EXTENSIONS OF A DUALIZING COMPLEX BY ITS RING:

COMMUTATIVE VERSIONS OF A CONJECTURE OF TACHIKAWA

LUCHEZAR L. AVRAMOV, RAGNAR-OLAF BUCHWEITZ, AND LIANA M. ŞEGA

Dedicated to Wolmer Vasconcelos on the occasion of his 65th birthday

Abstract. Let (R, 𝔪, k) be a commutative noetherian local ring with dualizing complex D^{R}, normalized by Ext_{R}^{depth(R)}(k, D^{R}) ≅ k. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) k-algebras of finite rank, we conjecture that if Ext_{R}^{n}(D^{R}, R) = 0 for all n > 0, then R is Gorenstein, and prove this in several significant cases.

Introduction

Let (R, 𝔪, k) be a local ring, that is, a commutative noetherian ring R with unique maximal ideal 𝔪 and residue field k = R/𝔪. We write edim R for the minimal number of generators of 𝔪 and set codepth R = edim R − depth R.

We assume that R has a dualizing complex, and let D^{R} denote such a complex shifted so that H_{i}(D^{R}) = 0 for i < 0 and H_{0}(D^{R}) ≠ 0; see Section 1 for more details on dualizing complexes. The ring R is Cohen-Macaulay (respectively, Gorenstein) if and only if D^{R} can be taken to be a dualizing module for R (respectively, to be the R-module R). Thus, when R is Gorenstein, Ext_{R}^{i}(D^{R}, R) = 0 holds trivially for all i > 0. We prove that vanishing of such Ext groups often implies that R is Gorenstein, as follows: Section 2. R is generically Gorenstein (e.g., R is reduced), and Ext_{R}^{i}(D^{R}, R) = 0

for all i ∈ [1, dim R].

Section 3. R is a homomorphic image of a generically Gorenstein ring with dualizing

complex, and Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1].

Section 4. R is Cohen-Macaulay, in the linkage class of a complete intersection

(e.g., codepth R ≤ 2), and Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1].

Section 5. R contains a regular sequence f with the property that 𝔪^{3} ⊆ (f), and

Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1].

Section 6. R is the special fiber of a finite flat local homomorphism that possesses

a Gorenstein fiber, and Ext_{R}^{1}(D^{R}, R) = 0.

Section 7. codepth R ≤ 3 and Ext_{R}^{i}(D^{R}, R) = 0 for all i ≫ 0. Section 8. R is Golod and Ext_{R}^{i}(D^{R}, R) = 0 for all i ≫ 0.

Date: March 15, 2003.

L.L.A. was partly supported by a grant from the NSF.

R.O.B. was partly supported by a grant from NSERC.

L.M.Ş. was a CMI Liftoff Mathematician for 2002. She thanks the University of Nebraska-Lincoln for hospitality during the Spring semester of 2002, and the University of Toronto for hosting a visit.

An effort to make sense of the disparate hypotheses leads us to pose the

Question. Does the vanishing of Ext_{R}^{i}(D^{R}, R) = 0 for (dim R + 1) consecutive positive values of i imply that R is Gorenstein?

A result of Foxby [10], see Section 1, implies that if Ext_{R}^{i}(D^{R}, R) vanishes for i = 1, …, dim R, then R is Cohen-Macaulay, so this hypothesis is implicit in the

Conjecture. If Ext_{R}^{i}(D^{R}, R) = 0 for all i > 0, then R is Gorenstein.

This is an analog of a 30 years old conjecture of Tachikawa [26] on (possibly noncommutative) algebras of finite rank over a field. In fact, if the hypotheses of both conjectures apply, then they are equivalent. The result of Section 5 is obtained by transposing the proof, due to Asashiba and Hoshino [1], [2], of one case of Tachikawa’s conjecture. On the other hand, our results prove that conjecture in new cases. Background material and more details on Tachikawa’s conjecture are given in Section 9.

Our arguments use a substantial amount of homological algebra for complexes of modules. Appendix A contains a synopsis of the basic constructions. In Appendix B we have collected miscellaneous results on complexes that play central roles in various proofs, but do not draw upon the framework of noetherian local algebra.

1. Dualizing complexes

Let (R, 𝔪, k) be a local ring. In this section we collect basic properties of dualizing complexes. A comprehensive treatment can be found in Foxby’s notes [10]. We refer to Appendix A for basic notation regarding complexes.

A complex of R-modules is said to be dualizing if it has finite homology and there is an integer d such that Ext_{R}^{d}(k, D) ≅ k and Ext_{R}^{i}(k, D) = 0 for i ≠ d. For completeness, we recall that this is the case if and only if R is a homomorphic image of a Gorenstein ring, where the “if” part is a deep recent result of Kawasaki [27]. In this paper, we always assume that R has a dualizing complex.

1.1. Let D^{R} denote a dualizing complex with inf H(D^{R}) = 0, and set d = depth R.

1.1.1. Every dualizing complex is quasi-isomorpic to a shift of D^{R}, cf. [10, 15.14].

1.1.2. sup H(D^{R}) = dim R − d, cf. [10, 15.18].

1.1.3. Ext_{R}^{i}(k, D^{R}) = 0 for i ≠ d and Ext_{R}^{d}(k, D^{R}) ≅ k, cf. [10, 15.18].

1.1.4. For every 𝔭 ∈ Spec R and for n = depth R − depth R_{𝔭} − dim(R/𝔭) there is a quasi-isomorphism Σ^{n}(D^{R})_{𝔭} ≃ D^{R_{𝔭}} of complexes of R_{𝔭}-modules, cf. [10, 15.17].

The next result slightly extends an observation by the referee.

1.2. Proposition. If 𝔭 is an associated prime ideal of R, then

Ext_{R}^{i}(D^{R}, R) ≠ 0 for i = dim(R/𝔭) − depth R

Proof. Since D^{R} has finite homology, Ext_{R}(D^{R}, R) localizes, cf. [10, 6.47]. This justifies the first isomorphism below:

Ext_{R}^{i}(D^{R}, R)_{𝔭} ≅ Ext_{R_{𝔭}}^{i} ((D^{R})_{𝔭}, R_{𝔭}) ≅ Ext_{R_{𝔭}}^{i} (Σ^{i}(D^{R_{𝔭}}), R_{𝔭})
≅ Ext_{R_{𝔭}}^{0}(D^{R_{𝔭}}, R_{𝔭})

EXTENSIONS OF A DUALIZING COMPLEX 3

The second isomorphism comes from 1.1.4 and the last one is due to the shift. Now inf H(D^{R_{𝔭}}) = 0 by 1.1.2, and hence it follows from [10, 6.42(3)] that

Ext_{R_{𝔭}}^{0}(D^{R_{𝔭}}, R_{𝔭}) ≅ Hom_{R}(H_{0}(D^{R_{𝔭}}), R_{𝔭})

Since H_{0}(D^{R_{𝔭}}) ≠ 0 by 1.1.2 and depth R_{𝔭} = 0, the module Hom_{R}(H_{0}(D^{R_{𝔭}}), R_{𝔭}) is nonzero, so we conclude Ext_{R}^{i}(D^{R}, R)_{𝔭} ≠ 0, and hence Ext_{R}^{i}(D^{R}, R) ≠ 0. □ 1.3. Corollary. If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R], then the ring R is Cohen-Macaulay. Proof. Choose 𝔭 ∈ Spec R with dim(R/𝔭) = dim R. Note that 𝔭 is associated to R and that dim(R/𝔭) − depth R = i, so Ext_{R}^{i}(D^{R}, R) ≠ 0 by the proposition. In view of our hypothesis, this implies i = 0, that is, R is Cohen-Macaulay. □

A special case of the property 1.1.2 reads: 1.4. The ring R is Cohen-Macaulay if and only if H_{i}(D^{R}) = 0 for all i ≠ 0. When this is the case, D^{R} can be taken to be a module, called a dualizing module of R.

For the properties of dualizing modules listed below we refer to [7, §3.3]. 1.5. Let R be a Cohen-Macaulay local ring. 1.5.1. The ring R has a dualizing module D if and only if there exists a finite homomorphism of rings Q → R with a Gorenstein local ring Q; when this is the case, every dualizing module is isomorphic to Ext_{Q}^{dim Q−dim R}(R, Q). 1.5.2. id_{R} D = dim R. 1.5.3. rank_{k}(D/𝔪D) = rank_{k}(Ext_{R}^{dim R}(k, R)). 1.5.4. Ass_{R} D = Ass R. 1.5.5. For each 𝔭 ∈ Spec R the module D_{𝔭} is dualizing for R_{𝔭}. 1.5.6. Each R-regular element f is also D-regular, and D/fD is dualizing for R/(f). 1.5.7. D ≅ R if and only if the ring R is Gorenstein.

2. Generically Gorenstein rings

A commutative ring R is said to be generically Gorenstein if the ring R_{𝔭} is Gorenstein for all 𝔭 ∈ Ass R. Our purpose in this section is to prove 2.1. Theorem. Let R be a generically Gorenstein local ring.

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R], then R is Gorenstein. Remark. Another proof of Theorem 2.1 was independently obtained by Huneke and Hanes [11, 2.2] when R is Cohen-Macaulay. The essential input comes from [11, 2.1], which implies that D^{R} ⊗_{R} D^{R} is a maximal Cohen-Macaulay module. The same conclusion can be obtained from Corollary B.4(3), that has weaker hypotheses.

We start with a criterion for a module to be free of rank 1. With a view towards later applications, we work in greater generality than needed here. 2.2. Lemma. Let P → Q be a finite homomorphism of commutative rings, and let N be a finite Q-module. If P is local, Ass_{P} (N ⊗_{Q} N) ∪ Ass_{P} Q ⊆ Ass P, and N_{𝔭} ≅ Q_{𝔭} as Q_{𝔭}-modules for every prime ideal 𝔭 ∈ Ass P, then N ≅ Q.

Proof. The second symmetric power S_{Q}^{2}(N) appears in a canonical epimorphism

π_{Q}^{N}: N ⊗_{Q} N ⟶ S_{Q}^{2}(N)

For each 𝔭 ∈ Ass P the homomorphism (π_{Q}^{N})_{𝔭} factors as a composition

(N ⊗_{Q} N)_{𝔭} ≅ N_{𝔭} ⊗_{Q_{𝔭}} N_{𝔭} ⟶^{π_{Q𝔭}^{N𝔭}} S_{Q_{𝔭}}^{2}(N_{𝔭}) ≅ (S_{Q}^{2}(N))_{𝔭}

with canonical isomorphisms. As the Q_{𝔭}-module N_{𝔭} is cyclic, the map π_{Q_{𝔭}}^{N_{𝔭}} is bijective, so (π_{Q}^{N})_{𝔭} is an isomorphism, hence (Ker(π_{Q}^{N}))_{𝔭} = 0. The inclusions

Ass_{P} (Ker(π_{Q}^{N})) ⊆ Ass_{P} (N ⊗_{Q} N) ⊆ Ass P

now imply Ker(π_{Q}^{N}) = 0, so π_{Q}^{N} is bijective.

Let 𝔫 be a maximal ideal of Q and set r = rank_{Q/𝔫}(N/𝔫N). Assuming r = 0, and localizing at 𝔫, we get N_{𝔫} = 0 by Nakayama’s Lemma. Since Q is finite as a P -module, 𝔫 ∩ P is the maximal ideal of P, hence 𝔫 contains 𝔭 ∈ Ass P. Thus, N_{𝔭} is a localization of N_{𝔫}, hence N_{𝔭} = 0, contradicting our hypothesis. We conclude that r ≥ 1. By naturality, the map π_{Q}^{N} induces an isomorphism

(N/𝔫N) ⊗_{Q/𝔫} (N/𝔫N) ≅ S_{Q/𝔫}^{2}(N/𝔫N)

of vector spaces over Q/𝔫. Comparing ranks, we get r = 1.

The ring Q, being a finite module over a local ring, is semilocal. Thus, if 𝔯 is its Jacobson radical, then Q/𝔯 is a finite product of fields. By what we have just proved, there is an isomorphism Q/𝔯Q ≅ N/𝔯N of Q/𝔯Q-modules. Nakayama’s Lemma implies that the Q-module N is cyclic. Choose an epimorphism κ: Q → N. For each 𝔭 ∈ Ass P we have N_{𝔭} ≅ Q_{𝔭} by hypothesis, so κ_{𝔭} is bijective, hence (Ker(κ))_{𝔭} = 0. Since Ass_{P} (Ker(κ)) is contained in Ass_{P} Q, and by hypothesis the latter set lies in Ass P, it follows that Ker(κ) = 0, so κ is an isomorphism. □

Proof of Theorem 2.1. By Corollary 1.3 the ring R is Cohen Macaulay, so in view of 1.4 we may assume that D^{R} is an R-module, D. Since id_{R} D = dim R, cf. 1.5.2, Corollary B.4 yields an isomorphism D ⊗_{R} D ≅ Hom_{R}(D^{∗}, D). This gives the first equality in the chain

Ass_{R}(D ⊗_{R} D) = Ass_{R} Hom_{R}(D^{∗}, D) = Supp_{R}(D^{∗}) ∩ Ass_{R} D ⊆ Ass R

where the second equality comes from a classical expression for the associator of a module of homomorphisms, and the last inclusion from property 1.5.4.

As R_{𝔭} is Gorenstein for each 𝔭 ∈ Ass R, we have D_{𝔭} ≅ R_{𝔭} by 1.5.5 and 1.5.7. Lemma 2.2 now yields D ≅ R, so R is Gorenstein by 1.5.7. □

3. Homomorphic images of generically Gorenstein rings

In this section, Q denotes a generically Gorenstein homomorphic image of a Gorenstein local ring.

Admitting slightly more vanishing, we extend the scope of Theorem 2.1.

3.1. Theorem. Assume R ≅ Q/(g), where g is a Q-regular set.

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1], then R is Gorenstein.

We start with a lemma that grew out of a discussion with Bernd Ulrich.

3.2. Lemma. If Q is Cohen-Macaulay, then R ≅ P/(g), where P is a generically Gorenstein, Cohen-Macaulay homomorphic image of a Gorenstein local ring and g is P -regular.

Proof. Being a homomorphic image of a Gorenstein ring, Q contains an ideal 𝔡 such that for every 𝔮 ∈ Spec Q, the ring Q_{𝔮} is Gorenstein if and only if 𝔮 ⊇ 𝔡. Let 𝔫 denote the maximal ideal of Q and set X = {𝔮 ∈ Spec Q | height 𝔮 = 1 and 𝔮 ⊇ 𝔡}. If card(g) = s > 1 then X is finite, so pick g_{1} ∈ (g) ∖ (𝔫(g) ∪ ⋃_{𝔮∈X} 𝔮) using prime avoidance. The ideal 𝔡′ = 𝔡 + (g_{1})/(g_{1}) defines the non-Gorenstein locus of Q′ = Q/(g_{1}) and has height 𝔡′ > 0, so Q′ is generically Gorenstein. Extend g_{1} to a minimal set g_{1}, …, g_{s} generating (g). This set is Q-regular. Thus, Q′ is a Cohen-Macaulay homomorphic image of a Gorenstein local ring, the images of g_{2}, …, g_{s} in Q′ form a Q′-regular set g′, and Q′/(g′) ≅ R. To obtain the desired result, iterate the procedure s − 1 times. □

Next we record an easy result on change of rings.

3.3. Lemma. Let P be a local noetherian ring, let M, N be finite P -modules, let g ∈ P be a P ⊕ M ⊕ N-regular element, and let n be a positive integer.

(1) If Ext_{P/(g)}^{n}(M/gM, N/gN) = 0, then Ext_{P}^{n} (M, N) = 0.

(2) If Ext_{P}^{i} (M, N) = 0 for i = n, n + 1, then Ext_{P/(g)}^{n}(M/gM, N/gN) = 0.

Proof. The canonical isomorphisms Ext_{P}^{i+1}(M/gM, N) ≅ Ext_{P/(g)}^{i}(M/gM, N/gN), holding for all i ≥ 0, show that the exact sequence of P -modules

0 → M ⟶^{g} M → M/gM → 0

yields for every i ≥ 0 an exact sequence

Ext_{P}^{i} (M, N) ⟶^{g} Ext_{P}^{i} (M, N) ⟶ Ext_{P/(g)}^{i}(M/gM, N/gN) ⟶ Ext_{P}^{i+1}(M, N)

This sequence establishes (2). It also shows that the hypothesis of (1) implies g Ext_{P}^{i} (M, N) = 0, so Nakayama’s Lemma yields the desired assertion. □

Proof of Theorem 3.1. By Corollary 1.3 the ring R is Cohen Macaulay, so Lemma 3.2 yields R ≅ P/(g), with g a P -regular element. From Lemma 3.3(1) and 1.5.6 we then get Ext_{P}^{i} (D^{P} , P) = 0 for all i ∈ [1, dim P]. Thus, the ring P is Gorenstein by Theorem 2.1, hence so is R. □

4. Well linked Cohen-Macaulay rings

In this section we describe a class of rings to which Theorem 3.1 applies. They are constructed by means of linkage, so we start by recalling some terminology.

Let Q be a Gorenstein local ring and 𝔞 an ideal in Q such that the ring Q/𝔞 is Cohen-Macaulay. An ideal 𝔟 of Q is said to be linked to 𝔞 if 𝔟 = (g) : 𝔞 and 𝔞 = (g) : 𝔟 for some Q-regular sequence g in 𝔞 ∩ 𝔟; when this is the case, we write 𝔟 ∼ 𝔞; the ring Q/𝔟 is Cohen-Macaulay, cf. Peskine and Szpiro [21, 1.3].

We say that an ideal 𝔟 is generically complete intersection, if for every 𝔮 ∈ Ass_{Q}(Q/𝔟) the ideal 𝔟_{𝔮} in Q_{𝔮} is generated by a Q_{𝔮}-regular sequence.

4.1. Theorem. Let R be a Cohen-Macaulay local ring of the form Q/𝔞, where Q is a Gorenstein local ring and 𝔞 is an ideal for which there is a sequence of links 𝔞 ∼ 𝔟_{1} ∼ ⋯ ∼ 𝔟_{s} ∼ 𝔟 with 𝔟 a generically complete intersection ideal.

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1], then R is Gorenstein.

The proof uses Theorem 3.1, as well as work of Huneke and Ulrich [16], [17].

Proof. The construction of generic links of 𝔟 in [17, 2.17(a)] provides a prime ideal 𝔯 in some polynomial ring Q[x], an ideal 𝔠 in the local ring Q[x]_{𝔯}, a regular set g′ in Q′ = Q[x]_{𝔯}/𝔠, and an isomorphism R ≅ Q′/(g′). The ring Q′ is Cohen-Macaulay because R is. By [16, 2.9(b)] the ideal 𝔠 is generically complete intersection along with 𝔟. This implies that Q′ is generically Gorenstein. It is also a residue ring of the Gorenstein ring Q[x]_{𝔯}, so Theorem 3.1 implies that the ring R is Gorenstein. □

The theorem covers the case when R is in the linkage class of a complete intersection, that is, in some Cohen presentation R̂ ≅ Q/𝔞 with Q a regular local ring, 𝔞 is linked to an ideal generated by a Q-regular sequence. Cohen-Macaulay rings of codimension at most 2 are of this type, cf. [21, 3.3], so we obtain:

4.2. Corollary. Let R be a Cohen-Macaulay local ring with codepth R ≤ 2. If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1], then R is Gorenstein. □

5. Cohen-Macaulay rings with short reductions

In this section we consider higher dimensional versions of artinian rings whose maximal ideal has a low degree of nilpotence.

5.1. Theorem. Let (R, 𝔪, k) be a local ring containing an R-regular sequence f with the property that 𝔪^{3} ⊆ (f).

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ∈ [1, dim R + 1], then R is Gorenstein.

Remark. In the special case when R is a k-algebra and rank_{k}(R) is finite, this is due to Asashiba [1]; a simplified proof is given by Asashiba and Hoshino [2]. A close reading of that proof shows that the hypothesis that R is a k-algebra can be avoided. We present that argument, referring to [1] and [2] whenever possible, and sketching modifications when necessary. A different proof of the theorem was obtained by Huneke, Şega, and Vraciu [15].

Proof. By Corollary 1.3 the ring R is Cohen Macaulay, so in view of 1.4 we may assume that D^{R} is an R-module, D. We may further assume 𝔪^{3} = 0 and Ext_{R}^{1}(D, R) = 0. Indeed, we get Ext_{R/(f)}^{1}(D/fD, R/(f)) = 0 by repeated applications of Lemma 3.3. Now note that D/fD is a dualizing module for R/(f) by 1.5.6, and that R and R/(f) are simultaneously Gorenstein.

For the rest of the proof we fix a free cover

(5.1.1)  0 ⟶ C ⟶ F ⟶ D ⟶ 0

Step 1 ([1, 2.2]). The module C is indecomposable.

Step 2 ([1, 2.3]). There is an equality 𝔪^{2} = (0 : 𝔪)_{R}.

The proofs of the two steps above do not use the hypothesis that R is an algebra. If M is an R-module, then we let ℓ(M) denote its length. The polynomial

[M] = ∑_{i⩾0} rank_{k}(𝔪^{i}M/𝔪^{i+1}M)t^{i} is called the Hilbert series of M.

Step 3 ([1, 2.4 and its proof]). There is an inequality ℓ(𝔪^{2}) ≤ 2.

If equality holds, then ℓ(C/𝔪C) = ℓ(𝔪/𝔪^{2}).

The original proof of Step 3 proceeds through computations with Hilbert series, using Steps 2 and 3 for rank counts in short exact sequences of k-vector spaces; one only needs to replace the latter by counts of lengths of R-modules.

The assertion of the following step is proved without restrictions on R.

Step 4 ([2, 2.1]). If ℓ(D/𝔪D) = 2, then there is an exact sequence

(5.1.2)  0 ⟶ Tor_{1}^{R}(D, D) ⟶ C ⊗_{R} D ⟶ Hom_{R}(D, D)

If ℓ(M) is finite, then for every x ∈ R a length count in the exact sequence

0 ⟶ (0 : x)_{M} ⟶ M ⟶^{x} M ⟶ M/xM ⟶ 0

yields ℓ((0 : x)_{M}) = ℓ(M/xM). Such equalities are used in the proof below.

Step 5 ([2, Proofs of 2.2–2.4 and 3.4]). The ring R is Gorenstein.

If 𝔪^{2} = 0, then 𝔪C = 0, so the isomorphism Ext_{R}^{1}(D, R) ≅ Ext_{R}^{2}(C, R) induced by (5.1.1) yields Ext_{R}^{2}(k, R) = 0, and hence R is Gorenstein.

If ℓ(𝔪^{2}) = 1, then (0 : 𝔪) ≅ k by Step 2, so R is Gorenstein.

By Step 3, for the rest of the proof we may assume ℓ(𝔪^{2}) = 2 and ℓ(C/𝔪C) = ℓ(𝔪/𝔪^{2}). From 1.5.3 and Step 2 we get ℓ(D/𝔪D) = 2. Since D is a faithfully injective R-module and Hom_{R}(D, D) ≅ R, we have

Hom_{R}(Tor_{1}^{R}(D, D), D) ≅ Ext_{R}^{1}(D, Hom_{R}(D, D)) ≅ Ext_{R}^{1}(D, R)

and hence Tor_{1}^{R}(D, D) = 0. Thus, from (5.1.2) we get an injection C ⊗_{R} D ↪ R. Since 𝔪^{2} ≠ 0, we may choose an element x ∈ 𝔪 ∖ (0 : 𝔪)_{R}. We now have

1 + ℓ(𝔪/𝔪^{2}) = ℓ(R) − 2
≥ ℓ(R) − ℓ(Rx) = ℓ((0 : x)_{R})
≥ ℓ((0 : x)_{C⊗_{R}D}) = ℓ((C ⊗_{R} D)/x(C ⊗_{R} D))
≥ ℓ((C ⊗_{R} D)/𝔪(C ⊗_{R} D)) = ℓ(C/𝔪C)ℓ(D/𝔪D)
= 2ℓ(𝔪/𝔪^{2})

As a consequence, we get ℓ(𝔪/𝔪^{2}) ≤ 1, hence ℓ(𝔪^{2}) ≤ 1, a contradiction. □

6. Artinian rings with Gorenstein deformations

Given a homomorphism of commutative rings P → Q, let D_{P} (Q) denote the Q-module Hom_{P} (Q, P) with the canonical action. The P -algebra Q is said to be Frobenius if Q is a finite free P -module and there is a Q-linear isomorphism D_{P} (Q) ≅ Q. For every prime ideal 𝔮 of P we set k(𝔮) = P_{𝔮}/𝔮P_{𝔮}.

6.1. Theorem. Let (P, 𝔭, k) be a local ring and let P → Q be a homomorphism of commutative rings, such that Q is a finite free P -module, and there exists a prime ideal 𝔮 of P for which the ring k(𝔮) ⊗_{P} Q is Gorenstein.

If R = Q/𝔭Q satisfies Ext_{R}^{1}(D_{k}(R), R) = 0, then the ring R is Gorenstein and the P -algebra Q is Frobenius.

6.2. Remark. The ring R in the theorem is a finite k-algebra. For such rings, X = Spec R is finite and R ≅ ∏_{𝔪∈X} R_{𝔪} as k-algebras. This yields the first isomorphism of R-modules in the next formula, the second one comes from 1.5.1:

D_{k}(R) ≅ ∐_{𝔪∈X} D_{k}(R_{𝔪}) ≅ ∐_{𝔪∈X} D^{R_{𝔪}}

For use in the proof we recall a couple of known facts, cf. e.g. [23, §14].

6.3. Let P → P′ be a homomorphism of commutative rings, set Q′ = P′ ⊗_{P} Q, and let P′ → Q′ be the induced homomorphism. In the composition

Q′ ⊗_{Q} D_{P} (Q) ⟶ P′ ⊗_{P} D_{P} (Q) ⟶ D_{P′}(Q′)

of canonical homomorphisms of Q′-modules the first arrow is always bijective. The second one is an isomorphism when Q is finite free over P.

6.4. Lemma. If (P, 𝔭, k) is a local ring and P → Q a homomorphism of commutative rings, then the following are equivalent.

(i) The P -algebra Q is Frobenius.

(ii) The P′-algebra Q′ = P′ ⊗_{P} Q is Frobenius for every local ring P′ and every

homomomorphism of rings P → P′. (iii) The P -module Q is finite free and the ring k(𝔮) ⊗_{P} Q is Gorenstein for

every 𝔮 ∈ Spec P. (iv) The P -module Q is finite free and the ring R = Q/𝔭Q is Gorenstein.

Proof. Consider first the special case P = k. The isomorphisms of Remark 6.2 then show that the ring R is Gorenstein if and only if R_{𝔪} is Gorenstein for every 𝔪, and the k-algebra R is Frobenius if and only if D_{k}(R_{𝔪}) ≅ R_{𝔪} for every 𝔪. The conditions at each 𝔪 are equivalent by 1.5.7, so (i) ⟺ (iv) whenever P is a field.

Under the hypothesis of (i) the second map in 6.3 is bijective, so (ii) holds. When (ii) holds the algebra k(𝔮)⊗_{P} Q over the field k(𝔮) is Frobenius. By the special case this implies that the ring k(𝔮) ⊗_{P} Q is Gorenstein, which is the assertion of (iii). It is clear that (iii) implies (iv). If (iv) holds, then from Remark 6.2 and 1.5.7 we obtain an isomorphism of R-modules k ⊗_{Q} D_{P} (Q) ≅ R. Nakayama’s Lemma then yields an epimorphism of Q-modules Q → D_{P} (Q). Over P both modules are finite free and have the same rank, so this map is bijective. □

Proof of Theorem 6.1. By Lemma 6.4, the P -algebra Q is Frobenius if and only if the ring R is Gorenstein. The induced map P/𝔮 → Q/𝔮Q turns Q/𝔮Q into a finite free P/𝔮-module; due to the isomorphism of rings (Q/𝔮Q)/𝔭(Q/𝔮Q) ≅ R, the same lemma implies that R is Gorenstein if and only the P/𝔮-algebra Q/𝔮Q is Frobenius.

Changing notation if necessary, for the rest of the proof we may assume that P is a domain with field of fractions K, and the ring K ⊗_{P} Q is Gorenstein. Setting D = D_{P} (Q), we now have isomorphisms of K ⊗_{P} Q-modules

K ⊗_{P} D ≅ D_{K}(K ⊗_{P} Q) ≅ K ⊗_{P} Q

given by 6.3 and Lemma 6.4. In view of Lemma 2.2, the desired isomorphism D ≅ Q will follow once we prove that the P -module D ⊗_{Q} D is torsion-free.

Let F ⟶^{≃} D be a resolution by free Q-modules, and let k ⟶^{≃} J be a resolution by injective P -modules. There are then chains of relations

\begin{aligned}R ⊗_{Q} F & ≅ (k ⊗_{P} Q) ⊗_{Q} F ≅ k ⊗_{P} F ≃ k ⊗_{P} D \\ & ≅ D_{k}(R) \end{aligned} (6.4.1)

Hom_{P} (D, J) ≃ Hom_{P} (D, k) ≅ Hom_{k}(k ⊗_{P} D, k) ≅ Hom_{k}(D_{k}(R), k)

≅ R (6.4.2)

where the quasi-isomorphisms result from the freeness of D over P, cf. (A.2.3) and (A.2.1), the isomorphism k ⊗_{P} D ≅ D_{k}(R) comes from 6.3, and all other maps are

EXTENSIONS OF A DUALIZING COMPLEX 9

canonical. Using (6.4.2) and (A.2.1) we get quasi-isomorphisms

Hom_{R}(R ⊗_{Q} F, R) ≅ Hom_{Q}(F, R) ≃ Hom_{Q}(F, Hom_{P} (D, J))
≅ Hom_{P} (F ⊗_{Q} D, J)

As (6.4.1) shows that R⊗_{Q}F is a free resolution of the R-module D_{k}(R), Proposition B.1 applied to G = F ⊗_{Q} D and J yields a strongly convergent spectral sequence

2E_{p,q} = Ext_{P}^{−p}(Tor_{−q}^{Q}(D, D), k) ⟹ Ext_{R}^{−p−q}(D_{k}(R), R)

with differentials ^{r}d_{p,q}: ^{r}E_{p,q} → ^{r}E_{p+r,q+r−1}. Since ^{2}E_{p,q} = 0 if p > 0 or q > 0, the spectral sequence defines an exact sequence

0 ⟶ Ext_{P}^{1} (D ⊗_{Q} D, k) ⟶ Ext_{R}^{1}(D_{k}(R), R)

which implies that Ext_{P}^{1} (D ⊗_{Q} D, k) vanishes. Because the P -module D ⊗_{Q} D is finite, we conclude that it is actually free, and so a fortiori torsion-free. □

7. Rings of small codepth

In this section we prove the theorem below, after substantial preparation. 7.1. Theorem. Let R be a local ring with codepth R ≤ 3.

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ≫ 0, then R is Gorenstein.

Let M, N be complexes with finite homology. 7.2.1. The R-modules Tor_{i}^{R}(M, N) and Ext_{R}^{i}(M, N) are finite for each n ∈ ℤ, and vanish for all n ≪ 0.

The preceding result is established by using standard arguments with spectral sequences. It ensures that the right hand sides of the equalities

I_{R}^{M}(t) = ∑_{n∈ℤ} rank_{k} ( Ext_{R}^{i}(k, M))t^{i}
P_{M}^{R}(t) = ∑_{n∈ℤ} rank_{k} ( Tor_{i}^{R}(M, k))t^{i}

are formal Laurent series. They are called, respectively, the Bass series and the Poincaré series of M, and are invariant under quasi-isomorphisms by A.3.3, A.3.4. 7.2.2. If F → M is a semiprojective resolution and m = sup H(M), then the module M′ = Coker(∂_{m+1}^{F}) satisfies P_{M}^{R}(t) − t^{m}P_{M′}^{R}(t) ∈ ℤ[t, t^{−1}].

Indeed, Tor_{i}^{R}(M, k) ≅ Tor_{i−m}^{R}(M′, k) holds for each i > m by Lemma B.5.

The next equality is due to Foxby, cf. [8, 4.1(a)] and [10, 15.18]. 7.2.3. I_{R}^{RHom_{R}(M,N)}(t) = P_{M}^{R}(t)I_{R}^{N}(t)

Once again, we focus upon the special case of dualizing complexes. 7.3. Set d = depth R and D = D^{R}. 7.3.1. I_{R}^{D}(t) = t^{d} holds by 1.1.3.

We recall another fundamental property of dualizing complexes. 7.3.2. Set M^{†} = RHom_{R}(M, D). The canonical map M → M^{††} is then a quasiisomorphism, cf. [10, 15.14]. For M = R this yields R ≃ RHom_{R}(D, D).

10 L. L. AVRAMOV, R.-O. BUCHWEITZ, AND L. M. ŞEGA

From the formulas and quasi-isomorphisms above one now gets: I_{R}^{R}(t) = I_{R}^{RHom_{R}(D,D)}(t) = P_{D}^{R}(t)I_{R}^{D}(t) = P_{D}^{R}(t)t^{d} (7.3.3) I_{R}^{M^{†}}(t) = P_{M}^{R}(t)I_{R}^{D}(t) = P_{M}^{R}(t)t^{d} (7.3.4) I_{R}^{M}(t) = I_{R}^{M^{††}}(t) = P_{M^{†}}^{R}(t)I_{R}^{D}(t) = P_{M^{†}}^{R}(t)t^{d} (7.3.5)

For the proof the theorem we need specific information on Poincaré series and Bass series of complexes. It suffices to have it for Poincaré series of modules: 7.4. Proposition. If R is a local ring that has a dualizing complex and c(t) is a polynomial in ℤ[t], then the following statements are equivalent:

(1) c(t)P_{M}^{R}(t) ∈ ℤ[t] for all finite R-modules M.

(2) c(t)P_{M}^{R}(t) ∈ ℤ[t, t^{−1}] for all complexes M with finite homology.

(3) c(t)I_{R}^{M}(t) ∈ ℤ[t, t^{−1}] for all complexes M with finite homology. Proof. Formulas (7.3.4) and (7.3.5) show that (2) and (3) are equivalent. Obviously,

(2) implies (1), and the converse follows from 7.2.2. □ We need several results on Poincaré series of modules over rings of small codepth.

7.5. Let (R, 𝔪, k) be a ring, such that codim R ≤ 3.

The next result is proved by Avramov, Kustin and Miller [6, 6.1 and 5.18]. 7.5.1. There exists a polynomial d(t) ∈ ℤ[t], such that (1 + t)d(t)P_{M}^{R}(t) ∈ ℤ[t] for every finite R-module M and (1 + t)d(t)P_{k}^{R}(t) = (1 + t)^{edim R}.

The table below gives a classification of the rings of codepth at most 3 obtained in [6, 2.1]; the values of d(t) are computed by Avramov [3, 3.5]. 7.5.2. If R is not complete intersection, then there exist integers l, m, p, q, r that determine the polynomial d(t) from 7.5.1 through the following table:

type codepth R  d(t)  restrictions
GO  2  1 − t − lt^{2}  l ≥ 1
TE  3  1 − t − lt^{2} − (m − l − 3)t^{3} − t^{5} m > l + 1 ≥ 3
B  3  1 − t − lt^{2} − (m − l − 1)t^{3} + t^{4} m > l + 1 ≥ 3
G(r)  3  1 − t − lt^{2} − (m − l)t^{3} + t^{4}  \begin{matrix}m > l + 1 ≥ 3 \\ l + 1 ≥ r ≥ 2 \end{matrix}
H(p, q)  3  1 − t − lt^{2} − (m − l − p)t^{3} + qt^{4}  \begin{matrix}m > l + 1 ≥ 3 \\ l ≥ p ≥ 0 \\ m − l ≥ q ≥ 0 \end{matrix}

As a consequence of [3, 3.6], we have: 7.5.3. If d(t) has 1 as a root, then R is of type H(l, l − 1) for some l ≥ 2 and d(t) = (1 + t)(1 − t)(1 − t − (l − 1)t^{2}).

Sun [25, Proof of 1.2, p. 61] provides additional information on the roots of d(t).

7.5.4. If d(t) has a real root r with 0 < r < 1, then this root is simple.

In the proof of the theorem we use a technique developed by Şega [24]. In order to apply it, we need an extension of Sun’s result.

7.6. Lemma. Let R be a local ring with codim R ≤ 3, which is not complete intersection, and let d(t) be the polynomial described in 7.5.1.

If p(t) ∈ ℤ[t] is an irreducible polynomial with constant term 1 and at least one negative coefficient, then p(t)^{2} does not divide d(t).

Proof. We assume that our assertion fails, note that it implies deg p(t) ≤ 2, and obtain a contradiction for each type of rings described in 7.5.2.

GO. We must have 1 − t − lt^{2} = p(t)^{2}, hence l = −1/4. This is absurd.

TE. Only 1 + t can be a linear factor of d(t), so p(t) = 1 + at ± t^{2} and d(t) = (1 + t)p(t)^{2} = 1 + ⋯ + t^{5}. However, by 7.5.2 the coefficient of t^{5} is −1.

B or G(r). Only 1 + t can be a linear factor of d(t). Thus, p(t) = 1 + at ± t^{2} and d(t) = p(t)^{2} = 1 + 2at + ⋯, yielding a = −1/2, a contradiction.

H(p, q). If p(t) = 1 + at + bt^{2}, then d(t) = p(t)^{2}, so we get a contradiction as above. If p(t) = 1 − at with a > 1, then d(t) has a double real root r = 1/a < 1, contradicting 7.5.4. Finally, if p(t) = 1−t, then d(t) = (1+t)(1−t)(1−t−(l−1)t^{2}) by 7.5.3, so 1 − t divides 1 − t − (l − 1)t^{2}, hence l = 1. This is impossible. □

Proof of Theorem 7.1. Using 7.5.1 and 7.4, choose d(t), r(t) ∈ ℤ[t] such that

(7.6.1)  I_{R}^{R}(t) = \frac{r(t)}{(1 + t)d(t)}

Set D = D^{R} and d = depth R. Our hypothesis means that H RHom_{R}(D, R) is bounded. It is degreewise finite by 7.2.1, so using 7.2.3 and (7.3.3) we get

I_{R}^{RHom_{R}(D,R)}(t) = P_{D}^{R}(t)I_{R}^{R}(t) = \frac{I_{R}^{R}(t)^{2}}{t^{d}} = \frac{r(t)^{2}}{t^{d}(1 + t)^{2}d(t)^{2}}

Referring again to 7.5.1 and 7.4, we now obtain

(7.6.2)  \frac{r(t)^{2}}{d(t)} = t^{d}(1 + t)((1 + t)d(t)I_{R}^{RHom_{R}(D,R)}(t)) ∈ ℤ[t, t^{−1}]

Let s(t) denote the product of all the irreducible in ℤ[t] factors of d(t) with constant term 1 and at least one negative coefficient, and set q(t) = d(t)/s(t).

If R is complete intersection, then it is Gorenstein, so there is nothing to prove. Else, (7.6.2) and 7.6 imply that s(t) divides r(t) in ℤ[t, t^{−1}], so (7.6.1) yields

(1 + t)q(t)I_{R}^{R}(t) = \frac{r(t)}{s(t)} ∈ ℤ[t, t^{−1}]

Since q(t) and I_{R}^{R}(t) have non-negative coefficients, the last formula implies I_{R}^{R}(t) ∈ ℤ[t, t^{−1}], hence id_{R} R < ∞. Thus, R is Gorenstein, as desired. □

8. Golod rings

Let (R, 𝔪, k) be a local ring. Serre proved a coefficientwise inequality

P_{k}^{R}(t) ≼ \frac{(1 + t)^{edim R}}{1 − ∑_{j=1}^{∞} rank H_{j}(K^{R})t^{j+1}}

of formal power series, where K^{R} denotes the Koszul complex on a minimal set of generators of 𝔪. If equality holds, then R is said to be a Golod ring.

The class of Golod rings is essentially disjoint from that of Gorenstein rings: if R belongs to their intersection, then R is a hypersurface ring, in the sense that its 𝔪-adic completion R̂ is the homomorphic image of a regular local ring by a principal ideal.

8.1. Theorem. Let R be a Golod local ring.

If Ext_{R}^{i}(D^{R}, R) = 0 for all i ≫ 0, then R is a hypersurface ring.

For the proof we recall a theorem of Jorgensen [18, 3.1]:

8.2. Let R be a Golod ring and L, M modules with finite homology.

If Tor_{i}^{R}(L, M) = 0 for all i ≫ 0, then L or M has finite projective dimension.

We need an extension of this result to complexes.

8.3. Proposition. Let R be a Golod ring and L, M complexes with finite homology. If Tor_{i}^{R}(L, M) = 0 for all i ≫ 0, then P_{L}^{R}(t) or P_{M}^{R}(t) is a Laurent polynomial.

Proof. Set l = sup H(L) and m = sup H(M), choose semiprojective resolutions E → L and F → M, then set L′ = Coker(∂_{l}^{E}) and M′ = Coker(∂_{m}^{F}). In view of Lemma B.5, our hypothesis yields Tor_{i}^{R}(L′, M′) = 0 for all i ≫ 0. By 8.2, this means that P_{L′}^{R}(t) or P_{M′}^{R}(t) is a polynomial, so 7.2.2 gives the desired assertion. □

Proof of Theorem 8.1. Set D = D^{R} and choose a semiprojective resolution F → D with inf F = 0. For all i ≫ 0 we have H_{i} Hom_{R}(F, R) = 0 by hypothesis, hence H_{i}(F ⊗_{R} D) = 0 by Proposition B.3(2), and thus P_{R}^{D}(t) ∈ ℤ[t, t^{−1}] by Proposition 8.3. Now (7.3.3) implies I_{R}^{R}(t) ∈ ℤ[t], hence id_{R} R < ∞, that is, R is Gorenstein. As noted before the statement of the theorem, R is then a hypersurface ring. □

9. Relations with Tachikawa’s conjecture

In this section A denotes a not necessarily commutative algebra of finite rank over a field k, and D_{k}(A) = Hom_{k}(A, k) has the canonical action of A on the left.

In [20] Nakayama proposed a now famous conjecture. The original statement was in terms of the injective resolution of A as a bimodule over itself, but Müller [19] proved that it is equivalent to the one-sided version below:

Conjecture (NC). If the left A-module A has an injective resolution in which each term is also projective, then A is selfinjective.

Considerable efforts notwithstanding, this still is one of the main open problems in the representation theory of Artin algebras. It has generated a significant body of research and has led to the study of several variants and to the formulation of equivalent forms, see Yamagata [28] for a general presentation of the subject. In particular, Tachikawa [26, Ch. 8] proved that the validity of Nakayama’s conjecture is equivalent to that of both statements below for all k-algebras of finite rank.

Conjecture (TC1). If Ext_{A}^{i}(D_{k}(A), A) = 0 for all i > 0, then A is selfinjective.

Conjecture (TC2). If the ring A is selfinjective and M is a finite A-module satisfying Ext_{A}^{i}(M, M) = 0 for all i > 0, then M is projective.

Now we turn to the case when A is commutative. In view of Remark 6.2, for the conjectures above one may assume that A is local. The validity of Nakayama’s Conjecture is then well-known and easy to see. However, both of Tachikawa’s Conjectures are open for commutative algebras. This apparent paradox is explained by the fact that the proof of the equivalence of Conjecture (NC) with Conjectures (TC1) and (TC2) involves intermediate non-commutative algebras.

Here are some instances in which Conjecture (TC1) is known to hold.

9.1. Let (A, 𝔯, k) be a commutative local algebra of finite rank over k.

Each set of hypotheses below implies that the algebra A is selfinjective:

9.1.1. edim A ≤ 2 and Ext_{A}^{i}(D_{k}(A), A) = 0 for i = 1, 2 (Hoshino [13]).

9.1.2. rank_{k}(A/xA) ≤ 2 for some x ∈ 𝔯 and Ext_{A}^{1}(D_{k}(A), A) = 0 (Zeng [29]).

9.1.3. rank_{k}(A/xA) ≤ 3 for some x ∈ 𝔯 and Ext_{A}^{i}(D_{k}(A), A) = 0 for i = 1, 2 (Asashiba and Hoshino [2]).

9.1.4. 𝔯^{3} = 0 and Ext_{A}^{1}(D_{k}(A), A) = 0 (Asashiba [1]).

The last result was extended in Theorem 5.1. Other results of this paper establish Conjecture (TC1) in new cases, some of them collected in the next theorem. We remark that Part (1) below improves 9.1.1 and implies 9.1.2.

9.2. Theorem. A commutative local algebra A of finite rank over a field k is selfinjective whenever one of the following assumptions is satisfied.

(1) edim A ≤ 2 and Ext_{A}^{1}(D_{k}(A), A) = 0.

(2) edim A ≤ 3 and Ext_{A}^{i}(D_{k}(A), A) = 0 for all i ≫ 0.

(3) A is Golod and Ext_{A}^{i}(D_{k}(A), A) = 0 for all i ≫ 0.

(4) A ≅ Q/(f), where Q is a reduced complete local ring, f is a Q-regular sequence, and Ext_{A}^{1}(D_{k}(A), A) = 0.

Proof. The first three parts are specializations of Corollary 4.2, Theorem 7.1, and Theorem 8.1, respectively. For (4), use Cohen’s Structure Theorem to present Q as a residue ring of a regular local ring modulo a radical ideal. Such an ideal is generically complete intersection, so Theorem 3.1 applies. □

Appendix A. Constructions with complexes

In this appendix we describe terminology and notation that are used throughout the paper. Proofs of the basic results listed below can be found, for instance, in [12], [22], or [10]. We let R denote an associative ring over which modules have left actions, and M denote a complex of R-modules of the form

⋯ ⟶ M_{i+1} ⟶^{∂_{i+1}^{M}} M_{i} ⟶^{∂_{i}^{M}} M_{i−1} ⟶ ⋯

We write b ∈ M to indicate that b belongs to M_{i} for some i ∈ ℤ, and |b| = n to state that b is in M_{n}. The n’th shift Σ^{n}M of M is the complex with

(Σ^{n}M)_{i} = M_{i−n}  and  ∂_{i}^{Σ^{n}M} = (−1)^{n}∂_{i−n}^{M} .

If N is a complex, then a morphism α: M → N is a sequence of R-linear maps α_{i}: M_{i} → N_{i}, such that ∂_{i}^{N}α_{i} = α_{i−1}∂_{i}^{M} holds for all i. It is a quasi-isomorphism if H_{i}(α) is bijective for each i; we use the symbol ≃ to identify quasi-isomorphisms.

Set inf M = inf{i | M_{i} ≠ 0} and sup M = sup{i | M_{i} ≠ 0}. The complex M is bounded above (respectively, below) if sup M (respectively, inf M) is finite. It is bounded when both numbers are finite. It is degreewise finite if each R-module M_{i} is finite. If H(M) is bounded and degreewise finite, then M has finite homology.

A.1.1. We let M_{<n} denote the subcomplex of M having (M_{<n})_{i} = 0 for i ≥ n and (M_{<n})_{i} = M_{i} for i < n, and we set M_{⩾n} = M/M_{<n}.

A.1.2. We let τ_{⩽n}(M) denote the residue complex of M with τ_{⩽n}(M)_{i} = 0 for i > n, τ_{⩽n}(M)_{n} = M_{n}/ Im(∂_{n+1}^{M}), and τ_{⩽n}(M)_{i} = M_{i} for i < n; the maps H_{i}(M) → H_{i}(τ_{⩽n}(M)) induced by the natural surjection are bijective for all i ≤ n.

A.2. Let L be a complex of right R-modules and M, N be complexes of R-modules. A complex Hom_{R}(M, N) is defined by the formulas

Hom_{R}(M, N)_{n} = ∏_{j−i=n} Hom_{R}(M_{i}, N_{j}) ∂(β) = ∂^{N} ∘ β − (−1)^{|β|}β ∘ ∂^{M}

and a complex L ⊗_{R} M defined by the formulas

(L ⊗_{R} M)_{n} = ∐_{h+i=n} L_{h} ⊗_{R} M_{i} ∂(a ⊗ b) = ∂^{L}(a) ⊗ b + (−1)^{|a|}a ⊗ ∂^{M}(b)

If λ: L → L′ is a quasi-isomorphism of complexes of right R-modules, μ: M → M′ is a quasi-isomorphism of complexes of R-modules, J is a bounded above complex of injective R-modules, and F is a bounded below complex of projective Rmodules, then the maps below are quasi-isomorphisms.

(A.2.1)  Hom_{R}(F, μ): Hom_{R}(F, M) ⟶ Hom_{R}(F, M′)
(A.2.2)  Hom_{R}(μ, J): Hom_{R}(M′, J) ⟶ Hom_{R}(M, J)
(A.2.3)  λ ⊗_{R} F: L ⊗_{R} F ⟶ L′ ⊗_{R} F

If, furthermore, ϕ: F → F′ is a quasi-isomorphism of bounded below complexes of projective R-modules, then the maps below are quasi-isomorphisms.

(A.2.4)  Hom_{R}(ϕ, M): Hom_{R}(F′, M) ⟶ Hom_{R}(F, M)
(A.2.5)  L ⊗_{R} ϕ: L ⊗_{R} F ⟶ L ⊗_{R} F′

We use derived functors of complexes, for which we recall some basic notions.

A.3. If H(M) is bounded below, then there exists a quasi-isomorphism F → M with F a bounded below complex of projective modules, called a semiprojective resolution of M; it can be chosen with inf F = inf H(M). Any two semiprojective resolutions of M are homotopy equivalent, so the complexes Hom_{R}(F, N) and L⊗_{R} F are defined uniquely up to homotopy. The symbols RHom_{R}(M, N) and L ⊗_{R}^{L} M denote any complex in the corresponding homotopy class. One sets

(A.3.1)  Ext_{R}^{i}(M, N) = H_{−i} RHom_{R}(M, N)
(A.3.2)  Tor_{i}^{R}(L, N) = H_{i}(L ⊗_{R}^{L} N)

If L → L′, M → M′ and N → N′ are quasi-isomorphisms, then so are

RHom_{R}(M, N) ⟶ RHom_{R}(M, N′) ⟵ RHom_{R}(M′, N′)
L ⊗_{R}^{L} M ⟶ L′ ⊗_{R}^{L} M ⟶ L′ ⊗_{R}^{L} M′

by (A.2.1) and (A.2.4), respectively by (A.2.1) and (A.2.4). Thus, for all i ∈ ℤ

(A.3.3)  Ext_{R}^{i}(M, N) ≅ Ext_{R}^{i}(M′, N′)
(A.3.4)  Tor_{i}^{R}(L, M) ≅ Tor_{i}^{R}(L′, M′)

A.3.5. We identify R-modules with complexes concentrated in degree zero. For modules the constructions above yield the classical derived functors.

Appendix B. Lemmas about complexes

The hypotheses here are the same as in Appendix A. In one form or another, the results collected below are probably known to (some) experts, but may not be easy to find in the literature. Furthermore, slight weakenings of the hypotheses may lead to false statements, so we are providing complete arguments.

We start with the computation of a classical spectral sequence.

B.1. Proposition. If G is a complex of R-modules and J is a complex of injective R-modules with sup J = 0, then there exists a spectral sequence having

2E_{p,q} = H_{p} Hom_{R}(H_{−q}(G), J)  and  ^{2}d_{p,q} = Hom_{R}(H_{−q}(G), ∂_{p}^{J})

and differentials acting by the pattern ^{r}d_{p,q}: ^{r}E_{p,q} → ^{r}E_{p−r,q+r−1} for all r ≥ 2.

If J or G is bounded below, then this sequence converges strongly to Hom_{R}(G, J), in the sense that for each n ∈ ℤ the group H_{n} Hom_{R}(G, J) has a finite, exhaustive, and separated filtration, the component of degree q of the associated graded group is isomorphic to ^{∞}E_{n−q,q}, and ^{r}E_{p,q} = ^{∞}E_{p,q} for all r ≫ 0 and all (p, q) ∈ ℤ × ℤ.

Proof. The filtration of J by its subcomplexes J_{⩽p} induces a filtration

F_{p} Hom_{R}(G, J) = Im ( Hom_{R}(G, J_{⩽p}) → Hom_{R}(G, J))

of Hom_{R}(G, J). The differentials ^{r}d_{p,q} in the resulting spectral sequence ^{r}E_{p,q} have the desired pattern for all r ≥ 0 and all (p, q) ∈ ℤ × ℤ. The isomorphisms

0E_{p,q} = (\frac{F_{p} Hom_{R}(G, J)}{F_{p−1} Hom_{R}(G, J)})_{p+q} ≅ Hom_{R}(G_{−q}, J_{p})

take ^{0}d_{p,q} to (−1)^{q} Hom_{R}(∂_{−q+1}^{G}, J_{p}). As each J_{p} is injective, the isomorphisms

1E_{p,q} = H_{−q} Hom_{R}(G, J_{p}) ≅ Hom_{R}(H_{−q}(G), J_{p})

transfer ^{1}d_{p,q} into Hom_{R}(H_{−q}(G), ∂_{p}^{J}). Thus, they lead to isomorphisms

2E_{p,q} ≅ H_{p} Hom_{R}(H_{−q}(G), J)

that take ^{2}d_{p,q} to Hom_{R}(H_{−q}(G), ∂_{p}^{J}).

If J or G is bounded below, then the filtration of Hom_{R}(G, J)_{n} by its subgroups (F_{p} Hom_{R}(G, J))_{n} is finite for each n ∈ ℤ; this implies strong convergence. □

Next comes a small variation on [5, 4.4].

B.2. Lemma. If E is a complex of right R-modules, G denotes the complex of R-modules Hom_{R}(E, R), and J is a complex of R-modules, then the map

θ^{EJ}: E ⊗_{R} J ⟶ Hom_{R}(G, J)
x ⊗ y ⟼ (γ ↦ (−1)^{|γ||y|}γ(x)y )

is a morphism of complexes. It is bijective if J is bounded above, E is bounded below, and each right module E_{h} is finite projective.

Proof. It is easily verified that θ^{EJ} is a morphism of complexes.

By definition, Hom_{R}(G, J)_{n} = ∏_{i−(−h)=n} Hom_{R}(G_{−h}, J_{i}) for each n ∈ ℤ. The boundedness hypotheses imply that the abelian groups Hom_{R}(G_{−h}, J_{i}) are trivial for almost all pairs (h, i) satisfying i − (−h) = n, so the product of these groups is equal to their coproduct. The homomorphism θ_{n}^{EJ}: (E ⊗_{R} J)_{n} → Hom_{R}(G, J)_{n} is thus the coproduct of the canonical evaluation maps E_{h} ⊗_{R} J_{i} → Hom_{R}(G_{−h}, J_{i}), each one multiplied by (−1)^{ni}. For every h ∈ ℤ the R-module G_{−h} = Hom_{R}(E_{h}, R) is finite projective, all the evaluation homomorphisms are bijective. □

Now we are ready to produce some canonical isomorphisms. Our approach was suggested by results in an early version of a paper of Huneke and Leuschke [14].

When N is a right R-module N^{∗} = Hom_{R}(N, R) carries the standard left action.

B.3. Proposition. Let E be a complex of finite projective right R-modules with inf E = 0, let J be a complex of R-modules, and form the composition of morphisms

ϑ^{EJ}: E ⊗_{R} J ⟶^{θ^{EJ}} Hom_{R}(Hom_{R}(E, R), J) ⟶^{Hom_{R}(Hom_{R}(ϵ,R),J)} Hom_{R}(N^{∗}, J)

where N = H_{0}(E), the map ϵ: E → N is canonical, and θ^{EJ} is from Lemma B.2. When J is bounded and each R-module J_{i} is injective the following hold.

(1) If H_{−i} Hom_{R}(E, R) = 0 for all i > 0, then ϑ^{EJ} is a quasi-isomorphism.

(2) If H_{−i} Hom_{R}(E, R) = 0 for all i ≫ 0, then H_{i}(E ⊗_{R} J) = 0 for all i ≫ 0.

(3) If sup J = 0 and there is number m ≥ 1 such that H_{−i} Hom_{R}(E, R) = 0 for all i ∈ [1, m], then for every i ≤ m + inf J there is an isomorphism

H_{i}(ϑ^{EJ}): H_{i}(E ⊗_{R} J) ⟶^{≅} H_{i} Hom_{R}(N^{∗}, J)

Proof. For G = Hom_{R}(E, R) Lemma B.2 yields an isomorphism of complexes

θ^{EJ}: E ⊗_{R} J ⟶^{≅} Hom_{R}(G, J)

(1) By hypothesis, Hom_{R}(ϵ, R): N^{∗} → G is a quasi-isomorphism, hence by A.2.2 so is Hom_{R}(Hom_{R}(ϵ, R), J); as θ^{EJ} is bijective, we are done.

(2) The hypothesis means that H(G) is bounded. The spectral sequence of Proposition B.1 shows that so is H Hom_{R}(G, J), which is isomorphic to H(E ⊗_{R} J).

(3) Proposition B.1 yields a strongly convergent spectral sequence

2E_{p,q} = H_{p} Hom_{R}(H_{−q}(G), J) ⟹ H_{p+q} Hom_{R}(G, J)

Note that inf J = −n for some integer n ≥ 0. Because G_{−q} = 0 for all q < 0, and H_{−q}(G) = 0 for all q ∈ [1, m], we get ^{2}E_{p,q} = 0 for all pairs (p, q) with p ∉ [−n, 0] or with q ≤ m, unless q = 0. It follows that if p + q ≤ m − n, then

∞E_{p,q} = ^{2}E_{p,q} ≅ \begin{cases}H_{p} Hom_{R}(N^{∗}, J) & when q = 0 \\ 0 & when q ≠ 0 \end{cases}

As a consequence, the edge map H_{p} Hom_{R}(G, J) → ^{2}E_{p,0} is bijective for p ≤ m−n. It follows from the construction of the spectral sequence in Proposition B.1 that the isomorphism above takes this edge map to H_{p} Hom_{R}(Hom_{R}(ϵ, R), J). □

The next corollary contains as special cases [11, 2.3] and part of [11, 2.1].

B.4. Corollary. Let L be a right R-module that has a projective resolution E by finite projective right R-modules, and let M be an R-module with id_{R} M = n < ∞.

If m ≥ 1 is a number such that Ext_{R}^{i}(L, R) = 0 for all i ∈ [1, m], then

(1) Ext_{R}^{i}(L^{∗}, M) = 0 for all i > max{0, n − m − 1}.

(2) Tor_{i}^{R}(L, M) = 0 for all i ∈ [1, m].

(3) If m ≥ n, then L ⊗_{R} M ≅ Hom_{R}(L^{∗}, M).

Proof. Let M → J be an injective resolution with J_{i} = 0 for i < −n. The last part of the proposition yields for all i ≤ m − n isomorphisms

Tor_{i}^{R}(L, M) ≅ Ext_{R}^{−i}(L^{∗}, M)

The desired assertions follow, since Tor_{j}^{R}(L, M) = 0 = Ext_{R}^{j}(L^{∗}, M) for j < 0. □

Here is a version of the familiar degree-shifting procedure.

B.5. Lemma. Let L be a complex of right modules with sup H(L) = l < ∞ and M a complex with sup H(M) = m < ∞. If E → L and F → M are semiprojective resolutions, then for the R-modules L′ = Coker(∂_{l}^{E}) and M′ = Coker(∂_{m}^{F}) there are isomorphisms Tor_{i}^{R}(L, M) ≅ Tor_{i−l−m}^{R}(L′, M′) for all i > l + m.

Proof. The quasi-isomorphism L ≃ τ_{⩽l}(L), cf. A.1.2, induces by (A.2.3) a quasiisomorphism L ⊗_{R} F_{<m} ≃ τ_{⩽l}(L) ⊗_{R} (F_{<m}). As a consequence, we get

sup H(L ⊗_{R} (F_{<m})) = sup H(τ_{⩽l}(L) ⊗_{R} (F_{<m}))
≤ sup(τ_{⩽l}(L) ⊗_{R} (F_{<m}))
= l + m − 1

The homology exact sequence of the exact sequence of complexes

0 → L ⊗_{R} (F_{<m}) ⟶ L ⊗_{R} F ⟶ L ⊗_{R} F_{⩾m} → 0

now yields isomorphisms H_{i}(L ⊗_{R} F) ≅ H_{i}(L ⊗_{R} F_{⩾m}) for all i > l + m. Since Σ^{−m}F is a projective resolution of M′, these isomorphisms can be rewritten as Tor_{i}^{R}(L, M) ≅ Tor_{i−m}^{R}(L, M′) for all i > l + m. Similar arguments applied to the resolution E → L yield Tor_{i−m}^{R}(L, M′) ≅ Tor_{i−l−m}^{R}(L′, M′) for all i > l + m. □

Acknowledgements

The authors want to thank Craig Huneke and Graham Leuschke for making an early version of [14] available to them, Bernd Ulrich for useful discussion concerning the material of Sections 4 and 3, and the referee for directing them to Proposition 1.2.

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Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U. S. A.

   E-mail address: avramov@math.unl.edu

Department of Mathematics, University of Toronto, Toronto ON M5S 3G3, Canada

   E-mail address: ragnar@math.utoronto.ca

Department of Mathematics, Purdue University, West Lafayette, In 47907, U. S. A. Current address: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley,

CA 94720

   E-mail address: lmsega@math.purdue.edu