algebra, which is independent of the choice of E. THEOREM.

RŽ On Tor# S, T . the operator x j is a G-deri¨ ation for 1 F j F c.

Proof. Consider a graded G-algebra E˜ s Q² X˜ :, with X˜ a new set of divided power variables in bijective, in degree preserving correspondence ˜x l x with those of X. This correspondence extends to an isomorphism of G-algebras E˜ mQ R s Eh. For each x g X, choose in E˜ an element xX such that xX m 1 s ­ Ž x ., and let ­˜ be the unique Q-linear G-derivation of E˜ with ­˜Ž ˜ x . s xX . It is easy to see that ­˜2 is a G-derivation. For each ˜ x g X˜ there are e j Ž ˜ x . g E˜ such that ­˜2 Ž ˜ x. s Ý j x j ejŽ ˜ x .. Let ˜t j be the unique Q-linear G-derivation of E˜ such that ˜t j Ž ˜ x. s ejŽ ˜ x . for all ˜ As ­˜2 and Ýcjs1 x j˜t j agree on the G-generators of the Q-algebra E, ˜ ˜x s X. these two endomorphisms of E˜ coincide. Thus, t jE s ˜t j mQ R is a G-derivation of E˜ mQ R s E, and hence HŽ t jE mR T . is a G-derivation of RŽ Tor# S, T .. As x j s HŽ t jE mR T . by Proposition 3.2, this proves our assertion. We say that g g Ext Rn Ž S, S . is primiti¨ e if g Ž tu. s 0 whenever t g R Ž ToriR Ž S, S . and u g Tornyi S, S . with 0 - i - n, and g Ž t Ž i. . s 0 for all RŽ t g Tor2 n S, S . with n ) 0 and i G 2. Primitives form a graded R-submodule Pr RU Ž S . : ExtUR Ž S, S ., and Pr R2 Ž S . s KerŽp RS ., where p RS is the homomorphism of R-modules 2

p RS : Ext 2R Ž S, S . ª Hom R Ž Tor1R Ž S, S . , Tor0R Ž S, S . . with p RS Ž g . Ž g . s g ? t. We denote PRU Ž S . the subalgebra of ExtUR Ž S, S . generated over RrannŽ S . by the central primiti¨ e elements of degree 2. The last theorem and Corollary 5.1 yield COROLLARY. If T is a DG G-algebra o¨ er R and M is a complex of R-modules such that ExtUQ Ž M, T . is noetherian o¨ er Q, then ExtUR Ž M, T . is noetherian o¨ er PRU ŽT .. 5.3. CI-Dimension. Let R be a noetherian local ring with residue field k. A quasi-deformation of R is a diagram of homomorphisms R ª RX ¤ Q of local rings, with R ª RX faithfully flat and RX ¤ Q surjective with kernel generated by a regular sequence. An R-module M / 0 is said to have finite CI-dimension, cf. w4x, if R has a quasi-deformation such that M mR RX has finite projective dimension over Q.

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THEOREM. If R is a local ring with residue field k and M is a finite R-module of finite CI-dimension, then the graded left module ExtUR Ž M, k . is finite o¨ er the k-subalgebra PRU Ž k . : ExtUR Ž k, k . generated by the central primiti¨ e elements of degree 2. Proof. We set S s k mR RX and make the identification ExtUR Ž k, k . mk S s ExtURX Ž S, S . through the canonical isomorphism due to the flatness of RX over R. We have ZRU Ž k .

mk S s ZRU Ž S . X

since the center of a tensor product of algebras over a field is equal to the tensor product of the centers of theX factors. The flatness of RX also RŽ identifies Tor# k, k . mk S and Tor R#Ž S, S . in a way compatible with products in Tor and the action of Ext, so Pr R Ž k .

mR

S s Ker Ž p Rk .

mR

S s Ker Ž p RSX . s Pr RX Ž S .

and we conclude that PRU Ž k .

mk S s PRU Ž S . . X

Identifying ExtUR Ž M, k . mk S with ExtURX Ž M mR RX , S ., and noting that by Corollary 5.2 the latter is a noetherian module over PRUX Ž S .. We conclude by flat descent that ExtUR Ž M, k . is noetherian over PRU Ž k .. REFERENCES 1. L. L. Avramov, Modules of finite virtual projective dimension, In¨ ent. Math. 96 Ž1989., 71]101. 2. L. L. Avramov and R.-O. Buchweitz, Homological algebra modulo a regular sequence, with special attention to codimension two, preprint, 1998. 3. L. L. Avramov, H.-B. Foxby, and S. Halperin, Differential graded homological algebra, in preparation. 4. L. L. Avramov, V. G. Gasharov, and I. V. Peeva, Complete intersection dimension, Publ. Math. I. H. E.S. 86 Ž1997., 67]114. 5. L. L. Avramov and S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, in ‘‘Algebra, Algebraic Topology, and Their Interactions’’ ŽJ.-E. Roos, Ed.., Lecture Notes in Math., Vol. 1183, pp. 1]27, Springer-Verlag, Berlin, 1986. 6. N. Bourbaki, ‘‘Algebre,’’ Chap. X, Algebre ` ` Homologique, Masson, Paris, 1980. 7. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 Ž1980., 35]64. 8. T. H. Gulliksen, A change of ring theorem with applications to Poincare ´ series and intersection multiplicity, Math. Scand. 34 Ž1974., 167]183. 9. V. Mehta, ‘‘Endomorphisms of Complexes and Modules over Golod Rings,’’ Thesis, University of California, Berkeley, CA, 1976. 10. J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 Ž1957., 14]25.