RINGS WITH AUSLANDER DUALIZING COMPLEXES

AMNON YEKUTIELI AND JAMES J. ZHANG

Abstract. A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently.

The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras.

We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes.

Contents

0. Introduction 1 1. Dualizing complexes 4 2. Auslander Dualizing Complexes 10 3. Rigid Dualizing Complexes 16 4. Dualizing Complexes over Graded Algebras 20 5. Graded Algebras with Some Commutativity Hypothesis 25 6. Noetherian Connected Filtrations 30 References 38

0. Introduction

Dualizing complexes play an essential role in the Serre-Grothendieck Duality Theory on schemes (see [RD]). The duality formalism was generalized to noncommutative rings by the first author, in order to answer some questions which arose in this context, mainly regarding local duality for noncommutative graded algebras (see [Ye1]). A version of Serre duality for noncommutative projective schemes was established using dualizing complexes (see Jørgensen’s [Jo2] and our [YZ1]). Dualizing complexes, and more generally derived categories, are powerful tools for proving abstract properties of noncommutative rings. For examples, consider the noncommutative graded versions of the Auslander-Buchsbaum Theorem, The Bass Theorem and the No-Holes Theorem for Bass numbers (see [Jo1] Theorems 3.2, 4.5, 4.6 and 4.8). Under the synonym “cotilting complexes,” dualizing complexes were studied by Miyachi [Mi2]. Cotilting bimodules occur often in papers on representations of finite-dimensional algebras, cf. [Ha].

Date: 8 September 1998.

1991 Mathematics Subject Classification. Primary: 16D90; Secondary: 16E30, 16L30, 16P60.

Both authors were supported by the US-Israel Binational Science Foundation. The second author was also supported by the NSF and Sloan Research Fellowship.

In this paper we will provide further evidence that the dualizing complex (Definition 1.1) is an effective tool for studying noncommutative rings. We are especially interested in those dualizing complexes which satisfy an extra homological condition called the Auslander property (Definition 2.1). The basic idea here is that if a statement holds for Auslander-Gorenstein rings in the sense of [Bj], then an appropriate version of the statement should hold for rings with Auslander dualizing complexes. The Gorenstein condition, i.e. the ring itself having finite injective dimension, is considered to be very restrictive; in contrast, having an Auslander dualizing complex is considered to be a mild condition.

There are a few ways to show existence of Auslander dualizing complexes. For example, if A is a connected graded algebra (over a field k) with enough normal elements, then A has an Auslander dualizing complex. Recall that a connected graded k-algebra A has enough normal elements if every graded prime factor A/𝔭 ≠ k has a nonzero normal element of positive degree. This class of rings has been studied recently by many algebraists, because of developments in quantum groups and noncommutative algebraic geometry.

In this paper we prove a diverse collection of results, whose common thread is that their proofs are based on the existence of an Auslander dualizing complex. Throughout k denotes a fixed base field, and a k-algebra means an associative algebra with 1.

First we generalize [GL] Theorem 1.6, by dropping the Gorenstein condition. Note that the hypothesis on gr A in next theorem is easy to check in practice (see Example 6.11), and we suspect it can even be weakened.

Theorem 0.1. Assume A is a normally separated filtered k-algebra such that gr A is a noetherian connected graded k-algebra with enough normal elements. Then Spec A is catenary.

This is proved after Corollary 6.22.

We generalize some results in [ASZ], two of which are:

Theorem 0.2. Assume A is a noetherian k-algebra with an Auslander dualizing complex. Then there is a step duality between the category Mod_{f} A of finitely generated left A-modules and the category Mod_{f} A^{∘} of finitely generated right A-modules.

Actually we prove a more general result involving two algebras – see Theorem 2.15. It follows that if the algebra A has an Auslander dualizing complex then the left and right Krull dimensions of A are finite.

Theorem 0.3. Let A be an Auslander-Gorenstein noetherian k-algebra. Assume A has a filtration such that gr A is an AS-Gorenstein noetherian connected graded k-algebra (e.g. if A is connected graded). Then A has an artinian self-injective ring of fractions.

For more details see Theorem 6.23.

The notion of characteristic variety Ch M of a module M was introduced in Dmodule theory (cf. [Co] p. 98). The next theorem generalizes a result of Gabber by dropping the Gorenstein condition. The possibility of making this generalization was suggested by Van den Bergh.

Theorem 0.4 (Purity of Characteristic Variety). Assume A is a filtered k-algebra such that gr A is a commutative connected graded affine k-algebra. If M is a finitely generated GKdim-pure left A-module, then the characteristic variety Ch M is pure.

Given an algebra A there might exist non-isomorphic Auslander dualizing complexes over it (see Example 2.3(c)). For various reasons (like functoriality) it is desirable to find dualizing complexes which are canonical in some sense. In the graded case, the balanced dualizing complex (Definition 4.2), introduced by the first author in [Ye1], is a natural choice. In the ungraded case, one should consider the rigid dualizing complex (Definition 3.1) introduced by Van den Bergh [VdB]. Rigid dualizing complexes (and balanced dualizing complexes in graded case) are uniquely determined up to a unique isomorphism. A balanced dualizing complex is always rigid (see [VdB] Proposition 8.2(2) and our Corollary 6.7). The next theorem on the functoriality of rigid dualizing complexes is a combination of Theorem 3.2 and Corollary 3.4. A homomorphism A → B of algebras is called finite if B is a finitely generated left and right A-module.

Theorem 0.5. Let A be a noetherian k-algebra. A rigid dualizing complex R_{A} over A is unique up to a unique isomorphism. If A → B is a finite homomorphism and R_{A}, R_{B} are rigid dualizing complexes over A, B respectively, then there is at most one morphism Tr_{B/A} : R_{B} → R_{A} compatible with the rigidity.

The existence of a dualizing complex is not automatic. A very effective criterion for existence of balanced complexes is given in [VdB] Theorem 6.3 (which is Theorem 4.6 here). Van den Bergh’s idea was to first prove the Local Duality Theorem, and then to show this duality is represented by a balanced dualizing complex. The Rees algebra allows us to transfer results on graded algebras to non-graded algebras (see Theorem 6.2).

We prove that Auslander rigid dualizing complexes exist for a large class of rings. First, the similar submodule condition on graded A-modules (see Definition 5.12) enables induction on GKdim. Combining Theorems 5.13 and 5.14 we get:

Theorem 0.6. Assume A is a noetherian connected graded k-algebra which has a balanced dualizing complex R and satisfies the similar submodule condition (e.g. A is FBN or has enough normal elements). Then the balanced dualizing complex R is graded Auslander.

Next, Theorem 6.2 says:

Theorem 0.7. Suppose A is a filtered k-algebra such that the associated graded algebra gr A is noetherian. If gr A has a graded Auslander balanced dualizing complex, then A has an Auslander rigid dualizing complex.

By results of Grothendieck, a commutative affine connected graded k-algebra has a graded Auslander balanced dualizing complex (this also follows from Theorem 0.6). Since factor rings of universal enveloping algebras of finite dimensional Lie algebras are filtered, and their associated graded algebras are commutative, Theorem 0.7 tells us that these algebras have Auslander rigid dualizing complexes. In the same way we may use Theorems 0.7 and 0.6 to show that many quantum algebras and their factor algebras have Auslander rigid dualizing complexes. A key step in the proof of Theorem 0.7 is the following theorem (see Theorem 5.1 for full details and proof).

Theorem 0.8. Let A be a noetherian connected graded k-algebra. Suppose t ∈ A is a nonzero homogeneous normal element of positive degree. Then A has a graded Auslander balanced dualizing complex if and only if so does A/(t).

As noted on [Zh] p. 399, the proof of [SZ] Lemma 6.1(ii) has a gap, and an alternative proof of the result (under extra hypotheses) is given in [Zh] Theorem 3.1. We now give a complete proof of [SZ] Lemma 6.1(ii) using Auslander dualizing complexes (the proof is at end of Section 5).

Proposition 0.9. Let A be a noetherian locally finite ℕ-graded k-algebra. If A is graded FBN, or A has enough normal elements, then GKdim M = Kdim M ∈ ℕ for every finitely generated left or right graded A-module M.

Here are some other results we prove:

1. Gabber’s Maximality Principle (Theorem 2.19).

2. Existence of double-Ext spectral sequence (Proposition 1.7).

3. The existence of an Auslander rigid dualizing complex is transferred to related algebras (Propositions 4.18 and 4.20, Corollaries 4.17, 5.10 and 5.11).

The canonical dimension, denoted by Cdim, is defined when A has an Auslander dualizing complex (Definition 2.9). It is an exact finitely partitive dimension function (Theorem 2.10). Local duality implies that the canonical dimension is symmetric in the graded case (Proposition 4.13). Therefore if A is a connected graded algebra with an Auslander balanced dualizing complex, the canonical dimension Cdim is exact, finitely partitive, and symmetric on graded modules. Note that the Krull dimension, denoted by Kdim, is exact and finitely partitive, but it is unknown whether it is symmetric. On the other hand Gelfand-Kirillov dimension, denoted by GKdim, is symmetric, but neither exact nor finitely partitive in general. Hence the canonical dimension is the better dimension function – at least in the graded or filtered case.

The study of dualizing complexes over noncommutative rings presents many interesting and subtle questions. We conclude the introduction by mentioning two of them:

Question 0.10. Which (noetherian, affine) k-algebras have (rigid) dualizing complexes?

Question 0.11. Is a rigid dualizing complex is always Auslander?

Acknowledgments. We thank M. Artin and M. Van den Bergh for many useful conversations and email correspondences.

1. Dualizing complexes

Let k be a field and let A be an associative k-algebra with 1. All A-modules will be by default left modules, and we denote by Mod A the category of left A-modules. Let A^{∘} be the opposite algebra, and let A^{e} := A ⊗ A^{∘} where ⊗ = ⊗_{k}. Thus an A^{e}-module M is, in the conventional notation, an A-A-bimodule _{A}M_{A} central over k. Most of our definitions and results have a left-right symmetry, expressible by the exchange A ↔ A^{∘}. Since these symmetries are evident we shall usually not mention them.

Let D(Mod A) be the derived category of A-modules, and let D^{⋆}(Mod A), for ⋆ = b, +, − or blank, be the full subcategories of bounded, bounded below, bounded above, or unbounded complexes respectively [RD].

Given another k-algebra B, the forgetful functor Mod(A ⊗ B^{∘}) → Mod A is exact, and so induces a functor D^{⋆}(Mod(A ⊗ B^{∘})) → D^{⋆}(Mod A). Now A ⊗ B^{∘} is a projective A-module, so any projective (resp. flat, injective) (A ⊗ B^{∘})-module is projective (resp. flat, injective) over A.

Consider k-algebras A, B, C. For complexes M ∈ D(Mod(A ⊗ B^{∘})) and N ∈ D(Mod(A ⊗ C^{∘})), with either M ∈ D^{−} or N ∈ D^{+}, there is a derived functor

R Hom_{A}(M, N) ∈ D(Mod(B ⊗ C^{∘})).

It is calculated by replacing M with an isomorphic complex in D^{−}(Mod(A ⊗ B^{∘})) which consists of projective modules over A; or by replacing N with an isomorphic complex in D^{+}(Mod(A ⊗ C^{∘})) which consists of injective modules over A. For full details see [RD] and [Ye1]. Note that for modules M and N, viewed as complexes concentrated in degree 0, one has

H^{q}R Hom_{A}(M, N) = Ext_{A}^{q}(M, N),

the latter being the usual Ext.

Because the forgetful functors Mod(A ⊗ B^{∘}) → Mod A etc. commute with R Hom_{A}(−, −) there is no need to mention them explicitly.

A complex N ∈ D^{+}(Mod A) is said to have finite injective dimension if there is an integer q_{0} with Ext_{A}^{q}(M, N) = 0 for all q > q_{0} and M ∈ Mod A.

For the rest of this section A denotes a left noetherian k-algebra and B denotes a right noetherian k-algebra. (For instance we could take A = B a two-sided noetherian algebra.) Observe that the algebra A ⊗ B^{∘} need not be left noetherian.

The subcategory Mod_{f} A of finitely generated A-modules is abelian and closed under extensions. Hence there is a full triangulated subcategory D_{f}(Mod A) ⊂ D(Mod A) consisting of all complexes with finitely generated cohomologies.

Dualizing complexes over commutative rings were introduced in [RD]. The noncommutative graded version first appeared in [Ye1], and we now give a slightly more general version.

Definition 1.1. Assume A and B are k-algebras, with A left noetherian and B right noetherian. A complex R ∈ D^{b}(Mod(A ⊗ B^{∘})) is called a dualizing complex if it satisfies the three conditions below:

(i) R has finite injective dimension over A and B^{∘}. (ii) R has finitely generated cohomology modules over A and B^{∘}. (iii) The canonical morphisms B → R Hom_{A}(R, R) in D(Mod B^{e}), and A →

R Hom_{B^{∘}}(R, R) in D(Mod A^{e}), are both isomorphisms. In case A = B, we shall say that R is a dualizing complex over A.

Condition (i) is equivalent to having an isomorphism R ≅ I ∈ D^{b}(Mod A ⊗ B^{∘}), where each I^{q} is injective over A and over B^{∘}.

Example 1.2. Suppose A is commutative and R is a dualizing complex in the sense of [RD]. If we consider R as a complex of bimodules, by identifying A = A^{∘}, then R is a dualizing complex in the sense of the definition above. According to [Ye3], if Spec A is connected, then any dualizing complex R′ over A is isomorphic to R ⊗_{A} P [n], where P is an invertible bimodule (not necessarily central!) and n ∈ ℤ.

Some easy examples of dualizing complexes over noncommutative rings are given in Example 2.3.

The next proposition offers an explanation of the name “dualizing complex.” The duality functors associated to R are the contravariant functors

D := R Hom_{A}(−, R) : D(Mod A) → D(Mod B^{∘})
D^{∘} := R Hom_{B^{∘}}(−, R) : D(Mod B^{∘}) → D(Mod A).

Proposition 1.3. Let R ∈ D(Mod(A ⊗ B^{∘})) be a dualizing complex.

1. For any M ∈ D_{f}(Mod A) one has DM ∈ D_{f}(Mod B^{∘}) and M ≅ D^{∘}DM.

2. The functors D and D^{∘} determine a duality, i.e., an anti-equivalence, of triangulated categories between D_{f}(Mod A) and D_{f}(Mod B^{∘}), restricting to a duality between D_{f}^{b}(Mod A) and D_{f}^{b}(Mod B^{∘}).

Proof. 1. This is slightly stronger than [Ye1] Lemma 3.5. By adjunction we get a functorial morphism M → D^{∘}DM. Since the functor D^{∘}D is way out in both directions and D^{∘}DA ≅ A by assumption, the claim follows from the reversed forms of [RD] Propositions I.7.1 and I.7.3.

2. Immediate from part 1, together with the fact that M ∈ D^{b}(Mod A) implies DM ∈ D^{b}(Mod B^{∘}).

Remark 1.4. The noetherian hypothesis can be relaxed – dualizing complexes can be defined over any coherent algebra A (see [Ye1] or [Mi1]). The category of finitely generated A-modules is then replaced by the category of coherent modules. Many definitions and results in our paper hold for coherent algebras, as can be easily checked.

Perhaps one can even work over an arbitrary algebra, using the category of coherent complexes, as defined by Illusie (see [SGA6] Exposé I).

Another direction to extend the theory is to allow k to be any commutative ring. In this case the derived category of bimodules should be D(DGMod(A ⊗_{k}^{L} B^{∘})), where A ⊗_{k}^{L} B^{∘} is a differential graded algebra. See [Ye3] Remark 1.12.

Remark 1.5. Miyachi proved a converse to Proposition 1.3(2): if there are contravariant triangle functors D(Mod A) → D(Mod B^{∘}) and D(Mod B^{∘}) → D(Mod A) which send ⨁ to ∏, preserve D^{b} and induce duality on D_{f}^{b}, then there is a dualizing complex in D(Mod(A ⊗ B^{∘})) (see [Mi2] Theorem 3.3).

Remark 1.6. There are examples of algebras A and B where there is a dualizing complex R ∈ D(Mod(A ⊗ B^{∘})), but there is no dualizing complex in D(Mod A^{e}); cf. [WZ]. The algebras A and B are necessarily not derived Morita equivalent, since given a tilting complex T ∈ D(Mod(B ⊗ A^{∘})), the complex R ⊗_{B}^{L} T ∈ D(Mod A^{e}) would be dualizing (cf. [Ye3]).

There are Grothendieck spectral sequences for the isomorphism of functors 1_{D_{f}^{b}(Mod A)} ≅ D^{∘}D and 1_{D_{f}^{b}(Mod B^{∘})} ≅ DD^{∘}. For modules they take this form:

Proposition 1.7. Let R ∈ D(Mod(A ⊗ B^{∘})) be a dualizing complex. Then there are convergent double-Ext spectral sequences

E_{2}^{p,q} := Ext_{B^{∘}}^{p}(Ext_{A}^{−q}(M, R), R) ⇒ M (1.8) for all M ∈ Mod_{f} A, and

E_{2}^{p,q} := Ext_{A}^{p}(Ext_{B^{∘}}^{−q}(N, R), R) ⇒ N (1.9)

for all N ∈ Mod_{f} B^{∘}.

Proof. By symmetry it suffices to consider (1.8) only. We can assume R is a bounded complex of bimodules with each R^{q} an injective module over A and B^{∘}. Given a nonzero finitely generated A-module M, define the complex

H := Hom_{B^{∘}}(Hom_{A}(M, R), R).

Then the adjunction homomorphism M → H is a quasi-isomorphism. Pick a positive integer d large enough so that R^{q} = 0 if |q| > d. Consider the decreasing filtration on H given by the subcomplexes

F^{p}H := Hom_{B^{∘}}(Hom_{A}(M, R), R^{≥p}).

Then F is an exhaustive filtration, and it determines the convergent spectral sequence (1.8).

Given M ∈ D^{+}(Mod A), there is a quasi-isomorphism M → I in D^{+}(Mod A), where each I^{q} is injective and Ker(I^{q} → I^{q+1}) ⊂ I^{q} is essential. Such I is unique (up to a non-unique isomorphism), and it is called the minimal injective resolution of M (cf. [Ye1] Lemma 4.2). If M has finite injective dimension then I is bounded.

The next two results are straightforward generalizations of [ASZ] Lemma 2.2 and Theorem 2.3, so the proofs are omitted.

Lemma 1.10. Let R ∈ D(Mod(A ⊗ B^{∘})) be a dualizing complex, and let I be the minimal injective resolution of R in D^{b}(Mod A). Let Z_{i} := Ker(I^{i} → I^{i+1}) and let M be a finitely generated left A-module. Then there exist f_{1}, …, f_{n} ∈ Hom(M, Z_{i}) such that for every N ⊂ ⋂_{j} Ker(f_{j}) the natural map Ext_{A}^{i}(M, R) → Ext_{A}^{i}(N, R) is zero; or equivalently, the natural map Ext_{A}^{i}(M/N, R) → Ext_{A}^{i}(M, R) is surjective.

Theorem 1.11. Let R ∈ D(Mod(A ⊗ B^{∘})) be a dualizing complex, let I be the minimal injective resolution of R in D^{b}(Mod A), and let Z_{i} := Ker(I^{i} → I^{i+1}). Then:

1. For every nonzero A-module M there is a nonzero submodule N ⊂ M which embeds in some Z_{i}.

2. Every indecomposable injective A-module appears in I.

We conclude this section with a discussion of dualizing complexes in D(Mod(A⊗ B^{∘})) when A is commutative. For a prime ideal 𝔭 ⊂ A let J_{A}(𝔭) be an injective hull of A/𝔭. Let us recall a result of Grothendieck.

Proposition 1.12 ([RD] Proposition V.7.3). Suppose A is a commutative noetherian ring and R ∈ D_{f}^{b}(Mod A) is a (central) dualizing complex. Let I be the minimal injective resolution of R in Mod A.

1. There is a function d : Spec A → ℤ such that

I^{q} ≅ ⨁_{d(𝔭)=q} J_{A}(𝔭).

2. If 𝔭 ⊂ 𝔮 are primes and 𝔮/𝔭 ⊂ A/𝔭 has height 1, then d(𝔭) = d(𝔮) − 1.

The function d is called a codimension function in [RD]. In our case we get:

Theorem 1.13. Suppose A is a commutative noetherian k-algebra, B is a right noetherian k-algebra, and R ∈ D(Mod(A ⊗ B^{∘})) is a dualizing complex. Let I be the minimal injective resolution of R in Mod A. Then:

1. There are functions d, r : Spec A → ℤ, with r ≥ 1 and constant on connected components of Spec A, s.t.

I^{q} ≅ ⨁_{d(𝔭)=q} J_{A}(𝔭)^{r(𝔭)}.

2. If 𝔭 ⊂ 𝔮 are primes of A and 𝔮/𝔭 ⊂ A/𝔭 has height 1, then d(𝔭) = d(𝔮) − 1.

3. A is catenary, and (if A ≠ 0) its Krull dimension is

Kdim A ≤ max{d(𝔭)} − min{d(𝔭)} < ∞.

4. B is an Azumaya A-algebra.

The proof of the theorem is after the next lemma.

Lemma 1.14. Assume in addition that A is local. Then there is an integer d such that Ext_{A}^{q}(M, R) = 0 and Ext_{B^{∘}}^{q}(N, R) = 0 for all q ≠ d, all finite length A-modules M and all finite length B^{∘}-modules N.

Proof. According to [Ye3] Proposition 5.4 (which works even when A ≠ B; cf. ibid. Proposition 2.5), left and right multiplications on R induce ring isomorphisms A ≅ End_{D(Mod(A⊗B^{∘}))}(R) ≅ Z(B). Moreover since A is noetherian and B^{∘} ≅ Ext_{A}^{0}(R, R) we see that B is a finite A-algebra. If N is an A-central (A ⊗ B^{∘})-module, then Ext_{B^{∘}}^{q}(N, R) is a central A-bimodule.

Denote by K the residue field of A. Let p_{0} := min{p | Ext_{A}^{p}(K, R) ≠ 0} and p_{1} := max{p | Ext_{A}^{p}(K, R) ≠ 0}. By induction on length we see that for every finite length A-module M, p_{0} = min{p | Ext_{A}^{p}(M, R) ≠ 0} and p_{1} = max{p | Ext_{A}^{p}(M, R) ≠ 0}.

Now take a nonzero finite length B^{∘}-module N. Since we can view N as a central A-bimodule, it follows that Ext_{B^{∘}}^{q}(N, R) is also a central A-bimodule for every q; and so it has finite length. Define q_{0} := min{q | Ext_{B^{∘}}^{q}(N, R) ≠ 0} and q_{1} := max{q | Ext_{B^{∘}}^{q}(N, R) ≠ 0}. In the E_{2}-page of the spectral sequence of Proposition 1.7 we have nonzero terms E_{2}^{p_{1},−q_{0}} = Ext_{A}^{p_{1}}(Ext_{B^{∘}}^{q_{0}}(N, R)) and E_{2}^{p_{0},−q_{1}} = Ext_{A}^{p_{0}}(Ext_{B^{∘}}^{q_{1}}(N, R)), that appear in the right-top and left-bottom corners respectively. The convergence of the spectral sequence forces p_{1} = q_{0} and p_{0} = q_{1}. Therefore we get d := p_{1} = q_{0} = q_{1} = p_{0}.

Proof of Theorem 1.13. 1. As observed in the proof of the lemma, B is a finite Aalgebra. Take a prime 𝔭 ⊂ A, and define B_{𝔭} := B⊗_{A}A_{𝔭} and R_{𝔭} := A_{𝔭}⊗_{A}R⊗_{B}B_{𝔭} ∈ D^{b}(Mod(A_{𝔭} ⊗ B_{𝔭}^{∘})). We claim that R_{𝔭} is a dualizing complex.

First note that the cohomologies H^{q} R are central A-bimodules. Since A → A_{𝔭} is flat and A_{𝔭} ⊗_{A} A_{𝔭} = A_{𝔭}, there are isomorphisms H^{q} R_{𝔭} ≅ A_{𝔭} ⊗_{A} H^{q} R ≅ H^{q} R ⊗_{B} B_{𝔭}. Therefore R_{𝔭} ≅ A_{𝔭} ⊗_{A} R ≅ R ⊗_{B} B_{𝔭} in D^{b}(Mod(A ⊗ B^{∘})). It is easy to see that the cohomology bimodules H^{q} R are finitely generated on both sides, R Hom_{A_{𝔭}}(R_{𝔭}, R_{𝔭}) ≅ B_{𝔭}^{∘} and R Hom_{B_{𝔭}^{∘}}(R_{𝔭}, R_{𝔭}) ≅ A_{𝔭}.

In order to verify that R_{𝔭} has finite injective dimension over B_{𝔭}^{∘} it suffices to show that Ext_{B_{𝔭}^{∘}}^{q}(N, R_{𝔭}) vanishes for all finitely generated B_{𝔭}^{∘}-modules N, for large q. Now we can write N ≅ N′ ⊗_{B} B_{𝔭} for some finitely generated B^{∘}-module N′, and then Ext_{B_{𝔭}^{∘}}^{q}(N, R_{𝔭}) ≅ Ext_{B^{∘}}^{q}(N′, R) ⊗_{A} A_{𝔭}. Likewise for the injective dimension overA_{𝔭}. So indeed R_{𝔭} is dualizing.

Since the multiplicity of the indecomposable injective J_{A}(𝔭) in I is measured by Ext_{A_{𝔭}}^{q}(k(𝔭), R_{𝔭}), where k(𝔭) is the residue field, the lemma says that J_{A}(𝔭) occurs in the complex I in only one degree, say d(𝔭). The fact that the multiplicity r(𝔭) is locally constant will be proved in part 4 below.

2. Choose a ∈ 𝔮 − 𝔭, so in the exact sequence

0 → (A/𝔭)_{𝔮} ⟶^{a} (A/𝔭)_{𝔮} → M → 0

the A_{𝔮}-module M has finite length. Applying Ext_{A}^{q}(−, R) to this sequence we obtain an exact sequence of finitely generated A_{𝔮}-modules

Ext_{A}^{q}((A/𝔭)_{𝔮}, R) ⟶^{a} Ext_{A}^{q}((A/𝔭)_{𝔮}, R) → Ext_{A}^{q+1}(M, R)

for each q. By Nakayama’s Lemma and part 1 we find that Ext_{A}^{q}((A/𝔭)_{𝔮}, R) = 0 unless q + 1 = d(𝔮). Hence d(𝔭) = d(𝔮) − 1 as claimed.

3. This follows trivially from 2.

4. Pick a prime ideal 𝔭 ⊂ A. Since R_{𝔭} is a dualizing complex in D(Mod(A_{𝔭} ⊗ B_{𝔭}^{∘})), the lemma tells us that that the functors Ext_{A_{𝔭}}^{d(𝔭)}(−, R_{𝔭}) and Ext_{B_{𝔭}^{∘}}^{d(𝔭)}(−, R_{𝔭}) are a duality between the categories of finite length modules over A_{𝔭} and B_{𝔭}^{∘}. Furthermore since this duality is A_{𝔭}-linear, it restricts to a duality between Mod_{f} A_{𝔭}/𝔭_{𝔭}^{n} and Mod_{f}(B_{𝔭}/𝔭_{𝔭}^{n}B_{𝔭})^{∘} for every n ≥ 1. Since Mod_{f} A_{𝔭}/𝔭_{𝔭}^{n} has an auto-duality, namely Hom_{A_{𝔭}}(−, J_{A}(𝔭)), it follows that there is an equivalence between Mod_{f} A_{𝔭}/𝔭_{𝔭}^{n} and Mod_{f} B_{𝔭}/𝔭_{𝔭}^{n}B_{𝔭}. Morita Theory says there is an isomorphism ϕ_{n} : B_{𝔭}/𝔭_{𝔭}^{n}B_{𝔭} →^{≃} M_{r(n)}(A_{𝔭}/𝔭_{𝔭}^{n}) for some number r(n).

Since the isomorphisms ϕ_{n} all arise from the same equivalence they can be made compatible, and in particular we have r(n) = r(𝔭) for all n. In the inverse limit we get Â_{𝔭} ⊗_{A} B ≅ M_{r(𝔭)}(Â_{𝔭}) as Â_{𝔭}-algebras.

Since 𝔭 was an arbitrary prime ideal of A, we conclude that the multiplication map B ⊗_{A} B^{∘} → End_{A}(B) is bijective. Therefore B is Azumaya over A. As a projective A-module, the rank of B at a prime 𝔭 is precisely r(𝔭)^{2}, so the function r is locally constant.

Remark 1.15. A complex such as I in Theorem 1.12 is called a residual complex. It actually depends functorially on R: I = ER, where E is the Cousin functor. Noncommutative variants of the Cousin functor are studied in [Ye2] and [YZ2]. In particular one can show that the complex I can be made into a complex of bimodules, where on the right it is the minimal injective resolution of R in Mod B^{∘}. It follows that R is an Auslander dualizing complex, as defined in Section 2. In [YZ2] we show that a more complicated version of Theorem 1.13 holds when A = B is a PI algebra.

Example 1.16. Let A be a noetherian commutative regular ring of infinite Krull dimension – see Nagata’s example [Na] Appendix, p. 203 Example 1. The complex R = A is a pointwise dualizing complex (see [RD] Section V.8), and R Hom_{A}(−, A) : D_{f}(Mod A) → D_{f}(Mod A) is a duality. However by Theorem 1.13, there is no dualizing complex in D^{b}(Mod(A ⊗ B^{∘})) for any right noetherian k-algebra B.

Example 1.17. Let A be the non-catenary noetherian commutative local ring of [Na] Appendix, p. 203 Example 2. Then again there is no dualizing complex in D^{b}(Mod(A ⊗ B^{∘})) for any right noetherian k-algebra B.

2. Auslander Dualizing Complexes

The basic ideas in this section already appear in [Ye2] Section 1 (which treats graded algebras).

Assume R ∈ D^{b}(Mod(A ⊗ B^{∘})) is a dualizing complex. Let M be a finitely generated A-module. The grade of M with respect to R is

j_{R;A}(M) := inf{q | Ext_{A}^{q}(M, R) ≠ 0} ∈ ℤ ∪ {∞}.

Similarly define j_{R;B^{∘}} for a B^{∘}-module.

We are ready to define the notion appearing in the title of the paper.

Definition 2.1. Let A and B be k-algebras, with A left noetherian and B right noetherian, and let R ∈ D(Mod(A ⊗ B^{∘})) be a dualizing complex. We say that R has the Auslander property, or that R is an Auslander dualizing complex, if

(i) for every finitely generated A-module M, integer q and B^{∘}-submodule N ⊂ Ext_{A}^{q}(M, R), one has j_{R;B^{∘}}(N) ≥ q.

(ii) the same holds after exchanging A and B^{∘}.

Note that the role of the algebras A and B^{∘} is symmetric. Also note that if R is an Auslander dualizing complex, then any shift R[n] is also an Auslander dualizing complex (the shift cancels out in the double dual).

Example 2.2. Let A be a commutative k-algebra and R a central dualizing complex over it. From Proposition 1.12 it is clear that R is Auslander. For a prime ideal 𝔭 one has j_{R;A}(A/𝔭) = d(𝔭).

Example 2.3. (a) If A is Gorenstein, i.e. the bimodule A has finite injective dimension on both sides, then R = A is a dualizing complex over A. If A is an Auslander-Gorenstein ring in the sense of [Bj], then R = A is an Auslander dualizing complex.

(b) If A is a finite k-algebra, i.e. rank_{k} A < ∞, then the bimodule A^{∗} := Hom_{k}(A, k) is injective on both sides, and R = A^{∗} is an Auslander dualizing complex over A. Clearly j_{R;A}(M) = 0 for all M.

(c) Let A be the matrix algebra ( \begin{matrix}k & V \\ 0 & k \end{matrix} ) where V is a finite rank k-module. Since A is hereditary it is also Gorenstein, so both A and A^{∗} are dualizing complexes, and if V ≠ 0 they are non-isomorphic. According to [ASZ] Example 5.4, the dualizing complex A is Auslander iff rank_{k} V ≤ 1. See also [Ye3] Section 3.

More examples of algebras with Auslander dualizing complexes are in Examples 6.11-6.15.

The next definition is taken from [MR] Section 6.8.4 (with a slight modification – we allow negative dimensions).

Definition 2.4. An exact dimension function is a function dim which assigns to each module M ∈ Mod_{f} A a value dim M in an ordered set containing −∞, ℝ and some infinite ordinals, and satisfies the following axioms:

(i) dim 0 = −∞.

(ii) For every short exact sequence 0 → M′ → M → M′′ → 0 one has dim M =

max{dim M′, dim M′′}. (iii) If 𝔭M = 0 for some prime ideal 𝔭, and M is a torsion A/𝔭-module, then

dim_{R} M ≤ dim_{R} A/𝔭 − 1.

Familiar examples of exact dimension functions are the Krull dimension Kdim and (sometimes) the Gelfand-Kirillov dimension GKdim.

Lemma 2.5. A function dim defined on Mod_{f} A and satisfying axioms (i)-(ii) (resp. axioms (i)-(iii)) extends uniquely to a function on Mod A, satisfying axioms (i)-(ii) (resp. axioms (i)-(iii)) and the axiom (iv) dim M = sup{dim M′ | M′ ⊂ M finitely generated}.

The proof of the lemma is standard.

Usually we will have a pair of dimension functions, one on Mod A and the other on Mod B^{∘}; when necessary we shall distinguish between them by writing dim_{A} and dim_{B^{∘}} respectively.

Definition 2.6. An exact dimension function dim is called finitely partitive if given a finitely generated module M there is a number l_{0}, such that for every chain of submodules M = M_{0} ⫌ M_{1} ⋯ ⫌ M_{l} with dim M_{i}/M_{i+1} = dim M one has l ≤ l_{0}.

In the next two definitions (taken from [ASZ] and [Ye2]) dim denotes a function on Mod A satisfying axioms (i), (ii) and (iv).

Definition 2.7. 1. A module M is called dim-pure if dim M′ = dim M for every

nonzero submodule M′ ⊂ M.

2. A module M is called dim-essentially pure if M contains an essential submodule which is pure.

3. A module M is called dim-critical if M ≠ 0, and dim M/M′ < dim M for every 0 ≠ M′ ⫋ M.

Definition 2.8. 1. Let M_{q}(dim) be the full subcategory of Mod A consisting of

modules M with dim M ≤ q, and let M_{q,f}(dim) := M_{q}(dim) ∩ Mod_{f}A.

2. Given a module M let Γ_{M_{q}(dim)}M ⊂ M be the largest submodule M′ ⊂ M such that dim M′ ≤ q.

Since M_{q}(dim) is a localizing subcategory of ModA the submodule Γ_{M_{q}(dim)}M is well defined (and in fact Γ_{M_{q}(dim)} is a left exact idempotent functor). The corresponding subcategories of Mod B^{∘} shall be denoted by M_{q}^{∘}(dim) and M_{q,f}^{∘}(dim).

Definition 2.9. Let M be a finitely generated A-module. The canonical dimension of M with respect to R is

Cdim_{R;A} M := −j_{R;A}(M) ∈ ℤ ∪ {−∞}.

Likewise define Cdim_{R;B^{∘}}.

The canonical dimension will not be an exact dimension function in general. However we have the following theorem, which generalizes results of Björk and Levasseur (the graded case was proved in [Ye2]).

Theorem 2.10. If R is an Auslander dualizing complex then Cdim_{R;A} is a finitely partitive exact dimension function.

The proof of this theorem appears later on in the section. The key step is:

[figure: p; q; p = −q; q = −j(M); p = d]

Figure 1. The E_{2} term of the spectral sequence (2.12) in the (p, q) plane

Lemma 2.11. Let 0 → M′ → M′ → M′′ → 0 be a short exact sequence of finitely generated A-modules. Then

j_{R;A}(M) = inf{j_{R;A}(M′), j_{R;A}(M′′)}.

Proof. The proof goes along the lines of the proofs in [Bj] and [Lev]. By Proposition 1.7 we have a convergent spectral sequence

E_{2}^{p,q} := Ext_{B^{∘}}^{p}(Ext_{A}^{−q}(M, R), R) ⇒ M, (2.12)

so there is a corresponding filtration (called the b-filtration in [Lev] Theorem 2.2)

M = F^{−d}M ⊃ F^{−d+1}M ⊃ ⋯ ⊃ F^{d+1}M = 0.

The Auslander condition tells us that E_{2}^{p,q} = 0 if p < −q. So the spectral sequence lives in a bounded region of the (p, q) plane: p ≥ −q, q ≤ −j_{A;R}(M) and p ≤ d (see Figure 1). The coboundary operator of E_{r} has bidegree (r, 1 − r) and r ≥ 2. We conclude that for every |p| ≤ d there is an exact sequence of A-modules

0 → \frac{F^{p}M}{F^{p+1}M} → E_{2}^{p,−p} → Q^{p} → 0 (2.13)

with Q^{p} a subquotient of ⨁_{r≥2} E_{2}^{p+r,−p+(1−r)} (cf. [Bj] Theorem 1.3 and [Lev] Theorem 2.2).

By the Auslander property it then follows that j_{R;A}(F^{p}M/F^{p+1}M) ≥ p for all M and p. Just as in [Bj] Proposition 1.6, one uses descending induction on p, starting at p = d + 1, to prove that j_{R;A}(F^{p}M) ≥ p for all p. This implies that

j_{R;B^{∘}}(Ext^{j_{R;A}(M)}(M, R)) = j_{R;A}(M).

Now continue exactly like in [Bj] Proposition 1.8.

We conclude that Cdim_{R;A} verifies axiom (ii). Axiom (i) holds trivially. By symmetry Cdim_{R;B^{∘}} also verifies axioms (i)-(ii).

Theorem 2.14. Suppose R ∈ D(Mod(A⊗B^{∘})) is an Auslander dualizing complex. Let M ∈ Mod_{f} A be nonzero and Cdim_{R} M = n. Then:

1. Ext_{A}^{−n}(M, R) is Cdim_{R}-pure of dimension n.

2. For each p, Ext_{B^{∘}}^{−p}(Ext_{A}^{−p}(M, R), R) is Cdim_{R}-pure of dimension p, or is 0.

3. For each p there is an exact sequence

0 → Γ_{M_{p−1}}M → Γ_{M_{p}}M → Ext_{B^{∘}}^{−p}(Ext_{A}^{−p}(M, R), R) → Q^{p} → 0,

functorial in M, where Cdim_{R} Q^{p} ≤ p − 2 and M_{p} = M_{p}(Cdim_{R}).

Proof. 1. Because the line q = −j_{R;A}(M) is on the boundary of the region of support of the spectral sequence (2.12), and the coboundary operator of E_{r} has bidegree (r, 1 − r), r ≥ 2, we see that for this value of q there is a bounded filtration E_{2}^{p,q} ⊃ E_{3}^{p,q} ⊃ ⋯, with E_{r}^{p,q}/E_{r+1}^{p,q} a subquotient of E_{r}^{p+r,q+(1−r)}. Now the abutment of the spectral sequence is concentrated on the line p = −q of the (p, q)-plane, so E_{r}^{p,q} = 0 for p > −q and r ≫ 0. By Lemma 2.11 we conclude that for q = −j_{R;A}(M) and p > −q,

j_{R;A}(Ext_{B^{∘}}^{p}(Ext_{A}^{−q}(M, R), R)) = j_{R;A}(E_{2}^{p,q}) ≥ p + 2

(cf. [Bj] formula (1.10)). Just like in [Bj] Proposition 1.11 it follows that Ext_{A}^{p}(Ext_{B^{∘}}^{p}(N, R), R) = 0 for p > j_{R;A}(M) and N = Ext_{A}^{j_{R;A}(M)}(M, R). So by [Bj] Proposition 1.9 we conclude that N is pure.

2. Take N := Ext_{A}^{−p}(M, R). Then Cdim_{R;B^{∘}} N ≤ p and part 1 applies.

3. By part 2, the sequence (2.13) and induction on p we see that Γ_{M_{p}}M = F^{−p}M.

For an integer q let M_{q} := M_{q}(Cdim_{R}) ⊂ Mod A be the localizing subcategory from Definition 2.8. The filtration by dimension of support {M_{q}} of Mod A is called the niveau filtration in commutative algebraic geometry. For each q the quotient category M_{q}/M_{q−1} is a locally noetherian abelian category, and the full subcategory M_{q,f}/M_{q−1,f} is noetherian (see [ASZ] Lemma 1.1). By symmetry we have corresponding localizing subcategories M_{q,f}^{∘} ⊂ M_{q}^{∘} ⊂ Mod B^{∘}.

Recall from [ASZ] Section 1 that two abelian categories C and D are said to be dual if they are anti-equivalent, i.e. if C is equivalent to the opposite category D^{∘}. Two categories C and D are said to be in step duality if there are filtrations by dense abelian subcategories

0 = C_{n_{0}−1} ⊂ C_{n_{0}} ⊂ ⋯ ⊂ C_{n_{1}} = C and 0 = D_{n_{0}−1} ⊂ D_{n_{0}} ⊂ ⋯ ⊂ D_{n_{1}} = D

such that the quotient categories C_{i} / C_{i−1} and D_{i} / D_{i−1} are dual for all i = n_{0}, …, n_{1}. Now Theorem 0.2 is a special case of:

Theorem 2.15. Suppose R ∈ D(Mod(A⊗B^{∘})) is an Auslander dualizing complex. Then Mod_{f} A and Mod_{f} B^{∘} are in step duality. More precisely, for every q the functors Ext_{A}^{q}(−, R) and Ext_{B^{∘}}^{q}(−, R) induce a duality between the quotient categories M_{q,f}/M_{q−1,f} and M_{q,f}^{∘}/M_{q−1,f}^{∘}.

Proof. Use Theorem 2.14 and the proof of [ASZ] Theorem 1.2.

Corollary 2.16. For each q the category M_{q,f}/M_{q−1,f} is artinian, i.e. every object has finite length.

Proof. By [ASZ] Lemma 1.1 the categories M_{q,f}/M_{q−1,f} and M_{q,f}^{∘}/M_{q−1,f}^{∘} are noetherian, hence by Theorem 2.15 they are also artinian.

At last here is:

Proof of Theorem 2.10. We already verified axioms (i) and (ii). The fact that Cdim_{R} is finitely partitive is immediate from Corollary 2.16, and this in turn easily implies axiom (iii) – cf. [MR] Corollary 8.3.6.

Corollary 2.17. Every finitely generated A-module has a Cdim_{R}-critical composition series.

Proof. By Theorem 2.10 and [MR] Proposition 6.2.20.

Let

d_{0} := inf{Cdim_{R} M | M ∈ Mod A, M ≠ 0}
d_{1} := Cdim_{R} A = sup{Cdim_{R} M | M ∈ Mod A}.

Corollary 2.18. Suppose R ∈ D(Mod(A⊗B^{∘})) is an Auslander dualizing complex. Then

Kdim M ≤ Cdim_{R} M − d_{0}

for all finitely generated A-modules M. In particular if Cdim_{R} M = d_{0} then M is artinian.

Proof. By induction, starting with q = d_{0}, Theorem 2.14 and Corollary 2.16 show that Kdim M ≤ q − d_{0} for all M ∈ M_{q,f}.

Here is a generalization of [Bj] Theorem 1.14.

Theorem 2.19. (Gabber’s Maximality Principle) Let A and B be k-algebras, with A left noetherian and B right noetherian, and let R ∈ D(Mod(A ⊗ B^{∘})) be an Auslander dualizing complex. Suppose N is a Cdim_{R}-pure A-module with Cdim_{R} N = n, and M is a finitely generated submodule. Then there is a unique maximal module M̃ such that M ⊂ M̃ ⊂ N, M̃ is finitely generated, and Cdim_{R} M̃/M ≤ n − 2.

Proof. Note that we do not assume N is finitely generated. The uniqueness is clear because Cdim_{R} is an exact dimension function. So it remains to show existence. If M̃ is any finitely generated submodule of N containing M, such that Cdim_{R} M̃/M ≤ n − 2, then Ext_{A}^{−n}(M, R) ≅ Ext_{A}^{−n}(M̃, R). Hence, by Theorem 2.14(3), the module M̃ embeds functorially into the finitely generated A-module Ext_{B^{∘}}^{−n}(Ext_{A}^{−n}(M, R), R). This implies there is a maximal such M̃.

The next two theorems generalize [GL] Theorems 1.4 and 1.6 by eliminating the Gorenstein and Cohen-Macaulay conditions. From here on we consider a single noetherian algebra A (i.e. A = B).

Definition 2.20. Let dim be an exact dimension function.

1. dim is called symmetric if dim_{A} M = dim_{A^{∘}} M for every bimodule M finitely generated on both sides.

2. dim is called weakly symmetric if dim_{A} M = dim_{A^{∘}} M for every bimodule M which is a subquotient of A.

Lemma 2.21. Let dim be a weakly symmetric exact dimension function.

1. Let M be a finitely generated dim-pure A-module, and let I := Ann_{A}(M). If dim M = dim A/I then A/I is dim-pure.

2. Let I be an ideal of A such that A/I is dim-pure and let 𝔮 be a prime ideal of A that is minimal over I. Then dim A/𝔮 = dim A/I.

Proof. This is completely analogous to [KL] 9.6 and 9.5.

Recall from [GL] that Spec A is said to have normal separation provided that for any pair prime ideals 𝔭 ⫋ 𝔮, the factor 𝔮/𝔭 contains a nonzero normal element of A/𝔭. Under the assumptions of the lemma we say that Tauvel’s height formula holds in A provided

height 𝔭 + dim A/𝔭 = dim A

for all primes 𝔭.

Theorem 2.22. Suppose that A is a noetherian k-algebra, R is an Auslander dualizing complex over A and Cdim_{R} is weakly symmetric. Let 𝔭 ⫋ 𝔮 be prime ideals of A with height 𝔮/𝔭 = 1. If there exists an element a ∈ 𝔮−𝔭 that is normal modulo 𝔭, then Cdim_{R} A/𝔭 = Cdim_{R} A/𝔮 + 1.

Proof. Use the proof of [GL] Theorem 1.4, but replace GKdim by Cdim_{R}, and use Lemma 2.21 instead of [Len] Lemmas 2 and 3.

Theorem 2.23. Suppose that A is a noetherian k-algebra, R is an Auslander dualizing complex over A and Cdim_{R} is weakly symmetric. If Spec A is normally separated, then A is catenary. If in addition A is prime, then Tauvel’s height formula holds.

Proof. The proof of [GL] Theorem 1.6 works here after we replace GKdim by Cdim_{R}.

We call attention to Question 3.15 regarding the possible symmetry of Cdim_{R}. The Macaulay property of [SZ] is adapted in the following way, to be used later

in the paper.

Definition 2.24. Suppose R is an Auslander dualizing complex over A. Let dim be an exact dimension function on finitely generated A-modules. If there is some integer c such that

dim M = Cdim_{R} M + c

for all M ∈ Mod_{f} A, then we say R is Macaulay with respect to dim, or that R is dim-Macaulay.

Note that if R = A, then “GKdim-Macaulay” is equivalent to “Cohen-Macaulay” as it is used in [Bj], [Lev], [ASZ], and [SZ]. This is because GKdim M + j_{R}(M) = c = dim A in this case.

Example 2.25. If A is a commutative affine k-algebra and R is any central dualizing complex over A, then R is Auslander and GKdim-Macaulay. In this case we also have Kdim M = GKdim M for all finitely generated A-modules M.

3. Rigid Dualizing Complexes

In this section we consider dualizing complexes which satisfy a special condition discovered by Van den Bergh [VdB]. Rigid dualizing complexes are unique and even functorial. Furthermore if R is an Auslander rigid dualizing complex then the canonical dimension Cdim_{R} is particularly well behaved (as examples indicate; see Question 3.15). By default A and B denote noetherian k-algebras.

First we shall need some more notation for bimodules. Suppose A and B are k-algebras. For an element a ∈ A we denote by a^{∘} ∈ A^{∘} the same element. Thus for a_{1}, a_{2} ∈ A, a_{1}^{∘} ·a_{2}^{∘} = (a_{2} ·a_{1})^{∘} ∈ A^{∘}. With this notation if M is a right A-module then the left A^{∘} action is a^{∘} · m = m · a, m ∈ M. The algebra A^{e} has an involution A^{e} → (A^{e})^{∘}, a_{1} ⊗ a_{2}^{∘} ↦ a_{2} ⊗ a_{1}^{∘} which allows us to regard every left A^{e}-module M as a right A^{e}-module in a consistent way:

(a_{1} ⊗ a_{2}^{∘}) · m = (a_{2} ⊗ a_{1}^{∘})^{∘} · m = m · (a_{2} ⊗ a_{1}^{∘}) = a_{1} · m · a_{2}.

Given an (A ⊗ B)-module M and a (B ⊗ A)-module N we define a mixed action of A^{e} ⊗ B^{e} on the tensor product M ⊗ N as follows. A^{e} acts on M ⊗ N by the outside action

(a_{1} ⊗ a_{2}^{∘}) · (m ⊗ n) := (a_{1} · m) ⊗ (n · a_{2}),

whereas B^{e} acts on M ⊗ N by the inside action

(b_{1} ⊗ b_{2}^{∘}) · (m ⊗ n) := (m · b_{2}) ⊗ (b_{1} · n).

By default we regard the outside action as a left action and the inside action as a right action. If A = B and M = N then the two actions by A^{e} on M ⊗ M are interchanged the involution m_{1} ⊗ m_{2} ↦ m_{2} ⊗ m_{1}. However for the sake of definiteness in this case, given an A^{e}-module L, Hom_{A^{e}}(L, M ⊗ M) shall refer to homomorphisms L → M ⊗M which are A^{e}-linear with respect to the outside action.

Definition 3.1 ([VdB] Definition 8.1). Suppose R is a dualizing complex over A. If there is an isomorphism

ϕ : R →^{≃} R Hom_{A^{e}}(A, R ⊗ R)

in D(Mod A^{e}), we call (R, ϕ) a rigid dualizing complex.

It is obvious that if R is rigid, then any shift R[n], for n ≠ 0, is no longer rigid. Van den Bergh proved that a rigid dualizing complex (R, ϕ) over A is unique, up to an isomorphism in D(Mod A^{e}) (see [VdB] Proposition 8.2). Below we extend this result by proving that rigid dualizing complexes are functorial, in a suitable sense.

Let A → B be a k-algebra homomorphism. Given M ∈ D^{+}(Mod B), N ∈ D^{+}(Mod A) and a morphism ψ : M → N in D(Mod A), ψ factors naturally through R Hom_{A}(B, N). This can be seen by replacing N with an injective resolution I in D^{+}(Mod A), and then we can take ψ to be a homomorphism of complexes. The image of ψ will then land inside Hom_{A}(B, I). The same fact is true for bimodules.

We say a k-algebra homomorphism A → B if finite if B is finitely generated as a left and as a right A-module.

Theorem 3.2. Let A → B be a finite k-algebra homomorphism. Suppose A and B have rigid dualizing complexes (R_{A}, ϕ_{A}) and (R_{B}, ϕ_{B}) respectively. Then there is at most one morphism ψ : R_{B} → R_{A} in D(Mod A^{e}) satisfying conditions (i) and (ii) below:

(i) ψ induces an isomorphism

R_{B} ≅ R Hom_{A}(B, R_{A}) ≅ R Hom_{A^{∘}}(B, R_{A})

in D(Mod A^{e}). (ii) The diagram

R_{B} ⟶^{ϕ_{B}} R Hom_{B^{e}}(B, R_{B} ⊗ R_{B})
ψ↓  ^{ψ⊗ψ}↓
R_{A} ⟶^{ϕ_{A}} R Hom_{A^{e}}(A, R_{A} ⊗ R_{A})

in D(Mod A^{e}) is commutative.

The theorem is proved after this lemma. Given a k-algebra homomorphism A → B, denote by Z_{B}(A) ⊂ B the centralizer of A.

Lemma 3.3. Let A → B be a finite k-algebra homomorphism. Suppose A and B have dualizing complexes R_{A} and R_{B} respectively, and ψ : R_{A} → R_{B} is a morphism in D(Mod A^{e}) satisfying condition (i) of the theorem. Then Hom_{D(Mod A^{e})}(R_{B}, R_{A}) is a free left and right Z_{B}(A)-module with basis ψ.

Proof. Denote by D := R Hom_{A}(−, R_{A}) and D^{∘} := R Hom_{A^{∘}}(−, R_{A}) the dualizing functors. By assumption R_{B} ≅ D^{∘}B. Applying the functor D we get isomorphisms of left Z_{B}(A)-modules

Hom_{D(Mod A^{e})}(R_{B}, R_{A}) ≅ Hom_{D(Mod A^{e})}(D^{∘}B, D^{∘}A)
≅ Hom_{D(Mod A^{e})}(A, B)
≅ Hom_{A^{e}}(A, B) ≅ Z_{B}(A).

Likewise for the right action.

Proof of the theorem. Assume ψ′ is another such isomorphism. According to the lemma above,

ψ′ = (b_{1} ⊗ 1)ψ = (1 ⊗ b_{2}^{∘})ψ

for suitable b_{i} ∈ Z_{B}(A)^{×}. So

ψ′ ⊗ ψ′ = (b_{1} ⊗ b_{2}^{∘})(ψ ⊗ ψ).

Now the diagram in condition (ii) consists of morphisms in D(Mod A^{e}). Since multiplications by b_{1} and b_{2}^{∘} are A^{e}-linear, we see that

(b_{1} ⊗ 1)ψ = (b_{1} ⊗ b_{2}^{∘})ψ ∈ Hom_{D(Mod A^{e})}(R_{B}, R_{A}).

Hence dividing by the unit b_{1} ⊗ 1, we see that b_{2} = 1.

Corollary 3.4. A rigid dualizing complex (R, ϕ) over A is unique up to a unique isomorphism.

Proof. Suppose (R′, ϕ′) is another rigid dualizing complex. We claim that there is an isomorphism ψ : R →^{≃} R′ in D(Mod A^{e}) which satisfies condition (ii) of the theorem. By Theorem 3.2 such ψ is unique.

To produce ψ, choose any isomorphism ψ′ : R →^{≃} R′, which we know exists by [VdB] Proposition 8.2. Then by Lemma 3.3 there are a_{i} ∈ Z(A)^{×} such that

(a_{1} ⊗ 1)^{−1}ψ′ = (1 ⊗ a_{2}^{∘})^{−1}ψ′ = ϕ′^{−1}(ψ′ ⊗ ψ′)ϕ.

The isomorphism

ψ := (a_{1} ⊗ 1)ψ′ = (1 ⊗ a_{2}^{∘})ψ′

will satisfy condition (ii).

Thus we may speak of the rigid dualizing complex of A (if it exists).

Lemma 3.3 can be sharpened when A = B.

Proposition 3.5. Let (R, ϕ) be a rigid dualizing complex over A. Then the two k-algebra homomorphisms

λ, ρ : Z(A) → End_{D(Mod A^{e})}(R)

from the center of A, namely left and right multiplication, are both isomorphisms, and are equal.

Proof. By Lemma 3.3 with A = B and ψ = 1, we see that λ and ρ are isomorphisms. Take a ∈ Z(A), and let a′ := ρ^{−1}λ(a) ∈ Z(A). Using the definition of the mixed action on R ⊗ R and the rigidification isomorphism ϕ, the commutation (or conjugation) of a across R is transferred to commutation of a′ across A. Since a′ does commute with A it follows that λ(a) = ρ(a) (and so in fact a′ = a).

We will often omit reference to the rigidifying isomorphism ϕ.

Corollary 3.6. If R is a rigid dualizing complex over A then for any q the cohomology bimodule H^{q}R is central over Z(A).

Definition 3.7. Let A → B be a finite homomorphism of k-algebras. Assume the rigid dualizing complexes (R_{A}, ϕ_{A}) and (R_{B}, ϕ_{B}) exist. If there is a morphism ψ satisfying the conditions of Theorem 3.2 then we call it the trace morphism and denote it by Tr_{B/A}.

The next corollary is obvious.

Corollary 3.8. Let A → B and B → C be finite k-algebra homomorphisms. Assume the rigid dualizing complexes (R_{A}, ϕ_{A}), (R_{B}, ϕ_{B}) and (R_{C}, ϕ_{C}) and the trace morphisms Tr_{B/A} and Tr_{C/B} exist. Then Tr_{C/A} exists too, and

Tr_{C/A} = Tr_{B/A} Tr_{C/B} .

The existence of the trace morphism allows to transfer good properties of R_{A} to R_{B}.

Proposition 3.9. Let A → B be a finite homomorphism of k-algebras, and assume the rigid dualizing complexes R_{A} and R_{B} and the trace morphism Tr_{B/A} exist.

1. Let C be any k-algebra. Then for M ∈ D(Mod(B ⊗ C^{∘})) there is a functorial isomorphism

R Hom_{B}(M, R_{B}) ≅ R Hom_{A}(M, R_{A})

in D(Mod(C ⊗ A^{∘})).

2. If R_{A} is Auslander then so is R_{B}.

3. For any B-module M, Cdim_{R_{A};A} M = Cdim_{R_{B};B} M.

4. If R_{A} is GKdim-Macaulay, then so is R_{B}.

5. Suppose A → B is surjective. If R_{A} is Kdim-Macaulay then so is R_{B}.

Proof. 1. We can assume R_{A} and R_{B} are complexes of injectives over A^{e} and B^{e} respectively, and Tr_{B/A} is a homomorphism of complexes. Then we get a functorial morphism R Hom_{B}(M, R_{B}) → R Hom_{A}(M, R_{A}) in D(Mod(C ⊗ A^{∘})). To prove it’s an isomorphism we can forget the C-module structure. Because the two functors are way-out in both directions (see [RD] Section I.7), and they send direct sums to direct products, it suffices to check for an isomorphism when M = B. But that’s given. 2, 3. Let M be a finitely generated B-module and N ⊂ Ext_{B}^{q}(M, R_{B}) a B^{∘}- submodule. Then by part 1, N ⊂ Ext_{A}^{q}(M, R_{A}) as A^{∘}-modules, and for every p, Ext_{B^{∘}}^{p}(N, R_{B}) ≅ Ext_{A^{∘}}^{p}(N, R_{A}) as A-modules. This proves the Auslander condition for B and the dimension equality for finitely generated B-modules.

4. Follows from part 3 and the fact GKdim_{A} M = GKdim_{B} M.

5. Similar to 4.

Definition 3.10. Suppose A has an Auslander rigid dualizing complex R. Then we denote the canonical dimension Cdim_{R} by Cdim.

Example 3.11. Suppose A is an affine k-algebra and finite over its center. Then we can find a smooth integral commutative k-algebra C (e.g. a polynomial algebra), and a finite homomorphism C → Z(A). Say Kdim C = n. Since Ω_{C/k}^{n}[n] is a rigid dualizing complex over C, it follows from [Ye3] Proposition 5.9 that

R := R Hom_{C}(A, Ω_{C/k}^{n}[n])

is a rigid dualizing complex over A.

Example 3.12. Let A be the algebra ( \begin{matrix}k & V \\ 0 & k \end{matrix} ) where V is a finite rank k-module. The rigid dualizing complex is A^{∗} := Hom_{k}(A, k). Now A is hereditary, hence Gorenstein, so the bimodule A is a dualizing complex. When V ≠ 0 the dualizing complexes A and A^{∗} are not isomorphic, so A is not rigid then.

Example 3.13. Let t_{1}, t_{2}, … be a countable sequence of commuting indeterminates and let C := k(t_{1}, t_{2}, …) be the field of rational functions. We claim that as k-algebra, C has no rigid dualizing complex. Since any dualizing complex over C has to be of the form R = C^{σ}[n] for an automorphism σ and an integer n (by [Ye3] Corollary 4.6 and Propositions 3.4 and 3.5), it suffices to prove that Ext_{C^{e}}^{i}(C, C^{e}) = 0 for all i. This follows from the next lemma with n = 0.

Lemma 3.14. Let D_{n} := C^{e}/I where I is the ideal generated by the elements f_{j} := x_{j} ⊗ 1 − 1 ⊗ x_{j} for j = 1, …, n. Then Ext_{C^{e}}^{i}(C, D_{n}) = 0 for all i, n ≥ 0.

Proof. Assume on the contrary that Ext_{C^{e}}^{i}(C, D_{n}) ≠ 0 for some i and n. Let i_{0} be the smallest such i. Since f_{n+1} is nonzero in the domain D_{n} there is a short exact sequence

0 → D_{n} ⟶^{f_{n+1}} D_{n} → D_{n+1} → 0

of C^{e}-modules. That induces an exact sequence

0 = Ext_{C^{e}}^{i_{0}−1}(C, D_{n+1}) → Ext_{C^{e}}^{i_{0}}(C, D_{n}) ⟶^{f_{n+1}} Ext_{C^{e}}^{i_{0}}(C, D_{n}).

But f_{n+1} annihilates the C^{e}-module C, which implies Ext_{C^{e}}^{i_{0}}(C, D_{n}) = 0, contradicting the choice of i_{0}.

We end the section with a basic question.

Question 3.15. Let R be a rigid dualizing complex and M an A-bimodule finitely generated on both sides. Is there a functorial isomorphism R Hom_{A}(M, R) ≅ R Hom_{A^{∘}}(M, R)? Or, at least, is Cdim_{R;A} M = Cdim_{R;A^{∘}} M?

For a partial answer turn to Section 6, where the presence of auxiliary filtrations allows us to take advantage of results in Sections 4-5 on graded algebras.

4. Dualizing Complexes over Graded Algebras

In this section we consider connected ℤ-graded k-algebras, namely algebras A = ⨁_{n≥0} A_{n} with A_{0} ≅ k and A_{n} a finitely generated k-module.

For such an algebra A let GrMod A be the category of ℤ-graded left modules, with degree 0 homomorphisms. For M, N ∈ GrMod A we write M(n) for the shifted module with M(n)_{i} = M_{n+i}, and

Hom_{A}^{gr}(M, N) := ⨁_{n∈ℤ} Hom_{GrMod A}^{gr}(M, N(n)).

There is a forgetful functor GrMod A → Mod A. Observe that Hom_{A}^{gr}(M, N) ⊂ Hom_{A}(M, N) with equality when M is finitely generated.

GrMod A is an abelian category with direct and inverse limits, enough injectives and enough projectives. Let D(GrMod A) be the derived category. The derived functors R Hom_{A}^{gr}(M, N) and M⊗_{A}^{L}N are calculated just as in the ungraded case, see Section 1, but using graded-projectives or graded-injectives.

We say M ∈ GrMod k is locally finite if each M_{n} is a finitely generated k-module. Let M^{∗} := Hom_{k}^{gr}(M, k). Denote by D_{lf}(GrMod A) the subcategory of complexes with locally finite cohomologies. Matlis duality says that M ≅ M^{∗∗} for M ∈ D_{lf}(GrMod A).

We denote by 𝔪 the augmentation ideal ⨁_{n>0} A_{n} of A, and we write Γ_{𝔪}M for the 𝔪-torsion submodule of a graded A-module M. There is a derived functor RΓ_{𝔪}, which is calculated by graded-injectives (see [Ye1]). The cohomology modules are H^{i}RΓ_{𝔪}M = lim_{n→} Ext_{A}^{i}(A/A_{≥n}, M). We write 𝔪^{∘} for the augmentation ideal of A^{∘}.

The definitions and results of the previous sections can all be translated to the graded category by adding the adjective “graded” where needed, like “graded dualizing complex,” “graded Auslander property” etc. The proofs of the graded variants of these results are identical to the ungraded ones, so there is no need to repeat them. In the rest of the paper we shall refer to such a result by writing something like “according to the graded variant of Theorem ...”.

Remark 4.1. Let G be any finitely generated abelian group. Fix an isomorphism G ≅ ℤ^{r} × T, where T is a finite group, and a basis g_{1}, …, g_{r} of ℤ^{r}. Let G_{+} be the semigroup generated by 0 and the elements g_{i} + t, 1 ≤ i ≤ r, t ∈ T. For g, g′ ∈ G we write g ≥ g′ if g − g′ ∈ G_{+}, and this defines a partial order on G. A G-graded k-algebra A is called connected if A = ⨁_{g∈G_{+}} A_{g}, A_{0} = k and each A_{g} is finitely generated as a module over k. The augmentation ideal of A is 𝔪 := ⨁_{g>0} A_{g}.

Note that the group homomorphism ϕ : G → ℤ sending g_{i} ↦ 1 makes A into a connected ℤ-graded algebra, with A_{n} = ⨁_{ϕ(g)=n} A_{g}, n ∈ ℤ.

It is not hard to see that all results in this paper which are stated for connected ℤ-graded algebras are also valid for connected G-graded k-algebras, for any G as above.

Throughout this section A and B are connected graded noetherian k-algebras.

Definition 4.2 ([Ye1] Definition 4.1). A balanced dualizing complex over A is a graded dualizing complex R such that

RΓ_{𝔪}R ≅ RΓ_{𝔪^{∘}}R ≅ A^{∗}

in D(GrMod A^{e}).

A balanced dualizing complex R is unique up to isomorphism in D(GrMod A^{e}), and its endomorphisms are just elements of k. Thus if we choose an isomorphism ϕ : RΓ_{𝔪}R →^{≃} A^{∗} in D(GrMod A^{e}), the pair (R, ϕ) is unique up to a unique isomorphism.

It had been known for some time (by [Ye1]) that balanced dualizing complexes exist for Artin-Schelter Gorenstein algebras, twisted homogeneous coordinate algebras and algebras finite over their centers. Recently additional existence results became available, due to the work of Van den Bergh. First recall the following definition taken from [AZ].

Definition 4.3. The condition χ holds for a noetherian connected graded k-algebra A if for every M ∈ GrMod_{f} A and integer i, Ext_{A}^{i}(k, M) is a finitely generated kmodule.

In view of [AZ] Proposition 3.1(3), this definition is equivalent to [AZ] Definition 3.2; and by [AZ] Proposition 3.11(2) it is equivalent to [AZ] Definition 3.7. The next lemma provides further characterization of the condition χ. Recall that a graded module M is said to be right bounded if M_{n} = 0 for n ≫ 0.

Lemma 4.4. Let A be a noetherian connected graded k-algebra and M ∈ GrMod_{f} A. Then the following are equivalent:

(i) Ext_{A}^{i}(k, M) is a finitely generated k-module for all i. (ii) H^{i}RΓ_{𝔪}M is right bounded for all i. (iii) H^{i}RΓ_{𝔪}M is an artinian A-module for all i.

Proof. (i) ⇔ (ii) is by [AZ] Corollary 3.6(3). (iii) ⇒ (ii) is immediate, since the socle of H^{i}RΓ_{𝔪}M is a finitely generated k-module, hence bounded. Finally assume (i), and let I be a minimal graded-injective resolution of M. From [Ye1] Lemma 4.3 it follows that Γ_{𝔪}I^{i} ≅ A^{∗} ⊗ Ext_{A}^{i}(k, M), which is artinian. Hence H^{i}RΓ_{𝔪}M is artinian.

In an earlier paper we proved the next theorem.

Theorem 4.5 ([YZ1] Theorem 4.2). Let A be a noetherian connected graded kalgebra. If A has a balanced dualizing complex then the condition χ holds for A and A^{∘}, and the functors Γ_{𝔪} and Γ_{𝔪^{∘}} have finite cohomological dimensions.

The converse, which is quite harder, was proved by Van den Bergh.

Theorem 4.6 ([VdB] Theorem 6.3). Let A be a noetherian connected graded kalgebra. Assume the condition χ holds for A and A^{∘}, and the functors Γ_{𝔪} and Γ_{𝔪^{∘}} have finite cohomological dimensions. Then

R_{A} := (RΓ_{𝔪}A)^{∗}

is a balanced dualizing complex.

Let us summarize some other known results related to the balanced dualizing complexes.

Theorem 4.7 (Local Duality). Let R be a balanced dualizing complex over a noetherian connected graded k-algebra A. Then for any graded k-algebra B and any M ∈ D(GrMod(A ⊗ B^{∘})) there is a functorial isomorphism

R Hom_{A}^{gr}(M, R) ≅ (RΓ_{𝔪}M)^{∗}.

This is proved by combining [VdB] Theorems 5.1 and 6.3. The theorem was first proved in [Ye1], but only for M ∈ D_{f}^{b}(GrMod A).

Proposition 4.8 ([VdB] Proposition 8.2(2)). A balanced dualizing complex R is rigid in the graded sense.

Remark 4.9. According to an exercise in [VdB] (whose only proof we know is quite involved), if I is a graded-injective A-module, then I has injective dimension ≤ 1 in Mod A. An immediate consequence of this fact is that a graded dualizing complex R over A is also an ungraded dualizing complex. The special case we need, namely that a balanced dualizing complex R is rigid in the ungraded sense, is proved by other means in Corollary 6.7.

Here is another result from [VdB]. Let us write 𝔪_{A^{e}} for the augmentation ideal of A^{e}, so 𝔪_{A^{e}} = 𝔪 ⊗ A^{∘} + A ⊗ 𝔪^{∘}.

Theorem 4.10 ([VdB] Corollary 4.8). Assume A has a balanced dualizing complex R. Let M ∈ D(GrMod A^{e}) have finitely generated cohomology modules on both sides. Then there is a functorial isomorphism

RΓ_{𝔪}M ≅ RΓ_{𝔪_{A^{e}}} M ≅ RΓ_{𝔪^{∘}}M.

We obtain the following interesting result:

Corollary 4.11. Let R be a balanced dualizing complex over A. Then there is a functorial isomorphism

R Hom_{A}(M, R) ≅ R Hom_{A^{∘}}(M, R)

for M ∈ D(GrMod A^{e}) with finitely generated cohomology modules on both sides.

We shall write Cdim_{A} instead of Cdim_{R;A} when R is the balanced dualizing complex, and when we are working in GrMod A. Since a balanced dualizing complex is rigid in the ungraded sense (by Corollary 6.7), this is consistent with Definition 3.10.

Definition 4.12. If A has a graded Auslander balanced dualizing complex R we say A is graded Auslander. Furthermore if dim is an exact dimension function on graded modules, and if R is graded dim-Macaulay, then we say A is graded Auslander dim-Macaulay.

According to [Ye1] Theorem 3.9, any two graded dualizing complexes R_{1}, R_{2} satisfy R_{2} ≅ R_{1} ⊗_{A} A^{σ}(m)[n], for an automorphism σ and an integers n, m. It follows that R_{1} is graded Auslander iff R_{2} is so. In particular, if A is graded Gorenstein, then A is graded Auslander-Gorenstein in the usual sense iff it is graded Auslander in the sense of Definition 4.12.

Taking cohomologies in the previous corollary we get:

Corollary 4.13. Suppose A is graded Auslander. Then Cdim is symmetric on graded modules. That is to say, if M is a graded A-bimodule, finitely generated on both sides, then Cdim_{A} M = Cdim_{A^{∘}} M.

If A is graded Auslander we have a bound on Krull dimension of graded modules:

Theorem 4.14. Suppose A is graded Auslander. Then

Kdim M ≤ Cdim M = sup{q | H^{q}RΓ_{𝔪}M ≠ 0} < ∞

for all finitely generated graded A-modules M.

Proof. By Theorem 4.7, if M ≠ 0, we have Cdim M = sup{q | H^{q}RΓ_{𝔪}M ≠ 0}. Next for a finitely generated graded A-module M the Krull dimension is the same when computed in GrMod A and in Mod A. And by the graded variant of Corollary 2.18 we get Kdim M ≤ Cdim_{R} M, since clearly d_{0} = 0.

Lemma 4.15. Let A → B be a finite homomorphism of noetherian connected graded k-algebras, with augmentation ideals 𝔪_{A}, 𝔪_{B}. Assume A satisfies condition χ. Then there is a functorial isomorphism

RΓ_{𝔪_{B}}M ≅ RΓ_{𝔪_{A}}M

for M ∈ D^{+}(GrMod B).

Proof. For any homomorphism A → B of graded algebras there is a functorial morphism RΓ_{𝔪_{B}}M → RΓ_{𝔪_{A}}M in D(GrMod A). By [AZ] Lemma 8.2, the extra assumptions guarantee that H^{p}RΓ_{𝔪_{B}}M → H^{p}RΓ_{𝔪_{A}}M is bijective for all p.

The following theorem is a generalization of [Jo1] Theorem 3.3.

Theorem 4.16. Let A → B be a finite homomorphism of graded k-algebras and let R_{A} be a balanced dualizing complex over A. Then:

1. B has a balanced dualizing complex R_{B}.

2. There is a morphism Tr_{B/A} : R_{B} → R_{A} in D(GrMod A^{e}), which satisfies conditions (i) and (ii) of Theorem 3.2.

Proof. From [VdB] Theorem 6.3 we know that A and A^{∘} satisfy χ, and Γ_{𝔪_{A}} and Γ_{𝔪_{A}^{∘}} = Γ_{𝔪_{A^{∘}}} have finite cohomological dimensions. So by Lemma 4.15 the same is true for B. Thus B has a balanced dualizing complex R_{B} ≅ (RΓ_{𝔪_{B}}B)^{∗}.

The morphism A → B in D(GrMod A^{e}) induces a morphism (RΓ_{𝔪_{A}}B)^{∗} → (RΓ_{𝔪_{A}}A)^{∗}, also in D(GrMod A^{e}). But (RΓ_{𝔪_{A}}B)^{∗} ≅ (RΓ_{𝔪_{B}}B)^{∗} and we get Tr_{B/A} : R_{B} → R_{A}. The isomorphism of functors RΓ_{𝔪_{A}} ≅ RΓ_{𝔪_{A}^{∘}} of [VdB] Corollary 4.8 shows that Tr_{B/A} is the same when calculated on the right, i.e., using RΓ_{𝔪_{A}^{∘}}.

Condition (i) of Theorem 3.2 is a consequence of local duality. To verify condition (ii) we again view A → B as a morphism in D(GrMod A^{e}). By [VdB] Theorems 4.7 and 5.1 we get a commutative diagram

(RΓ_{𝔪_{B}}B)^{∗} ⟶^{≅} (RΓ_{𝔪_{B^{e}}} B)^{∗} ⟶^{≅} R Hom_{B^{e}}(B, (RΓ_{𝔪_{B^{e}}} B^{e})^{∗})
↓  ↓  ↓
(RΓ_{𝔪_{A}}A)^{∗} ⟶^{≅} (RΓ_{𝔪_{A^{e}}} A)^{∗} ⟶^{≅} R Hom_{A^{e}}(A, (RΓ_{𝔪_{A^{e}}} A^{e})^{∗})

Finally by [VdB] Theorem 7.1

(RΓ_{𝔪_{A^{e}}} A^{e})^{∗} ≅ (RΓ_{𝔪_{A}}A)^{∗} ⊗ (RΓ_{𝔪_{A}^{∘}}A)^{∗},

and of course the same for B.

Applying the graded variant of Proposition 3.9 we obtain the following corollary.

Corollary 4.17. Let A and B be as in Theorem 4.16.

1. There is a functorial isomorphism

R Hom_{B}^{gr}(M, R_{B}) ≅ R Hom_{A}^{gr}(M, R_{A})

for all M ∈ D(GrMod B).

2. If A is graded Auslander then so is B.

3. Cdim_{A} M = Cdim_{B} M for M ∈ GrMod B.

4. If A is graded Auslander GKdim-Macaulay then so is B.

5. Suppose A → B is surjective. If A is graded Auslander Kdim-Macaulay, then so is B.

The next three propositions show that the graded Auslander property can be transferred from one algebra to a related algebra.

Proposition 4.18. Suppose A has a balanced dualizing complex. Let dim stand for either Kdim or GKdim.

1. Let 𝔞, 𝔟 be graded ideals. If the quotient algebras A/𝔞 and A/𝔟 are graded Auslander (resp. and graded dim-Macaulay), then so is A/𝔞𝔟.

2. Let 𝔞 be a nilpotent graded ideal of A. If A/𝔞 is graded Auslander (resp. and graded dim-Macaulay) then so is A.

3. If for every minimal graded prime ideal 𝔭 the quotient algebra A/𝔭 is graded Auslander (resp. and graded dim-Macaulay), then so is A.

Proof. 1. As usual we write D := R Hom_{A}^{gr}(−, R) where R is the balanced dualizing complex. We may assume 𝔞𝔟 = 0. Given a finitely generated graded module M consider the exact sequence 0 → 𝔟M → M → M/𝔟M → 0, and note that 𝔟M is an A/𝔞-module. For any i there is an exact sequence H^{i}D(M/𝔟M) → H^{i}DM → H^{i}D(𝔟M). Since A/𝔞 and A/𝔟 have the graded Auslander property, the subquotients of H^{i}D(M/𝔟M) and H^{i}D(𝔟M) have Cdim no more than i. Observe that here we are using Corollary 4.17. Hence by the long exact sequence of duality, submodules of H^{i}DM have Cdim no more than i. The assertion about the Macaulay property is clear.

2. Use part 1 and induction.

3. Let 𝔭_{1}, …, 𝔭_{m} be the minimal prime ideals of A. Then (∏_{i} 𝔭_{i})^{n} = 0 for some n, and we can use parts 1 and 2.

Proposition 4.19. Let A → B be a finite homomorphism of connected graded algebras, and assume B ≅ A ⊕ L as graded A-bimodules. If B has a balanced dualizing complex then so does A.

Proof. Let M be a finitely generated graded A-module. Then by Lemma 4.15 we get

RΓ_{𝔪_{B}}(B ⊗_{A} M) ≅ RΓ_{𝔪_{A}}(B ⊗_{A} M) ≅ RΓ_{𝔪_{A}}M ⊕ RΓ_{𝔪_{A}}(L ⊗_{A} M).

By Theorem 4.5 and Lemma 4.4 applied to B, we see that the graded B-module H^{i}RΓ_{𝔪_{B}}(B ⊗_{A} M) is right bounded, and vanishes for large i. Hence the same is true for the A-module H^{i}RΓ_{𝔪_{A}}M. Now apply Theorem 4.6.

We do not know if the under the assumptions of the proposition above the Auslander property can be transferred from B to A. However as shown to us by Van den Bergh this is true in a special case:

Proposition 4.20. Let G be a finite group of order prime to char k, acting on B by graded k-algebra automorphisms, and let A := B^{G} be the fixed ring. If B is graded Auslander then so is A.

Proof. Given a finitely generated graded A-module M and a graded A^{∘}-submodule N ⊂ Ext_{A}^{q}(M, R_{A}) we want to prove that Ext_{A^{∘}}^{p}(N, R_{A}) = 0 for all p < q. Let L := {b ∈ B | ∑_{g∈G} g(b) = 0}, so B = A ⊕ L. We have isomorphisms of graded A^{∘}-modules

Ext_{A}^{q}(M, R_{A}) ⊕ Ext_{A}^{q}(L ⊗_{A} M, R_{A}) ≅

Ext_{A}^{q}(B ⊗_{A} M, R_{A}) ≅ Ext_{B}^{q}(B ⊗_{A} M, R_{B})

that respect the G-action. Note that G acts trivially on Ext_{A}^{q}(M, R_{A}). Consider the graded B^{∘}-module N · B ⊂ Ext_{B}^{q}(B ⊗_{A} M, R_{B}). Clearly N · B = N + N · L. But if

n = ∑_{i} n_{i}l_{i} ∈ N ∩ (N · L)

with n_{i} ∈ N and l_{i} ∈ L, then

n = |G|^{−1} ∑_{g∈G} g (∑_{i} n_{i}l_{i}) = ∑_{i} n_{i} · |G|^{−1} ∑_{g∈G} g(l_{i}) = 0.

We conclude that N · B = N ⊕ N · L as graded A^{∘}-modules. Therefore

Ext_{A^{∘}}^{p}(N, R_{A}) ⊕ Ext_{A^{∘}}^{p}(N · L, R_{A}) ≅

Ext_{A^{∘}}^{p}(N · B, R_{A}) ≅ Ext_{B^{∘}}^{p}(N · B, R_{B}) = 0.

5. Graded Algebras with Some Commutativity Hypothesis

In this section we continue the discussion of balanced dualizing complexes over connected graded noetherian k-algebras, but now we look at algebras which have some commutativity hypothesis, like PI, FBN, or enough normal elements. The main result here is:

Theorem 5.1. Let A be a noetherian connected graded k-algebra. Suppose t ∈ A is a homogeneous normal element of positive degree, and let B := A/(t).

1. If B has a balanced dualizing complex, then so does A.

2. If in addition B is graded Auslander, then so is A.

3. If in addition B is graded Kdim-(resp. GKdim)-Macaulay, then so is A.

The “classical” case of this theorem, i.e. when B is Gorenstein and t is a regular element (i.e. a non-zero-divisor), is [Lev] Theorem 3.6. Part 1 of the theorem is a trivial consequence of Theorems 4.5 and 4.6, and [AZ] Theorem 8.8.

The proof of parts 2 and 3 appears after a series of lemmas. In these lemmas we assume that A has a balanced dualizing complex (by part 1) and B is graded Auslander. The modules M, N, … will be finitely generated graded by default. By Proposition 4.18 we can assume A is prime, hence t is a regular element. The same proposition tells us that

(5.2) A/(t^{n}) is graded Auslander for all n ≥ 1.

We denote by D the duality functor R Hom_{A}(−, R), where R is the balanced dualizing complex of A. Recall that according to Theorem 4.7, H^{−i}DM ≅ (R^{i}Γ_{𝔪}M)^{∗}. By definition Cdim M = sup{i | H^{−i}DM ≠ 0}, so trivially

(5.3) Cdim M ≤ max{Cdim M′, Cdim M/M′}

for all M′ ⊂ M.

Lemma 5.4. If M is t-torsion-free then Cdim M = Cdim M/tM + 1. If d = Cdim M then H^{−d}DM is t-torsion-free.

Proof. Let σ be the automorphism of A such that t · a = σ(a) · t, and let ^{σ}M be the corresponding twisted module. Then we have an exact sequence

0 → ^{σ^{−1}}M(−l) ⟶^{t·} M → M/tM → 0

where l is the degree of t. It is easy to see that H^{i}D(^{σ^{−1}}M(−l)) ≅ (H^{i}DM)^{σ^{−1}}(l), so there is a long exact sequence

H^{i}D(M/tM) → H^{i}DM ⟶^{·t} (H^{i}DM)^{σ^{−1}}(l) → H^{i+1}D(M/tM). (5.5)

If H^{i+1}D(M/tM) = 0 then by the graded Nakayama Lemma we get H^{i}DM = 0. Therefore

Cdim M/tM ≤ Cdim M ≤ Cdim M/tM + 1.

Now let d = Cdim M. We need to show that H^{−d}D(M/tM) = 0. If not, then Cdim M/tM = d, and hence also Cdim M/t^{n}M = Cdim t^{n−1}M/t^{n}M = d for all n ≥ 1. According to Proposition 4.18 the algebra A/(t)^{n} has the graded Auslander property. This implies that

Cdim H^{−d}D(t^{n−1}M/t^{n}M) = d and Cdim H^{−d+1}D(M/t^{n−1}M) ≤ d − 1.

Looking at the exact sequence

0 → H^{−d}D(M/t^{n−1}M) → H^{−d}D(M/t^{n}M)

→ H^{−d}D(t^{n−1}M/t^{n}M) ⟶^{ϕ} H^{−d+1}D(M/t^{n−1}M)

we see that ϕ cannot be an injection. Therefore

H^{−d}D(M/t^{n−1}M) ⫋ H^{−d}D(M/t^{n}M) ⊂ H^{−d}D(M).

But this is true for all n ≥ 1, contradicting with the noetherian property of H^{−d}D(M). The upshot is that H^{d}D(M/tM) = 0. Taking i = −d in (5.5) we conclude that H^{d}DM is t-torsion-free.

Lemma 5.6. Let N be the t-torsion submodule of M. Then

Cdim M = max{Cdim N, Cdim M/N}.

Proof. Let d be the right hand side. Trivially Cdim M ≤ d holds. Assume Cdim M < d. Dualizing the exact sequence 0 → N → M → M/N → 0 we obtain a long exact sequence

⋯ → H^{i}D(M/N) → H^{i}DM → H^{i}DN → H^{i+1}D(M/N) → H^{i+1}DM → ⋯

and taking i = −d − 1 we see that H^{−d}D(M/N) = 0, so Cdim M/N < d. Hence Cdim N = d. Also taking i = −d we see that H^{−d}DN ⊂ H^{−d+1}D(M/N). Now H^{−d}DN is t-torsion, yet by Lemma 5.4, H^{−d+1}D(M/N) is t-torsion-free. We conclude that H^{−d}DN = 0, which is a contradiction.

Lemma 5.7.

Cdim M = max{Cdim L, Cdim M/L}

for every L ⊂ M.

Proof. By (5.3) it remains to prove “≥”. Let N be the t-torsion submodule of M. By (5.2) we have Cdim N ≥ Cdim N ∩ L. On the other hand by Lemma 5.6, Cdim M = max{Cdim N, Cdim M/N} and Cdim L = max{Cdim N ∩ L, Cdim L/N ∩ L}. Hence it suffices to prove that Cdim M/N = max{Cdim L/N ∩ L, Cdim M/L}. So we may assume M = M/N is t-torsion-free.

If P := M/L is also t-torsion-free then we get a short exact sequence

0 → L/tL → M/tM → P/tP → 0.

Hence the assertion follows from Lemma 5.4.

In general let L′ ⊃ L be such that L′/L is the t-torsion submodule of M/L. So t^{n}L′ ⊂ L for some n. By Lemma 5.4,

Cdim L′/L ≤ Cdim L′/t^{n}L′ = Cdim L′ − 1

and hence, from the long exact sequence of duality, we get Cdim L = Cdim L′. Applying the previous paragraph and Lemma 5.6 we have

Cdim M = max{Cdim L′, Cdim M/L′}
= max{Cdim L, Cdim M/L′}
= max{Cdim L, Cdim L′/L, Cdim M/L′}
= max{Cdim L, Cdim M/L}

and we finish the proof.

Lemma 5.8. Suppose t^{n} kills the t-torsion submodule of M. Then

Cdim M ≤ Cdim M/t^{n}M + 1.

Proof. Let N be the t-torsion submodule of M and P := M/N. Then t^{n}M ≅ t^{n}P is an t-torsion-free submodule of M and P/t^{n}P ≅ M/(N ⊕ t^{n}M). By Lemmas 5.4 and 5.6,

Cdim M = max{Cdim N, Cdim P } = max{Cdim N, Cdim P/t^{n}P + 1}.

On the other hand by Lemma 5.7,

Cdim M/t^{n}M = max{Cdim N, Cdim P/t^{n}P }.

It remains to combine these equalities.

Proof of Theorem 5.1. As mentioned above part 1 is a consequence of Theorems 4.5 and 4.6, and [AZ] Theorem 8.8.

2. Recall that the Auslander condition says that if N ⊂ H^{−i}DM then Cdim N ≤ i. We can assume A is prime and t is regular. By Lemma 5.7 it suffices to prove that Cdim H^{−i}DM ≤ i for all i. According to (5.2) the inequality holds for t-torsion modules, so using the long exact sequence of duality we may assume that M is t-torsion-free.

Choose n such that t^{n} kills the t-torsion submodule of H^{−i}DM. The short exact sequence

0 → M ⟶^{t^{n}·} ^{σ^{n}}M(nl) → ^{σ^{n}}(M/t^{n}M)(nl) → 0

gives rise to a long exact sequence

⋯ → (H^{−i}DM)^{σ^{n}}(−nl) ⟶^{t^{n}·} H^{−i}DM → P = (H^{−i+1}D(M/t^{n}M))^{σ^{n}}(−nl) → · · · .

Since M/t^{n}M is t-torsion, P is also t-torsion, and every submodule of P has Cdim ≤ i − 1. So according to Lemma 5.8, Cdim H^{−i}DM ≤ (i − 1) + 1 = i.

3. Assume B is graded GKdim-Macaulay. Since A/(t^{n}) is graded GKdim-Macaulay for n ≥ 1 (by Proposition 4.18), it suffices to show that Cdim M = GKdim M for t-torsion-free modules. We know that in this case GKdim M = GKdim M/tM + 1, and by Lemma 5.4 this is true also for Cdim M.

Finally assume B is graded Kdim-Macaulay. Again we need only consider M which is t-torsion-free. It is clear that Kdim M ≥ Kdim M/tM + 1. Hence it follows from Lemmas 5.4 and 5.7 that Kdim M ≥ Cdim M. But by Theorem 4.14, Kdim M ≤ Cdim M.

Example 5.9. Let us consider a simple case of Theorem 7.1. Let A = B[t] where t is a central variable of degree 1. Let R_{B} be the balanced dualizing complex of B. Then the balanced dualizing complex of A is nothing but

R_{A} = R_{B} ⊗_{k[t]} Ω_{k[t]/k}^{1}[1] ≅ R_{B}[t](−1)[1]

where Ω_{k[t]/k}^{1} = k[t] · dt ≅ k[t](−1) as graded k[t]-modules. This follows from [VdB] Theorem 7.1, which states that if B, C are noetherian connected graded kalgebras with balanced dualizing complexes R_{B} and R_{C} respectively, and if B⊗C is noetherian, then R_{B} ⊗R_{C} is a balanced dualizing complex over B⊗C. By Theorem 5.1, A is graded Auslander and Kdim-(resp. GKdim-)Macaulay if and only if B is.

Corollary 5.10. Let A be a connected graded k-algebra with a balanced dualizing complex. Suppose that every graded prime quotient A/𝔭 satisfies one of the following conditions:

(i) A/𝔭 is graded Auslander (resp. and graded Kdim-(or GKdim)-Macaulay); or (ii) A/𝔭 has a normal element of positive degree. Then A is graded Auslander (resp. and graded Kdim-(or GKdim-)Macaulay)

Proof. Use Proposition 4.18, Theorem 5.1 and induction on Kdim A.

Recall that A has enough normal elements if every graded prime quotient A/𝔭 (except for 𝔭 = 𝔪) contains a normal element of positive degree. By Corollary 5.10, a noetherian connected graded algebra with enough normal elements is Auslander and GKdim-Macaulay. In the rest of this section we generalize this statement.

Corollary 5.11. Let A and B be noetherian connected graded k-algebras. Assume A has a balanced dualizing complex and is graded Auslander (resp. and graded Kdim(or GKdim)-Macaulay), and B has enough normal elements. Then A ⊗ B is a noetherian connected graded k-algebra with a balanced dualizing complex, and it is graded Auslander (resp. and graded Kdim(or GKdim)-Macaulay).

Proof. We first prove that A ⊗ B is noetherian by induction on Kdim B. Suppose {𝔭_{1}, ⋯, 𝔭_{n}} is the complete set of minimal graded primes of B. Then some product ∏_{i=1}^{t} 𝔭_{n_{i}} is zero. For any s ≥ 1 let W_{s} := ∏_{i=1}^{s−1} 𝔭_{n_{i}}/ ∏_{i=1}^{s} 𝔭_{n_{i}}, where W_{1} = B/𝔭_{n_{1}}, and let B_{s} = B/𝔭_{n_{s}}. Since W_{s} is a finitely generated B_{s}-module, A ⊗ W_{s} is a finitely generated A ⊗ B_{s}-module. Hence it suffices to show that each A ⊗ B_{s} is noetherian. This reduces to the case when B is prime. Hence we may assume B has a regular normal element t of positive degree. It is obvious that 1 ⊗ t is a regular normal element in A ⊗ B. By induction hypothesis, A ⊗ B/(t) is noetherian, and therefore A ⊗ B is noetherian by [ATV] Theorem 8.2.

The graded Auslander and Macaulay properties follow from the same inductive procedure and Theorem 5.1.

In [SZ] Theorem 3.10 it was shown that a connected graded PI algebra of finite injective dimension (i.e. a Gorenstein algebra) is graded Auslander-Gorenstein and graded GKdim-Macaulay. This was extended in [Zh] to an algebra A having enough normal elements. Corollary 5.11 (when A = k) generalizes these theorems by eliminating the Gorenstein condition. We extend the result further in Theorems 5.13 and 5.14 below.

Definition 5.12. Let R be a balanced dualizing complex over A.

1. We say two graded A-modules M and N are similar if there are isomorphisms M ≅ N and R Hom_{A}^{gr}(M, R) ≅ R Hom_{A}^{gr}(N, R) in D(GrMod k).

2. The algebra A satisfies the similar submodule condition if every nonzero, 𝔪torsion-free, finitely generated, graded A-module M has graded submodules N′ ⫋ N ⊂ M with N′ similar to N(−l) for some l > 0.

We remark that two complexes M, N ∈ D_{f}^{b}(GrMod A) are isomorphic in D(GrMod k) if and only if they have equal Hilbert functions, namely rank_{k}(H^{i}M)_{j} = rank_{k}(H^{i}N)_{j} for all i, j ∈ ℤ. This definition of similar submodule condition is equivalent to the definition given in [Zh] Section 2 when A is AS-Gorenstein, which is the case considered there. Also, as mentioned in [Zh] Section 2, there are algebras which do not satisfy the similar submodule condition.

Theorem 5.13. Assume A is a noetherian connected graded k-algebra and satisfies one of the following:

(i) A is a PI algebra. (ii) A is graded-FBN. (iii) A has enough normal elements. Then A has a balanced dualizing complex, and the similar submodule condition holds.

Note that if A is PI, then A is graded FBN and has enough normal elements.

Proof. The existence of a balanced dualizing complex follows Theorem 4.6, together with [AZ] Theorems 8.8 and 8.13. The statement about the similar submodule condition is proved like in [Zh] Section 2 (first paragraph and Proposition 2.3).

We now prove a generalized version of [Zh] Theorem 3.1.

Theorem 5.14. Let A be a noetherian connected graded k-algebra which has a balanced dualizing complex R and satisfies the similar submodule condition. Then

1. A is graded Auslander.

2. A is graded Kdim- and GKdim-Macaulay. More precisely

Cdim_{R} M = Kdim M = GKdim M

for every finitely generated graded left or right A-module M.

Proof. First we observe that [Zh] Lemma 2.2 holds (same proof), and hence Kdim M ≥ GKdim M and GKdim M < ∞ for every finitely generated graded A-module M. Therefore, replacing Ext^{i}(−, A) with Ext_{A}^{i}(−, R), the proof of [Zh] Theorem 3.1 can be copied verbatim. Let us just mention the key point of the proof. We prove by induction on GKdim M that for every finitely generated graded A-module (resp. A^{∘}-module) M:

(a) j_{R}(M) = − GKdim M.

(b) GKdim Ext_{A}^{j}(M, R) = GKdim M, where j = j_{R}(M).

(c) For every j_{R}(M) ≤ i ≤ 0 one has GKdim Ext_{A}^{i}(M, R) ≤ −i. This implies that A is graded Auslander and graded GKdim-Macaulay. By [Zh] Lemma 2.2, Kdim M ≥ GKdim M = Cdim M, and by Theorem 4.14, Kdim M ≤ Cdim M. Hence A is graded Kdim-Macaulay.

Theorem 0.6 is an immediate consequence of Theorems 5.13 and 5.14. Another consequence is Proposition 0.9 which we now prove.

Proof of Proposition 0.9. If A is connected graded, i.e., A_{0} = k, the assertion follows from Theorems 5.13 and 5.14. Now assume A is not connected. Let B := k + A_{≥1}, which is connected. Since A and B differ by a finite rank k-module, B is noetherian. It remains to verify the following statements.

(a) If M is a finitely generated graded A-module, then Kdim_{A} M = Kdim_{B} M and GKdim_{A} M = GKdim_{B} M.

(b) If A has enough normal elements, then so does B.

(c) If A is graded FBN, then so is B.

(a) The statement about GKdim is obvious because GKdim is determined by the degree of the Hilbert function of M (see [Zh] Lemma 2.2(1)). Next we consider Kdim. Clearly Kdim_{A} M = 0 if and only if M is a finitely generated k-module, if and only if Kdim_{B} M = 0. For higher dimension we consider the quotient category QGr A := GrMod A/ M_{0}, where M_{0} = M_{0}(Kdim) is the localizing subcategory consisting of Kdim = 0 modules (torsion modules in the terminology of [AZ]). For any M ∈ GrMod_{f} A one has

Kdim_{A} M = Kdim_{GrMod A} M = Kdim_{QGr A} M + 1.

Now since A and B differ by a finite rank k-module, QGr A is equivalent to QGr B by [AZ] Proposition 2.5, so Kdim_{QGr A} M = Kdim_{QGr B} M.

(b) Let 𝔭 be a graded prime ideal of B which is not 𝔪_{B} = B_{≥1}. Then A𝔭A and 𝔭 differ by a finitely generated k-module, and hence GKdim A/A𝔭A = GKdim B/𝔭. Let 𝔮 be a graded prime of A minimal over A𝔭A such that GKdim A/𝔮 = GKdim B/𝔭. Then the map B/𝔭 → A/𝔮 is injective, and bijective in positive degrees. Thus normal elements of positive degree in A/𝔮 are also normal elements in B/𝔭.

(c) The proof is similar to (b) and we leave it to the reader.

6. Noetherian Connected Filtrations

In this section we use filtrations to transfer results of Sections 4-5 on connected graded algebras to non-graded algebras. Throughout the section A denotes a noetherian k-algebra.

Suppose a k-module M is given an increasing filtration F = {F_{n}M}_{n∈ℤ} with ⋂_{n} F_{n}M = 0 and ⋃_{n} F_{n}M = M. The Rees module is the graded k[t]-module

Rees^{F} M := ⨁_{n∈ℤ} F_{n}M · t^{n} ⊂ M[t, t^{−1}]

where t is a central indeterminate. It is easy to check that

(Rees^{F} M)/(t − 1) · (Rees^{F} M) ≅ M

and

(Rees^{F} M)/t · (Rees^{F} M) ≅ gr^{F} M = ⨁_{n∈ℤ} F_{n}M/F_{n−1}M.

Now let F = {F_{n}A}_{n∈ℤ} be a filtration of A such that F_{n}A·F_{m}A ⊂ F_{n+m}A. The graded k-algebra Rees^{F} A is called the Rees algebra.

Definition 6.1. If the Rees algebra Rees^{F} A is a noetherian connected graded kalgebra then F is called a noetherian connected filtration.

Observe that Rees^{F} A is connected graded iff F_{0}A ≅ k, F_{−1}A = 0 and rank_{k} F_{n}A < ∞. If Rees^{F} A is noetherian then so is the associated graded algebra gr^{F} A. By [ATV] Theorem 8.2 the converse is also true – if gr^{F} A is noetherian then so is Rees^{F} A.

A filtration {F_{n}A} on an A-module M with F_{n}A·F_{m}M ⊂ F_{n+m}M gives a graded module Rees^{F} M over Rees^{F} A. We say {F_{n}A} is a good filtration if Rees^{F} M is a finitely generated (Rees^{F} A)-module.

The main result of this section is the following theorem. Part 1 is due to Van den Bergh ([VdB] Theorem 8.6), but there was a subtle flaw in his statement: the shift by −1 was missing.

Theorem 6.2. Let F be a noetherian connected filtration on A and let Ã := Rees^{F} A.

1. If Ã has a balanced dualizing complex R̃ then

R := (A ⊗Ã R̃ ⊗Ã A)[−1]

is a rigid dualizing complex over A.

2. If R̃ is graded Auslander then R is Auslander.

3. Suppose R̃ is graded Auslander. If R̃ is graded GKdim-Macaulay then R is GKdim-Macaulay.

The proof comes after this lemma.

Consider the functor π : GrMod k[t] → Mod k, M̃ ↦ M̃/(t − 1)M̃. Write Ã_{t} := A[t, t^{−1}].

Lemma 6.3. 1. The functor π is exact.

2. If Ĩ ∈ GrMod Ã is injective then πĨ ∈ Mod A is injective.

3. There is a functorial isomorphism

R Hom_{A}(πM̃, πÑ) ≅ πR Homgr ; Ã(M̃, Ñ)

for M̃, Ñ ∈ D_{f}^{b}(GrMod Ã).

Proof. 1, 2. π is a composition of the functors “localization by t” (−)_{t} : GrMod Ã → GrMod Ã_{t} and “taking degree 0 component” (−)_{0} : GrMod Ã_{t} → Mod A, both of which are exact. The second is even an equivalence, so injectives go to injectives. Since Ã_{t} is noetherian a standard argument shows that the Ã_{t}-module Ĩ_{t} is injective.

2. There is a functorial morphism ψ : πR Homgr ; Ã(M̃, Ñ) → R Hom_{A}(πM̃, πÑ). Fixing Ñ these are way-out right contravariant functors of M̃. By [RD] Proposition I.7.1(iv) – reversed – it’s enough to check that ψ is an isomorphism when M̃ = Ã, and then it’s trivial.

Proof of Theorem 6.2. 1. By Proposition 4.8 and the graded version of Corollary 3.6 each cohomology module H^{q}R̃ is k[t]-central (cf. Remark 6.6 below). Define

R̃_{t} := Ã_{t} ⊗Ã R̃ ⊗Ã Ã_{t} ∈ D(GrMod(Ã_{t})^{e}).

We see that the homomorphisms Ã_{t} ⊗Ã R̃ → R̃_{t} and R̃ ⊗Ã Ã_{t} → R̃_{t} are quasiisomorphisms.

Because Ã → A = πÃ is flat we also have

A ⊗Ã R̃[−1] ≅ R̃[−1] ⊗Ã A ≅ R ∈ D(Mod Ã^{e}).

Considering only Ã-modules we have R ≅ πR̃[−1], so by Lemma 6.2, R has finite injective dimension and finitely generated cohomologies over A. By symmetry this is true also over A^{∘}. Part 3 of the lemma implies that R Hom_{A}(R, R) ≅ πÃ = A, and likewise R Hom_{A^{∘}}(R, R) = A. We conclude that R is dualizing.

Now let’s prove R is rigid. There is a functorial isomorphism πM̃ ≅ (M̃_{t})_{0} for M̃ ∈ GrMod k[t], and the algebra A^{e} is ℤ^{2}-graded. Therefore we get

A^{e} ⊗Ã^{e} (R̃ ⊗ R̃) ≅ (A ⊗Ã R̃) ⊗ (R̃ ⊗Ã A) ≅ (R̃_{t})_{0} ⊗ (R̃_{t})_{0} ≅ (R̃_{t} ⊗ R̃_{t})_{(0,0)}.

The algebra (Ã_{t})^{e} is strongly ℤ-graded, and its degree 0 component is

(Ã_{t})_{0}^{e} ≅ A^{e}[s, s^{−1}] ≅ k[s, s^{−1}] ⊗ A^{e}

where s := t ⊗ t^{−1}. Applying (Ã_{t})^{e} ⊗Ã^{e} − to

R̃ ≅ R Homgr ; Ã^{e}(Ã, R̃ ⊗ R̃)

we obtain

R̃_{t} ≅ R Hom_{(Ã_{t})^{e}}^{gr}(Ã_{t}, R̃_{t} ⊗ R̃_{t})

and taking degree 0 components we get

R[1] ≅ R Hom_{(Ã_{t})_{0}^{e}}^{gr}(A, (R̃_{t} ⊗ R̃_{t})_{0})
≅ R Hom_{k[s,s^{−1}]⊗A^{e}}(A, k[s, s^{−1}] ⊗ (R[1] ⊗ R[1])).

But

k ≅ (k[s, s^{−1}] ⟶^{s−1} k[s, s^{−1}]) ∈ D(Mod k[s, s^{−1}]),

and A is a finitely presented A^{e}-module, so

R Hom_{k[s,s^{−1}]⊗A^{e}}(k ⊗ A, k[s, s^{−1}] ⊗ M) ≅ R Hom_{A^{e}}(A, M)[−1]

for any complex of A^{e}-modules M.

2. Let M̃ be a t-torsion-free finitely generated graded Ã-module. We claim that

j_{R;A}(πM̃) = j_{R̃;Ã}M̃ + 1. (6.4)

First by Lemma 6.3 we get for any q:

Ext_{A}^{−q+1}(πM̃, R) ≅ π Ext−q ; Ã(M̃, R̃), (6.5)

so j_{R;A}(πM̃) ≥ j_{R̃;Ã}M̃ + 1. Now take q := −j_{R̃;Ã}M̃ and write Ñ := Ext−q ; Ã(M̃, R̃). If πÑ = 0 then t^{l}Ñ = 0 for some l > 0. But in the Ext spectral sequence converging to M̃ (see proof of Theorem 2.10) the dominant term is Ext−q ; Ã^{∘}(Ñ, R̃) which is killed by t^{l}. We get CdimR̃ t^{l}M̃ < CdimR̃ M̃ = q, which is absurd since t^{l}M̃ ≅ M̃(−l).

Given finitely generated A-modules M ⊂ N, take any good filtration {F_{n}M}, and let F_{n}N := N ∩ F_{n}M, M̃ := Rees^{F} M and Ñ := Rees^{F} N. Since M̃ and Ñ are t-torsion-free we see that

Cdim_{R} M = CdimR̃ M̃ − 1 ≤ CdimR̃ Ñ − 1 = Cdim_{R} N.

Finally let L := Ext_{A}^{−q}(M, R). Then, with M̃ := Rees^{F} M and L̃ := Extq−1 ; Ã(M̃, R̃), we see that j_{R;A}L ≥ j_{R̃;Ã}L̃ + 1 ≥ −q, verifying the Auslander condition on one side. By symmetry it holds also on the other side.

3. By the proof of part 2, given a finitely generated A-module M one has Cdim_{R;A} M +1 = Cdim_{R̃;Ã} M̃ where M̃ := Rees^{F} M w.r.t. any good filtration {F_{n}M}. But because M̃_{t} ≅ M ⊗ k[t, t^{−1}] we also have

GKdim_{A} M + 1 = GKdimÃ_{t} M̃_{t} = GKdimÃ M̃.

Remark 6.6. It is not too hard to show that R could be chosen to be a k[t]-central complex of graded A-bimodules.

Observe that the rigid dualizing complex R has H^{q}R = 0 for q > 0, since H^{0}R̃ is t-torsion and H^{q}R̃ = 0 for q > 0.

Corollary 6.7. Suppose A is connected graded and R is a balanced dualizing complex over A. Then R is a rigid dualizing complex over A in the ungraded sense.

Proof. First let us note that for any graded A-module M we can define a filtration F_{n}M := ⨁_{i≤n} M_{i}. Then we have a functorial isomorphism of graded A[t]-modules

Rees^{F} M →^{≃} M[t].

In particular we get Ã ≅ A[t], so by Example 5.9 we know that R̃ = R[t](−1)[1] is the balanced dualizing complex of Ã. But then πR̃[−1] = πR[t](−1) ≅ R in D(Mod A^{e}), and this is rigid by Theorem 6.2(1).

The next corollary implies Theorem 0.7.

Corollary 6.8. Suppose A has a noetherian connected filtration F, and let Ā := gr^{F} A.

1. If Ā has a balanced dualizing complex R̄, then A has a rigid dualizing complex R.

2. If R̄ is graded Auslander then R is Auslander.

3. If R̄ is also graded GKdim-Macaulay, then R is GKdim-Macaulay.

Proof. According to Theorem 5.1, the Rees algebra Ã = Rees^{F} A inherits these properties from Ā. And by Theorem 6.2 they pass to A.

Corollary 6.9. Suppose A has a noetherian connected filtration F, and Ā := gr^{F} A satisfies either of the following:

(i) Ā is a PI algebra. (ii) Ā is graded-FBN. (iii) Ā has enough normal elements. Then A has an Auslander, GKdim-Macaulay, rigid dualizing complex.

Proof. Combine Corollary 6.8 and Theorems 5.13 and 5.14.

Here are some examples of algebras which admit noetherian connected filtrations.

Example 6.10. If A is a noetherian connected graded algebra, then the filtration F_{n}A = ⨁_{i≤n} A_{i} is a noetherian connected filtration.

Example 6.11. Suppose A is generated by elements x_{1}, …, x_{n}, and for every i ≠ j there is some relation

x_{j}x_{i} = q_{i,j}x_{i}x_{j} + a_{i,j}x_{i} + b_{i,j}x_{j} + c_{i,j} (6.12)

with q_{i,j}, a_{i,j}, b_{i,j}, c_{i,j} ∈ k. Let V := k + ∑ k · x_{i} ⊂ A and define a filtration F_{n}A := V^{n}. Then gr^{F} A is a quotient of the skew polynomial algebra k_{q}[x_{1}, …, x_{n}], so F is a noetherian connected filtration. Furthermore gr^{F} A has enough normal elements (namely the x_{i}); so Corollary 6.9 holds. It is easy to check that the relations (6.11) are satisfied in the following classes of algebras:

(i) Commutative affine algebras. (ii) Weyl algebras, enveloping algebras of finite dimensional Lie algebras and their

quotients. (iii) Most classes of quantum algebras listed in [GL].

Recall that a homomorphism f : A → B is called finite if B is a finite left and right A-module. f is centralizing if B = A · Z_{B}(A). Thus f is finite centralizing iff there exist b_{1}, …, b_{m} ∈ Z_{B}(A) such that B = ∑ A · b_{i}.

Lemma 6.13. Suppose f : A → B is a finite centralizing homomorphism and F is a noetherian connected filtration on A. Then there is a noetherian connected filtration F on B such that f preserves the filtrations and Rees^{F} (f) : Rees^{F} A → Rees^{F} B is finite.

Proof. Let b_{1}, …, b_{m} be elements of B which commute with A and B = ∑ A · b_{i}. Choose elements a_{i,j,l} ∈ A such that b_{i}b_{j} = ∑_{l} a_{i,j,l}b_{l}. Let n_{0} > 0 be large enough such that all a_{i,j,l} are in F_{n_{0}}A. Define

F_{n}B := F_{n}A · 1 + ∑_{i} F_{n−n_{0}}A · b_{i} ⊂ B.

Clearly this is a connected filtration. Since the elements (1, b_{1}, …, b_{m}) determine a surjective bimodule homomorphism

Rees^{F} A ⊕ (Rees^{F} A)^{m}(−n_{0}) ↠ Rees^{F} B

we see that Rees^{F} B is noetherian.

Example 6.14. If A is an affine k-algebra finite over its center, then there is a finite centralizing homomorphism k[t_{1}, …, t_{n}] → A from a commutative polynomial algebra. By the lemma and Example 6.10, A has a noetherian connected filtration. Thus A satisfies the assumptions of Corollary 6.9.

Example 6.15. Here is an example of a prime PI algebra A which is not finite over its center yet has a noetherian connected filtration (Schelter’s Example, [Ro] p. 492 Exercise 27). Let t, λ_{1}, λ_{2} be commuting indeterminates of degree 1. Define

\begin{aligned}C̃ & := ℚ[\sqrt{2}, \sqrt{3}, λ_{1}, λ_{2}, t] \\ C̃_{1} & := ℚ[\sqrt{6}, t\sqrt{2} + λ_{1}, λ_{2}, λ_{2}\sqrt{2}, t] \\ C̃_{2} & := ℚ[\sqrt{6}, t\sqrt{3} + λ_{1}, λ_{2}, λ_{2}\sqrt{2}, t] \\ M̃ & := C̃_{1}λ_{2} + C̃_{1}λ_{2}\sqrt{2} ⊂ C̃ \\ Ã & := [\begin{matrix}C̃_{1} & M̃ \\ M̃ & C̃_{2} \end{matrix}] . \end{aligned}

Then Ã is a noetherian graded algebra and Ã_{0} is finite over k := ℚ. The quotient A := Ã/(t−1) acquires a filtration {F_{n}A}, and if we modify it by setting F_{0}A := ℚ this becomes a connected filtration. But A is not finite over its center.

Question 6.16. Does every noetherian affine PI k-algebra admit a noetherian connected filtration? This seems to be a hard question. A similar one was posed by M. Lorenz over ten years ago (see [Lo] p. 436).

In Section 3 we found that rigid dualizing complexes are sometimes functorial w.r.t. finite algebra homomorphisms, via the trace morphism. Here is such an instance:

Theorem 6.17. Let A → B be a finite centralizing homomorphism. Suppose A has a noetherian connected filtration F and gr^{F} A has a balanced dualizing complex. Then A and B have rigid dualizing complexes R_{A} and R_{B} respectively, and the trace morphism Tr_{B/A} : R_{B} → R_{A} of Definition 3.7 exists.

Proof. By Lemma 6.13 we get a finite homomorphism of graded algebras Ã = Rees^{F} A → B̃ = Rees^{F} B. So according to Theorem 4.16 the trace morphism Tr_{B̃/B̃} : RB̃ → RÃ exists. Now apply the functor π and use Theorem 6.2.

For applications of this result see Proposition 3.9.

Let σ be a k-algebra automorphism A. Recall that A^{σ} is the invertible bimodule with basis e satisfying e · a = σ(a) · e. A noetherian connected graded k-algebra B is called AS-Gorenstein if B satisfies χ and the bimodule B has finite injective dimension on both sides. Here “AS” stands for Artin-Schelter. We say B is ASregular if B is AS-Gorenstein and gl. dim B < ∞.

Proposition 6.18. Suppose A has a noetherian connected filtration F and gr^{F} A is AS-Gorenstein (resp. AS-regular). Then the following statements hold.

1. A is a Gorenstein (resp. regular) algebra.

2. The rigid dualizing complex of A is R = A^{σ}[n] where n is an integer and σ is some k-algebra automorphism of A.

3. If gr^{F} A is graded Auslander (resp. and graded GKdim-Macaulay), then A is Auslander-Gorenstein (resp. and Cohen-Macaulay) in the sense of [Bj].

4. Let B = A/𝔞 be any quotient algebra, M any B-module and q an integer. Then the twisted module Ext_{A}^{q}(M, A)^{σ} is a B^{∘}-module.

Proof. 1, 2. Let n be the injective dimension of gr^{F} A, and let Ã := Rees^{F} A. By [Lev] Theorem 3.6 the injective dimension of Ã is n + 1. Since Ã satisfies χ it is AS-Gorenstein. So the balanced dualizing complex of Ã is R̃ = Ãσ̃(d)[n + 1] for some graded automorphism σ̃ and for some integer d. By Theorem 6.2 the rigid dualizing complex of A is R = πÃσ̃[n]. Since Ãσ̃ = H^{−n−1}R̃ this is k[t]-central; so σ̃(t) = t. We see there is an induced automorphism σ of A and R = A^{σ}[n].

If gr A has finite global dimension, then so does A.

3. This follows from 2 and Theorems 5.1 and 6.2.

4. It follows from Theorem 6.17 and Proposition 3.9(1) that

Ext_{A}^{q}(M, A)^{σ} ≅ Ext_{A}^{q−n}(M, A^{σ}[n]) ≅ Ext_{A}^{q−n}(M, R) ≅ Ext_{B}^{q−n}(M, R_{B})

where R_{B} is the rigid dualizing complex of B.

Example 2.3 shows that Proposition 6.18(2,3) might fail even if A is Gorenstein. The next example shows that σ could be nontrivial.

Example 6.19. Let A be the quantum plane k_{q}[x, y] := k⟨x, y⟩/(yx − qxy) for q ∈ k^{×} with q^{2} ≠ 1. The automorphism σ in Proposition 6.18(4) is σ(x) = qx, σ(y) = q^{−1}y (cf. [Ye1] Examples 6.21 and 7.14). Consider the ideal I = A·f·A where f := x−y, which is not normal. An easy computation shows that B ≅ k[ϵ] with ϵ^{2} = 0, and ϵ ≡ x ≡ y (mod I). Now consider the graded A^{e}-module N := Ext_{A}^{2}(B, A). One has N^{σ}(−2) ≅ B^{∗} as A^{e}-modules, so N is killed by qx − q^{−1}y = σ(f) ∈ A^{∘}, and hence N can not be a B^{∘}-module.

Example 6.20. Let A be the Weyl algebra k⟨x, y⟩/(xy−yx−1). Take the standard filtration F_{n}A = (k + kx + ky)^{n}. Then the Rees algebra Ã is generated by x, y, t with t central and xy − yx = t^{2}. Ã is an Artin-Schelter regular algebra of global dimension 3, so its balanced dualizing complex is Ãσ̃(−3)[3] for some automorphism σ̃. Let B̃ := Ã/(t^{2}), which is a commutative AS-Gorenstein algebra. As in [Ye1] Theorem 7.18 we find that σ̃ = 1. Therefore the rigid dualizing complex of A is A[2]. Observe that Cdim A = 2 = GKdim A. In [Ye4] we prove the more general statement that if C is any smooth integral commutative k-algebra of dimension n, char k = 0 and A := D(C) is the ring of differential operators, then the rigid dualizing complex of A is A[2n].

Suppose A has a noetherian connected filtration F. A two-sided good filtration on a bimodule M is a filtration {F_{n}M} such that F_{n}A·F_{m}M ⊂ F_{n+m}M, F_{n}M ·F_{m}A ⊂ F_{n+m}M and Rees^{F} M is a finitely generated (Rees^{F} A)-module on both sides.

Proposition 6.21. Assume A has a noetherian connected filtration and gr A has a balanced dualizing complex. Let R be the rigid dualizing complex of A. If a bimodule M has a two-sided good filtration then

R Hom_{A}(M, R) ≅ R Hom_{A^{∘}}(M, R).

Proof. Let M̃ := Rees^{F} M. According to Corollary 4.17 there is an isomorphism

R HomÃ(M̃, R̃) ≅ R HomÃ^{∘}(M̃, R̃)

in D(GrMod A^{e}), where R̃ is the balanced dualizing complex of Ã. Since M̃ is k[t]-central we can apply the functor π.

Recall the notion of weakly symmetric dimension function (Definition 2.20).

Corollary 6.22. Assume A has an Auslander rigid dualizing complex, and a noetherian connected filtration such that gr A has a balanced dualizing complex. Then Cdim is weakly symmetric.

Proof. As can be readily verified, the class of A-bimodules which admit two-sided good filtrations is closed under submodules, quotients and finite direct sums. Given a bimodule M which is a subquotient of A, Proposition 6.21 applies and hence Cdim_{A} M = Cdim_{A^{∘}} M.

We can now give the

Proof of Theorem 0.1. By Corollary 6.9, A has an Auslander rigid dualizing complex R, and by Corollary 6.22, Cdim_{R} is weakly symmetric (this also follows from the GKdim-Macaulay property). Now use Theorem 2.23.

The next theorem has the same conclusions as [ASZ] Theorem 6.1, but our assumptions are much more focused. Let 𝔫 be the prime radical of A. Recall that 𝔫 is said to be weakly invariant w.r.t. an exact dimension function dim if dim 𝔫 ⊗_{A} M < dim A/𝔫 = dim A for every finitely generated A-module M with dim M < dim A (and the same for right modules); cf. [MR] 6.8.13. A ring is called quasi-Frobenius if it is artinian and self injective.

Theorem 6.23. Let A be an Auslander-Gorenstein noetherian k-algebra of injective dimension n. Assume A has a filtration such that gr A is an AS-Gorenstein noetherian connected graded k-algebra. Then

1. The prime radical 𝔫 is weakly invariant.

2. If 𝔭 is a minimal prime then Cdim A/𝔭 = n.

3. A has a quasi-Frobenius ring of fractions.

Proof. By Proposition 6.18 the rigid dualizing complex of A is R = A^{σ}[n]. According to Corollary 6.22, Cdim_{R} is weakly symmetric. Now the function denoted δ in [ASZ] Theorem 6.1 coincides with Cdim_{R}, so all assumptions of that theorem hold.

The next theorem is due to Gabber in the case when gr A is Auslander-Goren-stein, and an elegant proof was communicated to us by Van den Bergh. We extend the result by dropping the Gorenstein condition.

Theorem 6.24. Let A be a filtered k-algebra such that gr A is a noetherian connected graded k-algebra with graded Auslander balanced dualizing complex. Given a Cdim-pure A-module M, there is a good filtration {F_{n}M} on it such that gr^{F} M is Cdim-pure.

Proof. The basic idea is to start with an arbitrary good filtration on M and to modify it to get purity.

Let Ã be the Rees algebra of A, and let R̃ be its balanced dualizing complex. So R̃ has the graded Auslander property.

Let n := Cdim M + 1. Choose any good filtration F′ on M and let M̃ := Rees^{F′} M. Since M̃ is t-torsion-free and M = πM̃, by (6.4) we have Cdim M̃ = n. If M̃′ ⊂ M̃ is any nonzero graded submodule then because πM̃′ ⊂ M, and because M is pure, we see that Cdim M̃′ = n. Thus M̃ is pure.

Set

E(M̃) = Ext−n ; Ã^{∘}(Ext−n ; Ã(M, R̃), R̃).

As in Theorem 2.14(3) there is an exact sequence

0 → M̃ → E(M̃) → Q̃ → 0 (6.25) where Cdim Q̃ ≤ n − 2. Consider the module Ñ which is the t-saturation of M̃ in E(M̃), i.e.

Ñ := {x ∈ E(M̃) | t^{i}x ∈ M̃ for some i ≥ 0}.

Since Ñ/M̃ is t-torsion it follows that M = πM̃ → πÑ is bijective; but the filtration F on πÑ may be different from F′. Observe that Ñ is t-torsion-free, so by lifting back the filtration we obtain Ñ ≅ Rees^{F} πÑ. By (6.25) we get Cdim Ñ/M̃ ≤ n−2, which implies that E(M̃) → E(Ñ) is bijective. Therefore by changing the good filtration on M from F′ to F, we can assume that in (6.25) the module Q̃ is ttorsion-free.

Having done so we get a short exact sequence

0 → π_{0}M̃ → π_{0}E(M̃) → π_{0}Q̃ → 0

where π_{0} denotes the functor M ↦ M/tM. Hence in order to prove that gr M = π_{0}M̃ is pure it suffices to prove that π_{0}E(M̃) is pure. Now, using the duality functor D = R Homgr ; Ã(−, R̃) we have L̃ := Ext−n ; Ã(M̃, R̃) = H^{−n}DM̃, which is ttorsion-free by Lemma 5.4. Therefore by the same lemma, Cdim π_{0}L̃ = n − 1. Now E(M̃) = H^{−n}D^{∘}L̃, and by formula (5.5) when i = −n we get

π_{0}H^{−n}D^{∘}L̃ ⊂ (H^{−(n−1)}D^{∘}π_{0}L̃)(−1).

But by Theorem 2.14, the module H^{−(n−1)}D^{∘}π_{0}L̃ is pure of dimension n − 1.

If gr A is a commutative affine k-algebra and M is a finitely generated A-module, define I(M) ⊂ gr A to be the prime radical of Ann_{gr A} gr^{F} M for some good filtration F on M. By [MR] Proposition 8.6.17, I(M) is independent of the choice of good filtration. The characteristic variety of M is defined to be

Ch(M) = Spec gr A/I(M)

(cf. [Co] p. 98 for the case when A is a Weyl algebra).

Proof of Theorem 0.4. Recall that a variety is called pure if all its irreducible components have the same dimension. The support of a finitely generated module N over a commutative affine k-algebra B is pure iff N is Kdim-pure. But for a commutative algebra Kdim_{B} = GKdim_{B} = Cdim_{B}. Here we take B := gr A and N := gr^{F} M.

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A. Yekutieli: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

   E-mail address: amnon@wisdom.weizmann.ac.il

J.J. Zhang: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195, USA

   E-mail address: zhang@math.washington.edu