View of Symmetry in the vanishing of Ext over Gorenstein rings

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SYMMETRY IN THE VANISHING OF EXT OVER GORENSTEIN RINGS

CRAIG HUNEKE and DAVID A. JORGENSEN^∗

Abstract

We investigate symmetry in the vanishing of Ext for finitely generated modules over local Goren- stein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modulesMandNover an AB ringR, Extⁱ_R(M, N)=0 for all i0 if and only if Extⁱ_R(N, M)=0 for alli0.

Introduction

LetR be a local Gorenstein ring and letM andN denote finitely generated R-modules. This paper is concerned with the relation between the vanishing of all higher Ext_R(M, N)and the vanishing of all higher Ext_R(N, M). As a means of investigation we concentrate on the more natural duality between the vanishing of all higher Ext_R(M, N)and the vanishing of all higher Tor modules where eitherM orN is replaced by its dualM^∗(:= Hom_R(M, R)) orN^∗.

Our interest in this topic came about in part from the following striking result proved recently by Avramov and Buchweitz [2, Thm. III]. SupposeM andNare finitely generated modules over a complete intersectionR. Then the following are equivalent:

(1) Tor^R_i (M, N)=0 for alli0 (2) Extⁱ_R(M, N)=0 for alli0 (3) Extⁱ_R(N, M)=0 for alli0.

Their proof relies heavily on the use of certain affine algebraic sets associated to MandN, called support varieties. In their paper [2], Avramov and Buchweitz raise the question of what class of rings satisfy these equivalences for all finitely generated modulesMandN. They point out this class lies somewhere between

∗The first author was partially supported by the NSF and the second author was partially supported by the NSA. This work was done while the second author was visiting Kansas University.

He thanks KU for their generous support.

Received October 1, 2001; in revised form June 25, 2002.

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complete intersections and local Gorenstein rings, but mention that they do not know whether this class is equal to either complete intersections or Gorenstein rings. In this paper we introduce a class of local Gorenstein rings, which we callAB rings, and prove that AB rings satisfy the property that for all finitely generated modulesM andN, the following are equivalent (Theorem 4.1):

(1) Extⁱ_R(M, N)=0 for alli0 (2) Extⁱ_R(N, M)=0 for alli0.

Regular local rings are AB rings, and ifRis an AB ring, thenR/(x1, . . . , xc)is also anAB ring wheneverx¹, . . . , xcis a regular sequence (see Proposition 3.3).

This implies that complete intersections are AB rings. Even when restricted to the case of a complete intersection, our proof of the above equivalence avoids the use of support varieties, and in some ways is more direct than the methods of [2]. We also prove that local Gorenstein rings of minimal possible multiplicity are AB rings, for the strong reason that over such rings (except when the embedding dimension is 2) all large Ext_R(M, N)vanish if and only if eitherMorNhas finite projective dimension. See Theorem 3.6 for a precise statement. These rings are not complete intersections in general, so that in particular the class of AB rings is strictly larger than that of complete intersections. In Theorem 3.8 and Proposition 4.4 we show that there also exist AB ringsR which are not complete intersections and over which there are finitely generated modulesM andN such that Extⁱ_R(M, N) = 0 for all i >dimReven though bothM andNhave infinite projective dimension over R.

An AB ringRis a local Gorenstein ring defined by the property that there is a constantC, depending only on the ring, such that if Extⁱ_R(M, N) = 0 for all i 0, then Ext_Rⁱ(M, N) = 0 for all i > C. As far as we know everyGorenstein ring is an AB ring; we have been unable to find an example which is not. The name ‘AB’ stands for both Auslander-Bridger and Avramov- Buchweitz.

The paper is organized as follows. In Section 1 we give some preliminary and straightforward results concerning the relationship of Ext and Tor. In Section 2 we prove a basic result concerning what holds over an arbitrary local Gorenstein ring. Specifically, ifMandNare finitely generated maximal Cohen-Macaulay modules over a local Gorenstein ringR, then the following are equivalent:

(1) Tor^R_i (M, N)=0 for alli0, (2) Extⁱ_R(M, N^∗)=0 for alli0, and (3) Extⁱ_R(N, M^∗)=0 for alli0.

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Section 3 introduces AB rings, details their basic properties, and gives the main examples. In Section 4 we prove the main theorem of symmetry in the vanishing of Ext over AB rings. Section 5 contains some independent observa- tions concerning what the vanishing of Ext means. In particular we relate the vanishing ofd consecutive Ext modules (d being the dimension of the ring) to the Cohen-Macaulay property of a related tensor product. We include some questions in a final section.

1. Preliminaries

In this section we set notation and discuss some basic facts which will be used throughout the paper.

Unless otherwise stated, we will assumeR to be a local Gorenstein ring.

Also,M andN will denote finitely generatedR-modules. For anR-module Mwe letM^∗denote its dual Hom_R(M, R). IfMis maximal Cohen-Macaulay then it is also reflexive, meaningM^∗∗M (assumingRis Gorenstein).

By acomplete intersection we mean a local ring whose completion with respect to the maximal ideal is the quotient of a regular local ring by a regular sequence.

For a local ringR, we let embdimRdenote itsembedding dimension.

Syzygies and Conversions forExtand Tor

SupposeMis anR-module. Then fori ≥0 we letMi denote imagefi, where fi is theith differential in a minimal free resolution

F: · · · →F2−→f2 F1−→f1 F0−→f0 M →0

ofM. TheseMi are thenon-negative syzygiesof M. They are unique up to isomorphism, or if one considers a non-minimal resolution any two are stably isomorphic.

Now suppose thatM is a maximal Cohen-MacaulayR-module. Let G: · · · →G2−→g² G1−→g¹ G0→M^∗→0

a minimal free resolution of its dualM^∗. SinceM^∗is maximal Cohen-Macau- lay, the dual sequence

G^∗: 0→M^∗∗→G^∗0 g^∗1

−→G^∗1 g2^∗

−→G^∗2 → · · ·

is exact. Using the fact thatM is reflexive, we can spliceFandG^∗together, getting the doubly infinite long exact sequence

C(M): · · · →F

22−→f2 F

11−→f1 F

00→G

−1

∗0 g^∗1

−→ G

−2

∗1 g^∗2

−→ G

−3

∗2→ · · ·.

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(Note the degree convention.) Fori ≤ −1 we setMi := image(g^∗_−i). These are thenegative syzygiesofM. They are unique up to isomorphism. Note that Mi is again maximal Cohen-Macaulay for alliwhenMis.

We now list some properties of the long exact sequencesC(M).

Lemma1.1. LetM be a finitely generated maximal Cohen-MacaulayR- module, and letNbe a finitely generatedR-module.

(1) C(M)^∗C(M^∗)

(2) (Mi)^∗(M^∗)−ifor alli.

(3) Tor^R_i (M, N)H_i(C(M)⊗N)fori≥1.

(4) Extⁱ_R(M, N)H_−i−1(C(M)^∗⊗N)fori≥1.

(5) For fixedt ≥3and for1≤i≤t−2we have (i) Extⁱ_R(M−t, N)Tor^R_t_−i−₁(M^∗, N)

and

(ii) Tor^R_i (M−t, N)Ext^t_R⁻ⁱ⁻¹(M^∗, N).

Proof. Conditions (1)–(3) are straightforward. Condition (5) follows easily from (1)–(4), so only Condition (4) needs some explanation. The critical fact we need to show is that for any complex of freeR-modulesF, Hom_R(F, N) and Hom_R(F, R)⊗RN are isomorphic as complexes: write

F: · · · →Fi+1−−→fi+1 Fi −→fi Fi−1→ · · ·,

where theFiare freeR-modules. The natural mapshi : Hom_R(Fi, R)⊗RN → Hom_R(Fi, N)given byf ⊗n→ {a→f (a)n}are isomorphisms sinceFiis free. It is easy to check that the diagram

Hom_R(Fi, R)⊗RN−−−−−−−−→^fⁱ⁺¹^∗ ^⊗N Hom_R(Fi+1, R)⊗RN

↓^hⁱ ↓^hⁱ⁺¹ Hom_R(Fi, N) −−−−−−−−→^Hom^(fⁱ⁺¹^,N) Hom_R(Fi+1, N) is commutative, and this establishes our fact.

SupposeMandNareR-modules withMmaximal Cohen-Macaulay. Then Extⁱ(M, R)=0 for alli≥1, and so by shifting along the short exact sequences 0→Nn+1→Gn→Nn →0 (withGnfree) we obtain isomorphisms (1.2) Extⁱ_R(M, N)Extⁱ⁺ⁿ_R (M, Nn)

fori ≥1 andn≥0.

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The Change of Rings Long Exact Sequences of Extand Tor

Suppose thatSis a commutative ring,xis a non-zerodivisor ofS andR := S/(x). LetM andN beR-modules. Then we have thechange of rings long exact sequence of Ext [6, 11.65]

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... ... ...

Ext¹_R(M, N)←−Ext²_S(M, N)←−Ext²_R(M, N)←− Ext⁰_R(M, N)←−Ext¹_S(M, N)←−Ext¹_R(M, N)←−0, and thechange of rings long exact sequence of Tor [6, 11.64]

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... ... ...

Tor^R₁(M, N)−→Tor^S₂(M, N)−→Tor^R₂(M, N)−→ Tor^R₀(M, N)−→Tor^S₁(M, N)−→Tor^R₁(M, N)−→0.

2. Vanishing of Ext and Tor over Arbitrary Local Gorenstein Rings In this section we prove what type of duality between the vanishing of Ext and Tor holds over arbitrary local Gorenstein rings. It is possible an even stronger result is true, as we discuss in Section 4, but the main result of this section is what is true ‘on the surface’. In particular, Theorem 2.1 states that one can flip the arguments in vanishing Ext modules ‘up to duals’.

Theorem 2.1. Let R be a local Gorenstein ring, and let M and N be finitely generated maximal Cohen-MacaulayR-modules. Then the following are equivalent:

(1) Tor^R_i (M, N)=0for alli0, (2) Extⁱ_R(M, N^∗)=0for alli0, and (3) Extⁱ_R(N, M^∗)=0for alli0.

Proof. Suppose we have shown that (1) and (2) are equivalent. By repla- cing (1) by the equivalent condition that Tor^R_i (N, M) = 0 for alli 0, we see then that (1) is equivalent to (3). Hence it suffices to prove (1) and (2) are equivalent, and for this we only need to assume that N is maximal Cohen-Macaulay.

We induce upon the dimension ofR, sayd. Ifd =0, then Extⁱ_R(M, N^∗)is the Matlis dual of Tor^R_i (M, N), so the result is immediate in this case.

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Now assume thatd >0. Choosex ∈R a non-zerodivisor onMd,N,N^∗, andR. We have

Tor^R_i (M, N)=0 for alli0;

⇐⇒ Tor^R_i (Md, N)=0 for alli0;

and from the long exact sequence of Tor coming from the short exact sequence 0→N −→^x N →N/xN →0 and Nakayama’s lemma,

⇐⇒ Tor^R_i (Md, N/xN)=0 for alli0;

and by the standard isomorphisms Tor^R_i (Md, N/xN) Tor^R/(x)_i (Md/xMd, N/xN),

⇐⇒ Tor^R/(x)_i (Md/xMd, N/xN)=0 for alli0;

by the induction hypothesis,

⇐⇒ Extⁱ_R/(x)(Md/xMd, (N/xN)^∗)=0 for alli0;

and now sinceNis maximal Cohen-Macaulay andRis Gorenstein,N^∗/xN^∗ (N/xN)^∗(where the second module is Hom_R/(x)(N/xN, R/(x))), thus,

⇐⇒ Extⁱ_R/(x)(Md/xMd, N^∗/xN^∗)=0 for alli0;

by the isomorphisms [5] Extⁱ_R(Md, N^∗/xN^∗)Extⁱ_R/(x)(Md/xMd, N^∗/xN^∗),

⇐⇒ Extⁱ_R(Md, N^∗/xN^∗)=0 for alli0;

and now from the long exact sequence of Ext coming from the short exact sequence 0→N^∗−→^x N^∗→N^∗/xN^∗→0 and Nakayama’s lemma,

⇐⇒ Extⁱ_R(Md, N^∗)=0 for alli0;

⇐⇒ Extⁱ_R(M, N^∗)=0 for alli 0.

Remark2.2. SupposeM andN are maximal Cohen-Macaulay modules over the local Gorenstein ringR. Then

Extⁱ_R(M, N)Extⁱ_R(N^∗, M^∗).

This isomorphism can be seen as follows: suppose thati = 1. AsM and Nare maximal Cohen-Macaulay, they are reflexive, so short exact sequences 0 → N → T → M → 0 dualize to short exact sequences 0 → M^∗ → T^∗ →N^∗ →0 and vice-versa. The Yoneda definition of Ext¹then gives the isomorphism.

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Fori >1 we have

Extⁱ_R(M, N)Ext¹_R(Mi−1, N) Ext¹_R(N^∗, M_i^∗₋1) Ext¹_R(N^∗, (M^∗)−i+1) Extⁱ_R(N^∗, M^∗)

by thei=1 case by (2) of Lemma 1.1 by (1.2).

Presumably, one can directly prove this remark using the Yoneda definition of Ext and the fact that bothMandN are maximal Cohen-Macaulay.

Below is an example showing that the hypothesis thatNis maximal Cohen- Macaulay in the equivalence of (1) and (2) in Theorem 2.1 cannot be dropped.

Example2.3. LetRbe the 3-dimensional hypersurfacek[[W, X, Y, Z]]/ (WX− Y Z), and set M := k and N := coker

_w

xy z

. Then pd_RN = 1 (but pd_RN^∗ = ∞), so we have Tor^R_i (M, N) = 0 for all i 0. However, Ext⁴_R(M, N^∗) = 0. By what is shown in the next section, R is an AB ring.

If it were the case that Extⁱ_R(M, N^∗)is zero for alli 0 then, asRis AB, Proposition 3.2 shows then that Extⁱ_R(M, N^∗) = 0 for alli >dimR, which would be a contradiction.

3. AB Rings

LetRbe a commutative ring. We define the Ext-indexofRto be sup{n|Extⁱ_R(M, N)=0 for alli > nand Extⁿ_R(M, N)=0}, where the sup is taken over all pairs of finitely generatedR-modules(M, N) with Extⁱ_R(M, N)=0 for alli0.

Deﬁnition3.1. IfRis a local Gorenstein ring of finite Ext-index, we say thatRis anAB ring.

We will prove that all complete intersections are AB rings. More generally, it is obvious thatRis an AB ring if Rˆ is (whereRˆ is the completion ofR), and we show (3.3) that ifRis an AB ring andx1, . . . , xcis a regular sequence, thenR/(x1, . . . , xc)is also an AB ring. The class of AB rings also includes local Gorenstein rings of ‘minimal’ multiplicity embdim(R)−dim(R)+2 (see 3.6).

Proposition3.2. Suppose thatRis an AB ring. Then theExt-index ofR equalsdimR.

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Proof. Let ndenote the Ext-index ofR andd the dimension ofR. Let x¹, . . . , xd be a maximalR-sequence. SetM := R/(x¹, . . . , xd)andN = k, the residue field ofR. Then Extⁱ_R(M, N) = 0 fori > d and Ext^d_R(M, N) k=0. Hencen≥d.

Suppose thatn > d. There exists a pair of finitely generatedR-modules (M, N) such that Extⁱ_R(M, N) = 0 for i > nand Extⁿ_R(M, N) = 0. We have the isomorphisms Extⁱ⁺_R¹((Md)−d−1, N) Ext^i−d_R (Md, N) Extⁱ_R(M, N)fori > d. Hence Extⁱ((Md)−d−1, N) = 0 for i > n+1 and Extⁿ⁺¹((Md)−d−1, N)= 0, which contradicts the definition ofn. Therefore n=d.

We of course have the dual notion of Tor-index. IfR is local Gorenstein with finite Tor-index, then it is also equal to dimR, by an argument analogous to that of 3.2.

Another related property of rings we are interested in is the following. We say that Ext^∗_R(M, N)has agap of lengthtif for somen≥0, Extⁱ_R(M, N)=0 forn+1≤i≤n+t, but Extⁿ_R(M, N)and Ext^n+t+_R ¹(M, N)are both nonzero.

We have the analogous notion of gap for Tor^R_∗(M, N). (We allow gaps of length 0.) We set

Ext-gap(R):=sup

t ∈N

Ext^∗_R(M, N)has a gap of lengtht for finiteR-modulesM andN

,

and

Tor-gap(R):=sup

t ∈N

Tor^R_∗(M, N)has a gap of lengtht for finiteR-modulesM andN

.

We say thatR is Ext-boundedif it has finite Ext-gap. Similarly, we sayR is Tor-boundedif it has finite Tor-gap.

We list some elementary properties involving finite Ext-index, Tor-index, Ext-boundedness and Tor-boundedness for local Gorenstein rings.

Proposition3.3. Let x be a non-zerodivisor of thed-dimensional local Gorenstein ringR. Then

(1) Ris an AB ring if and only ifR/(x)is an AB ring.

(2) Rhas finiteTor-index if and only ifR/(x)does.

(3) RisExt-bounded if and only ifR/(x)is.

(4) RisTor-bounded if and only ifR/(x)is.

Proof. (1). Suppose thatRis an AB ring. LetMandNbe finitely generated R/(x)-modules such that Extⁱ_R/(x)(M, N) = 0 for all i 0. By the change

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of rings long exact sequence of Ext (1.3) we conclude that Extⁱ_R(M, N)= 0 for all i 0, and so Extⁱ_R(M, N) = 0 for all i > d. Looking at (1.3) again, we see that Extⁱ_R/(x)(M, N) Extⁱ⁺_R/(x)² (M, N) fori > d −1. But as Extⁱ_R/(x)(M, N) = 0 for alli 0, we have Extⁱ_R/(x)(M, N) = 0 for all i > d−1. HenceR/(x)is an AB ring.

Now suppose thatR/(x)is an AB ring, and letM andN be finitely generated R-modules such that Extⁱ_R(M, N) = 0 for alli 0. We have the isomorphisms

Extⁱ_R(M, N)Ext^i−d_R (Md, N)Extⁱ_R(Md, Nd)

which are valid fori > d, the second one being that of (1.2). The short exact sequence 0 →Nd −→x Nd →Nd/xNd →0 gives rise to the the long exact sequence of Ext

(3.3.1)

· · · →Extⁱ_R(Md, Nd)−→^x Extⁱ_R(Md, Nd)→Extⁱ_R(Md, Nd/xNd)→ · · ·. Since Extⁱ_R(Md, Nd)= 0 for alli 0, we see that Extⁱ_R(Md, Nd/xNd)=0 for alli 0. We have the isomorphisms [5]

(3.3.2) Extⁱ_R/(x)(Md/xMd, Nd/xNd)Extⁱ_R(Md, Nd/xNd)

for alli ≥ 0. Hence Extⁱ_R/(x)(Md/xMd, Nd/xNd) = 0 for alli 0, which means that Extⁱ_R/(x)(Md/xMd, Nd/xNd) =0 for alli > d−1, sinceR/(x) is an AB ring. Therefore Extⁱ_R(Md, Nd/xNd) = 0 for all i > d −1. By (3.3.1) and Nakayama’s Lemma, we conclude that Extⁱ_R(Md, Nd)=0 for all i > d−1, and so Extⁱ_R(M, N)=0 for alli > d. ThereforeRis an AB ring.

The proof of (2) is exactly analogous to the proof of (1), using (1.4) and a long exact sequence of Tor this time.

(3). Assume thate := Ext-gap(R) < ∞. LetM andN be finitely gener- atedR/(x)-modules such that Extⁱ_R/(x)(M, N) = 0 forn≤ i ≤ n+e+1, somen ≥ 1. The change of rings long exact sequence of Ext (1.3) shows that Extⁱ_R(M, N) = 0 forn+1 ≤ i ≤ n+e+1. Since Ext-gap(R) = e we have Extⁱ_R(M, N) = 0 for alli > n. Another look at (1.3) shows that Extⁱ_R/(x)(M, N)Extⁱ⁺_R/(x)² (M, N)for alli > n−1. Since Extⁱ_R/(x)(M, N)= 0 fori =n, n+1, we see then that Extⁱ_R/(x)(M, N)=0 fori > n−1 Hence Ext-gap(R/(x))≤e+1.

Now assume thate :=Ext-gap(R/(x)) <∞. Suppose thatM andN are finitely generatedR-modules with Extⁱ_R(M, N)=0 forn≤i ≤n+d+e+1, somen≥1. We have Extⁱ_R(Md, Nd)Extⁱ_R(M, N)=0 forn+d ≤i≤n+ d+e+1. Therefore, from (3.3.1), we get Extⁱ_R(Md, Nd/xNd)=0 forn+d ≤

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i≤n+d+e. Equivalently, Extⁱ_R/(x)(Md/xMd, Nd/xNd)=0 forn+d≤i≤ n+d+e. Since Ext-gap(R/(x))=e, Extⁱ_R/(x)(Md/xMd, Nd/xNd)=0 for all i≥n+d, which implies, by (3.3.1) and Nakayama’s lemma, Extⁱ_R(Md, Nd)= 0 for alli ≥ n+d, which means Extⁱ_R(M, N) = 0 for alli ≥ n. Therefore Ext-gap(R)≤d+e+1.

The proof of (4) is similar to the proof of (3).

Theorem3.4. Assume thatRis a local Gorenstein ring. Then (1) Ris an AB ring if and only if it has finiteTor-index;

(2) RisExt-bounded if and only if it isTor-bounded;

(3) ifRisExt-bounded, then it is an AB ring.

Proof. We first prove (1). Choose a maximal regular sequence inR and letI be the ideal generated by this sequence. Proposition 3.3 states thatRis an AB ring if and only ifR/I is an AB ring, andRhas finite Tor-index if and only ifR/I has finite Tor-index. Hence it suffices to prove (1) in caseR is 0-dimensional. In this case Extⁱ_R(M, N)is the Matlis dual of Tor^R_i (M, N^∗)so that the vanishing of one implies the vanishing of the other. This proves (1).

Statement (2) is proved in a similar manner, using Proposition 3.3.

We prove (3). AssumeR is Ext-bounded. We prove thatR has finite Tor- index. Letd denote the dimension ofR and e := Ext-gap(R), and suppose that for finiteR-modulesM andN, Tor^R_i (M, N) =0 for alli 0. Letb = d−depthM so thatMbis maximal Cohen-Macaulay. Choosenlargest such that Tor^R_n(Mb, N)=0. Usingt =e+n+3 in (5)(i) of Lemma 1.1, we have Extⁱ_R((M_b^∗)−e−n−3, N) Tor^R_e+n+₂_−i(Mb, N)= 0 for 1≤i ≤e+1. Hence we have a gap of zero Ext larger thane. Therefore Extⁱ_R((M_b^∗)−e−n−3, N)=0 for alli ≥1, which forcesn=0. Thus Tor^R_i (M, N)=0 for alli > d.

The following Corollary is an almost immediate consequence of Proposi- tion 3.4, as regular local rings are clearly Ext-bounded.

Corollary 3.5. Let R be a local Gorenstein ring. If R is a complete intersection, thenRisExt-bounded. In particular,Ris an AB ring.

Proof. SinceR &→ ˆR is a faithfully flat extension, Extⁱ_R(M, N) = 0 if and only if Extⁱ_R_ˆ(M,ˆ N)ˆ =0 and soRis an AB ring ifRˆis. Therefore we may without loss of generality assume thatR =S/(x1, . . . , xc)whereSis a regular local ring andx1, . . . , xcis anS-regular sequence. By Proposition 3.3 it suffices to prove thatSis Ext-bounded. But this is trivial as every finitely generated module overShas projective dimension≤ dimS, so that Extⁱ_R(M, N) = 0 fori >dimSand Ext-gaps can occur of length no longer than dimS−2.

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It could be thatalllocal Gorenstein rings are AB rings; we have no counter- example. The class of AB rings is strictly bigger than the class of complete intersections as the next theorem proves, albeit for rather strong reasons. (See, however, Theorem 3.8 and Propostion 4.4.)

Theorem3.6. Let (R,ᒊ, k)be a local Gorenstein ring with multiplicity equal toembdimR−dimR+2. Assume thatembdimR > dimR+2(so thatRis not a complete intersection). Then for finitely generatedR-modules M andN,Extⁱ_R(M, N) = 0for alli 0if and only if eitherM orN has finite projective dimension. In particular,Ris an AB ring.

Proof. We induce ond :=dimR.

d = 0. In this case by duality we have that Tor^R_i (M, N^∗) = 0 for all i 0. Replace M by a high enough syzygy such that Tor^R₃(M, N^∗) = Tor^R₄(M, N^∗) = Tor^R₅(M, N^∗) = 0. Now the following lemma, 3.7, says that eitherM orN is free.

d > 0. By replacingM andN by syzygies we can assume they are both maximal Cohen-Macaulay. Choose a minimal generatorxof the maximal ideal ofRsuch that the multiplicity ofR/(x)is embdimR/(x)−dimR/(x)+2, andxis a non-zerodivisor on bothM andN. Once again we use the fact that Extⁱ_R(M, N)=0 for alli0 if and only if Extⁱ_R/(x)(M/xM, N/xN)=0 for alli0. By induction eitherM/xMorN/xNhas finite projective dimension overR/(x). But then eitherM orN has finite projective dimension overR.

Lemma3.7. Let(R,ᒊ, k)be a0-dimensional local Gorenstein ring with multiplicity embdimR + 2. Assume embdimR > 2 (so that R is not a complete intersection). LetM andN be finitely generatedR-modules. Then Tor³_R(M, N)=Tor⁴_R(M, N)=Tor⁵_R(M, N)=0if and only if eitherM orN is free.

Proof. Let ndenote the embedding dimension of R. Assume thatM is not free. Ifkis a summand ofMi for any 0 ≤i ≤4, then we get right away thatN is free, since Tor^R₁(Mi, N) = Tor^R_i+₁(M, N) = 0 would then imply Tor^R₁(k, N)=0. Therefore assumekis not a summand ofMifor all 0≤i ≤4.

ReplaceM by its first syzygy. Then asM ⊆ ᒊF, forF a free module, we haveᒊ²M =0 (sinceᒊ³=0). Letbi denote theith Betti number ofM and s:=dim_kᒊM. Then, as in Lescot’s paper [3, Lemma 3.3],Mis 3-exceptional andb1=nb0−s,b2=b0(n²−1)−snandb3=b0(n³−2n)−s(n²−1). Now suppose thatN is also not free. Also replaceNby its first syzygy, so that_ᒊ²N = 0. Write_ᒊN k^d. We have a short exact sequence 0 →k^d → N →k^c →0, wherecis the minimal number of generators ofN. Applying M ⊗R to this short exact sequence and using the fact that Tor¹_R(M, N) = Tor²_R(M, N) = Tor³_R(M, N) = 0 we getcb2 = db1 andcb3 = db2. Letting

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α := d/cwe can write these equations asb2 = αb1andb3 = αb2 = α²b1. Substituting for thebi we get

b0(n²−1)−sn=α(nb0−s) b0(n³−2n)−s(n²−1)=α²(nb0−s).

After rearranging we arrive at

b0(n²−αn−1)=s(n−α) b0(n³−α²n−2n)=s(n²−α²−1).

Now cross multiplying, cancelling off theb⁰s terms, and simplifying we are left with the conditionα²−nα+1 = 0. This says thatα ∈ Qis algebraic overZ. Henceαis an integer, and the only choice isα = 1. But thenn= 2, which is a contradiction. HenceNmust be free.

There also exist AB rings which are not complete intersections and which have multiplicity>embdimR−dimR+2:

Theorem 3.8. Suppose that (R,ᒊR, k) and (S,ᒊS, k) are local rings essentially of finite type over the same fieldk. Set A := (R⊗k S)P, where P :=ᒊ_R⊗kS+R⊗kᒊ_S. If bothRandSare Gorenstein thenAis Gorenstein, and the multiplicity ofAis>embdimA−dimA+2. IfRis an AB ring and Sis the quotient of a regular local ring by a regular sequence (so thatSis a complete intersection), thenAis an AB ring;Ais not a complete intersection ifRis not.

Proof. Suppose thatRandSare Gorenstein. ObviouslyAis Noetherian, being a localization of a finitely generatedk-algebra. By applying [7] we have, moreover, that it is Gorenstein. Proposition 4.3 of the next section shows that there exist finitely generatedA-modulesM andN both of infinite projective dimension overAsuch that Extⁱ_A(M, N)=0 for alli >dimA. It follows from Theorem 3.6 that the multiplicity ofAis larger than embdimA−dimA+2.

Suppose now thatRis an AB ring and thatSis the quotient of the regular local ring(T ,ᒊ_T, k)by theT-regular sequencex:=x1, . . . , xc. ClearlyAis isomorphic to(R⊗k T )P/(1⊗x)whereP := ᒊ_R⊗k T +R ⊗k ᒊ_T and 1⊗xis the regular sequence 1⊗x1, . . . ,1⊗xc. Hence by Proposition 3.3,A is an AB ring if and only if(R⊗kT )P is an AB ring. But now going modulo 1⊗y¹, . . . ,1⊗yd where d := dimT and y¹, . . . , yd a regular system of parameters ofT shows, again by Proposition 3.3, that(R⊗k T )P is an AB ring if and only ifRis an AB ring.

Recall that a local ring(A,ᒊ, k)is a complete intersection precisely when dim_kH1(A)=embdimA−dimA,

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whereH1(A)is the first Koszul homology module on a system of generators for the maximal ideal ofA. In the current situation we have dim_kH¹(A) = dim_kH1(R)+dim_kH1(S), embdimA=embdimR+embdimS, and dimA= dimR+dimS. Hence ifRis not a complete intersection, then dim_kH1(R) >

embdimR−dimR, so that dim_kH1(A) >embdimA−dimA. 4. Vanishing of Ext and Tor over AB Rings

In this section we prove that AB rings are a class which gives the duality of vanishing Ext discussed in the introduction. Our main theorem states:

Theorem4.1. Suppose thatRis an AB ring, and letM andN be finitely generatedR-modules. Then

Extⁱ_R(M, N)=0 for all i 0 if and only if

Extⁱ_R(N, M)=0 for all i0.

Proof. First assume the theorem is true if bothM and N are maximal Cohen-Macaulay. For the general case, take syzygiesMmandNn(m, n≥0) of M and N, respectively, which are maximal Cohen-Macaulay. We have Extⁱ_R(M, N)=0 for alli 0 if and only if Extⁱ_R(Mm, N)=0 for alli 0 and by (1.2) this is equivalent to Extⁱ_R(Mm, Nn) = 0 for alli 0. Thus Extⁱ_R(M, N)=0 for alli0 if and only if Extⁱ_R(Mm, Nn)=0 for alli 0, and so the theorem holds generally.

Now suppose thatMandNare maximal Cohen-Macaulay and Extⁱ_R(M, N)

=0 for alli0. Then for allt ≥1, Extⁱ_R(M−t, N)=0 for alli0. SinceR is an AB ring, it follows from Proposition 3.2 that for allt ≥1 and alli > d := dim(R), Extⁱ_R(M−t, N) = 0. However, Extⁱ_R(M−t, N) Ext¹_R(Mi−t−1, N). Hence for allt ≥1 and alli > d, Ext¹_R(Mi−t+1, N)= 0. By varyingi and t, we obtain that Ext¹_R(M−t, N) = 0 for allt ≥ 1. Therefore by (5)(i) of Lemma 1.1, Tor^R_t−₂(M^∗, N) = Tor^R_t−₂(N, M^∗) = 0 for allt ≥ 3. Applying Theorem 2.1 then shows that Extⁱ_R(N, M)=0 for alli 0.

As an immediate corollary, we have an analogue of Theorem 2.1

Corollary 4.2. Suppose that R is an AB ring, and let M and N be finitely generated maximal Cohen-MacaulayR-modules. Then the following are equivalent:

(1) Tor^R_i (M, N)=0for alli0, (2) Extⁱ_R(M^∗, N)=0for alli0, and

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(3) Extⁱ_R(N^∗, M)=0for alli0.

Proof. Due to the natural symmetry in Tor, it suffices to prove just the equivalence between (1) and (2), and for this we only need to assume thatM is maximal Cohen-Macaulay.

We have

Tor^R_i (M, N)=0 for alli 0⇐⇒Tor^R_i (N, M)=0 for alli0

⇐⇒Extⁱ_R(N, M^∗)=0 for alli0 by 2.1

⇐⇒Extⁱ_R(M^∗, N)=0 for alli 0 by 4.1

If we strengthen the condition on the symmetry in the vanishing of Ext we get an equivalent definition of AB ring.

Proposition4.3. Let(R,ᒊ, k)be a local ring of dimensiond. Then the following are equivalent:

(1) Ris an AB ring;

(2) For finitely generatedR-modulesM andN,Extⁱ_R(M, N)=0fori > d if and only ifExtⁱ_R(N, M)=0fori > d.

Proof. (1) ⇒ (2). This is simply Theorem 4.1 coupled with Proposi- tion 3.2.

For (2)⇒(1), we get right away thatRis Gorenstein from the hypothesis withM :=RandN :=k.

For the remainder of the proof we will use the following fact. If M and N are finitely generated R-modules, then Extⁱ_R(M, N) Extⁱ⁺ⁿ_R (M, Nn) fori > d and n ≥ 0. This follows from the isomorphisms Extⁱ_R(M, N) Ext^i−d_R (Md, N)Ext^i−d+n(Md, Nn)Extⁱ⁺ⁿ_R (M, Nn)fori > dandn≥0, where the middle isomorphism is formula (1.2) of Section 1.

Suppose that Extⁱ_R(M, N)=0 for alli > d+n, some fixedn≥1. Then we have

Extⁱ_R(Mn, N)=0 fori > d;

⇒ Extⁱ_R(N, Mn)=0 fori > d by hypothesis;

⇒ Extⁱ⁺ⁿ_R (N, Mn)=0 fori > d;

⇒ Extⁱ_R(N, M)=0 fori > d by the isomorphism above;

⇒ Extⁱ_R(M, N)=0 fori > d by hypothesis.

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Our final proposition in this section is an observation that there are circum- stances other than where one module has finite projective dimension or where the ring is a complete intersection in which all large Ext modules vanish.

Proposition-Example 4.4. Let(R,ᒊR, k)and(S,ᒊS, k) be two local Gorenstein rings essentially of finite type over the same fieldk, and letMR

be a finitely generatedR-module andNS a finitely generatedS-module. Set A:=(R⊗kS)P whereP :=(ᒊ_R⊗kS+R⊗kᒊ_S),M :=(MR⊗kS)P and N :=(R⊗kNS)P. ThenAis a local Gorenstein ring,M has finite projective dimension overAif and only ifMRhas finite projective dimension overRand similarly forN, andExtⁱ_A(M, N)=0for alli >dimA.

Proof. We remark again thatAis also Gorenstein by [7].

For the second statement, we induce ond := dimA = dimR +dimS. Suppose thatAhas dimension 0. In this case duality yields Extⁱ_A(M, N)^∗ Tor^A_i (M, N^∗). Thus it suffices to prove Tor^A_i (M, N^∗)=0 for alli >0.

We first claim that Hom_R⊗_k_S(MR⊗k S, R⊗kS) Hom_R(MR, R)⊗k S. To see this, note that these modules are naturally isomorphic ifMR is free R-module. In general, letR^{m ρ}−→Rⁿ →MR →0 be a presentation ofMR

overR. LetA := R⊗k SandM := MR⊗k S. This yields a presentation (A)^m→(A)ⁿ→M→0 ofMoverA. We obtain a commutative diagram

0−→ Hom_A(M, A) −→ Hom_A((A)ⁿ, A) −→ Hom_A((A)^m, A)

↓ ↓

0−→Hom_R(MR, R)⊗kS−→Hom_R(Rⁿ, R)⊗kS−→Hom_R(R^m, R)⊗kS, where the first row is exact and the vertical arrows are isomorphisms. To establish the claim we only need to know that the bottom row is exact, but this follows from the fact that 0 → Hom_R(MR, R) → Hom_R(Rⁿ, R) → Hom_R(R^m, R)is an exact sequence ofk-modules andSis flat as ak-module.

Localizing the isomorphism in the claim above atP, we see that the A- moduleM^∗:=Hom_A(M, A)comes from theR-module Hom_R(MR, R). Sim- ilarlyN^∗ := Hom_A(N, A)comes from theS-module Hom_S(NS, S). Hence there is no distinction between proving Tor^A_i (M, N^∗) = 0 for all i > 0 and proving Tor^A_i (M, N)=0 for alli >0. We will prove the latter.

Let(F, f )be anR-free resolution ofMR. ThenFis an exact sequence of k modules, and since Sis flat as ak-module, F⊗k S is an exact sequence, ofR⊗kS-modules. Thus(F⊗k S)P is anA-free resolution ofM. To show that Tor^A_i (M, N) = 0 for alli > 0 we will simply show that the complex (F⊗k S)P ⊗AN is acyclic (meaning the homology is zero except in degree zero).

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For alliwe have a commutative diagram

(Fi⊗kS)⊗R⊗kS(R⊗kNS)−−−−−−−−−−→^(fⁱ^{⊗S)⊗(R⊗N}^S⁾ (Fi−1⊗kS)⊗R⊗kS(R⊗kNS)

↓ ↓

Fi ⊗kNS −−−−−−−−−−→fi⊗NS Fi−1⊗kNS,

where the vertical arrows are the natural isomorphisms. Hence(F⊗kS)⊗R⊗kS

(R⊗kNS)andF⊗kNSare isomorphic complexes ofR⊗kS-modules. Since NSis flat as ak-module, the latter is acyclic, and therefore so is(F⊗kRS)⊗R⊗kS

(R⊗kNS). Finally, localizing atP we get that(F⊗k S)P ⊗AN is acyclic, and this finishes the proof in thed =0 case.

Now without loss of generality assume that dimR >0. From the discussion above we know thatM1((MR)1⊗kS)P. Letxbe a non-zerodivisor on both (MR)1andR. Thenx⊗1 is a non-zerodivisor onM1,AandN, and we have

A/(x⊗1)(R/(x)⊗kS)P, M1/(x⊗1)M1((MR)1/x(MR)1⊗kS)P

and N/(x⊗1)N (R/(x)⊗kNS)P. Hence by induction we have that

Extⁱ_A/(x⊗₁₎(M1/(x⊗1)M1, N/(x⊗1)N)=0

for alli > d−1. Now (3.3.1) and (3.3.2) show that Extⁱ_A(M1, N)=0 for all i > d−1, which means that Extⁱ_A(M, N)=0 for alli > d.

5. What does the vanishing of Ext mean?

Many of the results in this section are closely related to the work of Auslander and Bridger. See [1], and the writeup [4] of the contents of [1]. However, none of the results below is explicitly in these works, and we found they gave us a better understanding of what the vanishing of Ext means.

The natural mapsM^∗⊗RN→Hom_R(M,N)andM⊗RN^∗→Hom_R(M,N)^∗ Assume thatM is maximal Cohen-Macaulay. From the short exact sequence 0→M¹→F →M →0 we get the dual short exact sequence 0→M^∗→ F^∗→M1^∗→0, and these yield a commutative diagram

M^∗⊗RN −−−→^α F^∗⊗RN −−−→ M1^∗⊗RN −−−→0

↓^f⁰ ↓^g ↓^f¹ 0−−−→Hom_R(M, N)−−−→Hom_R(F, N)−−−→^β Hom_R(M¹, N),

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where the vertical arrows are the natural mapsM^∗⊗RN →Hom_R(M, N) given byφ⊗n→ {m→φ(m)n}. Note thatgis an isomorphism (sinceF is free), kerα Tor^R₁(M1^∗, N)and cokerβ Ext¹_R(M, N). From this diagram one easily deduces the following three facts.

(1) kerf1cokerf0,

(2) Tor^R₁(M1^∗, N)kerf0, and (3) Ext¹_R(M, N)cokerf1.

Building a diagram as above for each of the exact sequences 0→Mi+1→ Fi →Mi →0 and using the corresponding three facts as above, we see that we have exact sequences

0→Ext¹_R(Mi−2, N)→M_i^∗⊗RN

→Hom_R(Mi, N)→Ext¹_R(Mi−1, N)→0, and

0→Tor^R₁(M_i+^∗ 1, N)→M_i^∗⊗RN

→Hom_R(Mi, N)→Tor^R₁(M_i+^∗2, N)→0. Fori≥2 the first exact sequence can be written as

(5.1) 0→Extⁱ⁻_R¹(M, N)→M_i^∗⊗RN

→Hom_R(Mi, N)→Extⁱ_R(M, N)→0. An immediate observation is

Proposition 5.2. Let R be a local Gorenstein ring, and let M and N be finitely generated R-modules with M maximal Cohen-Macaulay. Then Extⁱ_R(M, N)=0for alli 0if and only if the natural mapsM_i^∗⊗RN → Hom_R(Mi, N)are isomorphisms for alli 0.

Note also that building exact sequences (5.1) for arbitrarily large negative syzygies ofM, and then splicing the resulting exact sequences together, we obtain a doubly infinite long exact sequence

(5.3) · · · →M_i^∗⊗RN →Hom_R(Mi, N)

→M_i+i^∗ ⊗RN →Hom_R(Mi+1, N)→ · · ·. Now suppose thatNis maximal Cohen-Macaulay and that Ext¹_R(M, N)= 0. From the short exact sequence 0→M1→F →M →0 we get the short