ABSOLUTE, GORENSTEIN, AND TATE TORSIONMODULES

ALINA IACOB

Abstract. We show that there is an Avramov-Martsinkovskytype exact sequence with T̂ or , Gtor , and Tor . We prove that if

R is a Gorenstein ring then the modules T̂ orR

n (M,N), n ≥ 1 canbe computed using either a complete resolution of MR or using acomplete resolution of RN . We show that over a Gorenstein ring aleft R-module N is Gorenstein flat if and only if Gtor R

1 (−, N) = 0.We also show that over commutative Gorenstein rings the modules

T̂ orR

n (M,−) can be computed by the combined use of a flat reso-lution and a Gorenstein flat resolution of M .

1. Introduction

In this article we continue the development of the branch of relativehomological algebra that is called Gorenstein homological algebra, butwith an emphasis on the part that deals with the derived functors ofthe tensor product functor.

Our main results are first to show the existence of exact sequences con-necting the absolute torsion functors, the Gorenstein torsion functorsand the Tate torsion functors. These exact sequences are like those ofAvramov and Martsinkovsky in [1] for the extension functors. But weuse a completely different approach from theirs (which was restrictedto finitely generated modules).

The material presented here deals with several relative derived functors:

• Gtor Rn (M,−) defined via a Gorenstein projective resolution of

M• gtor R

n (M,−) defined via a Gorenstein flat resolution of M

• T̂ orR

n (M,−) defined via a complete resolution of M

• T̂ orG,Pn (M,−) = Hn+1(M(u)⊗R−) where M(u) is the mapping

cone of a chain map u : P. → G. induced by idM , with P.2000 Mathematics Subject Classification. 16E10; 16E30.

1

2 ALINA IACOB

a deleted projective resolution and G. a deleted Gorensteinprojective resolution of M (see Section 2 for definitions).

• torRn (M,−) = Hn+1(M(v)⊗R −) with M(v) the mapping cone

of a chain map v : F. → D. induced by idM , where F. isa deleted flat resolution and D. is a deleted Gorenstein flatresolution of M .

• TorR

n (−, N) defined via a complete resolution of N .

We consider a two-sided noetherian ring R. We prove first (Propo-sition 1) that when M has finite Gorenstein projective dimension we

have T̂ orR

n (M,−) ' T̂ orG,Pn (M,−) for any n ≥ 1. Using this we show

that for each left R-module N there is an exact sequence:

. . . → Gtor R2 (M, N) → T̂ or

R

1 (M, N) → Tor R1 (M, N)(1)

→ Gtor R1 (M,N) → 0

If on the other hand N has finite Gorenstein projective dimension thena similar procedure gives an exact sequence:

. . . → H2(M ⊗R G.) → TorR

1 (M,N) → Tor R1 (M,N) →(2)

H1(M ⊗R G.) → 0

where G. is now a deleted Gorenstein projective resolution of N .

Over a Gorenstein ring every module has finite Gorenstein projectivedimension, so we have both exact sequences. Also in this situation thesame Gorenstein derived functors of − ⊗R − for given M , N can becomputed using either a Gorenstein projective resolution of M or aGorenstein projective resolution of N i.e.:

Hn(M ⊗R G.) = Gtor Rn (M, N), for any n ≥ 0(3)

([2], Theorem 12.2.2)

Using (3) and comparing the exact sequences (1) and (2) it is natural

to ask if T̂ orR

n (M,N) ' TorR

n (M, N). We prove (Theorem 2) that this

is true, i.e. the Tate torsion functors T̂ orR

n (M, N) can be computedusing either a complete resolution of M or using a complete resolution

of N . Then we use balancedness of T̂ or to show that T̂ or commuteswith direct limits (Propositions 4 and 5).

We also use the exact sequence (1) to show that over Gorenstein rings,Gorenstein flat modules can be defined in terms of the vanishing ofGorenstein torsion functors (Proposition 2). This result seems to add

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 3

evidence that the somewhat mysterious definition of Gorenstein flatmodules is the correct one.

In Section 5 we prove that over a commutative Gorenstein ring the

Tate torsion functors T̂ orR

n (M,N), n ≥ 1 can also be computed by thecombined use of a flat and a Gorenstein flat resolution of M . More

precisely we show (Proposition 10) that T̂ orR

n (M, N) ' torRn (M,N)

for any R-modules M , N , for any n ≥ 1.

2. Preliminaries

Let R be an associative ring with 1.

We recall first the definition of Gorenstein projective modules.

Definition 1 ([2], Def. 10.2.1). An R-module M is said to be Goren-stein projective if there is a Hom(−, P roj) exact exact sequence

. . . → P1 → P0 → P 0 → P 1 → . . .

of projective modules such that M = Ker(P 0 → P 1).

The Gorenstein flat modules are defined by the tensor product functor−⊗R −.

Definition 2 ([2], Def. 10.3.1). A left R-module N is said to be Goren-stein flat if there exists an Inj ⊗R − exact exact sequence

. . . → F1 → F0 → F 0 → F 1 → . . .

of flat left R-modules such that N = Ker(F 0 → F 1).

Definition 3 ([2], pp. 167). Let P be a class of R-modules. For anR-module M a morphism φ : P → M where P ∈ P is a P-precover ofM if Hom(P ′, P ) → Hom(P ′,M) → 0 is exact for any P ′ ∈ P.

If moreover, any morphism f : P → P such that φ = φ ◦ f is anautomorphism of P then φ : P → M is called a P-cover of M .

Definition 4 ([2], Def. 8.1.2). A P-resolution of M is a Hom(P ,−)exact complex P : . . . → P2 → P1 → P0 → M → 0 (not necessarilyexact) with each Pi ∈ P.

We note that P is a P-resolution of M if and only if P0 → M , P1 →Ker(P0 → M), and Pi → Ker(Pi−1 → Pi−2) for i ≥ 2 are P-precovers.

Definition 5 ([2], pp. 169). A P-resolution P : . . . → P2 → P1 →P0 → M → 0 such that P0 → M , P1 → Ker(P0 → M), and Pi →Ker(Pi−1 → Pi−2) for i ≥ 2 are P-covers is called minimal.

4 ALINA IACOB

If P is a class of R-modules that contains all the projective R-modulesthen any P-precover is a surjective map. Therefore any P resolutionof M is an exact complex in this case.

If P : . . . → P1 → P0 → M → 0 is a P resolution of M then we referto the complex P. : . . . → P1 → P0 → 0 as a deleted resolution of M .A P-resolution of an R-module M is unique up to homotopy ([2], pg.169) so it can be used to compute derived functors.

By a Gorenstein projective resolution of M we mean a resolution inthe sense of Definition 4 when P is the class of Gorenstein projectivemodules.

Definition 6 ([3], pp. 1920). Let M be a right R-module that has aGorenstein projective resolution G. Then Gtor R

n (M, N) = Hn(G.⊗R

N) for each left R-module N , for any n ≥ 0, where G. is the deletedGorenstein projective resolution of M .

We can also compute left derived functors of M ⊗R N using resolu-tions by Gorenstein flat modules. We denote these gtor R

i (M,N) todistinguish them from the functors Gtor R

i (M,N).

Definition 7 ([2], pp. 299). If M is a right R-module that has aGorenstein flat resolution F then gtor R

n (M,N) = Hn(F. ⊗R N) foreach left R-module N , for each n ≥ 0. (F. is the deleted Gorensteinflat resolution of M)

A ring R is said to be Gorenstein if it is both left and right noetherianand has finite self injective dimension on both sides. Over a Goren-stein ring every module has a finite Gorenstein projective resolutionand a finite Gorenstein flat resolution ([2], Theorem 11.5.1 and Theo-rem 11.7.3).

By [2] Theorem 12.2.2, if R is a Gorenstein ring then the modulesGtor R

n (M, N) can also be computed using a Gorenstein projective res-olution of the left R-module N .

Again by [2], Theorem 12.2.2, if R is a Gorenstein ring then for eachn ≥ 0 gtor R

n (M, N) = Hn(M⊗R V.) where V. is a deleted Gorensteinflat resolution of N .

If R is Gorenstein then gtor Rn (−,−) ∼= Gtor R

n (−,−) for any n ≥ 0 ([2],pg. 299).

Remark 1. H. Holm studied the Gorenstein torsion functors over ar-bitrary rings. For more results on Gtor and gtor see [3].

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 5

3. An Avramov-Martsinkovsky type exact sequence withT̂ or , Tor and Gtor

Let R be a two-sided noetherian ring and let M be a right R-modulethat has a Gorenstein projective resolution. Such a resolution can beused to compute left derived functors Gtor R

i (M,N) of M ⊗R N . Thereare obvious natural maps Tor R

i (M,N) → Gtor Ri (M, N) for all i ≥ 0,

and Tor R0 (M, N) ' Gtor R

0 (M, N).

The main result of this section is showing the existence of an exactsequence:

. . . → Gtor R2 (M, N) → T̂ or

R

1 (M, N) → Tor R1 (M, N)

→ Gtor R1 (M,N) → 0

when M has finite Gorenstein projective dimension.

This comes to show that the Tate Tor (T̂ or ) measures the “difference”between the absolute Tor and the Gorenstein Tor (Gtor). In particularit shows that Tor R

i (M,−) → Gtor Ri (M,−) is an isomorphism for all

i ≥ 1 if and only if T̂ orR

i (M,−) = 0 for all i ≥ 1.

We recall first the following:

Definition 8. A complex T is totally acyclic if it is exact, each moduleof T is projective, and Hom(T, Q) is exact for any projective R-moduleQ.

The Tate torsion functors are defined by means of a complete resolutionof M :

Definition 9 ([1]). A complete resolution of an R-module M is a dia-

gram Tu−→ P → M where P → M is a deleted projective resolution of

M , T is a totally acyclic complex, u is a morphism of complexes andun is bijective for all n À 0.

If M has such a complete resolution Tu−→ P → M then for each left

R-module N , T̂ orR

n (M,N) = Hn(T⊗R N), for any n ∈ Z.

Definition 10 ([2], Definition 4.1). A module M has finite Gorensteinprojective dimension if there is a Gorenstein projective resolution of Mof the form

0 → Gn → Gn−1 → . . . → G0 → M → 0

If n is the least with this property then we set Gor proj dim M = n.

6 ALINA IACOB

We show first (Proposition 1) that if M is a right R-module with

Gor proj dim M < ∞ then the modules T̂ orR

n (M, N), n ≥ 1 canalso be computed by the combined use of a projective and a Goren-stein projective resolution of M .

Let P be a projective resolution and G be a Gorenstein projectiveresolution of M . Let u : P.→ G. be a map of complexes induced byidM and let M(u) be its mapping cone. For each left R-module N , we

define T̂ orG,P

n (M, N) by the equality

T̂ orG,P

n (M,N) = Hn+1(M(u)⊗R N), for any n ≥ 1

We showed ([6], pp. 392) that if P,P′ are two projective resolutions ofM , G,G′ are two Gorenstein projective resolutions of M , u : P.→ G.and v : P′. → G′. are two maps of complexes induced by idM then

M(u) ∼ M(v). So T̂ orG,P

n (M,−) is well defined.

Proposition 1. If Gor proj dim M < ∞ then T̂ orR

n (M, N) ∼= T̂ orG,P

n (M, N)for any left R-module N , for any n ≥ 1.

Proof. Let . . . → P1f1−→ P0

f0−→ M → 0 be a projective resolution of M .

Since Gor proj dim M = g < ∞ it follows that C = Ker fg−1 is Goren-stein projective ([4], Theorem 2.20). Thus there is a Hom(−, P roj)exact exact complex T,

T = . . . → Tg+2dg+2−−→ Tg+1

dg+1−−→ Tgdg−→ Tg−1 → . . .

with each Tn projective, such that C = Im dg.

Then

P = . . . → Tg+1dg+1−−→ Tg

dg−→ Pg−1fg−1−−→ . . . → P1 → P0 → M → 0

is a projective resolution of M .

Since Hom(T, Pj), j ≥ 0 is exact there are homomorphisms u0, u1, . . . , ug−1

that make the diagram commutative:

T : . . . Tg+1 Tg Tg−1 . . .

P : . . . Tg+1 Tg Pg−1 . . .

//

ÂÂÂ ÂÂ ÂÂ ÂÂ Â

 Â Â Â Â

//dg+1

 Â Â ÂÂÂ

ÂÂÂ ÂÂÂÂ ÂÂ Â

//dg

²²Â    Â

ug−1

//dg−1

// //dg+1

//dg

//fg−1

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 7

T1 T0 T−1 . . .

P1 P0 0 . . .

//

²²Â   Â

u1

//

²²Â  ÂÂÂ

u0

//

²²Â   Â

//

// // // //

So Tu−→ P → M is a complete resolution of M .

Let D = Imd0. Then D is Gorenstein projective and there is a com-mutative diagram

0 C Tg−1 Tg−2 . . .

0 C Pg−1 Pg−2 . . .

//

 Â  ÂÂÂ

 Â  ÂÂÂ

//

²²Â   ÂÂ

ug−1

//dg−1

²²Â  ÂÂÂÂ

ug−2

//

// // //fg−1

//

. . . T1 T0 D 0

. . . P1 P0 M 0

//

²²Â   ÂÂ

u1

//d1

²²Â   ÂÂ

u0

//d0

²²Â  Â

u

//

// //f1

//f0

//

with both rows exact complexes. Consequently the mapping cone is anexact complex.

The mapping cone has the exact subcomplex 0 → C∼−→ C → 0.

Forming the quotient we get an exact complex:

G = 0 → Tg−1 → Pg−1 ⊕ Tg−2 → . . . → P1 ⊕ T0 → P0 ⊕D → M → 0

Since K = Ker(P0⊕D → M) has finite projective dimension it followsthat G is a Gorenstein projective resolution of M .

There is a map of complexes e : P → G

. . . Tg+1 Tg Pg−1 . . .

. . . 0 Tg−1 Pg−1 ⊕ Tg−2 . . .

//

²²

//dg+1

²²

dg

//dg

²²

eg−1

//fg−1

// // // //

8 ALINA IACOB

. . . P1 P0 M 0

. . . P1 ⊕ T0 P0 ⊕D M 0

//

²²

e1

//f1

²²

e0

//f0

//

// // // //

with ej : Pj → Pj ⊕ Tj−1, ej(x) = (x, 0), 1 ≤ j ≤ g − 1and e0 : P0 → P0 ⊕D, e0(x) = (x, 0).

So for each n ≥ 1 we have T̂ orG,P

n (M,N) = Hn+1(M(e)⊗R N), whereM(e) is the mapping cone of e : P.→ G..Let T = . . . → Tg+1 → Tg → . . . → T1 → T0 → D → 0.

We show that M(e) and T[1] are homotopically equivalent.

There is a map of complexes α : T[1] → M(e) with

α′ : D → P0 ⊕D, α′(x) = (0, x).

αj : Tj → Pj+1 ⊕ Tj ⊕ Pj.

αj(x) = (0, x,−uj(x)) for any x ∈ Tj, 1 ≤ j ≤ g − 1.

αg−1 : Tg−1 → Tg−1 ⊕ Pg−1, αg−1(x) = (x,−ug−1(x)), ∀x ∈ Tg−1;

αg+j = −idTg+jif j ≥ 1 is odd; αg+j = idTg+j

if j ≥ 0 is even.

There is also a map of complexes l : M(e) → T[1]:

l′ : P0 ⊕D → D, l′(x, y) = y, ∀(x, y) ∈ P0 ⊕D.

lj : Pj+1 ⊕ Tj ⊕ Pj → Tj, lj(x, y, z) = y, ∀(x, y, z) ∈ Pj+1 ⊕ Tj ⊕ Pj

lg−1 : Tg−1 ⊕ Pg−1 → Tg−1, lg−1(x, y) = x, ∀(x, y) ∈ Tg−1 ⊕ Pg−1.

lg+j = −idTg+jif j ≥ 1 is odd; lg+j = idTg+j

if j ≥ 0 is even.

We have l ◦ α = idT[1] and α ◦ l ∼ idM(e).

(A chain homotopy between α ◦ l and IdM is given by the maps:

χ′ : P0 ⊕D → P1 ⊕ T0 ⊕ P0, χ′(x, y) = (0, 0,−x)

χj : Pj+1 ⊕ Tj ⊕ Pj → Pj+2 ⊕ Tj+1 ⊕ Pj+1, χj(x, y, z) = (0, 0,−x),1 ≤ j ≤ g − 3

χg−2 : Pg−1 ⊕ Tg−2 ⊕ Pg−2 → Tg−1 ⊕ Pg−1, χg−2(x, y, z) = (0,−x).)

So Hn+1(M(e)⊗RN) ' Hn+1(T[1]⊗RN) ⇔ T̂ orG,P

n (M,N) ' T̂ orR

n (M, N),for any RN , for any n ≥ 1. ¤

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 9

We can prove now the existence of an Avramov-Martsinkovsky type

exact sequence with T̂ or , Tor , and Gtor .

Theorem 1. Let M be a right R-module with Gor proj dim M < ∞.For each left R-module N there is an exact sequence:

. . . → Gtor R2 (M, N) → T̂ or

R

1 (M, N) → Tor R1 (M, N)

→ Gtor R1 (M,N) → 0

Proof. Let P be a projective resolution of M , G be a Gorenstein pro-jective resolution of M and let u : P → G be a chain map induced byidM .

P : . . . P1 P0 M 0

G : . . . G1 G0 M 0

//

²²Â   ÂÂÂ

u1

//

²²Â   ÂÂÂ

u0

//

 Â Â  Â

ÂÂÂ ÂÂ ÂÂÂÂ Â

//

// // // //

Both P and G are exact complexes so the mapping cone M(u) : . . . →G2 ⊕ P1 → G1 ⊕ P0 → G0 ⊕M → M → 0 is also exact. M(u) has the

exact subcomplex M : 0 → Mid−→ M → 0. Forming the quotient we

obtain the mapping cone M(u) of u : P.→ G.. Thus M(u) is exact.

The sequence 0 → G. → M(u) → P.[1] → 0 is split exact in eachdegree so for each left R-module N we have an exact sequence of com-plexes 0 → G. ⊗R N → M(u) ⊗R N → P.[1] ⊗R N → 0. Thereforewe have a long exact sequence:

. . . → Hn+1(G.⊗R N) → Hn+1(M(u)⊗R N) → Hn+1(P.[1]⊗R N)→ Hn(G.⊗R N) → Hn(M(u)⊗R N) → . . .

Since M(u) is exact and the functor − ⊗R N is right exact it followsthat H1(M(u)⊗R N) = H0(M(u)⊗R N) = 0.

So the exact sequence above is:

. . . → H2(G.⊗R N) → H2(M(u)⊗R N) → H2(P.[1]⊗R N)→ H1(G.⊗R N) → 0 → H1(P.[1]⊗R N) → H0(G.⊗R N) → 0

This gives us the exact sequence

(4) . . . → Gtor R2 (M, N) → T̂ or

G,P

1 (M, N) → Tor R1 (M, N)

→ Gtor R1 (M,N) → 0

10 ALINA IACOB

By Proposition 1 we have T̂ orG,P

n (M, N) ' T̂ orR

n (M,N), for any n ≥ 1.So we obtain the desired long exact sequence. ¤

Corollary 1. Let R be a Gorenstein ring. The following are equivalentfor a right R-module M :

(1) proj dim M < ∞(2) T̂ or

R

n (M,−) = 0 for all n ∈ Z

Proof. 1) ⇒ 2) If proj dim M < ∞ then a complete resolution of Mis 0 → P → M where P is a bounded projective resolution of M . So

T̂ orR

n (M,−) = 0 for any n ∈ Z.

2) ⇒ 1) Since R is Gorenstein we have Gor proj dim M < ∞. Theexact sequence

. . . → Gtor R2 (M, N) → T̂ or

R

1 (M, N) → Tor R1 (M, N)

→ Gtor R1 (M,N) → 0

gives that Tor Rn (M,−) → Gtor R

n (M,−) is an isomorphism for all n ≥1. By [2], Proposition 12.3.3, proj dim M < ∞. ¤

We use Theorem 1 to show that over Gorenstein rings, Gorenstein flatmodules can be characterized in terms of the vanishing of the Goren-stein torsion functors.

Proposition 2. Let R be a Gorenstein ring. For a left R-module Nthe following are equivalent:

(1) N is Gorenstein flat.(2) Gtor R

n (−, N) = 0, for any n ≥ 1.(3) Gtor R

1 (−, N) = 0.

Proof. 1) ⇒ 2) N is a Gorenstein flat left R-module so N ' lim−→Ci for

some inductive system ((Ci), (fji)) with each Ci a finitely generatedGorenstein projective left R-module ([2], Theorem 10.3.8(4)).

Let M be any right R-module and let G. be a deleted Gorensteinprojective resolution of M . We have G.⊗R (lim−→Ci) ' lim−→ (G.⊗R Ci).

Then Hn(G. ⊗R (lim−→Ci)) ' Hn(lim−→ (G. ⊗R Ci)) ' lim−→ (Hn(G. ⊗R

Ci)) = 0 ⇔ Gtor Rn (M, N) = 0 for any n ≥ 1.

2) ⇒ 3) Straightforward.

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 11

3) ⇒ 1) Let L be a right R-module with proj dim L < ∞. By

Corollary 1 T̂ orR

n (L,N) = 0, for any n ≥ 1. The exact sequence

. . . → Tor R2 (L,N) → Gtor R

2 (L,N) → T̂ orR

1 (L,N) → Tor R1 (L,N)

→ Gtor R1 (L,N) → 0

gives us Tor R1 (L,N) ' Gtor R

1 (L,N) = 0 (by hypothesis).

Since Tor R1 (L,N) = 0 for any LR with proj dim L < ∞ it follows that

N is Gorenstein flat (by [2], Theorem 10.3.8). ¤

Theorem 2. Let R be a Gorenstein ring and let N be a left R-module.The following are equivalent:

(1) Gor flat dim N ≤ r(2) Gtor R

i (M,N) = 0 for any right R-module M , for any i ≥ r+1.

Proof. 1) ⇒ 2) Since Gor flat dim N ≤ r there is a Gorenstein flatresolution of N , F : 0 → Fr → . . . → F1 → F0 → N → 0. Thenthe complex M ⊗R F. has length r and we have Gtor R

i (M,N) 'gtor R

i (M, N) = Hi(M ⊗R F.) = 0 for i > r.

2) ⇒ 1) If L is a right R-module with proj dim L < ∞ then T̂ orR

n (L,−) =0 for any n ∈ Z. The exact sequence

. . . → Gtor Rr+2(L, N) → T̂ or

R

r+1(L,N) → Tor Rr+1(L,N)

→ Gtor Rr+1(L,N) → T̂ or

R

r (L,N) → . . .

gives Tor Ri (L,N) = 0 for any i ≥ r+1, for any L with proj dim L < ∞.

By [2], Proposition 11.7.5, Gor flat dim N ≤ r. ¤

So over Gorenstein rings,

Gor flat dim N = Sup{i ∈ N0|Gtor Ri (M,N) 6= 0 for some MR}

Proposition 3. If R is a Gorenstein ring then Gor flat dim N ≤Gor proj dim N for any left R-module N .

Proof. Since R is Gorenstein we have Gor proj dim N = r < ∞.

So N has a Gorenstein projective resolution.

0 → Gr → Gr−1 → . . . → G1 → G0 → N → 0

Then Gtor Rr+i(M,N) = 0 for any MR, for any i ≥ 1. Thus Gor flat dim N ≤

r. ¤

12 ALINA IACOB

4. Balance of T̂ or

R is a left and right noetherian ring.

Let N be a left R-module with Gor proj dim N < ∞. Then we

can define Tate torsion functors TorR

n (−, N) by means of a completeresolution of N : if V → P → N is a complete resolution of N then for

each MR and for each n ∈ Z let TorR

n (M,N) = Hn(M ⊗R V).

The modules TorR

n (M, N), n ≥ 1 can also be computed by the com-bined use of a projective and a Gorenstein projective resolution of N .More precisely: if P. and G. are deleted projective and respectivelyGorenstein projective resolutions of N and v : P. → G. is a mapof complexes induced by idN then a similar argument to the proof of

Proposition 1 gives us TorR

n (M,N) = Hn+1(M ⊗R M(v)) for any rightR-module M , for any n ≥ 1.

By arguments similar to those in Section 3 (proof of Theorem 1) weobtain the exact sequence:

. . . → H2(M ⊗R G.) → TorR

1 (M, N) → Tor R1 (M, N)

→ H1(M ⊗R G.) → 0

If R is Gorenstein then Hn(M ⊗R G.) = Gtor Rn (M, N), for any n ≥ 1

([2], Theorem 12.2.2).

Thus for a Gorenstein ring R we have the exact sequence

(5) . . . → Gtor R2 (M, N) → Tor

R

1 (M, N) → Tor R1 (M, N)

→ Gtor R1 (M,N) → 0

By comparing the exact sequences (1) and (5) it is natural to ask if, over

Gorenstein rings, we have TorR

n (−,−) ' T̂ orR

n (−,−) for any n ≥ 1.We show (Theorem 3) that this is true.

Theorem 3. If R is a Gorenstein ring then T̂ orR

n (M, N) ' TorR

n (M, N)for any right R-module M , any left R-module N , for all n ∈ Z.

Proof. - We prove first that T̂ orR

n (M,N) ' TorR

n (M,N) for all MR,

RN , and for all n ≥ 1.

Let g = Gor proj dim M . R is Gorenstein, so g < ∞.

• Case g = 0

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 13

Since Gtor Ri (M,−) = 0 for any i ≥ 1, the exact sequence

. . . → Gtor R2 (M,−) → T̂ or

R

1 (M,−) → Tor R1 (M,−) → Gtor R

1 (M,−) → 0

gives us T̂ orR

i (M,N) ' Tor Ri (M, N) for any left R-module N .

R is Gorenstein so we also have the exact sequence:

. . . → Gtor R2 (M,N) → Tor

R

1 (M,N) → Tor R1 (M,N) → Gtor R

1 (M, N) → 0

with Gtor Ri (M,−) = 0, ∀i ≥ 1. It follows that Tor

R

i (M,N) ' Tor Ri (M, N),

for any RN , for any i ≥ 1.

Hence T̂ orR

i (M, N) ' TorR

i (M, N) ' Tor Ri (M, N) for any RN , for

any i ≥ 1 (similarly, T̂ orR

i (M, N) ' TorR

i (M, N) if RN Gorensteinprojective).

• Case g ≥ 1

R is Gorenstein so there is an exact sequence 0 → M → L → C → 0with C a Gorenstein projective right R-module and with proj dim L <∞ ([2], Remark 11.5.10).

Let N be any left R-module and let V be a complete resolution of N .Each Vn is projective so 0 → M ⊗R V → L ⊗R V → C ⊗R V → 0 isan exact sequence of complexes.

We have the long exact sequence:

. . . → Hn+1(L⊗RV) → Hn+1(C⊗RV) → Hn(M⊗RV) → Hn(L⊗RV) → . . .

Since each Kn = Ker(Vn → Vn−1) is a Gorenstein projective left R-module and inj dim L < ∞ it follows that Tor1(L,Kn) = 0 ([2],Theorem 10.3.8) ∀n ∈ Z. Therefore L⊗R V is an exact complex. So

(6) Hn(M ⊗R V) ' Hn+1(C ⊗R V) ⇔ TorR

n (M, N) ' TorR

n+1(C, N)

for any n ∈ Z, for any RN .

Since R is Gorenstein, for any RN there is an exact sequence 0 → L′ →C ′ → N → 0 with C ′ a Gorenstein projective left R-module and withproj dim L′ < ∞.

If MR is any right R-module and T is a complete resolution of M then0 → T ⊗R L′ → T ⊗R C ′ → T ⊗R N ′ → 0 is an exact sequence ofcomplexes. Therefore we have a long exact sequence:

. . . → Hn+1(T⊗RL′) → Hn+1(T⊗RC ′) → Hn+1(T⊗RN ′) → Hn(T⊗RL′) → . . .

14 ALINA IACOB

Since T⊗R L′ is an exact complex it follows that

(7) Hn(T⊗R C ′) ' Hn(T⊗R N ′) ⇔ T̂ orR

n (M,N) ' T̂ orR

n (M, C ′),

for any MR, for any n ∈ Z.

By (6) we have TorR

i (M,N) ' TorR

i+1(C,N) ' T̂ orR

i+1(C,N) (since

C ∈ Gor Proj). Then by (7), T̂ orR

i+1(C, N) ' T̂ orR

i+1(C, C ′). So

TorR

i (M,N) ' T̂ orR

i+1(C, C ′),∀i ≥ 1.

By (7), T̂ orR

i (M, N) ' T̂ orR

i (M, C ′) ' TorR

i (M, C ′) since C ′ ∈ Gor Proj.

By (6), TorR

i (M, C ′) ' TorR

i+1(C,C ′). So T̂ orR

i (M, N) ' TorR

i+1(C, C ′) 'T̂ or

R

i+1(C,C ′) (since C is Gorenstein projective).

Thus

(8) TorR

i (M, N) ' T̂ orR

i (M,N) for all MR, RN and all i ≥ 1.

• Case n = −k with k ≥ 0

We consider again the short exact sequence 0 → M → L → C → 0with C Gorenstein projective and with proj dim L < ∞.

Let F = . . . → F1 → F0 → F−1 → . . . be a complete resolution of C(C = Ker(F0 → F−1)) and let Ci = Ker(Fi → Fi−1). If N is any leftR-module and V is a complete resolution of N then 0 → C ⊗R V →F0 ⊗R V → C−1 ⊗R V → 0 is an exact sequence of complexes andF0 ⊗R V is exact (since F0 is projective). The long exact sequence:

. . . → Hn+1(F0⊗RV) → Hn+1(C−1⊗RV) → Hn(C⊗RV) → Hn(F0⊗RV) → . . .

gives us

TorR

n (C, N) ' TorR

n+1(C−1, N), for any n ∈ Z

Similarly,

(9) TorR

n (C, N) ' TorR

n+p(C−p, N)

By (6) and (9) we have

(10) TorR

n (M, N) ' TorR

n+p(C−p−1, N).

for any RN , any n ∈ Z, and any p ≥ 1.

R is Gorenstein, so for any RN we have an exact sequence 0 → L′ →C ′ → N → 0 with C ′ Gorenstein projective and with proj dim L′ < ∞.

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 15

Let G = . . . → G1 → G0 → G−1 → . . . be a complete resolution ofC ′ (C ′ = Ker(G0 → G1)) and let C ′

i = Ker(Gi → Gi−1). Then C ′i is

Gorenstein projective for any i ∈ Z.

A similar argument to the one above gives us:

(11) T̂ orR

n (M, C ′) ' T̂ orR

n+1(M, C ′−1)

for any MR and any n ∈ Z.

By (7) and (11) we have T̂ orR

n (M, N) ' T̂ orR

n+1(M, C ′−1).

Similarly,

(12) T̂ orR

n (M,N) ' T̂ orR

n+p(M,C ′−p)

for any MR, any n ∈ Z, and any p ≥ 1.

By (10), TorR

−k(M, N) ' TorR

1 (C−k−2, N) ' T̂ orR

1 (C−k−2, N) (sinceC−k−2 is Gorenstein projective). Then by (12) we have

T̂ orR

1 (C−k−2, N) ' T̂ orR

k+2(C−k−2, C′−k−1)

So TorR

−k(M,N) ' T̂ orR

k+2(C−k−2, C′−k−1)

By (12), T̂ orR

−k(M, N) ' T̂ orR

1 (M,C ′−k−1) ' Tor

R

1 (M,C ′−k−1) (since

C ′−k−1 is Gorenstein projective). Then by (10) we have Tor

R

1 (M, C ′−k−1) '

TorR

k+2(C−k−2, C′−k−1). Thus T̂ or

R

−k(M, N) ' TorR

k+2(C−k−2, C′−k−1).

So

TorR

−k(M,N) ' T̂ orR

k+2(C−k−2, C′−k−1) ' Tork+2(C−k−2, C

′−k−1) ' T̂ or−k(M, N)

for all MR, RN and all k ≥ 0.

¤

Corollary 2. Let R be a Gorenstein ring. The following are equivalentfor a left R-module L:

(1) proj dim L < ∞(2) T̂ or

R

n (−, L) = 0 for any n ∈ Z

Proof. 1) ⇒ 2) By Theorem 3, T̂ orR

n (−, L) ' TorR

n (−, L). Sinceproj dim L < ∞ a complete resolution of L is 0 → P → L when-

ever P is a bounded projective resolution of L. So TorR

n (−, L) = 0 forany n ∈ Z.

16 ALINA IACOB

2) ⇒ 1) For each NR we have an exact sequence:

. . . → Gtor R2 (N,L) → T̂ or

R

1 (N, L) → Tor R1 (N, L) → Gtor R

1 (N,L) → 0

Since T̂ orR

n (N, L) = 0,∀n ≥ 1, it follows that Gtor Rn (N, L) ' Tor R

n (N,L)for any n ≥ 1, for any NR.

For N ∈ Gor F lat we have Tor Rn (N,L) ' Gtor R

n (N,L) ' gtor Rn (N,L) =

0 ∀n ≥ 1.

By [2], Proposition 11.5.9, proj dim L < ∞. ¤

Using balancedness of T̂ or we can prove now that over Gorenstein rings

T̂ or commutes with direct limits.

Proposition 4. Let R be a Gorenstein ring. For any left R-moduleN and any inductive system ((Mi), (fji)) of right R-modules we have

T̂ orR

n (lim−→Mi, N) ' lim−→ T̂ orR

n (Mi, N), for any n ∈ Z.

Proof. Since R is a Gorenstein ring it follows that Gor proj dim N <∞. So N has a complete resolution U. We have (lim−→Mi) ⊗R U 'lim−→ (Mi ⊗R U).

Then Hn((lim−→Mi)⊗R U) ' Hn(lim−→ (Mi ⊗R U)) ' lim−→Hn(Mi ⊗R U).

So T̂ orR

n (lim−→Mi, N) ' lim−→ T̂ orR

n (Mi, N) (by Theorem 3). ¤

A similar argument gives:

Proposition 5. If R is Gorenstein then T̂ orn(M, lim−→Ni) ' lim−→ T̂ orR

n (M,Ni)

for any right R-module M , any n ∈ Z and any inductive system (Ni)i∈I

of left R-modules.

5. Computing T̂ orR

n (M, N) n ≥ 1 by the combined use of aflat and a Gorenstein flat resolution of M

Let R be a commutative noetherian ring and let M be an R-modulethat has a Gorenstein flat resolution. If F. is a deleted flat resolution,G. is a deleted Gorenstein flat resolution of M , and v : F.→ G. is amap of complexes induced by idM then for each R-module N and foreach n ≥ 1 let

torRn (M, N) = Hn+1(M(v)⊗R N)

(by [6], pp. 392, torRn (M,−) is well defined).

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 17

We prove that for a commutative Gorenstein ring R these are the Tate

torsion functors T̂ orR

n (M, N), for n ≥ 1.

We note first that a similar argument to the proof of Theorem 1 showsthe existence of an Avramov-Martsinkovsky type exact sequence withTor, gtor, and tor :

Proposition 6. Let R be a commutative noetherian ring and let M bean R-module that has a Gorenstein flat resolution. For each R-moduleN there is an exact sequence:

. . . → gtor R2 (M, N) → tor

R1 (M, N) → Tor R

1 (M, N) → gtor R1 (M,N) → 0

We use this result to prove that:

Proposition 7. If R is a commutative Gorenstein ring then the fol-lowing are equivalent for an R-module L:

(1) proj dim L < ∞(2) tor

Rn (L,−) = 0 for any n ≥ 1.

Proof. 1) ⇒ 2) R is Gorenstein, so flat dim L < ∞. Let F : 0 →Fn → . . . → F1 → F0 → L → 0 be a minimal Flat resolution of L (inthe sense of Definition 5 when P is the class of flat R-modules; by [2],Theorem 7.4.4 such a resolution always exists). Let C0 = Ker(F0 →L), Ci = Ker(Fi → Fi−1) for i ≥ 1. Since Ci is cotorsion (by [2],Lemma 5.3.25) and flat dim Ci < ∞, we have Ext 1

R(G,Ci) = 0 forany Gorenstein flat module G, for any i ≥ 0 ([2], Corollary 10.4.27).So Hom(G,F) is exact for any Gorenstein flat module G. Thus F is aGorenstein flat resolution of L. Since the exact sequence of complexes0 → F. → M(id) → F.[1] → 0 is split exact in each degree, foreach R-module N we have an exact sequence of complexes 0 → F.⊗R

N → M(id) ⊗R N → F.[1] ⊗R N → 0. The associated long exactsequence: . . . Hn+1(F.[1] ⊗R N) → Hn(F. ⊗R N) → Hn(M(id) ⊗R

N) → Hn(F.[1] ⊗R N) → . . . gives us Hn+1(M(id) ⊗R N) = 0 ⇔tor

Rn (L,N) = 0 for any n ≥ 1, for any R-module N .

2) ⇒ 1) Since torRn (L,−) = 0, the exact sequence . . . → tor

R1 (M, N) →

Tor R1 (L,N) → gtor R

1 (L,N) → 0 gives Tor Rn (L,−) ' gtor R

n (L,−).For N Gorenstein flat we obtain Tor R

n (L,N) ' gtor Rn (L,N) = 0 for

any n ≥ 1. Since Tor Rn (N,L) ' Tor R

n (L,N) = 0 for any n ≥ 1, forany Gorenstein flat R-module N it follows that proj dim L < ∞ ([2],Proposition 11.5.9). ¤

18 ALINA IACOB

The main result of this section (Proposition 10) shows that when R is

commutative Gorenstein we have torRn (M, N) ' T̂ or

R

n (M, N) for anyR-modules M , N , for any n ≥ 1.

The proof uses the following property of the functors torRn (−,−) (Propo-

sition 8):

if R is Gorenstein then a Hom(Gor F lat,−) exact se-quence 0 → M ′ → M → M ′′ → 0 gives a long exactsequence:

. . . → torR2 (M ′′,−) → tor

R1 (M ′,−) → tor

R1 (M,−) → tor

R1 (M ′′,−) → 0

as well as a similar result for the functors T̂ orR

n (−,−) (Proposition 9):

if R is Gorenstein then a Hom(Gor Proj,−) exact se-quence 0 → M ′ → M → M ′′ → 0 gives a long exactsequence:

. . . → T̂ orR

2 (M ′′,−) → T̂ orR

1 (M ′,−) → T̂ orR

1 (M,−) → T̂ orR

1 (M ′′,−) → 0

The proofs of Propositions 8 and 9 use the following result:

If P , C are two precovering classes closed under finite direct sums suchthat Proj ⊂ P ⊂ C and 0 → M ′ → M → M ′′ → 0 is a Hom(C,−) ex-act sequence of R-modules then there is an exact sequence of complexes0 → M(u) → M(ω) → M(v) → 0 which is split exact in each degree,with u : F′. → G′. (ω : F. → G. and v : F′′. → G′′. respectively)a map of complexes induced by idM ′ (idM , idM ′′ respectively), whereF′. (F., F′′. respectively) is a deleted P resolution of M ′ (M , M ′′

respectively) and G′. (G., G′′. respectively) is a deleted C resolutionof M ′ (M , M ′′ respectively) ([5], proof of Proposition 1).

Proposition 8. If R is a Gorenstein ring and 0 → M ′ → M →M ′′ → 0 is a Hom(Gor F lat,−) exact sequence of R-modules then forany R-module N there is an exact sequence:

. . . → torR2 (M ′′, N) → tor

R1 (M ′, N) → tor

R1 (M,N) → tor

R1 (M ′′, N) → 0

Proof. Since R is Gorenstein, Gor F lat is precovering. Proj ⊂ Flat ⊂Gor F lat, Flat and Gor F lat are closed under finite direct sums, sowe have an exact sequence of complexes:

0 → M(u) → M(ω) → M(v) → 0

where u : F′.→ G′. (ω : F.→ G. and v : F′′.→ G′′. respectively) isa map of complexes induced by idM ′ (idM , idM ′′ respectively), F′. (F.,

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 19

F′′. respectively) is a deleted flat resolution of M ′ (M , M ′′ respectively)and G′. (G., G′′. respectively) is a deleted Gorenstein flat resolutionof M ′ (M , M ′′ respectively).

The sequence 0 → M(u) → M(ω) → M(v) → 0 is split exact in eachdegree, so for each N we have an exact sequence

0 → M(u)⊗R N → M(ω)⊗R N → M(v)⊗R N → 0.

The associated long exact sequence

. . . → Hn+1(M(v)⊗R N) → Hn(M(u)⊗R N) → Hn(M(ω)⊗R N)→ Hn(M(v)⊗R N) → . . .

is the sequence

. . . → tor2(M′′, N) → tor1(M

′, N) → tor1(M, N) → tor1(M′′, N) → 0

¤

A similar argument shows:

Proposition 9. If R is a Gorenstein ring and 0 → M ′ → M → M ′′ →0 is a Hom(Gor Proj,−) exact sequence of R-modules then for anyR-module N there is an exact sequence:

. . . → T̂ orR

2 (M ′′, N) → T̂ orR

1 (M ′, N) → T̂ orR

1 (M,N)

→ T̂ orR

1 (M ′′, N) → 0

We can prove now:

Proposition 10. If R is a commutative Gorenstein ring then torRn (M,N) '

T̂ orR

n (M,N) for any R-modules M and N , any n ≥ 1.

Proof. Let M be an R-module.

R is Gorenstein so there is an exact sequence 0 → L → P → M → 0with P → M a Gorenstein projective precover and with proj dim L <∞ ([2], Theorem 11.5.1). By [2], Lemma 11.7.7, there is an exactsequence 0 → L → C → K → 0 such that K is flat and C is cotorsionwith finite projective dimension.

We consider the following pushout diagram:

20 ALINA IACOB

0 0

0 L P M 0

0 C F M 0

K K

0 0

²²Â  ÂÂ

²²Â  ÂÂ

//

²²Â  ÂÂ

//

²²Â  ÂÂ

//

 Â Â ÂÂ

 Â ÂÂÂÂ

//

//

²²Â  ÂÂ

//

²²Â  ÂÂ

// //

²²Â  ÂÂ

²²Â  ÂÂ

F is Gorenstein flat since K and P are.

So there is an exact sequence 0 → C → F → M → 0 with F Gorensteinflat, and C cotorsion with finite projective dimension.

Since flat dim C < ∞ and C is cotorsion the sequence is Hom(Gor F lat,−)exact ([2], Corollary 10.4.27). By Proposition 8, for each R-module Nwe have an exact sequence:

. . . → torR2 (M, N) → tor

R1 (C,N) → tor

R1 (F,N) → tor

R1 (M,N) → 0

By Proposition 7, torRn (C,N) = 0 for any n ≥ 1. So

(13) torRn (M, N) ' tor

Rn (F, N) for any n ≥ 1.

Since proj dim C < ∞ the sequence 0 → C → F → M → 0 isHom(Gor Proj,−) exact. So (by Proposition 9) we have an exactsequence:

. . . → T̂ orR

2 (M,N) → T̂ orR

1 (C, N) → T̂ orR

1 (F,N) → T̂ orR

1 (M,N) → 0

ABSOLUTE, GORENSTEIN, AND TATE TORSION MODULES 21

We have T̂ orR

n (C,−) = 0 for any n ≥ 1 (by Corollary 1). It followsthat

(14) T̂ orR

n (M, N) ' T̂ orR

n (F, N) for any n ≥ 1.

F is Gorenstein flat, so Gtor Rn (F,−) = 0 for any n ≥ 1.

The long exact sequence

. . . → Gtor R2 (F, N) → T̂ or

R

1 (F, N) → Tor R1 (F, N) → Gtor R

1 (F,N) → 0

gives us T̂ orR

n (F, N) ' Tor Rn (F, N) for any n ≥ 1.

The exact sequence

. . . → Gtor R2 (F, N) → tor

R1 (F, N) → Tor R

1 (F, N) → Gtor R1 (F,N) → 0

gives us torRn (F, N) ' Tor R

n (F,N) for any n ≥ 1.

So

(15) T̂ orR

n (F,N) ' torRn (F, N) ' Tor R

n (F,N)

for any left R-module N , for any n ≥ 1.

By (13), (14), (15) we have torRn (M, N) ' T̂ or

R

n (M,N) for any M , N ,for any n ≥ 1. ¤

Acknowledgement

I thank the referee for many useful suggestions.

References

[1] L.L. Avramov and A. Martsinkovsky. Absolute, Relative and Tate cohomologyof modules of finite Gorenstein dimension. Proc. London Math. Soc., 3(85):393–440, 2002.

[2] E.E. Enochs and O.M.G. Jenda. Relative Homological Algebra. Walter deGruyter, 2000. De Gruyter Expositions in Math; 30.

[3] H. Holm. Gorenstein derived functors. Proc. of Amer. Math. Soc., 132(7):1913–1923, 2004.

[4] H. Holm. Gorenstein homological dimensions. J. Pure and Appl. Alg., 189:167–193, 2004.

[5] A. Iacob. Balance in generalized Tate cohomology. Communications in Algebra,33(6):2009–2024, 2005.

[6] A. Iacob. Generalized Tate cohomology. Tsukuba Journal of Mathematics,29(2):389–404, 2005.

22 ALINA IACOB

Department of Mathematics and Statistics, University of North Car-olina at Wilmington, Wilmington, North Carolina 28403 USA, Email:iacoba@uncw.edu