GENERALIZED TATE COHOMOLOGY ALINA IACOB

Abstract. We consider two classes of left R-modules, P and C, such that P ⊂ C. If the module M has a P-resolution and a C-resolution then for any module N and n ≥ 0 we define genn d C,P (M, N ) and show that eralized Tate cohomology modules Ext we get a long exact sequence connecting these modules and the modules ExtCn (M, N ) and ExtPn (M, N ). When C is the class of Gorenstein projective modules, P is the class of projective modules and when M has a complete resolution we show that the modules n d C,P (M, N ) for n ≥ 1 are the usual Tate cohomology modules Ext and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate con d R (M, N ) can be computed using either a complete homology Ext resolution of M or a complete injective resolution of N . And so, using our dual result, we obtain Avramov and Martsinkovsky’s exact sequence under hypotheses different from theirs.

1. Introduction We consider two classes of left R-modules P, C such that Proj ⊂ P ⊂ C, where Proj is the class of projective modules. Let M be a left Rmodule. Let P be a deleted P-resolution of M, C a deleted Cresolution of M (see Section 2 for definitions), let u : P → C be a chain map induced by IdM , and M(u) the associated mapping cone. d n (M, N) by We define the generalized Tate cohomology module Ext C,P n n+1 d the equality ExtC,P (M, N) = H (Hom(M(u), N)), for any n ≥ 0 n d C,P and any left R-module N. We show that Ext (M, −) is well-defined. We also show that there is an exact sequence connecting these modules and the modules ExtnC (M, N) and ExtnP (M, N):

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(1)

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1

d (M, N) → ... 0 → ExtC1 (M, N) → ExtP1 (M, N) → Ext C,P

2000 Mathematics Subject Classification. Primary 16E05; Secondary 18G25. 1

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ALINA IACOB

We prove (Proposition 1) that when we apply this procedure to C = Gor P roj, P = P roj, over a left noetherian ring R, for an R-module d n (M, N) are the M with Gor proj dimM = g < ∞, the modules Ext C,P usual Tate cohomology modules for any n ≥ 1. In this case our exact sequence (1) becomes L.L. Avramov and A. Martsinkovsky’s exact sequence ([1], th. 7.1): d 1 (M, N) → ... 0 → ExtG1 (M, N) → ExtR1 (M, N) → Ext R d Rg (M, N) → 0 → ExtGg (M, N) → ExtRg (M, N) → Ext

Our proof works in a more general case, for any module M of finite Gorenstein projective dimension, whether finitely generated or not. There is also a dual result (Theorem 1). If Gor inj dim N = d < ∞ then the dth cosyzygy H of an injective resolution of N is a Gorenstein injective module. So there exists an exact sequence E : ... → E1 → E0 → E−1 → E−2 → ... of injective modules such that Hom(I, E) is exact for any injective left R-module I and H = Ker (E0 → E−1 ) We call such sequence a complete injective resolution of N. We show that a complete injective resolution of N is unique up to homotopy. n def For each left R-module M and for each n ∈ Z let ExtR (M, N) = H n (Hom(M, E)). A dual argument of the proof of Proposition 1 shows the existence of an exact sequence 0 → Ext1GI (M, N) → Ext1R (M, N) → 1 ExtR (M, N) → Ext2GI (M, N) → ... → ExtdGI (M, N) → ExtdR (M, N) → d ExtR (M, N) → 0 where ExtiGI (M, N) are the right derived functors of Hom(M, N), computed using a right Gorenstein injective resolution of N. If Gor proj dim M < ∞ then ExtiG (M, N) ≃ ExtiGI (M, N), for all i ≥ 0 ([4], Theorem 3.6). So in this case we obtain an exact sequence 1

0 → Ext1G (M, N) → Ext1R (M, N) → ExtR (M, N) → . . . n

We prove (Theorem 2) that over Gorenstein rings we have ExtR (M, N) d n (M, N) for all left R-modules M, N, for any n ∈ Z. Thus, over ≃ Ext R Gorenstein rings there is a new way of computing the Tate cohomology. 2. Preliminaries Let R be an associative ring with 1 and let P be a class of left Rmodules. Definition 1. [3] For a left R-module M a morphism φ : P → M where P ∈ P is a P-precover of M if Hom(P ′, P ) → Hom(P ′ , M) → 0 is exact for any P ′ ∈ P.

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3

Definition 2. A P-resolution of a left R-module M is a complex P : ... → P1 → P0 → M → 0 (not necessarily exact) with each Pi ∈ P and such that for any P ′ ∈ P the complex Hom(P ′ , P) is exact. Throughout the paper we refer to the complex P : ... → P1 → P0 → 0 as a deleted P resolution of M.

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We note that a complex P as in Definition 2 is a P-resolution if and only if P0 → M, P1 → Ker (P0 → M) and Pi → Ker (Pi−1 → Pi−2 ) for i ≥ 2 are P-precovers. If P contains all the projective left R-modules then any P-precover is a surjective map and therefore any P-resolution is an exact complex. A P-resolution of a left R-module M is unique up to homotopy ([3], pg. 169) and so it can be used to compute derived functors. Definition 3. Let M be a left R-module that has a P-resolution P : ... → P1 → P0 → M → 0. Then ExtnP (M, N) = H n (Hom(P , N)) for any left R-module N and any n ≥ 0, where P is the deleted resolution.

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We prove the existence of the exact sequence (1). Let P, C be two classes of left R-modules such that P roj ⊂ P ⊂ C where P roj is the class of projective modules. Let M be a left Rmodule that has both a P-resolution P : ... → P1 → P0 → M → 0 and a C-resolution C : ... → C1 → C0 → M → 0. Pi ∈ P ⊂ C so Hom(Pi , C) is an exact complex for any i ≥ 0. It follows that there are morphisms Pi → Ci making P : ... →P1 −→ P0 −→ M u1 u0 ?

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C: ... →C1 −→ C0 −→ M

−→ 0

−→ 0

into a commutative diagram. Let u : P → C, u = (ui )i≥0 be such a chain map induced by IdM and let M(u) be the associated mapping cone. Since 0 → C → M(u) → P[1] → 0 is exact and both P and C are exact complexes, the exactness Id of M(u) follows. M(u) has the exact subcomplex 0 → M − → M → 0. Forming the quotient, we get an exact complex, M(u), which is the mapping cone of the chain map u : P → C (P and C being the deleted P, C-resolutions). The sequence 0 → C → M(u) → P [1] → 0 is split exact in each degree, so for any left R-module N we have an exact sequence of complexes 0 → Hom(P [1], N)) → Hom(M(u), N) →

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Hom(C , N) → 0 and therefore an associated cohomology exact sequence: ... → H n (Hom(M(u), N)) → H n (Hom(C , N)) → H n+1(Hom(P [1], N)) → H n+1 (Hom(M(u), N)) → H n+1(Hom(C , N)) → ... Since M(u) is exact and the functor Hom(−, N) is left exact, it follows that H 0 (Hom(M(u), N)) = H 1 (Hom(M(u), N)) = 0. We have H 0 (Hom(C , N)) ≃ Hom(M, N) and H 1 (Hom(P [1], N)) ≃ Hom(M, N). So, the long exact sequence above is: 0 → Hom(M, N) → Hom(M, N) → 0 → H 1 (Hom(C , N)) → H 2 (Hom(P [1], N)) → H 2 (Hom(M(u), N)) → ... After factoring out the exact sequence 0 → ∼ Hom(M, N) − → Hom(M, N) → 0 we obtain the exact sequence (1):

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d 1 (M, N) → ... 0 → ExtC1 (M, N) → ExtP1 (M, N) → Ext C,P d C,P (M, −) is well We prove that the generalized Tate cohomology Ext defined. Let P, C be two classes of left R-modules such that P ⊂ C. Let P, P’ be two P-resolutions of M and let C, C’ be two C-resolutions of M. f2

f1

f0

g2

g1

g0

f′

f′

f′

2 1 0 P’ : ... − → P1′ − → P0′ − → M →0

P : ... − → P1 − → P0 − → M → 0,

g′

g′

g′

0 1 2 → M →0 → C0′ − → C1′ − C’ : ... −

C : ... − → C1 − → C0 − → M → 0,

There exist maps of complexes u : P → C and v : P’ → C’, both δ3 δ2 δ1 induced by IdM . M(u) : ... → C3 ⊕ P2 − → C2 ⊕ P1 − → C1 ⊕ P0 − → δ′

δ′

δ

δ′

0 1 2 3 C0 ⊕ M − → M → 0 and M(v) : ... → C3′ ⊕ P2′ − → → C1′ ⊕ P0′ − → C2′ ⊕ P1′ −

δ′

0 → M → 0 (with δn (x, y) = (gn (x) + un−1(y), −fn−1 (y)) for C0′ ⊕ M − ′ n ≥ 1, δ0 (x, y) = g0 (x) + y, δn′ (x, y) = (gn′ (x) + vn−1 (y), −fn−1 (y)) for ′ ′ n ≥ 1, δ0 (x, y) = g0 (x) + y) are the associated mapping cones.

δ

δ

δ

1 3 2 → C0 → 0 (with M(u) : ... → C3 ⊕ P2 − → C2 ⊕ P1 − → C1 ⊕ P0 −

δ′

δ′

3 2 δ1 (x, y) = g1 (x) + u0 (y)) and M(v) : ... → C3′ ⊕ P2′ − → C2′ ⊕ P1′ − →

δ′

1 C1′ ⊕ P0′ − → C0′ → 0 (with δ1′ (x, y) = g1′ (x) + v0 (y)) are the mapping cones of u : P → C and v : P′ → C′ .

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Since the exact sequence of complexes 0 → C → M(u) → P[1] → 0 is split exact in each degree, for each R F we have an exact sequence: 0 → Hom(F, C) → Hom(F, M(u)) → Hom(F, P[1]) → 0. If F ∈ P ⊂ C then both complexes Hom(F, C) and Hom(F, P[1]) are exact, so the exactness of Hom(F, M(u)) follows.

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Each Pi ∈ P, so by the above, the complex Hom(Pi , M(u)) is exact. Id

Let M denote the complex 0 → M − → M → 0. The exact sequence of complexes 0 → M → M(u) → M(u) → 0 is split exact in each degree. Consequently the sequence 0 → Hom(Pi , M ) → Hom(Pi, M(u)) → Hom(Pi, M(u)) → 0 is exact for any i ≥ 0. Since both Hom(Pi , M(u)) and Hom(Pi , M ) are exact complexes, it follows that (2)

Hom(Pi , M(u)) is an exact complex,

for any i ≥ 0. The identity map IdM induces maps of complexes h : P → P’ and k : C → C’ .

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Both v ◦ h : P → C’ and k ◦ u : P → C’ are maps of complexes induced by IdM , so v ◦ h and k ◦ u are homotopic. Hence there exists ′ si ∈ Hom(Pi , Ci+1 ), i ≥ 0 such that v0 ◦ h0 − k0 ◦ u0 = g1′ ◦ s0 and ′ vn ◦ hn − kn ◦ un = gn+1 ◦ sn + sn−1 ◦ fn for any n ≥ 1.

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Then ω : M(u) → M(v) defined by ω : C0 → C0′ , ω = k0 , ωn : ′ Cn+1 ⊕ Pn → Cn+1 ⊕ Pn′ , ωn (x, y) = (kn+1(x) − sn (y), hn (y)) for any n ≥ 0, is a map of complexes. The identity map IdM also induces maps of complexes l : P’ → P , t : C’ → C . Then t ◦ v : P’ → C and u ◦ l : P’ → C are homotopic.

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′ ⊕ So we have a map of complexes ψ : M(v) → M(u) where ψn : Cn+1 Pn′ → Cn+1 ⊕ Pn is defined by ψn (x, y) = (tn+1 (x) − sn (y), ln (y)), n ≥ 0 (with sn : Pn′ → Cn+1 such that un ◦ ln − tn ◦ vn = sn−1 ◦ fn′ + gn+1 ◦ sn , ∀n ≥ 1, u0 ◦ l0 − t0 ◦ v0 = g1 ◦ s0 ) and ψ : C0′ → C0 , ψ = t0 .

We prove that ψ ◦ ω is homotopic to IdM (u) . Since t ◦ k : C → C is a chain map induced by IdM , we have t ◦ k ∼ IdC . So there exist maps βi ∈ Hom(Ci , Ci+1 ), i ≥ 0 such that t0 ◦ k0 − Id = g1 ◦ β0 and ti ◦ ki − Id = βi−1 ◦ gi + gi+1 ◦ βi , ∀i ≥ 1.

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Let χ0 : C0 → C1 ⊕ P0 , χ0 (x) = (β0 (x), 0), ∀x ∈ C0 . Then δ1 ◦ χ0 (x) = δ1 (β0 (x), 0) = g1 (β0 (x)) + u0 (0) = (t0 ◦ k0 − Id)(x) = (ψ ◦ ω − Id)(x), ∀x ∈ C0 . We have δ1 ◦ (ψ0 ◦ ω0 − χ0 ◦ δ1 − Id) = δ1 ◦ ψ0 ◦ ω0 − (δ1 ◦ χ0 ) ◦ δ1 − δ1 = t0 ◦ k0 ◦ δ1 − (t0 ◦ k0 − Id) ◦ δ1 − δ1 = 0. Let r0 : P0 → C1 ⊕ P0 , r0 = (ψ0 ◦ ω0 − Id − χ0 ◦ δ1 ) ◦ e0 with e0 : P0 → C1 ⊕ P0 , e0 (y) = (0, y). We have δ1 ◦ r0 = δ1 ◦ (ψ0 ◦ ω0 − Id − χ0 ◦

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ALINA IACOB

δ1 ) ◦ e0 = 0. Since r0 ∈ Ker Hom(P0 , δ1 ) = Im Hom(P0 , δ2 ) (by (2)) it follows that r0 = δ2 ◦ γ1 for some γ1 ∈ Hom(P0 , C2 ⊕ P1 ). Hence (ψ0 ◦ ω0 − Id − χ0 ◦ δ1 )(0, y) = δ2 (γ1 (y)). Also we have (ψ0 ◦ ω0 − Id − χ0 ◦ δ1 )(x, 0) = ψ0 (ω0 (x, 0)) − (x, 0) − χ0 (δ1 (x, 0)) = ψ0 (k1 (x), 0) − (x, 0) − χ0 (g1 (x)) = ((t1 ◦ k1 − Id − β0 ◦ g1 )(x), 0) = ((g2 ◦ β1 )(x), 0) = δ2 (β1 (x), 0). So (ψ0 ◦ ω0 − Id − χ0 ◦ δ1 )(x, y) = δ2 ◦ χ1 (x, y) where χ1 : C1 ⊕ P0 → C2 ⊕P1 , χ1 (x, y) = (β1 (x), 0)+γ1(y). Hence ψ0 ◦ω0 −Id = χ0 ◦δ1 +δ2 ◦χ1 . Similarly, there exists χi ∈ Hom(Ci ⊕ Pi−1 , Ci+1 ⊕ Pi ) such that ψi ◦ ωi − Id = χi ◦ δi+1 + δi+2 ◦ χi+1 , ∀i ≥ 1. Thus ψ ◦ ω ∼ IdM (u) . Similarly, ω ◦ ψ ∼ IdM (v) . Then H n (Hom(M(v), N)) ≃ H n (Hom(M(u), N)) for any R N, for any n ≥ 0. Remark 1. The proof above does not depend on P, C containing all the projective R-modules. It works for any two classes P, C of left R-modules such that P ⊂ C. And even without assuming that P, C contain the projectives we still get an Avramov-Martsinkovsky type sequence. Let P, C be two classes of left R-modules such that P ⊂ C. If the R-module M has a P-resolution P and a C-resolution C then IdM induces a chain map u : P → C and we have an exact sequence of complexes 0 → C → M(u) → P [1] → 0 which is split exact in each degree, so 0 → Hom(P [1], N) → Hom(M(u), N) → Hom(C , N) → 0 is still exact for any R-module N. Its associated long exact sequence is: 0 → H 0 (Hom(M(u), N)) → ExtC0 (M, N) → 0 c C,P ExtP0 (M, N) → Ext (M, N) → ExtC1 (M, N) → ... n c (M, N) = H n+1(Hom(M(u), N)) for any n ≥ 0). (with Ext

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C,P

Example 1. Let R = Z, P = the class of projective Z-modules, T = the class of torsion free modules (so P ⊂ T ), M = Z/2Z , N = Z/2Z . A 2 π P-resolution of M is 0 → Z − →Z− → Z/ 2Z → 0. A T -resolution of M is P ∞ ϕ b2 → Z b2 − 0 → 2Z → Z/2Z → 0, with ϕ αi · 2i = a0 . There is a map

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i=0

of complexes u : P → T (P , T are the deleted P, T -resolutions) b2 ⊕ Z → Z b 2 → 0 is exact. and the mapping cone M(u) : 0 → Z → 2Z Since the class T of torsion free Z-modules coincides with the class of flat Z-modules and P ⊂ T , M(u) is an exact sequence of flat Zmodules. We have Hom(Z/2Z , Q/Z ) ≃ Z/2Z . So Z/2Z is pure injective and therefore cotorsion. It follows that Hom(M(u), Z/2Z ) is an exc n (Z/2Z , Z/2Z ) = 0 for all n. So, in this act complex and therefore Ext C,P

GENERALIZED TATE COHOMOLOGY

7

case, the exact sequence 0 → ExtT1 (Z/2Z , Z/2Z ) → ExtZ1 (Z/2Z , Z/2Z ) → c 1 (Z/2Z , Z/2Z ) → Ext 2 (Z/2Z , Z/2Z ) → ... is 0 → Ext 1 (Z/2Z , Z/2Z ) Ext T ,P

T

T

→ ExtZ1 (Z/2Z , Z/2Z ) → 0 with ExtZ1 (Z/2Z , Z/2Z ) ≃ Z/2Z .

3. Avramov-Martsinkovsky’s exact sequence For the rest of the article R denotes a left noetherian ring (unless otherwise specified) and R-module means left R-module. For unexplained terminology and notation please see [1] and [3]. Proposition 1 below shows that when P is the class of projective Rmodules, G is the class of Gorenstein projective R-modules and M is an R-module of finite Gorenstein projective dimension, the modules d n (M, N) are the usual Tate cohomology modules for any n ≥ 1. Ext G,P We recall first the following:

Definition 4. ([1]) A complete resolution of an R-module M is a diu π π agram T − →P− → M where P − → M is a projective resolution of M, T is a totally acyclic complex, u is a morphism of complexes and un u π is bijective for all n ≫ 0. If T − → P − → M is such a complete resolution of M then for each left R-module N and for each n ∈ Z the d n (M, N) is defined by the equality usual Tate cohomology module Ext R n n d ExtR (M, N) = H (Hom(T, N)). Proposition 1. If M is an R-module with Gor proj dim M < ∞ then d n (M, N) ≃ Ext d n (M, N) for any for each R-module N we have Ext G,P R n ≥ 1. Proof. Let g = Gor proj dim M. We start by constructing a complete resolution of M. fg−1

i

fg−2

f1

π

If 0 → C − → Pg−1 −−→ Pg−2 −−→ ... → P1 − → P0 − → M → 0 is a partial projective resolution of M then C is a Gorenstein projective module ([5], Theorem 2.20). Hence there exists an exact sequence d−2

d−1

d

0 T : ... → P −2 −−→ P −1 −−→ P 0 − → P 1 → ... of projective modules such that C = Ker d0 and Hom(T, P ) is an exact complex for any projective R-module P . In particular Hom(T, R) is exact. Since each P n is a projective module and Hn (T) = 0 = Hn (T∗ ) for any integer n, the complex T is totally acyclic.

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ALINA IACOB d−2

d−1

Since C = Im d−1 = Ker fg−1 and ... → P −2 −−→ P −1 −−→ C → 0 is d−2

fg−1

i◦d−1

exact, the complex P : ... → P −2 −−→ P −1 −−−→ Pg−1 −−→ Pg−2 → f1

π

... → P1 − → P0 − → M → 0 is a projective resolution of M. d−1

d

dg−2

d

dg−1

0 1 T : ... →P −1 −−→ P 0 − → → P1 − ... → P g−2 −−→P g−1 −−→ P g →... ug−1 ug−2 u1 u0

i◦d−1

?

fg−1

?

?

fg−2

P : ... →P −1 −−−→ Pg−1−−→ Pg−2 −−→ ... → P1

f1

?

− → P0

− →

?

0 →...

Since Pg−1 is projective, the complex Hom(T, Pg−1) is exact. We have i ◦ d−1 ∈ Ker Hom(d−2 , Pg−1 ) = Im Hom(d−1 , Pg−1 ). So there exists ug−1 ∈ Hom(P 0, Pg−1 ) such that i ◦ d−1 = ug−1 ◦ d−1 . Similarly there exist ug−2 , ..., u0 that make the diagram commutative. Since u : T → P (with u0 , u1 , ..., ug−1 as above and un = IdP g−1−n for n ≥ g) is a morphism of complexes, un is bijective for n ≥ g, T is a totally acyclic complex and P → M is a projective resolution of M, it follows that u π T− →P− → M is a complete resolution of M. We use now the projective resolution P and the complete resolution T to construct a Gorenstein projective resolution of M. Let D = Im dg−1 . Then D is a Gorenstein projective module ([5], Obs. 2.2) and there is a commutative diagram: d

d

dg−2

d

dg−1

0 1 2 0 → C →P 0 − → → → P1 − P2 − ... → P g−2 −−→ P g−1 −−→ D − →0 ug−1 ug−2 ug−3 u1 u0 u

?

fg−1

?

fg−2

?

?

fg−3

0 → C →Pg−1−−→ Pg−2 −−→ Pg−3 −−→ ... → P1

?

f1

− → P0

π

− →

?

M− →0

with u defined by: u(dg−1 (x)) = π(u0 (x)). Since both rows are exact complexes, the associated mapping cone C : ∆

δ

δg−1

δ

0 1 0→C− → C ⊕P0 − → Pg−1 ⊕ P 1 − → Pg−2 ⊕ P 2 → ... → P1 ⊕ P g−1 −−→

β

P0 ⊕ D − → M → 0 is also an exact complex. ∼

C has the exact subcomplex: 0 → C − → C → 0. Forming the quotient δ

δ

0 1 → Pg−1 ⊕ P 1 − → complex, we get an exact complex: 0 → 0 → P 0 −

δg−1

β

Pg−2 ⊕ P 2 → ... → P1 ⊕ P g−1 −−→ P0 ⊕ D − → M → 0.

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Let L be a Gorenstein projective module. Since proj dim Ker β < ∞, we have ExtR1 (L, Ker β) = 0 ([5], Proposition 2.3). The sequence 0 → Ker β → P0 ⊕ D → M → 0 is exact, so we have the associated exact sequence: 0 → Hom(L, Ker β) → Hom(L, P0 ⊕ D) → Hom(L, M) → ExtR1 (L, Ker β) = 0. Thus P0 ⊕ D → M is a Gorenstein projective precover. Similarly P1 ⊕ P g−1 → Ker β is a Gorenstein projective precover, ..., P 0 → Ker δ1 is a Gorenstein projective precover, so G : 0 → P 0 → Pg−1 ⊕ P 1 → Pg−2 ⊕ P 2 → ... → P0 ⊕ D → M → 0 is a Gorenstein projective resolution of M. There is a map of complexes e : P → G d−2

d−1

... →P −2−−→P −1−−→Pg−1 d−1 eg−1 ?

... →0

?

δ

fg−1

−−→ ... →P1 e1

?

δ

?

f1

− → P0 e0 δg−1

?

π

− →M→0 β

0 1 → Pg−1 ⊕ P 1− → ... →P1 ⊕ P g−1 −−→P0 ⊕ D− →M→0 − → P0 −

with e0 : P0 → P0 ⊕ D, e0 (x) = (x, 0) ej : Pj → Pj ⊕ P g−j , ej (x) = (x, 0)

1≤ j ≤g−1

P is a projective resolution of M, G is a Gorenstein projective resolution of M and e : P → G is a chain map induced by IdM , so d n (M, N) = H n+1(Hom(M(e), N)), ∀n ≥ 0, where M(e) is the Ext G,P mapping cone of e : P → G .

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d−2

d−1

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dg−2

dg−1

Let T : ... −−→ P −1 −−→ P 0 → ... → P g−2 −−→ P g−1 −−→ D → 0. We prove that M(e) and T[1] are homotopically equivalent. There is a map of complexes α : T[1] → M(e) with α0 : P 0 → P 0 ⊕ Pg−1 , α0 (x) = (x, −ug−1 (x)) ∀x ∈ P 0 ; αj : P j → Pg−j ⊕P j ⊕Pg−j−1 , αj (x) = (0, x, −ug−j−1(x)), ∀x ∈ P j , 1 ≤ j ≤g−1 α′ : D → P0 ⊕ D, α′ (x) = (0, x) ∀x ∈ D; αj = −IdP j if j ≤ −1 is odd; αj = IdP j if j ≤ −1 is even. There is also a map of complexes l : M(e) → T[1]: l0 : P 0 ⊕ Pg−1 → P 0 l0 (x, y) = x ∀(x, y) ∈ P 0 ⊕ Pg−1 lj : Pg−j ⊕ P j ⊕ Pg−j−1 → P j lj (x, y, z) = y ∀(x, y, z) ∈ Pg−j ⊕ P j ⊕ Pg−j−1 1 ≤ j ≤ g − 1 l′ : P 0 ⊕ D → D l′ (x, y) = y ∀(x, y) ∈ P 0 ⊕ D lj = −IdP j if j ≤ −1 is odd; lj = IdP j if j ≤ −1 is even.

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We have l ◦ α = IdT[1] and α ◦ l ∼ IdM (e)

(3)

(a chain homotopy between α ◦ l and IdM is given by the maps: χ0 : P0 ⊕ D → P1 ⊕ P g−1 ⊕ P0 , χ0 (x, y) = (0, 0, −x) χj : Pj ⊕ P g−j ⊕ Pj−1 → Pj+1 ⊕ P g−j−1 ⊕ Pj , χj (x, y, z) = (0, 0, −x), 1≤j ≤g−2 χg−1 : Pg−1 ⊕ P 1 ⊕ Pg−2 → P 0 ⊕ Pg−1 , χg−1 (x, y, z) = (0, −x)) By (3) we have H n+1 (Hom(M(e), N)) ≃ H n+1 (Hom(T [1], N)) that is d n (M, N) = Ext d n (M, N), for any R N, for all n ≥ 1. Ext  G,P R

Corollary 1. (Avramov-Martsinkovsky) Let M be an R-module with Gor proj dim M = g < ∞. For each R-module N there is an exact d 1 (M, N) → ... → sequence: 0 → ExtG1 (M, N) → ExtR1 (M, N) → Ext R d n (M, N) → ... → Ext g (M, N) → ExtGn (M, N) → ExtRn (M, N) → Ext R R d Rg (M, N) → 0. Ext

Proof. By (1) there is an exact sequence: 0 → ExtG1 (M, N) → ExtR1 (M, N) d 1 (M, N) → .... → Ext G,P

d i (M, N) ≃ Ext d i (M, N), ∀i ≥ 1. By Proposition 1 we have Ext G,P R

Since Extg+i G (M, N) = 0, ∀i ≥ 1 the exact sequence above gives d 1 (M, N) → ... → us: 0 → ExtG1 (M, N) → ExtR1 (M, N) → Ext R d n (M, N) → ... → Ext g (M, N) → ExtGn (M, N) → ExtRn (M, N) → Ext R R d g (M, N) → 0. Ext  R

4. Computing the Tate cohomology using complete injective resolutions The classical groups ExtnR (M, N) can be computed using either a projective resolution of M or an injective resolution of N. In this section d nR (M, N). We we want to prove an analogous result for the groups Ext note that we cannot use a straightforward modification of the proof in classical case. This is basically because the associated double complex in our case is not a first (or third) quadrant one and so we cannot use the usual machinery of spectral sequences. We start by defining a complete injective resolution. Let N be an R-module with Gor inj dim N = d < ∞.

GENERALIZED TATE COHOMOLOGY f0

11

fd−1

f1

If 0 → N → E 0 − → E1 − → ... → E d−1 −−→ H → 0 is a partial injective resolution of N, then H is a Gorenstein injective module ([5], Theorem 2.22). Hence there exists a Hom(Inj, −) exact sequence d

d

d−1

d

d−2

2 1 0 E : ... → E2 − → E1 − → E0 − → E−1 −−→ E−2 −−→ ...

of injective modules such that E is exact and H = Ker d0 ([3], 10.1.1). We say that E is a complete injective resolution of N. i def For each module R M and each i ∈ Z let ExtR (M, N) = H i (Hom(M, E)). We prove that any two complete injective resolutions of N are homotopically equivalent. g−1

g0

g1

′ −1 g−1

0 g′

0 Let E′ : ... → I −1 −−→ I 0 − → I1 − → I 2 → ... and E : ... → I −−→ I − → g 1 1 I − → ... be two complete injective resolutions of N corresponding to two injective resolutions, N and N , of N (H = Ker g0 = Img−1 is the ′ dth cosyzygy of N and H = Ker g0′ = Img−1 is the dth cosyzygy of N ).

If H is the injective resolution of H obtained from N and H is the injective resolution of H obtained from N then H and H are homotopically equivalent (since the two injective resolutions of N, N and N , are homotopically equivalent). Since E ′ : 0 → H → I 0 → I 1 → ... is an injective resolution of H ′ it follows that E ′ and H are homotopically equivalent. Similarly E : 0 → H → I 0 → I 1 → ... is homotopically equivalent to H. Then, by ′ the above, E ′ and E are homotopically equivalent. So there exist chain ′ ′ maps u : E ′ → E and v : E → E ′ (u defined by u ∈ Hom(H, H), uj ∈ j Hom(I j , I ), j ≥ 0 and v defined by v ∈ Hom(H, H) and vj ∈ j Hom(I , I j )), there exist β ∈ Hom(I 0 , H), βj ∈ Hom(I j , I j−1 ), j ≥ 1 such that v ◦ u − Id = β ◦ i (where i : H → I 0 is the inclusion map), and v0 ◦ u0 − Id = β1 ◦ g0 + i ◦ β, vj ◦ uj − Id = gj−1 ◦ βj + βj+1 ◦ gj , ∀j ≥ 1. Since E ′′ : ... → I −2 → I −1 → H → 0 is an injective resolvent of H −2 −1 ([2], 1.3) and E ′′ : ... → I → I → H → 0 is an injective resolvent of H, u ∈ Hom(H, H) induces a map of complexes u : E ′′ → E ′′ , u = (uj )j≤−1 . Similarly, there is a map of complexes v : E ′′ → E ′′ , v = (vj )j≤−1 , induced by v ∈ Hom(H, H).

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ALINA IACOB

Since I 0 is injective and g−1 : I −1 → H is an injective precover, there exists β0 ∈ Hom(I 0 , I −1 ) such that β = g−1 ◦ β0 . So v0 ◦ u0 − Id = β1 ◦ g0 + i ◦ β = β1 ◦ g0 + g−1 ◦ β0 . We have g−1 ◦ (v−1 ◦ u−1 − Id − β0 ◦ g−1 ) = 0 ⇔ Im(v−1 ◦ u−1 − Id − g−2 β0 ◦ g−1 ) ⊂ Ker g−1 . Since I −1 is injective and I −2 −−→ Ker g−1 is an injective precover, there is β−1 ∈ Hom(I −1 , I −2 ) such that v−1 ◦ u−1 − Id − β0 ◦ g−1 = g−2 ◦ β−1 . Similarly, there exist βj ∈ Hom(I j , I j−1), ∀j ≤ −1 such that vj ◦ uj − Id = βj+1 ◦ gj + gj−1 ◦ βj , ∀j ≤ −1. Thus v ◦ u ∼ IdE . Similarly u ◦ v ∼ IdE . Hence H i(Hom(M, E)) ≃ H i (Hom(M, E)) for any R M, for all i ∈ Z. n

So ExtR (−, N) is well-defined. If N is a deleted injective resolution of N, G is a deleted Gorenstein injective resolution of N and v : G → N is a chain map induced by IdN then a dual argument of the proof of Theorem 1 shows that the cohomology of Hom(M, M(v)) gives us the functor ExtR (M, N) and that there is an exact sequence

.

.

.

. 1

0 → Ext1GI (M, N) → Ext1R (M, N) → ExtR (M, N) → Ext2GI (M, N) → d ... → ExtdGI (M, N) → ExtdR (M, N) → ExtR (M, N) → 0 where ExtiGI (M, N) = H i (Hom(M, G )) for any i ≥ 0.

.

If Gor proj dim M < ∞ then ExtiG (M, N) ≃ ExtiGI (M, N) for any i ≥ 0 ([4], Theorem 3.6). Thus we have: Theorem 1. Let N be an R-module with Gor inj dim N = d < ∞. For each R-module M with Gor proj dim M < ∞ there is an exact sequence: 1

0 → ExtG1 (M, N) → ExtR1 (M, N) → ExtR (M, N) → ... n d Rn (M, N) Theorem 2 shows that over Gorenstein rings ExtR (M, N) ≃ Ext for any left R-modules M and N, for any n ∈ Z. n d n (M, N) Theorem 2. If R is a Gorenstein ring then ExtR (M, N) ≃ Ext R for any R-modules M, N for any n ∈ Z.

Proof. Let g = Gor proj dim M and d = Gor inj dim N. R is a Gorenstein ring, so g < ∞ ([3], Corollary 11.5.8) and d < ∞ (this follows from [3], Theorem 11.2.1).

GENERALIZED TATE COHOMOLOGY

13

We are using the notations of Proposition 1 and Theorem 1. d n (M, N) ≃ • We prove first that if M is Gorenstein projective then Ext R n ExtR (M, N) for any n ∈ Z. u

Since M is Gorenstein projective we have a complete resolution T − → π P− → M with T n = P n , ∀n ≥ 0 and un = idP n , ∀n ≥ 0. So d n (M, N) ≃ Ext n (M, N) ∀n ≥ 1 Ext R R

(4)

We have the exact sequence (by Theorem 1): 1

0 → ExtG1 (M, N) → ExtR1 (M, N) → ExtR (M, N) → ExtG2 (M, N) → ... Since ExtGi (M, N) = 0, ∀i ≥ 1 it follows that i

ExtR (M, N) ≃ ExtRi (M, N), ∀i ≥ 1

(5)

i d Ri (M, N) ≃ Ext i (M, N), By (4) and (5) we have ExtR (M, N) ≃ Ext R for all i ≥ 1.

• Case n ≤ 0 Let n = −k, k ≥ 0. Let E be a complete injective resolution of N. d−2

d−1

d

d

0 1 Since T : ... → P −2 −−→ P −1 −−→ P 0 − → P1 − → P 2 → ... is exact with each P i projective and such that Hom(T, Q) is exact for any projective module Q, it follows that M i = Imdi is a Gorenstein projective module for any i ∈ Z ([5], Obs. 2.2).

Let M 1 = Imd1 . Since 0 → M → P 1 → M 1 → 0 is exact and all the terms of E are injective modules, we have an exact sequence of complexes 0 → Hom(M 1 , E) → Hom(P 1, E) → Hom(M, E) → 0 and therefore an associated long exact sequence:

(6)

... → H i (Hom(P 1, E)) → H i (Hom(M, E)) → H i+1 (Hom(M 1 , E)) → H i+1(Hom(P 1 , E)) → ...

Since a complete injective resolution E of N is exact and P 1 is projective, the complex Hom(P 1 , E) is exact. Then, by (6), we have

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ALINA IACOB i

i+1

H i(Hom(M, E)) ≃ H i+1 (Hom(M 1 , E)) ⇔ ExtR (M, N) ≃ ExtR (M 1 , N) for any R N, for any i ∈ Z. Similarly, i

i+k+1

ExtR (M, N) ≃ ExtR

(7)

(M k+1 , N)

for any R N for all i ∈ Z where M k+1 = Imdk+1 ∈ Gor P roj. Since R is a Gorenstein ring there is an exact sequence 0 → G′ → L′ → N → 0 with proj dim L′ < ∞ and G′ a Gorenstein injective module ([3], Exercise 6, pp.277). Since each term of a complete resolution T is a projective module, we have an exact sequence of complexes 0 → Hom(T, G′ ) → Hom(T, L′ ) → Hom(T, N) → 0 and therefore an associated long exact sequence:

(8)

... → H i (Hom(T, G′)) → H i (Hom(T, L′ )) → H i(Hom(T, N)) → H i+1 (Hom(T, G′ )) → H i+1(Hom(T, L′ )) → ...

Since proj dim L′ < ∞ it follows that Hom(T, L′ ) is an exact complex ([5], Proposition 2.3). Then, by (8), we have H i (Hom(T, N)) ≃ H i+1(Hom(T, G′ )) that is d i (M, N) ≃ Ext d i+1 (M, G′ ) Ext R R

(9)

for any i ∈ Z and for any R M. g−2

g−1

g0

g1

→ E1 − → E 2 → ... be a Let E : ... → E −2 −−→ E −1 −−→ E 0 − complete injective resolution of the Gorenstein injective module G′ (G′ = Ker g0 = Im g−1 ) and let Gi = Ker gi . We have (same argument as above) d i (M, N) ≃ Ext d i+k+1 (M, G−k ), ∀i ∈ Z Ext R R

(10)

for any R M, where G−k = Ker g−k . By (7),

−k 1 d 1 (M k+1 , N), ExtR (M, N) ≃ ExtR (M k+1 , N) ≃ ExtR1 (M k+1 , N) ≃ Ext R 1 k+1 d (since M is Gorenstein projective). Then, by (10), ExtR (M k+1 , N) ≃

d Rk+2 (M k+1 , G−k ) ≃ Ext k+2 (M k+1 , G−k ). Ext R −k

So ExtR (M, N) ≃ ExtRk+2 (M k+1 , G−k ).

By (10), d −k (M, N) ≃ Ext d 1 (M, G−k ) ≃ Ext 1 (M, G−k ) ≃ Ext 1 (M, G−k ), Ext R R R R

GENERALIZED TATE COHOMOLOGY

15 1

(since M is Gorenstein projective). Then, by (7), ExtR (M, G−k ) ≃ k+2 ExtR (M k+1 , G−k ) ≃ ExtRk+2 (M k+1 , G−k ). d −k (M, N) ≃ Ext k+2 (M k+1 , G−k ) ≃ Ext −k (M, N) for any k ∈ So Ext R R R Z, k ≥ 0. n d n (M, N) for any n ∈ Z, if M is Gorenstein Hence ExtR (M, N) ≃ Ext R projective. n d n (M, N) for any n ∈ Z, if N is GorenSimilarly, ExtR (M, N) ≃ Ext R stein injective.

• Case g = Gor proj dim M ≥ 1 R is a Gorenstein ring, so there is an exact sequence 0 → M → L → C ′ → 0 with proj dim L < ∞ and C ′ a Gorenstein projective module (the same argument used in [6], Corollary 3.3.7, gives this result for R-modules). Since proj dim L < ∞ it follows that Hom(L, E) is an exact complex.

(11)

Since 0 → M → L → C ′ → 0 is exact and each term of E is an injective module we have an exact sequence of complexes 0 → Hom(C ′ , E) → Hom(L, E) → Hom(M, E) → 0 and therefore an associated long exact sequence: ... → H n (Hom(C ′ , E)) → H n (Hom(L, E)) → H n (Hom(M, E)) → H n+1 (Hom(C ′, E)) → H n+1 (Hom(L, E)) → ... By (11) we have H n (Hom(L, E)) = 0 ∀n ∈ Z. So H n (Hom(M, E)) ≃ H n+1 (Hom(C ′, E)) n n+1 ⇔ ExtR (M, N) ≃ ExtR (C ′ , N)

(12)

for any R N, for any n ∈ Z. n n+1 d n+1 (C ′ , N) (since C ′ ∈ Gor P roj) So ExtR (M, N) ≃ ExtR (C ′ , N) ≃ Ext R for any R N, for all n ∈ Z.

d n+1 (C ′ , N) ≃ Ext d n+2 (C ′ , G′ ) ∀n ∈ Z. (where 0 → G′ → By (9) Ext R R L′ → N → 0 is exact, G′ ∈ Gor Inj, L ∈ L) n

n+2

′ ′ d Hence ExtR (M, N) ≃ Ext R (C , G ) ∀n ∈ Z.

d Rn (M, N) ≃ Ext d Rn+1 (M, G′ ) ≃ ExtRn+1 (M, G′ ) (since G′ is By (9) Ext n+1 Gorenstein injective), for all n ∈ Z. Then, by (12) ExtR (M, G′ ) ≃

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ALINA IACOB

n+2 d n+2 (C ′ , G′) (since C ′ is Gorenstein projective) ExtR (C ′ , G′ ) ≃ Ext R for all n ∈ Z. n d n+2 (C ′ , G′ ) ≃ Ext d n (M, N)∀n ∈ Z. Hence ExtR (M, N) ≃ Ext R R



Remark 2. Theorem 2 shows that over Gorenstein rings there is a new way of computing the Tate cohomology, i.e. by using a complete injective resolution of N.

In a subsequent publication we hope to show how we can exploit this procedure to gain new information about Tate cohomology modules. Theorem 1 together with Theorem 2 give us the following result: Let R be a Gorenstein ring, let N be an R-module with Gor inj dim N = d < ∞. For each R-module M there is an exact sequence: d 1 (M, N) → ... → Ext d (M, N) 0 → ExtG1 (M, N) → ExtR1 (M, N) → Ext R R d d → Ext (M, N) → 0. R

Theorem 2 allows us to give an easy proof of the existence of a long exact sequence of Tate cohomology associated with any short exact sequence 0 → M ′ → M → M ′′ → 0. Theorem 3. Let R be a Gorenstein ring. Let 0 → M ′ → M M ′′ → 0 be an exact sequence of R-modules. For any R-module there exists a long exact sequence of Tate cohomology modules ... d n (M ′′ , N) → Ext d n (M, N) → Ext d n (M ′ , N) → Ext d n+1 (M ′′ , N) Ext R R R R ...

→ N → →

Proof. Let E be a complete injective resolution of N. Then, by Theod Rn (M, N) ≃ H n (Hom(M, E)) for any R M and any n ∈ Z. rem 2, Ext

Since 0 → M ′ → M → M ′′ → 0 is exact and each term of E is an injective module, we have an exact sequence of complexes: 0 → Hom(M ′′ , E) → Hom(M, E) → Hom(M ′ , E) → 0. Its associated cohomology exact sequence is the desired long exact sequence.  Remark 3. J. Asadollahi and Sh. Salarian also have a proof of the claim of Theorem 2 in a recent preprint (Gorenstein Local Cohomology Modules) of theirs.

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17

Acknowledgement The author would like to thank Professor Edgar Enochs for his numerous advices and guidance in putting together this paper. References [1] L.L. Avramov and A. Martsinkovsky. Absolute, Relative and Tate cohomology of modules of finite Gorenstein dimension. Proc. London Math. Soc., 3(85):393– 440, 2002. [2] E.E. Enochs and O.M.G. Jenda. Gorenstein injective and projective modules. Mathematische Zeitschrift, (220):611–633, 1995. [3] E.E. Enochs and O.M.G. Jenda. Relative Homological Algebra. Walter de Gruyter, 2000. De Gruyter Exposition in Math; 30. [4] H. Holm. Gorenstein derived functors. Proc. of Amer. Math. Soc., 132(7):1913– 1923, 2004. [5] H. Holm. Gorenstein homological dimensions. J. Pure and Appl. Alg., 189:167– 193, 2004. [6] J.R. Garc´ıa Rozas. Covers and evelopes in the category of complexes of modules. C R C Press LLC, 1999. Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027 USA, Email: [email protected].