RING HOMOMORPHISMS AND FINITE GORENSTEIN DIMENSION

Luchezar L. Avramov and Hans-Bjørn Foxby

Contents

Introduction

1. Homological algebra

2. Dualizing complexes

3. Dualizing equivalences

4. Gorenstein dimension

5. Relative dualizing complexes: Properties

6. Relative dualizing complexes: Proofs

7. Bass numbers of local homomorphisms

8. Quasi-Gorenstein homomorphisms References

Introduction

We study the local structure of ring homomorphisms, and for the rest of the paper make the convention that all rings are commutative and noetherian.

Given a homomorphism φ: R ⟶ S, various numerical invariants have been attached in [8], [5], [6], to its localizations φ_{𝔮}: R_{𝔮∩R} ⟶ S_{𝔮} at prime ideals 𝔮 of S. Some of these numbers, like dimension, depth, or type, express quantitative characteristics of φ; others, like Cohen–Macaulay defect, or complete intersection defect, capture its qualitative aspects. An upshot of the work in [4–8] is the realization that when fd φ_{𝔮} is finite – that is, the R–module S_{𝔮} has finite flat dimension – at all 𝔮, then the properties of the homomorphism φ control the transfer of local properties between the rings R and S.

This point of view has put on common ground many phenomena perceived earlier as different, and treated accordingly. The new perspective has also led to the determination of large classes of ring homomorphisms, like the Gorenstein or Cohen–Macaulay ones, with remarkable stability properties. Even in the classical framework of flat homomorphisms, our approach has produced new results, like the solutions in [5] and [6] of a series of open “localization” problems for Cohen–Macaulay properties, going back to Grothendieck.

In the present paper we extend the study of homomorphisms by weakening the homological assumption on the maps. Namely, we replace the condition on the flat dimension of

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L.L.A. was partially supported by National Science Foundation Grant No. DMS-9102951.

H.B.F. was partially supported by the Danish Natural Science Research Council and the Research Academy of Denmark. .

the R–modules S_{𝔮} by one on the finiteness of their Gorenstein dimension, or G-dimension. This concept was initially defined for finite modules by Auslander [1]. It is a finer invariant than the classical projective dimension: they are equal when the latter is finite, but the Gorenstein dimension may be finite without the projective one being so. In fact, over a Gorenstein ring of finite Krull dimension all finite modules have finite G-dimension, a result which is doubtless responsible for the name of this homological dimension.

As we want to impose – at least locally – a finiteness condition on the G-dimension of the R–module S, it is crucial to have an operational notion for not necessarily finite R–modules. While a concept of G-dimension is introduced in that generality in [2], its properties are not sufficiently developed for the purpose at hand. Therefore we take a different approach, based on a relative version of the Cohen Structure Theorem for complete local rings.

It is proved in [8] that each local homomorphism to a complete local ring is the composition of a local flat extension with regular closed fiber and a surjective homomorphism. Using such a Cohen factorization R ⟶ R′ ⟶ Ŝ of the map R ⟶ Ŝ induced by a local homomorphism φ: R ⟶ S, we say φ has finite G-dimension, and write G-dim φ < ∞, if the R′–module Ŝ has finite G-dimension. This calls for a proof that the concept is independent of the choice of the factorization. The argument is given in Section 4, where some of the more immediate consequences of the finiteness of G-dim φ are also established.

In Section 5 we describe our main tool for the study of local homomorphisms of finite G-dimension: the dualizing complex for a local homomorphism φ. When G-dim φ is finite we present a roster of properties of this complex, closely paralleling those of dualizing complexes for local rings. Thus, φ has a dualizing complex when S is complete, or when both R and S have such complexes; dualizing complexes are unique up to isomorphism and translation in the derived category D(S) of S–modules; they localize when R has Gorenstein formal fibers; etc. The proofs of those results are presented in Section 6.

In [4] we defined a local homomorphism φ: (R, 𝔪) ⟶ (S, 𝔫) to be Gorenstein at 𝔫 if fd φ is finite and the Bass numbers of both rings essentially coincide: μ_{R}^{i+depth R} = μ_{S}^{i+depth S} for all i ∈ ℤ. Weakening the homological condition to finite G-dimension, we obtain a larger class of local homomorphisms, which we call quasi-Gorenstein at 𝔫. By using a normalized dualizing complex for such a φ, we define a formal Laurent series with non-negative integer coefficients I_{φ}(t) , the Bass series of φ, and establish a product formula

I_{s}(t) = I_{R}(t) I_{φ}(t)

where I_{R}(t) = ∑_{i∈ℤ} μ_{R}^{i}t^{i} is the usual Bass series of R. In the case of finite flat dimension, such a formula is established in [9], with a very different definition of I_{φ}(t) . Our present approach provides a better hold on the Bass series of a homomorphism, and allows us to prove it has no “gaps,” solving a problem raised in [4]. This result has the practical consequence, that now (quasi-)Gorenstein homomorphisms can be defined by the corresponding homological condition, and an equality μ_{R}^{i+depth R} = μ_{S}^{i+depth S} for a single integer i > 0.

The preceding results are proved in Section 7, where we also give a characterization of quasi-Gorenstein homomorphisms which does not refer to the finiteness of their Gdimension. They are precisely the local homomorphisms which base change dualizing complexes properly: if D is dualizing for R, then the derived tensor product D ⊗_{R}^{L} S is dualizing for S. In Section 8 we study homomorphisms of noetherian rings all of whose localizations are quasi-Gorenstein, and prove that this class enjoys essentially all the stability properties of Gorenstein homomorphisms, established in [4] and [6].

On a couple of occasions in this Introduction we have mentioned concepts defined in terms of derived categories of modules. In the body of the paper derived categories provide both the language and the proof techniques for most results, so in Section 1 we fix some terminology and notation. In Section 2, for the reader’s and our own convenience, we briefly recall a few standard properties of dualizing complexes for local rings.

In Section 3 we prove that when R is local with dualizing complex D, the functors D ⊗_{R}^{L} − and RHom_{R}(D, −) of D(R) establish an equivalence of the full subcategories F(R) and I(R) of D(R), consisting of complexes isomorphic to bounded complexes of flat and injective modules, respectively. This is a vast generalization of a theorem of Sharp [19] on the equivalence of the categories of finite modules of finite projective and finite injective dimension over a Cohen–Macaulay local ring with dualizing module. For our purposes, even the generalized form of such an equivalence is insufficient, so we extend it further to full subcategories G(R) and J (R) of D(R). The category G(R), which contains both F(R) and the full subcategory of finite R–modules of finite G-dimension, provides a particularly flexible environment for the study of homomorphisms of finite G-dimension.

1. Homological algebra

(1.1) Complexes. A complex M of R–modules, or R–complex, is a sequence of R–linear homomorphisms {∂_{n}: M_{n} ⟶ M_{n−1}}_{n∈ℤ} such that ∂_{n}∂_{n+1} = 0 for all n. (We only use subscripts and all differentials have degree −1.) The infimum, supremum, and amplitude of M are defined by inf M = inf{n ∈ ℤ | H_{n}(M) ≠ 0}, sup M = sup{n ∈ ℤ | H_{n}(M) ≠ 0}, and amp M = sup M − inf M. For s ∈ ℤ we denote by Σ^{s}M the complex with (Σ^{s}M)_{n} = M_{n−s} and ∂_{n}^{Σ^{s}M} = (−1)^{s}∂_{n−s}^{M}. If N is an R–module then we also denote by N the complex of R–modules with N_{n} = 0 for n ≠ 0 and N_{0} = N.

If N is an R–complex, then a morphism α: M ⟶ N is a sequence of R–linear homomorphisms α_{n}: M_{n} ⟶ N_{n}, such that ∂_{n}^{N}α_{n} = α_{n−1}∂_{n}^{M} for n ∈ ℤ. A quasi-isomorphism is a morphism α such that H_{n}(α) is an isomorphism for all n; we indicate quasi-isomorphisms by ≃, while ≅ is our notation for isomorphisms of complexes (and thereby of modules).

(1.2) Derived functors. The derived category of the category of R–modules, cf. [22] or [16], is denoted by D(R). Isomorphisms in D(R) are labeled with ≃ (as a morphism of complexes is a quasi-isomorphism if and only if its image in D(R) is an isomorphism, no notational confusion arises). We use ∼ to denote isomorphisms up to translation. If H_{i}(M) = 0 for i ≠ n, then M ∼ H_{n}(M).

We write D^{f}(R) for the full subcategory of D(R) consisting of complexes M with H_{n}(M) a finite R–module for each n ∈ ℤ. Also, D_{+}(R), D_{–}(R), D_{b}(R), D_{0}(R), denote the full subcategories defined by H_{n}(M) = 0 for, respectively, n ≪ 0, n ≫ 0, |n| ≫ 0, n ≠ 0. For a subcategory S(R) ⊆ D(R) we set S^{f}(R) = S ∩ D^{f}(R), etc. Obvious equivalences identify D_{0}(R) with the category of R–modules, and D_{0}^{f}(R) with that of finite R–modules.

The left derived functor of the tensor product functor of R–complexes is denoted by − ⊗_{R}^{L} −, and the right derived functor of the homomorphism functor of R–complexes is denoted by RHom_{R}(−, −) (no boundedness conditions are imposed on the arguments, due to the existence of appropriate resolutions, cf. [10], [21]). Thus, for arbitrary M, N ∈ D(R) there are complexes M ⊗_{R}^{L} N and RHom_{R}(M, N) which are defined uniquely up to isomorphism in D(R), and possess the expected functorial properties. As usual, we set Tor_{n}^{R}(M, N) = H_{n}(M ⊗_{R}^{L} N) and Ext_{R}^{n}(M, N) = H_{−n}(RHom_{R}(M, N)) for n ∈ ℤ. These are the classical notions when M and N are modules.

When φ: R ⟶ S is a ring homomorphism, M is an R–complex, and N is an S–complex, standard spectral sequence arguments, cf. e.g. [3; (4.7.I.Proof)], yield:

(1.2.1) If M ∈ D_{+}^{f}(R) and N ∈ D_{+}^{f}(S), then M ⊗_{R}^{L} N ∈ D_{+}^{f}(S). (1.2.2) If M ∈ D_{+}^{f}(R) and N ∈ D_{–}^{f}(S), then RHom_{R}(M, N) ∈ D_{–}^{f}(S).

The following result is useful in locating quasi-isomorphisms.

(1.2.3) Lemma. Let D ∈ D_{b}^{f}(R) have H(D_{𝔭}) ≠ 0 for all 𝔭 ∈ Spec R, and let M ∈ D_{b}(R). (a) If D ∈ D_{0}^{f}(R), then inf(D ⊗_{R}^{L} M) = inf M and sup RHom_{R}(D, M) = sup M.

(b) If α: M ⟶ N is a morphism in D_{b}(R), such that D ⊗_{R}^{L} α or RHom_{R}(D, α) is invertible, then α is an isomorphism.

Proof. By [14; (2.2), (2.1.1)] there are inequalities

sup M − sup D ≤ sup RHom_{R}(D, M) ≤ sup M − inf D .

Choosing a faithfully injective module I, note that H(Hom_{R}(M, I)) ≅ Hom_{R}(H(M), I) implies sup M = − inf Hom_{R}(M, I) and inf M = − sup Hom_{R}(M, I). Thus, the isomorphism Hom_{R}(D ⊗_{R}^{L} M, I) ≅ RHom_{R}(D, Hom_{R}(M, I)) translates the preceding inequalities into

inf M + inf D ≤ inf(D ⊗_{R}^{L} M) ≤ inf M + sup D .

The two sets of inequalities immediately imply (a). A standard mapping cone argument reduces (b) to the claim that if RHom_{R}(D, M) or D ⊗_{R}^{L} M is exact, then M is exact. In the first case the first inequality yields sup M ≤ −∞. In the second case the last inequality gives ∞ ≤ inf M. Either inequality means that M is exact. □

(1.3) Homological dimensions. In D(R) we consider the full subcategories F(R), I(R), and P(R), consisting of complexes isomorphic to bounded complexes of – respectively – flat, injective, or projective modules. We note that P^{f}(R) = F^{f}(R), cf. [3; (2.10.F)], and that P (R) = F (R) and P_{0}(R) = F_{0}(R) when dim R < ∞, cf. [15; (11.8), (11.9)]. Furthermore, F_{0}(R), I_{0}(R), and P_{0}(R) are equivalent – respectively – to the (full) subcategories of modules of finite flat, injective, or projective dimension.

If M ∈ D_{b}(R), F ∈ F(R), I, I′ ∈ I(R), and P ∈ P(R), then

(a) M ⊗_{R}^{L} F ∈ D_{b}(R) ;  (b) RHom_{R}(M, I) ∈ D_{b}(R) ;  (c) RHom_{R}(P, M) ∈ D_{b}(R) ;
(d) I ⊗_{R}^{L} F ∈ I(R) ; (e) RHom_{R}(I, I′) ∈ F(R) .

(1.4) Canonical morphisms. Several canonical morphisms in D(R′) are associated with a homomorphism of rings R ⟶ R′, complexes K, M ∈ D(R), and K′, M′, N′ ∈ D(R′).

Without comment we use the associativity and adjointness isomorphisms

(K ⊗_{R}^{L} M′) ⊗_{R′}^{L} N′ ≃ K ⊗_{R}^{L} (M′ ⊗_{R′}^{L} N′) ;
RHom_{R′}(K ⊗_{R}^{L} M′, N′) ≃ RHom_{R}(K, RHom_{R′}(M′, N′)) ,

and their special cases

(K ⊗_{R}^{L} R′) ⊗_{R′}^{L} N′ ≃ K ⊗_{R}^{L} N′;
RHom_{R′}(K ⊗_{R}^{L} R′, N′) ≃ RHom_{R}(K, N′) .

They easily yield an isomorphism

(1.4.1)  RHom_{R′}(K′ ⊗_{R}^{L} M, N′) ≃ RHom_{R′}(K′, RHom_{R}(M, N′)) .

Although the evaluation morphisms

ω_{KM′N′}: RHom_{R}(K, M′) ⊗_{R′}^{L} N′ ⟶ RHom_{R}(K, M′ ⊗_{R′}^{L} N′) ;
θ_{KM′N′}: K ⊗_{R}^{L} RHom_{R′}(M′, N′) ⟶ RHom_{R′}(RHom_{R}(K, M′), N′) ,

are not always invertible, by [3; (4.4)] the following hold for K ∈ D_{+}^{f}(R):

(1.4.2) ω_{KM′N′} is an isomorphism when M′ ∈ D_{–}(R′), and N′ ∈ F(R′) or K ∈ P(R). (1.4.3) θ_{KM′N′} is an isomorphism when M′ ∈ D_{b}(R′), and N′ ∈ I(R′) or K ∈ P(R).

We also systematically use the biduality morphism

δ_{MK}: M ⟶ RHom_{R}(RHom_{R}(M, K), K)

and the homothety morphism

χ_{M}: R ⟶ RHom_{R}(M, M) .χ_{M}: R ⟶ RHom_{R}(M, M) .

(1.5) Numerical and formal invariants. Let R be a local ring with residue field k.

If M ∈ D_{+}^{f}(R), then the Betti number β_{i}^{R}(M) = rank_{k} Tor_{i}^{R}(M, k) is finite for all i and vanishes for i ≪ 0 by (1.2.1). Thus, P_{M}^{R}(t) = ∑_{i∈ℤ} β_{i}^{R}(M)t^{i} is a formal Laurent series, known as the Poincaré series of M. By Nakayama, the order of P_{M}^{R}(t) is equal to inf M.

Similarly, when M ∈ D_{–}^{f}(R) the Bass number μ_{R}^{i}(M) = rank_{k} Ext_{R}^{i}(k, M) is finite for all i and vanishes for i ≪ 0 by (1.2.2). The formal Laurent series I_{R}^{M}(t) = ∑_{i∈ℤ} μ_{R}^{i}(M)t^{i} is known as the Bass series of M. We set μ_{R}^{i} = μ_{R}^{i}(R) and I_{R}(t) = I_{R}^{R}(t) .

The formal invariants of M determine membership in P(M) or I(M), cf. e.g. [3; (5.5)]:

(1.5.1) M ∈ D_{+}^{f}(R) is in P(M) if and only if P_{M}^{R}(t) is a Laurent polynomial. Furthermore, M ∼ R if and only if P_{M}^{R}(t) = t^{d} for some d ∈ ℤ. (1.5.2) M ∈ D_{–}^{f}(R) is in I(M) if and only if I_{R}^{M}(t) is a Laurent polynomial.

A homomorphism φ: R ⟶ S is said to be local if the ring S is local, and the closed fiber k ⊗_{R} S of φ is nontrivial. The next result generalizes [15; (10.8.a), (10.4.a)].

1.5.3 Lemma. Let φ: R ⟶ S be a local homomorphism φ: R ⟶ S.

(a) If M ∈ D_{+}^{f}(R), N ∈ D_{+}^{f}(S) and L = M ⊗_{R}^{L} N, then P_{L}^{S}(t) = P_{M}^{R}(t) P_{N}^{S}(t) .

(b) If M ∈ D_{+}^{f}(R), N ∈ D_{–}^{f}(S) and L = RHom_{R}(M, N), then I_{S}^{L}(t) = P_{M}^{R}(t) I_{S}^{N}(t) .

Proof. Let ℓ denote the residue field of S. A first sequence of isomorphisms

(M ⊗_{R}^{L} N) ⊗_{S}^{L} ℓ ≃ M ⊗_{R}^{L} (N ⊗_{S}^{L} ℓ) ≃ (M ⊗_{R}^{L} k) ⊗_{k} (N ⊗_{S}^{L} ℓ)

and the Künneth formula yield Tor_{∗}^{S}((M ⊗_{R}^{L} N), ℓ) ≅ Tor_{∗}^{R}(M, k) ⊗_{k} Tor_{∗}^{S}(N, ℓ), which gives (a). Formula (1.4.1) starts a second sequence of isomorphisms

RHom_{S}(ℓ, RHom_{R}(M, N)) ≃ RHom_{S}(ℓ ⊗_{R}^{L} M, N) ≃ RHom_{S}((M ⊗_{R}^{L} k) ⊗_{k} ℓ, N)
≃ Hom_{k}(M ⊗_{R}^{L} k, RHom_{S}(ℓ, N)) ≃ Hom_{k}(M ⊗_{R}^{L} k, k) ⊗_{k} RHom_{S}(ℓ, N) ,

hence Ext_{S}^{∗}(ℓ, RHom_{R}(M, N)) ≅ Hom_{k}(Tor_{∗}^{R}(M, k), k)⊗_{k} Ext_{S}^{∗}(ℓ, N), producing (b). □

2. Dualizing complexes

In this section (R, 𝔪, k) is a local ring with maximal ideal 𝔪 and residue field k = R/𝔪.

The Cohen–Macaulay defect of R is the positive integer cmd R = dim R−depth R. The first positive Bass μ_{R}^{i} number appears at i = depth R; known as the type of R, it is denoted type R. We write ν_{R}M for the minimal number of generators of a finite R–module M.

Definition. An R–complex D is said to be dualizing for R if the homothety morphism χ_{D}: R → RHom_{R}(D, D) is invertible, and D ∈ I^{f}(R).

We record some important properties of dualizing complexes [16], cf. also [17] and [15], with a dual purpose in mind. On the one hand, they appear in several proofs. On the other, they presage properties of the relative dualizing complexes of Section 5.

(2.1) Example. A ring R is Gorenstein if and only if the R–module R is a dualizing complex, cf. [16; (V.10.1)], [15; (8.22)].

(2.2) Completion. A complex D ∈ D(R) is dualizing for R if and only if D⊗_{R} R̂ is dualizing for R̂, cf. [16; (V.3.5)], [15; (10.15)].

(2.3) Existence. A homomorphic image of a Gorenstein ring has a dualizing complex. In particular, each complete ring has a dualizing complex, cf. [16; (V.10.4)], [15; (8.25)].

(2.4) Uniqueness. If D, D′ ∈ D(R) are dualizing complexes for R, then D ∼ D′, cf. [16; (V.3.1)], [15; (8.15)].

(2.5) Size. If D is a dualizing complex for R, then amp D = cmd R and ν_{R} H_{i}(D) = type R for i = inf D cf. [15; (8.12), (10.6)].

We say a dualizing complex D for R is normalized if inf D = depth R (this convention differs from the one in [4; p. 1031]); by (2.4) such a complex is unique up to isomorphism.

(2.6) Formal invariants. A complex D ∈ D_{b}^{f}(R) is dualizing for R if and only if I_{R}^{D}(t) = t^{d} for some d ∈ ℤ, cf. [16; (V.3.4)], [15; (8.5)].

When D is normalized, I_{R}^{D}(t) = 1 and P_{D}^{R}(t) = I_{R}(t) , cf. [15; (10.3), (10.5)].

(2.7) Biduality. If D ∈ D(R) is dualizing for R, then the biduality morphism δ_{MD}: M ⟶ RHom_{R}(RHom_{R}(M, D), D) is invertible for each M ∈ D_{+}^{f}(R), cf. [16; (V.2.1)], [15; (8.3)].

(2.8) Localization. If D is a dualizing complex for R, then for each 𝔭 ∈ Spec R the complex D_{𝔭} ∈ D(R_{𝔭}) is dualizing for R_{𝔭}, cf. [16; (V.8.1)], [15; (8.6.a)].

Recall that the fiber of a homomorphism of rings φ: R ⟶ S at a prime ideal 𝔭 in R is the ring k(𝔭) ⊗_{R} S, where k(𝔭) = R_{𝔭}/𝔭R_{𝔭} is the residue field of the local ring R_{𝔭}. The fibers of the 𝔪-adic completion map R ⟶ R̂ are called the formal fibers of R.

(2.9) Formal fibers. A ring with a dualizing complex has Gorenstein formal fibers, cf. [16; (V.10.1)], [15; (10.15)].

The next property appears undocumented; in view of (5.1.a), it is a special case of (5.10).

(2.10) Closed fiber. If φ: R ⟶ S is a flat local homomorphism, R is Gorenstein, and E is a dualizing complex for S, then the complex k ⊗_{R}^{L} E is dualizing for k ⊗_{R} S.

A homomorphism φ: R ⟶ S has finite flat dimension, denoted fd φ < ∞, if S admits a finite resolution by flat R–modules. A local homomorphism φ is Gorenstein at 𝔫, cf. [4], if fd φ < ∞ and μ_{R}^{i+depth R} = μ_{S}^{i+depth S} for i ∈ ℤ. By [4; (4.2)], cf. also (8.3) below, a flat homomorphism is Gorenstein if and only if the ring S/𝔪S is Gorenstein. The relevance of this notion in the present context comes from the next result, proved in [4; (5.1)].

(2.11) Base change. Let φ: R ⟶ S be a local homomorphism. The following conditions on D ∈ D_{–}^{f}(R) are equivalent.

(i) D is dualizing for R, and φ is Gorenstein at 𝔫.

(ii) D ⊗_{R}^{L} S is dualizing for S, and φ has finite flat dimension.

(2.12) Finite ascent. Let φ: R → S be a finite homomorphism of local rings. If D is a dualizing complex for R then RHom_{R}(S, D) is one for S, cf. [16; (V.10.2)], [15; (8.4)].

3. Dualizing equivalences

In this section R denotes a local ring with dualizing complex D.

We exhibit an equivalence of the categories F(R) and I(R), which is provided by the restriction of endofunctors of the entire derived category D(R). The existence of such an equivalence leads to natural extensions of both subcategories considered above.

(3.1) Gorenstein categories. Let G(R) denote the full subcategory of D_{b}(R), consisting of those complexes M for which D ⊗_{R}^{L} M ∈ D_{b}(R) and the canonical morphism

γ_{M}: M ⟶ RHom_{R}(D, D ⊗_{R}^{L} M) ,

induced by m ↦ (d ↦ (−1)^{|m||d|}d ⊗ m), is an isomorphism.

Similarly, let J (R) denote the full subcategory of D_{b}(R), consisting of those complexes M for which RHom_{R}(D, M) ∈ D_{b}(R) and the canonical morphism

ι_{M}: D ⊗_{R}^{L} RHom_{R}(D, M) ⟶ M ,

induced by d ⊗ α ↦ (−1)^{|d||α|}α(d), is an isomorphism.

(3.2) Theorem. If D is a dualizing complex for R, then there is a commutative diagram

D(R) ⟶^{D⊗_{R}^{L}−} ; ⟵_{RHom_{R}(D,−)} D(R)
⋃|  ⋃|
G(R) \overrightarrow{⟵} J (R)
⋃|  ⋃|
F(R) \overrightarrow{⟵} I(R)

in which the vertical inclusions are full embeddings, and the unlabeled horizontal arrows are quasi-inverse equivalences of categories.

Furthermore, for C, E ∈ D_{b}(R) the following hold:

(g) D ⊗_{R}^{L} C ∈ J (R) implies C ∈ G(R) ; (j) RHom_{R}(D, E) ∈ G(R) implies E ∈ J (R) ;
(f) D ⊗_{R}^{L} C ∈ I(R) implies C ∈ F(R) ; (i) RHom_{R}(D, E) ∈ F(R) implies E ∈ I(R) .

Remark. If R is Gorenstein, then it is clear that G(R) = J (R) = D_{b}(R), and (3.2) contains the well known fact that then also F(R) = I(R).

Proof. If C ∈ F(R), then D ⊗_{R}^{L} C ∈ D_{b}(R) by (1.3.a), and ω_{DDC} is a isomorphism by (1.4.2). The commutative diagram

C ⟶^{γ_{C}} RHom_{R}(D, D ⊗_{R}^{L} C)
↓^{≃}  ^{≃}↑^{ω_{DDC}}
R ⊗_{R}^{L} C ⟶_{χ_{D}⊗_{R}^{L}C}^{≃} RHom_{R}(D, D) ⊗_{R}^{L} C

shows that γ_{C} is an isomorphism. Similarly, if E ∈ I(R), then RHom_{R}(D, E) ∈ D_{b}(R) by (1.3.b), and θ_{DDE} is an isomorphism by (1.4.3), and from the commutative diagram

D ⊗_{R}^{L} RHom_{R}(D, E) ⟶^{ι_{E}}  E
θ_{DDE} ↓^{≃}  ^{≃}↑
RHom_{R}(RHom_{R}(D, D), E) ⟶_{RHom_{R}(χ_{D},E)}^{≃} RHom_{R}(R, E)

we see that ι_{E} is invertible. We have established the embeddings of categories. For the remainder of the proof, C and E denote complexes in D_{b}(R). Set F = D ⊗_{R}^{L} C, and consider the morphisms ι_{F} : D ⊗_{R}^{L} RHom_{R}(D, F) ⟶ F and

γ_{C} : C ⟶ RHom_{R}(D, F). The induced morphism

F = D ⊗_{R}^{L} C ⟶^{D⊗_{R}^{L}γ_{C}} D ⊗_{R}^{L} RHom_{R}(D, F)

satisfies ι_{F} (D ⊗_{R}^{L} γ_{C}) = 1_{F}, hence we see that

(∗) ι_{F} is an isomorphism if and only if D ⊗_{R}^{L} γ_{C} is one.

Starting with C ∈ G(R), the definition of G(R) shows that C, F, and RHom_{R}(D, F) are in D_{b}(R), and that D ⊗_{R}^{L} γ_{C} is an isomorphism. Thus, ι_{F} is an isomorphism, hence F ∈ J (R). We have shown that D ⊗_{R}^{L} − restricts to a functor G(R) ⟶ J (R).

Set B = RHom_{R}(D, E), and consider the morphisms γ_{B} : B ⟶ RHom_{R}(D, D ⊗_{R}^{L} B) and ι_{E}: D ⊗_{R}^{L} B ⟶ E. The induced morphism

RHom_{R}(D, D ⊗_{R}^{L} B) ⟶^{RHom_{R}(D,ι_{E})} RHom_{R}(D, E) = B

satisfies RHom_{R}(D, ι_{E}) γ_{B} = 1_{B}, hence (∗∗) γ_{B} is an isomorphism if and only if RHom_{R}(D, ι_{E}) is one.

Arguing as above, we see that E ∈ J (R) implies B ∈ G(R), so that RHom_{R}(D, −) restrict to a functor J (R) ⟶ G(R). The fact that those functors between G(R) and J (R) are quasi-inverse equivalences is built in the definitions of these categories.

If C ∈ F(R), then D⊗_{R}^{L} C ∈ I(R) by (1.3.d); if E ∈ I(R), then RHom_{R}(D, E) ∈ F(R) by (1.3.e). Thus, D ⊗_{R}^{L} − and RHom_{R}(D, −) restrict to functors between F(R) and I(R). They are quasi-inverse because this is true for their extensions to G(R) and J (R).

It remains to establish the last four assertions.

If F = D ⊗_{R}^{L} C is in J (R), then the complexes F and RHom_{R}(D, F) are homologically bounded, and ι_{F} is an isomorphism, hence by (∗) so is D ⊗_{R}^{L} γ_{C}. As D satisfies the assumption of (1.2.3.b), cf. (2.8), we conclude that γ_{C} is invertible, hence C is in G(R). This proves (g). The argument for (j) is similar, using (∗∗).

If D ⊗_{R}^{L} C is in I(R), then C ∈ G(R) in view of the commutative diagram and (g). This yields C ≃ RHom_{R}(D, D ⊗_{R}^{L} C) ∈ F(R), as desired. The proof of (i) is similar. □

(3.3) Lemma. The following conditions are equivalent.

(i) The ring R is Cohen–Macaulay.

(ii) amp(D ⊗_{R}^{L} M) = amp M for all M ∈ G(R).

(iii) amp RHom_{R}(D, M) = amp M for all M ∈ J (R).

(iv) amp D = 0.

Proof. (i) ⟺ (iv) is clear (and contained in (2.5)).

(ii) ⟹ (iv): set M = R.

(iii) ⟹ (iv): set M = D.

(iv) ⟹ (ii). For M ∈ G(R) we have sup M = sup RHom_{R}(D, D⊗_{R}^{L}M) = sup(D⊗_{R}^{L}M), with the second equality coming from (1.2.3.a), which also provides inf M = inf(D ⊗_{R}^{L} M).

(iv) ⟹ (iii) is similar. □

When R is Cohen–Macaulay, the single non-zero homology module of a dualizing complex D, defined uniquely up to isomorphism, is known as the canonical module of R.

(3.4) Proposition. Let R be Cohen–Macaulay ring with canonical module D.

An R–module M is in G_{0}(R) if and only if Tor_{i}^{R}(D, M) = 0 = Ext_{R}^{i}(D, D ⊗_{R} M) for i > 0, and the canonical map M ⟶ Hom(D, D ⊗_{R} M) is bijective.

An R–module M is in J_{0}(R) if and only if Ext_{R}^{i}(D, M) = 0 = Tor_{i}^{R}(D, Hom_{R}(D, M)) for i > 0, and the canonical map D ⊗_{R} Hom_{R}(D, M) ⟶ M is bijective.

Proof. We prove the first assertion, and leave the second one to the reader.

Note that the morphism α: RHom_{R}(D, D ⊗_{R} M) ⟶ Hom_{R}(D, D ⊗_{R} M) is invertible if and only if Ext_{R}^{i}(D, D ⊗_{R} M) = 0 for i > 0 and that the morphism β: D ⊗_{R}^{L} M ⟶ D ⊗_{R} M is invertible if and only if Tor_{i}^{R}(D, M) = 0 for i > 0. Furthermore, H(α) H(RHom_{R}(D, β)) H(γ_{M}) is the canonical map γ′: M ⟶ Hom(D, D ⊗_{R} M).

Assume M ∈ G_{0}(R), so that γ_{M} is an isomorphism. By (3.3) the Tor’s vanish, hence β is an isomorphism. Thus, RHom_{R}(D, β)γ_{M}: M ⟶ RHom_{R}(D, D ⊗_{R} M) is invertible. It follows that the Ext’s vanish, implying α is invertible, hence γ′ is bijective.

Conversely, the vanishing of Ext’s implies that α is an isomorphism. The vanishing of Tor’s implies that D ⊗_{R}^{L} M is bounded, and that β is invertible. Now from the bijectivity of γ′ we get that of H(γ_{M}), and thus γ_{M} is an isomorphism. □

(3.5) Remark. It follows from the first part of the proposition and [11; (4.4), (5.3), (5.4)] that a module M over a Cohen–Macaulay ring R is in G_{0}(R) if and only if there exists an exact sequence 0 ⟶ G_{n} ⟶ ⋯ ⟶ G_{0} ⟶ M ⟶ 0, such that for each i the module G_{i} is Gorenstein flat in the sense of Enochs et al. Similarly, the second part of the proposition and [11; (4.5)] imply that M is in J (R) if and only if it there exists an exact sequence 0 ⟶ M ⟶ J_{0} ⟶ ⋯ ⟶ J_{n} ⟶ 0, such that for each i the module J_{i} is Gorenstein injective.

Combining (3.2) and (3.4), we obtain the following

(3.6) Corollary. When R is Cohen–Macaulay with canonical module D there are commutative diagrams of categories of R–modules

D^{0}(R) ⟶^{D⊗_{R}−} ; ⟵_{Hom_{R}(D,−)} D^{0}(R)_{⋃|  ⋃|}_{G^{0}(R) \overrightarrow{⟵} J^{0}(R)  and}_{⋃|  ⋃|}_{F^{0}(R) \overrightarrow{⟵} I^{0}(R)} D f0(R) ⟶^{D⊗_{R}−} ; ⟵_{Hom_{R}(D,−)} D f0(R)
⋃|  ⋃|
G f 0(R) \overrightarrow{⟵} J f 0(R)
⋃|  ⋃|
F f 0(R) \overrightarrow{⟵} I f 0(R)

in which the vertical inclusions are full embeddings, and the unlabeled horizontal arrows are quasi-inverse equivalences of categories. □

Remark. In view of (3.4), the corollary can be obtained as a direct consequence of [13; (1.4), (2.1)]. Furthermore, loc. cit. establishes an analog of the last part of (3.2).

The equivalence above between F_{0}^{f}(R) and I_{0}^{f}(R) is due to Sharp, [19; (2.9)].

Finally, we record some stability properties of G(R).

(3.7) Proposition. Let φ: R ⟶ S be a local homomorphism and let N ∈ D(S). (a) N ∈ G(R) if and only if N ⊗_{S} Ŝ ∈ G(R̂).

(b) If φ is Gorenstein, then N ∈ G(R) if and only N ∈ G(S).

(c) If S ⟶ S′ is a local flat extension, then N ∈ G(R) if and only if N ⊗_{S} S′ ∈ G(R).

(d) If 𝔮 ∈ Spec S and 𝔭 = 𝔮 ∩ R, then N ∈ G(R) implies N_{𝔮} ∈ G(R_{𝔭}).

Proof. When φ is Gorenstein, E = D⊗_{R}^{L}S is a dualizing complex for S by (2.11), hence (b) results from the canonical isomorphisms D ⊗_{R}^{L} N ≃ E ⊗_{S}^{L} N and RHom_{R}(D, D ⊗_{R}^{L} N) ≃ RHom_{S}(E, E ⊗_{S}^{L} N).

(c) follows by faithful flatness from the canonical isomorphisms H(D ⊗_{R}^{L} N) ⊗_{S} S′ ≅ H(D ⊗_{R}^{L} N ⊗_{S} S′) and RHom_{R}(D, D ⊗_{R}^{L} N) ⊗_{S} S′ ≃ RHom_{R}(D, D ⊗_{R}^{L} N ⊗_{S} S′).

By (c), N is in G(R) if and only if N ⊗_{S} Ŝ is in G(R), and by (b) the last condition is equivalent to N ⊗_{S} Ŝ being in G(R̂). This establishes (a).

As D_{𝔭} is a dualizing complex for R_{𝔭} by (2.8), the canonical isomorphisms D_{𝔭} ⊗_{R_{𝔭}}^{L}N_{𝔮} ≃ (D ⊗_{R}^{L} N)_{𝔮} and RHom_{R_{𝔭}}(D_{𝔭}, D_{𝔭} ⊗_{R_{𝔭}}^{L}N_{𝔮}) ≃ RHom_{R}(D, D ⊗_{R}^{L} N)_{𝔮} yield (d). □

L. L. AVRAMOV AND H.–B. FOXBY 11

4. Gorenstein dimension

In this section R denotes a local ring.

The dimension discussed below is introduced for modules by Auslander [1], [2]. (4.1) Gorenstein dimension. A finite R–module M has G-dimension 0 if Ext_{R}^{i}(M, R) = 0 = Ext_{R}^{i}(Hom_{R}(M, R), R) for i > 0, and the canonical map M ⟶ Hom_{R}(Hom_{R}(M, R), R) is bijective. It is of G-dimension at most n if there exists an exact sequence

0 ⟶ G_{n} ⟶ G_{n−1} ⟶ … ⟶ G_{1} ⟶ G_{0} ⟶ M ⟶ 0

in which G_{j} has G-dimension 0 for 0 ≤ j ≤ n; in this case we write G-dim_{R} M ≤ n.

For each finite R–module M the following hold: (4.1.1) If pd_{R} M < ∞, then G-dim_{R} M ≤ pd_{R} M, cf. [2; (3.14)]. (4.1.2) If G-dim_{R} M < ∞, then G-dim_{R} = depth R − depth_{R} M, cf. [2; (4.13.b)]. (4.1.3) If G-dim_{R} M < ∞, then G-dim_{R} M = − inf RHom_{R}(M, R), cf. [2; (4.13.a)]. (4.1.4) If R ⟶ S is a flat local homomorphism, then G-dim_{S}(M ⊗_{R} S) = G-dim_{R} M. (4.1.5) G-dim_{R_{𝔭}} M_{𝔭} ≤ G-dim_{R} M for each 𝔭 ∈ Spec R, cf. [2; (4.15)]. (4.1.6) R is Gorenstein if and only if G-dim_{R} M < ∞ for all finite M, cf. [2; (4.20)].

A connection with G(R) is provided by the following result of Foxby, cf. [24; (2.7)]: (4.1.7) If R has a dualizing complex, then G-dim_{R} M < ∞ if and only if M ∈ G(R). (4.2) Factorizations. A factorization of a local homomorphism φ: (R, 𝔪) ⟶ (S, 𝔫) is a commutative triangle of local homomorphisms

R′

φ̇ ↗  ↘ φ′
R ⟶_{φ} S .

with φ̇ flat and φ′ surjective. A factorization is said to be regular, respectively, Gorenstein, if the local ring R′/𝔪R′ has the corresponding property. It is clear that (4.2.1) φ has a regular factorization if is essentially of finite type.

A Cohen factorization is a regular factorization with a complete local ring R′. It often exists, by a relative form [8; (1.1)] of Cohen’s Structure Theorem for complete local rings: (4.2.2) For each local homomorphism φ, the composition φ̀: R → Ŝ of φ and the 𝔫-adic completion map S ⟶ Ŝ has a Cohen factorization.

By [7; (2.7)] or [8; (3.3)], Cohen and regular factorizations reflect the finiteness of fd φ: (4.2.3) If fd φ is finite, then pd_{R′} Ŝ is finite in each regular factorization R ⟶ R′ → Ŝ of φ̀; conversely, if pd_{R′} Ŝ is finite in some Cohen factorization of φ̀, then fd φ is finite.

The next theorem establishes a corresponding result for G-dimension.

(4.3) Theorem. For a local homomorphism φ: R ⟶ S the following are equivalent.

(i) G-dim_{R′} Ŝ is finite for some Cohen factorization R ⟶ R′ → Ŝ of φ̀. (ii) G-dim_{R′} Ŝ is finite for some Gorenstein factorization R ⟶ R′ → Ŝ of φ̀. (iii) G-dim_{R′} Ŝ is finite for each Gorenstein factorization R ⟶ R′ → Ŝ of φ̀. (iv) Ŝ belongs to G(R̂).

When φ has a Gorenstein factorization R ⟶ R′′ ⟶ S, they are also equivalent to

(ii’) G-dim_{R′′} S is finite. When R has a dualizing complex, they are also equivalent to

(iv’) S belongs to G(R).

Proof. The implications (iii) ⟹ (i) ⟹ (ii) are clear.

By (4.1.4), conditions (ii) and (iii) do not change if we replace R ⟶ R′ ⟶ Ŝ with the Gorenstein factorization R ⟶ R̂′ ⟶ Ŝ. Once this is done, we see from (3.7.b) that (iv) is equivalent to the condition Ŝ ∈ G(R̂′). By (4.1.7) this inclusion is equivalent to the finiteness of G-dim_{R′} Ŝ. It follows that (ii) ⟹ (iv) ⟹ (iii).

If R ⟶ R′′ ⟶ S is a Gorenstein factorization of φ, then R ⟶ R̂′′ ⟶ Ŝ is one of φ̀, hence (ii) ⟺ (ii’) by (4.1.4).

If R has a dualizing complex, then (iv) ⟺ (iv’) by (3.7.a). □

Definition. A local homomorphism φ: R ⟶ S which satisfies the equivalent conditions of the theorem is said to have finite Gorenstein dimension, denoted G-dim φ < ∞.

Some basic properties of such homomorphisms follow easily from (4.1) and (4.3).

(4.4.1) Gorenstein source. If R is Gorenstein, then so is the ring R′ in any Cohen factorization of φ̀, hence by (4.1.6) each local homomorphism φ: R ⟶ S has G-dim φ < ∞.

(4.4.2) Finite flat dimension. If fd φ < ∞, then G-dim φ < ∞ by (4.2.3) and (4.1.1).

(4.4.3) Completion. G-dim φ < ∞ if and only if G-dim φ̀ < ∞, if and only if G-dim φ̂ < ∞.

(4.4.4) Finite homomorphism. When φ is finite, G-dim φ < ∞ if and only if G-dim_{R} S < ∞. Indeed, by (4.1.7) the condition Ŝ ∈ G(R̂) is equivalent to the finiteness of G-dimR̂ Ŝ. By (4.1.4) the latter is tantamount to the finiteness of G-dim_{R} S.

The property of φ to have finite flat dimension localizes: If fd φ < ∞, then for each prime ideal 𝔮 in S the induced local homomorphism φ_{𝔮}: R_{𝔮∩R} ⟶ S_{𝔮} satisfies fd φ_{𝔮} < ∞. For the Gorenstein dimension we have the following partial results.

(4.5) Proposition. If φ: R ⟶ S is a local homomorphism with G-dim φ < ∞, then G-dim φ_{𝔮} < ∞ for each 𝔮 ∈ Spec S under each one of the following conditions:

(1) φ is essentially of finite type; or

(2) R has Gorenstein formal fibers.

Proof. Fix 𝔮, and set 𝔭 = 𝔮 ∩ R.

Under condition (1), φ has a regular factorization R ⟶ R′ ⟶ S in which R′ is the localization of a ring of polynomials over R, at a prime ideal lying above 𝔪. If 𝔭′ = 𝔮 ∩ R′, then the sequence R_{𝔭} ⟶ R′_{𝔭′} ⟶ S_{𝔮} is a regular factorization of φ_{𝔮}, and we have G-dim_{R′_{𝔭′}} S_{𝔮} ≤ G-dim_{R′} S by (4.1.5). Thus, G-dim φ_{𝔮} is finite by (4.3).

By the same result, we see that under condition (2) it suffices to show that Ŝ ∈ G(R̂) implies (S_{𝔮}) ∈ G((R_{𝔭}) ). Choose a prime ideal 𝔮^{∗} lying over 𝔮 and set 𝔭^{∗} = 𝔮^{∗} ∩ R̂. By (3.7.d) and (3.7.a) we have (Ŝ_{𝔮^{∗}}) ∈ G((R̂_{𝔭^{∗}}) ). By hypothesis, the flat local homomorphism R_{𝔭} ⟶ R̂_{𝔭^{∗}} has a Gorenstein closed fiber, and hence so does its completion (R_{𝔭}) ⟶ (R̂_{𝔭^{∗}}) . Thus, from (3.7.b) we see that (Ŝ_{𝔮^{∗}}) ∈ G((R_{𝔭}) ). The desired assertion now follows from (3.7.c), due to the identification of (Ŝ_{𝔮^{∗}}) with (S_{𝔮}) ⊗_{(S_{𝔮})}ˆ (Ŝ_{𝔮^{∗}}) . □

Next we look at the behavior of Gorenstein categories under descent.

(4.6) Proposition. Let R be a local ring with a dualizing complex, and let φ: R ⟶ S be a local homomorphism.

(a) If G-dim φ is finite then F(S) ⊆ G(R) and I(S) ⊆ J (R).

(b) If fd φ is finite then F(S) ⊆ F(R) and I(S) ⊆ I(R).

Proof. Let D ∈ I^{f}(R) be dualizing for R.

Consider first a complex M ∈ F(S). When fd φ is finite, the canonical isomorphism − ⊗_{R}^{L} M ≃ (− ⊗_{R}^{L} S) ⊗_{S}^{L} M and (1.3.a) yield M ∈ F(R).

When G-dim φ is finite, by (4.3) we have S ∈ G(R). On the one hand, this implies that D ⊗_{R}^{L} S ∈ D_{b}(S), hence ω_{D(D⊗_{R}^{L}S)M} is invertible by (1.4.2), and that amp(D ⊗_{R}^{L} M) = amp((D⊗_{R}^{L} S)⊗_{S}^{L} M) is finite. On the other hand, it implies γ_{S}: S ⟶ RHom_{R}(D, D⊗_{R}^{L} S) is an isomorphism, hence γ_{S} ⊗_{R}^{L} M is one. Now the commutative diagram

S ⊗_{S}^{L} M = M  ⟶^{γ_{M}}  RHom_{R}(D, D ⊗_{R}^{L} M)
γ_{S}⊗_{R}^{L}M↓^{≃}  ^{≃}↓
RHom_{R}(D, D ⊗_{R}^{L} S) ⊗_{S}^{L} M ⟶_{ω_{D(D⊗^{L}_{R}S)M}}^{≃} RHom_{R}(D, (D ⊗_{R}^{L} S) ⊗_{S}^{L} M)

shows that γ_{M} is an isomorphism. We have proved that M is in G(R).

The proofs for a complex M ∈ I(S) are similar, using (1.3.b) and (1.4.3). □

(4.7) Proposition. Let ψ: Q ⟶ R and φ: R ⟶ S be local homomorphisms, with fd φ finite. If G-dim ψ < ∞, then G-dim φψ < ∞. The converse holds when φ is flat.

Proof. In view of (4.4.3) the assumption on G-dim ψ is invariant under completion; that on fd φ has the same property, cf. e.g. [8; (3.3)]. Thus, we may assume Q has a dualizing complex. The previous proposition then shows that S ∈ F(R) ⊆ G(Q), and hence G-dim φψ is finite by (4.3). The converse follows from (3.7.c), in view of of (4.3). □

(4.8) Remark. We do not know whether the composition of local homomorphisms of finite G-dimension has the same property. The construction of [6; (4.4)] shows that this is equivalent to the validity of the following property of G-dimension: If Q ⟶ R ⟶ S are surjective homomorphisms of local rings, such that G-dim_{Q} R and G-dim_{R} S are both finite, then G-dim_{Q} S is finite.

Extending this to finite modules, we raise the

Question. Let Q ⟶ R be a finite homomorphism of local rings, and let M be a finite R–module. If G-dim_{Q} R and G-dim_{R} M, are finite, then is G-dim_{Q} M finite?

The argument for (4.7) easily generalizes to yield a positive answer when pd_{R} N is finite, and another case is established in (7.11), but the general case appears to be open.

5. Relative dualizing complexes: Properties

In this section φ: (R, 𝔪) ⟶ (S, 𝔫) denotes a local homomorphism.

When M is a finite R–module, we write codim_{R} M for the height of its annihilator ideal, and grade_{R} M for the maximal length of an R–regular sequence contained in this ideal. The dimension, depth, Cohen–Macaulay defect, and type of φ are defined from a Cohen factorization R ⟶ R′ ⟶ Ŝ of φ̀, cf. (4.2), as follows:

dim φ = dim R′ − dim R − codim_{R′} Ŝ;
depth φ = depth S − depth R;
cmd φ = dim φ − depth φ;
type φ = νŜ Ext_{R′}^{d}(Ŝ, R′) for d = depth R′ − depth S .

It is proved in [8; (2.1)] that the right hand side of the first (and hence, the third) formula is independent of the factorization; the corresponding result for the last formula is established in [6; (7.1)]. Unlike the prototype invariants for rings, the dimension, depth, or Cohen– Macaulay defect of a homomorphism may be negative, and its type may be zero.

Definition. An S–complex C is dualizing for φ if the homothety χ_{C}: S → RHom_{S}(C, C) is invertible, C ∈ D_{b}^{f}(S), and D′ ⊗L ; R̂ (C ⊗_{S} Ŝ) ∈ I(Ŝ) for a dualizing complex D′ for R̂.

By applying (a) below to the local structure homomorphism ℤ_{(p)} ⟶ S, where p = char S/𝔫, one sees that the properties which follow specialize to the correspondingly numbered in Section 2 properties of absolute dualizing complexes:

(5.1) Examples. (a) If R is Gorenstein, then C ∈ D(S) is dualizing for φ if and only if it is dualizing for S: this follows from (2.1).

(b) The identity homomorphism 1_{R} has a dualizing complex, namely R.

(c) If ρ: R ⟶ R^{∗} is the completion map for the adic topology defined by a proper ideal 𝔞 in R, then R^{∗} is a dualizing complex for ρ: as ρ̂ = 1R̂, this follows from (b) and (5.2).

(5.2) Completion. Let C be an S–complex.

(a) C is dualizing for φ if and only if C ⊗_{S} Ŝ ∈ D(Ŝ) is dualizing for φ̀ and/or φ̂.

(b) If R has a dualizing complex D, then C is dualizing for φ if and only if χ_{C} is invertible, C ∈ D_{b}^{f}(S), and D ⊗_{R}^{L} C ∈ I(S).

Proof. (a) By faithful flatness, χ_{C} is an isomorphism if and only if χ_{C⊗_{S} Ŝ} is one, and C ∈ D_{b}^{f}(S) if and only if C ⊗_{S} Ŝ ∈ D_{b}^{f}(Ŝ).

(b) By (2.2) D′ = D ⊗_{R} R̂ is dualizing or R̂. Since D′ ⊗L ; R̂ (C ⊗_{S} Ŝ) ≃ (D ⊗_{R}^{L} C) ⊗_{S} Ŝ as Ŝ–complexes, D′ ⊗L ; R̂ (C ⊗_{S} Ŝ) ∈ I(Ŝ) if and only if D ⊗_{R}^{L} C ∈ I(S) by [3; (5.5.I)]. □

The proofs of the following theorems are collected at the end of Section 6.

Convention for Theorems (5.3) through (5.9): G-dim φ is finite.

(5.3) Existence. If both R and S have dualizing complexes, or if φ has a Gorenstein factorization, then φ has a dualizing complex.

In particular, each homomorphism to a complete ring has a dualizing complex.

Dualizing complexes are actually constructed in (6.1.b), (6.5), and (6.7) below.

(5.4) Uniqueness. If C, C′ ∈ D(S) are dualizing for φ, then C ∼ C′.

(5.5) Size. Let C be a dualizing complex for φ.

If R ⟶ R′ ⟶ Ŝ is a Gorenstein factorization for φ̀, then

amp C = cmd φ + codim_{R′} Ŝ − grade_{R′} Ŝ ≥ cmd φ

with equality if fd φ < ∞ or if R is Cohen–Macaulay; in the latter case, amp C = cmd S.

For i = inf C there are equalities ν_{S} H_{i}(C) = type φ = \frac{type S}{type R} .

We do not know if cmd φ ≥ 0 when G-dim φ is finite, unless fd φ < ∞, cf. [8; (3.6)]. We say a dualizing complex C for φ is normalized if inf C = depth S −depth R; by (5.4)

such a complex is unique up to isomorphism. (When R is Gorenstein, C is also dualizing for S by (5.1.a), but as such is not normalized unless R is artinian.)

(5.6) Formal invariants. If C is a normalized dualizing complex for φ, then

P_{C}^{S}(t) = \frac{I_{S}(t)}{I_{R}(t)}  and  I_{S}^{C}(t) =I_{R}(t) .

(5.7) Biduality. If C ∈ D(S) is dualizing for φ, then the biduality morphism δ_{NC}: N ⟶ RHom_{S}(RHom_{S}(N, C), C) is invertible for each N ∈ D_{b}^{f}(S) with N ⊗_{S} Ŝ ∈ G(R̂).

If R has a dualizing complex, then δ_{NC} is an isomorphism for N ∈ D_{b}^{f}(S) ∩ G(R).

Relative dualizing complexes do not always localize smoothly. Indeed, Ferrand and Raynaud [12; (4.2.i)] have constructed a one-dimensional local domain R, such that R̂ has a minimal prime 𝔮 with R̂_{𝔮} non-Gorenstein. From (5.1.c) we see that C = R̂ is a dualizing complex for ρ: R → R̂. On the other hand, as R_{(0)} is a field and the ring R_{𝔮} is not Gorenstein, (5.1.a) shows that C_{𝔮} = R̂_{𝔮} is not a dualizing complex for ρ_{𝔮}: R_{(0)} ⟶ R̂_{𝔮}.

Thus, the failure of the formal fibers of R to be Gorenstein is an obstruction for the dualizing complex of φ to localize properly. The following result proves it is the only one.

(5.8) Localization. If C ∈ D(S) is a dualizing complex for φ, then for each 𝔮 ∈ Spec S the complex C_{𝔮} ∈ D(S_{𝔮}) is dualizing for φ_{𝔮}, under any one of the following conditions:

(1) φ is essentially of finite type; or

(2) R has Gorenstein formal fibers.

(5.9) Formal fibers. If R has Gorenstein formal fibers and φ has a dualizing complex, then S has Gorenstein formal fibers.

(5.10) Closed fiber. If φ is flat and C ∈ D(S) is a (normalized) dualizing complex for φ, then the complex k ⊗_{R}^{L} C is (normalized) dualizing for k ⊗_{R} S.

Theorems (5.11) through (5.13) involve a second local homomorphism, ψ: Q ⟶ R.

(5.11) Base change. When G-dim ψ < ∞ and A ∈ D_{b}^{f}(R) the following are equivalent. (i) A is dualizing for ψ, and φ is Gorenstein at 𝔫. (ii) A ⊗_{R}^{L} S is dualizing for φψ, and φ has finite flat dimension.

(5.12) Finite ascent. If G-dim ψ < ∞, G-dim φψ < ∞, the homomorphism φ is finite, and the complex A ∈ D(R) is dualizing for ψ, then RHom_{R}(S, A) is dualizing for φψ.

(5.13) Composition. If G-dim ψ < ∞, fd φ < ∞, A ∈ D(R) is dualizing for ψ, and C ∈ D(S) is dualizing for φ, then A ⊗_{R}^{L} C ∈ D(S) is dualizing for φψ.

6. Relative dualizing complexes: Proofs

In this section φ: (R, 𝔪) ⟶ (S, 𝔫) denotes a local homomorphism.

The following result is pivotal for most of the arguments to follow.

(6.1) Theorem. Assume G-dim φ < ∞ and D is a (normalized) dualizing complex for R.

(a) C ∈ D(S) is (normalized) dualizing for φ if and only if D ⊗_{R}^{L} C is (normalized) dualizing for S.

(b) E ∈ D(S) is dualizing for S if and only if RHom_{R}(D, E) is dualizing for φ.

We precede its proof by

(6.2) Lemma. Assume that R has a dualizing complex, and let C and E be S–complexes. When G-dim φ < ∞, respectively, fd φ < ∞, the following hold.

(a) If C is dualizing for φ, then C ∈ G(R), respectively, C ∈ F(R).

(b) If E is dualizing for S, then E ∈ J (R), respectively, E ∈ I(R).

Proof. Let D be a dualizing complex for R.

(a) By definition, D ⊗_{R}^{L} C is in I(S), hence is in J (R) by (4.6.a), respectively, in I(R) by (4.6.b). Now C ∈ G(R), respectively, C ∈ F(R), from (3.2.g), respectively, (3.2.f).

(b) follows directly from (4.6.a), respectively, from (4.6.b). □

Proof of Theorem (6.1). (a’) Assume that C is dualizing for φ. The complex E = D ⊗_{R}^{L} C is in I(S) by (5.2.b), and in D^{f}(S) by (1.2.1). On the other hand, C ∈ G(R) by (6.2.a), hence γ_{C}: C ⟶ RHom_{R}(D, E) is an isomorphism. Thus, the commutative diagram

S  ⟶^{χ_{E}}  RHom_{S}(E, E)
χ_{C} ↓^{≃}  ^{≃}↓
RHom_{S}(C, C) ⟶_{RHom_{S}(C,γ_{C})}^{≃} RHom_{S}(C, RHom_{R}(D, E)) ,

in which the unnamed isomorphism is as in (1.4.1), implies that χ_{E} is invertible, so that E is dualizing for S.

(b’) Assume that E is dualizing for S, set C = RHom_{R}(D, E), and note that C ∈ D_{b}^{f}(S) by (1.2.2) and (1.3.b). On the other hand, (6.2.b) yields E ∈ J (S), hence ι_{E}: D⊗_{R}^{L}C ⟶ E is an isomorphism, and thus D ⊗_{R}^{L} C ∈ I(S). Finally, the commutative diagram

S  ⟶^{χ_{C}}  RHom_{S}(C, C)
χ_{E} ↓^{≃}  ^{≃}↓
RHom_{R}(E, E) ⟶_{RHom_{C}(ι_{E},E)}^{≃} RHom_{S}(D ⊗_{R}^{L} C, E)

shows that χ_{C} is invertible, hence C is dualizing for φ.

(a”) Assume that E = D ⊗_{R}^{L} C is a dualizing complex for S. By applying consecutively (6.2.b) and (3.2.g), we see that C belongs to G(R). It follows that C is isomorphic to RHom_{R}(D, E), which is a dualizing complex for φ by the part of (b) established in (b’).

(b”) If C = RHom_{R}(D, E) is dualizing for φ, then (6.2.a) and (3.2.j) yield E ∈ J (R), hence D ⊗_{R}^{L} C ≃ E. By the established part of (a), E is dualizing for S.

Finally, to get (a) in the form which involves normalizations, it suffices to remark that inf(D ⊗_{R}^{L} C) = inf D + inf C = depth R + inf C. □

(6.3) Lemma. If G-dim φ < ∞, the R–complex D is dualizing for R, the S–complex E is dualizing for S, and the S–complex C is dualizing for φ, then C ∼ RHom_{R}(D, E).

Proof : It suffices to prove the relations C ≃ RHom_{R}(D, D ⊗_{R}^{L} C) ∼ RHom_{R}(D, E). The first one is due to the inclusion C ∈ G(R), established in (6.2.a). The second one is induced by the relation D ⊗_{R}^{L} C ∼ E, which expresses the uniqueness (2.4) of dualizing complexes for S, since D ⊗_{R}^{L} C is one by (6.1.a). □

(6.4) Lemma. A complex C ∈ D(S) is dualizing for φ if and only if the complex C ⊗_{S} Ŝ ∈ D(Ŝ) is dualizing for φ̀ and/or for φ̂.

Proof. By the faithful flatness of Ŝ over S, the isomorphism H(C ⊗_{S} Ŝ) ≅ H(C) ⊗_{S} Ŝ shows C ⊗_{S} Ŝ ∈ D_{b}^{f}(Ŝ) if and only if C ∈ D_{b}^{f}(S). When this holds, χ_{C⊗_{S}} Ŝ is invertible together with χ_{C} ⊗_{S} Ŝ due to the commutative square

Ŝ ⟶^{χ_{C}⊗_{S} Ŝ} RHom_{S}(C, C) ⊗_{S} Ŝ
∥  ^{≃}↓
Ŝ ⟶_{χ_{C⊗_{S} _{S}}} RHomŜ(C ⊗_{S} Ŝ, C ⊗_{S} Ŝ) .

By faithful flatness, the latter morphism is invertible together with χ_{C}. □

(6.5) Lemma. If φ is finite and G-dim φ < ∞, then RHom_{R}(S, R) is dualizing for φ.

Proof. By the preceding lemma, we may assume that R and S are complete, and then choose a dualizing complex D for R. Note the canonical isomorphisms

RHom_{R}(S, R) ≃ RHom_{R}(S, RHom_{R}(D, D)) ≃ RHom_{R}(D, RHom_{R}(S, D)) .

As RHom_{R}(S, D) is a dualizing complex for S by (2.12), conclude by (6.1.b). □

(6.6) Lemma. If ψ: Q ⟶ R is a local homomorphism which is Gorenstein at 𝔪, and G-dim φ < ∞, then C ∈ D_{b}^{f}(S) is dualizing for φ if and only if it is dualizing for φψ.

Proof. As above, we may assume that Q has a dualizing complex, B. By (2.11) the complex D = B ⊗_{Q}^{L} R is dualizing for R. Using (3.7.b) and (4.3), we see that G-dim φψ is finite. In view of the isomorphism D ⊗_{R}^{L} C ≃ B ⊗_{Q}^{L} C, the assertion follows from (6.1.a). □

Combining the last two lemmas, we get:

(6.7) Lemma. If R ⟶ R′ ⟶ S is a Gorenstein factorization of φ and G-dim φ < ∞, then RHom_{R′}(S, R′) is a dualizing complex for φ. □

The proofs of the results of Section 5 follow a logical – not numerical – order.

Proof of Theorem (5.3): apply (6.1.b) and (6.7). □

Proof of Theorem (5.7). By (6.4) the complex C ⊗_{S} Ŝ is dualizing for φ̂.

The commutative triangle

RHom_{S}(RHom_{S}(N, C), C) ⊗_{S} Ŝ
δ_{NC}⊗_{S} Ŝ ↗  ↘≃
N ⊗_{S} Ŝ ⟶_{δ_{(N⊗_{S} _{S}) (C⊗_{S} _{S})}} RHomŜ(RHomŜ(N ⊗_{S} Ŝ, C ⊗_{S} Ŝ), C ⊗_{S} Ŝ)

and faithful flatness show that we may assume R and S are complete.

So let D and E be dualizing complexes for R and S, respectively. By (6.3) we may further assume that C = RHom_{R}(D, E). Set K = D ⊗_{R}^{L} S and L = D ⊗_{R}^{L} N. The canonical

isomorphisms RHom_{R}(D, L) ⟶^{≃} RHom_{S}(K, L), RHom_{S}(L, E) ⟶^{≃} RHom_{S}(N, C), and

C ⟶^{≃} RHom_{S}(K, E) induce a commutative diagram

N  ⟶^{δ_{NC}}  RHom_{S}(RHom_{S}(N, C), C)
γ_{N} ↓^{≃}  ^{≃}↓
RHom_{R}(D, L)  RHom_{S}(RHom_{S}(L, E), C)
↓^{≃}  ^{≃}↓
RHom_{S}(K, L)  RHom_{S}(RHom_{S}(L, E), RHom_{S}(K, E))
RHom_{S}(K,δ_{LE})↓^{≃}  ^{≃}↓
RHom_{S}(K, RHom_{S}(RHom_{S}(L, E), E)) ⟶^{≃} RHom_{S}(K ⊗_{S}^{L} RHom_{S}(L, E), E) .

Thus, δ_{NC} is an isomorphism.

The last assertion of the theorem follows from (3.7.a). □

Proof of Theorem (5.4). Since C′ ⊗_{S} Ŝ and C ⊗_{S} Ŝ are both dualizing for φ̂: R̂ ⟶ Ŝ, (6.3) yields C′ ⊗_{S} Ŝ ∼ C ⊗_{S} Ŝ. It follows that

RHom_{S}(C′, C) ⊗_{S} Ŝ ≃ RHomŜ(C′ ⊗_{S} Ŝ, C ⊗_{S} Ŝ) ∼ RHomŜ(C ⊗_{S} Ŝ, C ⊗_{S} Ŝ) ≃ Ŝ .

Faithful flatness shows that H(RHom_{S}(C′, C)) is concentrated in a single degree, where it is isomorphic to S. Thus, RHom_{S}(C′, C) ∼ S. It remains to note that in the sequence

C′ ⟶^{δ_{C′C}} RHom_{S}(RHom_{S}(C′, C), C) ∼ RHom_{S}(S, C) ≃ C .

the morphism δ_{C′C} is invertible by (5.7). □

Proof of Theorem (5.8.1). We have G-dim φ_{𝔮} < ∞ by (4.5.1). Let R ⟶ R′ ⟶ S be a regular factorization of φ as in the proof of that proposition. By (6.7) the complex RHom_{R′}(S, R′) is dualizing for φ, hence C ∼ RHom_{R′}(S, R′) by (5.4). The last relation localizes to C_{𝔮} ∼ RHom_{R′_{𝔭′}}(S_{𝔮}, R′_{𝔭′}), where 𝔭′ = 𝔮 ∩ R′. Since R_{𝔭} ⟶ R′_{𝔭′} ⟶ S_{𝔮} is a regular factorization of φ_{𝔮}, by (6.7) again we conclude that C_{𝔮} is dualizing for φ_{𝔮}. □

Proof of Theorem (5.13). By (4.4.2) and (4.7), all three homomorphisms ψ, φ, and φψ have finite G-dimension. We may assume that Q has a dualizing complex B. By (6.2.a) we have C ∈ F(R), hence A ⊗_{R}^{L} C ∈ D_{b}^{f}(S) by (1.2.1) and (1.3.a). On the other hand, by (6.1.a) B ⊗_{Q}^{L} A is dualizing for R. The isomorphism (B ⊗_{Q}^{L} A) ⊗_{R}^{L} C ≃ B ⊗_{Q}^{L} (A ⊗_{R}^{L} C) and another application of (6.1.a) show that A ⊗_{R}^{L} C is dualizing for φψ. □

Proof of Theorem (5.11). We may assume all rings are complete, and choose dualizing complexes B for Q and D for R.

(i) ⟹ (ii). The complex D ⊗_{R}^{L} S is dualizing for S by (2.11). Furthermore, as fd φ is finite, we have S ≃ RHom_{R}(D, D ⊗_{R}^{L} S) by (3.2), hence S is a dualizing complex for φ by (6.1.b). Now (5.13), shows that A ⊗_{R}^{L} S is dualizing for φψ.

(ii) ⟹ (i). By (6.1.a) the complex B ⊗_{Q}^{L} (A ⊗_{R}^{L} S) is dualizing for S. Since it is isomorphic to (B ⊗_{Q}^{L} A) ⊗_{R}^{L} S, we see from (2.11) that φ is Gorenstein at 𝔫 and B ⊗_{Q}^{L} A is dualizing for R. Applying once more (6.1.a), we conclude that A is dualizing for ψ. □

Proof of Theorem (5.8.2) and Theorem (5.9). Fix 𝔮 ∈ Spec S, set 𝔭 = 𝔮 ∩ R, and note that the induced homomorphism φ_{𝔮}: R_{𝔭} ⟶ S_{𝔮} has finite G-dimension by (4.5.2).

In the special case when R has a dualizing complex D, the S–complex E = D ⊗_{R}^{L} C is dualizing for S by (6.1.a), and C ∼ RHom_{R}(D, E) by (6.3). It follows that C_{𝔮} ∼ RHom_{R_{𝔭}}(D_{𝔭}, E_{𝔮}). By (2.8), D_{𝔭} and E_{𝔮} are dualizing for R_{𝔭} and S_{𝔮}, respectively. As G-dim φ_{𝔮} < ∞, we conclude by (6.1.b) that C_{𝔮} is dualizing for φ_{𝔮}.

In general, we remark that C ⊗_{S} Ŝ is a dualizing complex for φ̂ by (6.4), choose a prime ideal 𝔮^{∗} ∈ Spec Ŝ lying over 𝔮, and set 𝔭^{∗} = 𝔮^{∗} ∩ R̂. By the special case, the complex (C ⊗_{S} Ŝ)_{𝔮^{∗}} is dualizing for φ_{𝔮^{∗}}: R̂_{𝔭^{∗}} ⟶ Ŝ_{𝔮^{∗}}. At this point, we have a commutative square

R_{𝔭} ⟶^{φ_{𝔮}} S_{𝔮}
ρ↓  ↓^{σ}
R̂_{𝔭^{∗}} \overrightarrow{φ_{𝔮^{∗}}} Ŝ_{𝔮^{∗}}

of local homomorphisms, in which σ is flat, ρ is flat with Gorenstein closed fiber, G-dim φ_{𝔮} is finite (observed above), and G-dim φ_{𝔮^{∗}} is finite (from (4.4.3) and (4.5.2)).

The Ŝ_{𝔮^{∗}}–complex (C ⊗_{S} Ŝ)_{𝔮^{∗}} is dualizing for φ_{𝔮^{∗}}ρ by (6.6). As it is isomorphic to C_{𝔭} ⊗_{S_{𝔮}} Ŝ_{𝔮^{∗}}, and φ_{𝔮^{∗}}ρ = σφ_{𝔮}, we see that the last complex is dualizing for σφ_{𝔮}. Since σ is flat, (5.11) applies and shows that C_{𝔮} is dualizing for φ_{𝔮}, and the ring Ŝ_{𝔮^{∗}}/𝔮Ŝ_{𝔮^{∗}} is Gorenstein. It is a localization of the formal fiber k(𝔮) ⊗_{S} Ŝ. In choosing 𝔮^{∗} we had the liberty of picking it as the contraction of any prime ideal of k(𝔮) ⊗_{S} Ŝ, so we conclude that this formal fiber is Gorenstein. □

Proof of Theorem (5.6). We may assume that φ = φ̂, and choose a normalized dualizing complex D for R. By (6.1.a) the complex E = D ⊗_{R}^{L} C is normalized dualizing for S. By (1.5.3.a) we have P_{E}^{S}(t) = P_{D}^{R}(t) P_{C}^{S}(t) , and (2.6) translates the preceding equality into I_{S}(t) = I_{R}(t) P_{C}^{S}(t) , as desired. Furthermore, as C ∈ G(R) by (6.2.a), we have C ≃ RHom_{R}(D, E). Thus, (1.5.3.b) and (2.6) yield I_{S}^{C}(t) = P_{D}^{R}(t) I_{S}^{E}(t) = I_{R}(t) . □

Proof of Theorem (5.5). We may assume that S and R are complete, and let D be a dualizing complex for R.

We start with a Gorenstein factorization R ⟶ R′ ⟶ S of φ. By (6.7), the complex RHom_{R′}(S, R′) is dualizing for φ. By the uniqueness (5.4) of such complexes, we may assume C = RHom_{R′}(S, R′). Note that inf C = depth S − depth R′ by (4.1.3) and (4.1.2).

As sup C = − grade_{R′} S by the homological characterization of grade, we get

amp C = depth R′ − depth S − grade_{R′} S .

On the other hand, since R ⟶ R′ is flat with Gorenstein closed fiber, we have dim R′ − dim R + depth R = depth R′, hence

cmd φ = (dim R′ − dim R − codim_{R′} Ŝ) − (depth S − depth R)
= depth R′ − depth S − codim_{R′} Ŝ .

Thus, we obtain amp C − cmd φ = codim_{R′} S − grade_{R′} S ≥ 0, as desired.

If R is Cohen–Macaulay, then so is the ring R′, hence codim_{R′} S = grade_{R′} S, so amp C = cmd φ. By (6.1.a) the complex D ⊗_{R}^{L} C is dualizing for S, hence amp C = amp(D ⊗_{R}^{L} C) = cmd S by (3.3) and (2.5).

From now on we assume that R ⟶ R′ ⟶ S is a Cohen factorization, cf. (4.2.2). If fd φ is finite, then by (4.2.3) the projective dimension of the R′–module S is finite, hence codim_{R′} S = grade_{R′} S by [6; (2.5)], and so amp C = cmd φ.

As noted above, i = inf C is equal to depth S − depth R′, so that H_{i}(C) = Ext_{R′}^{−i}(S, R′). By definition, type φ is equal to the minimal number of generators of the last S–module, hence ν_{S} H_{i}(C) = type φ. On the other hand, the equality ν_{S} H_{i}(C) = type S/ type R follows immediately from the expression for P_{C}^{S}(t) obtained in (5.6). □

Proof of Theorem (5.10). We assume the rings are complete, cf. (6.4), and choose dualizing complexes D and E for R and S. As RHom_{R}(k, D) ∼ k by (2.6), we have

k ⊗_{R}^{L} RHom_{R}(D, E) ≃ RHom_{R}(RHom_{R}(k, D), E)
∼ RHom_{R}(k, E)
≃ RHom_{S}(k ⊗_{R} S, E) ,

where the first isomorphism is due to (1.4.3), which applies because E is in I(R) by (4.6.b). As k ⊗_{R}^{L} RHom_{R}(D, E) ∼ k ⊗_{R}^{L} C by (6.3), and the complex RHom_{S}(k ⊗_{R} S, E) is dualizing for k ⊗_{R} S by (2.12). If furthermore C is normalized, then inf(k ⊗_{R}^{L} C) = inf C = depth S − depth R = depth k ⊗_{R} S, hence k ⊗_{R}^{L} C is normalized as well. □

Proof of Theorem (5.12). Again, we may assume all rings complete, and take dualizing complexes B and D for Q and R, respectively. Consider now the canonical isomorphisms

RHom_{R}(S, RHom_{Q}(B, D)) ≃ RHom_{R}(S ⊗_{Q}^{L} B, D) ≃ RHom_{Q}(B, RHom_{R}(S, D)) ,

where the first one is given by (1.4.1). As RHom_{R}(S, D) is a dualizing complex for S by (2.12), and G-dim φψ is finite by assumption, we see that the isomorphic complexes above are dualizing for φψ. It remains to note that A ∼ RHom_{Q}(B, D) by (6.3). □

7. Bass numbers of local homomorphisms

In this section φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) denotes a local homomorphism.

If G-dim φ is finite, then so is G-dim φ̂ by (4.4.3), and the latter homomorphism has a normalized dualizing complex C′ by (5.3).

Definition. The number μ_{φ}^{i} = β_{i}^{S}(C′) is called the i′th Bass number of φ, and the formal Laurent series I_{φ}(t) is called the Bass series of φ.

Clearly, μ_{φ}^{i} = μi ; φ̀ = μi ; φ̂. The equality in the next theorem is simply a restatement of the first equality in (5.6). The inequalities come from the fact that I_{φ}(t) has order equal to inf C = depth S − depth R.

(7.1) Theorem. When G-dim φ is finite there is an equality of formal Laurent series

I_{S}(t) = I_{R}(t) I_{φ}(t) .

In particular, μ_{R}^{i+depth R} ≤ μ_{S}^{i+depth S} for each i ∈ ℤ. □

Now we compare the present Bass invariants with those introduced in [9].

(7.2) Homotopy fiber. The homotopy fiber F (φ) of φ is k ⊗_{R}^{L} S, equipped with its natural structure of differential graded algebra (which is unique up to isomorphism in the subcategory of D(R) generated by DG R–algebras and their morphisms). The “Bass series” of φ is defined in [9; p. 512] to be the “series” I_{F (φ)}(t) = ∑_{i∈ℤ} rank_{ℓ} Ext_{F (φ)}^{i}(ℓ, F (φ))t^{i}, where quotation marks indicate that the ranks involved need not be finite.

Assume that fd φ < ∞. It follows from [9; (5.1)] that then I_{F (φ)}(t) is actually a formal Laurent series, and satisfies I_{S}(t) = I_{R}(t) I_{F (φ)}(t) . As in this case G-dim φ < ∞ by (4.4.2), comparison of this equation with that of the theorem yields I_{F (φ)}(t) = I_{φ}(t) .

The following example shows that (7.1) extends part of [9; (5.1)] in an essential way.

(7.3) Example. Let (Q, 𝔮) be a Gorenstein local ring, let x and y be Q–regular sequences such that 0 ≠ (x) ⊆ 𝔮(y), and let φ be the canonical map R = Q/(x) ⟶ Q/(y) = S. As G-dim φ < ∞ by (4.4.1), we have I_{S}(t) = I_{R}(t) I_{φ}(t) by (7.1). On the other hand, it is shown in [9; (5.5), (5.7)] that I_{F (φ)}(t) is a formal Laurent series, but I_{S}(t) ≠ I_{R}(t) I_{F (φ)}(t) .

Next we show that the Bass series of homomorphisms exhibits the same rigidity as that of local rings. When fd φ is finite, it is proved in [4; (4.4)] that conditions (i) and (ii) below are equivalent, and the question is raised in [4; (3.11)] of their equivalence with (iii). Shida [20] has obtained an affirmative answer when R is Cohen–Macaulay, or when cmd φ ≤ 1. We get a positive answer it in the wider framework of finite G-dimension.

(7.4) Theorem. If G-dim φ is finite, then the following conditions are equivalent. (i) I_{φ}(t) = t^{depth S−depth R}. (ii) I_{φ}(t) is a Laurent polynomial. (iii) μ_{φ}^{i} = 0 for some i > depth S − depth R.

Proof. We may assume φ has a a normalized dualizing complex C, and set d = inf C.

(iii) ⟹ (ii). Let i > d be such that μ_{φ}^{i} = 0. By [17; (II.2.4.1)] or [15; (2.3.c)], C ∈ D_{+}^{f}(S) is isomorphic to a complex F of finite free S–modules with F_{n} = 0 for n < d, and ∂(F_{n}) ⊆ 𝔫F_{n−1} for each n ∈ ℤ. We then have rank_{k} F_{n} = β_{n}^{S}(F) = β_{n}^{S}(C) = μ_{φ}^{n}, and hence F_{i} = 0. Thus, C ≃ F′ ⊕ F′′, where F′ = F_{<i} ≠ 0 and F′′ = F_{>i}. Consequently:

S ≅ Ext_{S}(C, C) ≅ Ext_{S}(F′, F′) ⊕ Ext_{S}(F′, F′′) ⊕ Ext_{S}(F′′, F′) ⊕ Ext_{S}(F′′, F′′) .

The S–module S is indecomposable and Ext_{S}(F′, F′) ≠ 0, hence Ext_{S}(F′′, F′′) = 0. This means that F′′ = 0 or, in other words, that μ_{φ}^{n} = 0 for n > i.

(ii) ⟹ (i). Our assumption means that C is in P(S), hence by (1.4.2) we have

S ≃ RHom_{S}(C, C) ≃ C^{∗} ⊗_{S}^{L} C ,

where C^{∗} = RHom_{S}(C, S). The latter complex is in D_{b}^{f}(S) by (1.3.c) and (1.2.2), hence (1.5.3.a) gives 1 = P_{S}^{S}(t) = P_{C^{∗}}^{S}(t) P_{C}^{S}(t) . This implies P_{C}^{S}(t) = t^{m} for some m ∈ ℤ. As the order of P_{C}^{S}(t) is equal to d = depth S − depth R, it follows that m = d. □

Definition. A local homomorphism φ which has finite G-dimension and satisfies the equivalent conditions of (7.4) is said to be quasi-Gorenstein at 𝔫.

As an immediate consequence of (7.1) and (7.4) we have

(7.5) Theorem. The following condition are equivalent when φ has finite G-dimension.

(i) φ is quasi-Gorenstein at 𝔫.

(ii) μ_{R}^{i+depth R} = μ_{S}^{i+depth S} for each i ∈ ℤ.

(iii) μ_{R}^{i+depth R} = μ_{S}^{i+depth S} for some i > 0. □

Condition (iii) cannot be weakened any further: equality in (ii) holds trivially for i < 0, and the next example shows that equality for i = 0 needs not imply equalities for i > 0.

(7.6) Example. The composition R ⟶ R′ = R[[X, Y]] ⟶ R[[X, Y]]/X(X, Y) = S is a flat local homomorphism with non-Gorenstein closed fiber F = k[[X, Y]]/X(X, Y), hence μ_{F}^{i} > 0 for i ≥ 0. Since μ_{F}^{0} = 1, and I_{S}(t) = I_{R}(t) I_{F} (t) by (7.1) and (5.10), we see that μ_{R}^{depth R} = μ_{S}^{depth S} and μ_{R}^{i+depth R} < μ_{S}^{i+depth S} for each i > 0. More precisely, we have

I_{S}(t) = I_{R}(t) \frac{1 + t − t^{2}}{1 − t − t^{2}} ,

either by expressing I_{F} (t) from [23; Satz 8], or by noting that φ: R′ ⟶ S is Golod by [18; Theorem 3] and of finite flat dimension, and applying [9; (5.7.b)].

In view of (4.4.2) and (4.4.1), the following remarks result by comparison of definitions.

(7.7.1) φ is Gorenstein at 𝔫 if and only if it is quasi-Gorenstein at 𝔫 and fd φ is finite.

(7.7.2) The following conditions are equivalent: (i) R and S are Gorenstein; (ii) R is Gorenstein and φ is quasi-Gorenstein at 𝔫; (iii) S is Gorenstein and G-dim φ is finite.

(7.7.3) φ is quasi-Gorenstein at 𝔫 if and only if φ̀ and/or φ̂ is quasi-Gorenstein at 𝔫Ŝ.

Next we describe the homomorphisms which properly base change dualizing complexs.

(7.8) Theorem. The following conditions are equivalent.

(i) φ is quasi-Gorenstein at 𝔫.

(ii) S is a dualizing complex for φ, and φ has finite G-dimension.

(iii) D′ ⊗L ; R̂ Ŝ is a dualizing complex for Ŝ if D′ is one for R̂.

If R has a dualizing complex D, they are also equivalent to (iii’) D ⊗_{R}^{L} S is a dualizing complex for S.

Proof. (iii) ⟺ (iii’) by (2.2) and (5.2.a).

By (7.7.3), (5.2.a) and (4.4.3), conditions (i) and (ii) are invariant under passage from φ to φ̂, so for the rest of the proof we may assume that R has a dualizing complex D, and φ has a normalized dualizing complex C. Now (ii) ⟹ (iii’) by (6.1.a).

(iii’) ⟹ (ii). As D ⊗_{R}^{L} S is a dualizing complex for S, it has bounded homology along with S and, and produces a commutative diagram

S ⟶_{≃}^{χ_{D⊗^{L}_{R}S}} RHom_{S}(D ⊗_{R}^{L} S, D ⊗_{R}^{L} S)
∥  ^{≃}↑
S ⟶_{γ_{S}} RHom_{R}(D, D ⊗_{R}^{L} S)

which shows that γ_{S} is invertible. This implies S ∈ G(R), thus G-dim φ is finite by (4.3). Now (6.1.b) applies and shows that RHom_{R}(D, D ⊗_{R}^{L} S) is a dualizing complex for φ. Using once more the isomorphism γ_{S}, we see that S has the same property.

(i) amounts by (7.1) to I_{S}^{C}(t) = 1, which by (1.5.1) means C ∼ S, which is (ii). □

(7.9) Corollary. Assume φ is quasi-Gorenstein at 𝔫 and R has a dualizing complex. An S–complex M is in G(S), respectively, J (S), if and only if it is in G(R), respectively, J (R).

Proof. By (7.8) the complex E = D⊗_{R}^{L} S is dualizing for S, hence the functors D⊗_{R}^{L} − and E ⊗_{S}^{L} −, respectively, RHom_{R}(D, −) and RHom_{S}(E, −), are naturally equivalent. □

(7.10) Corollary. When a local homomorphism ψ: Q ⟶ R is quasi-Gorenstein at 𝔪, then G-dim φ < ∞ if and only if G-dim φψ < ∞.

Proof. By (7.7.3) and (4.4.3) we may assume all rings are complete, so by (4.3) we have to prove that R ∈ G(Q) is equivalent to S ∈ G(Q). This follows from (7.9). □

We now recall the useful result [2; (4.32)] of Auslander and Bridger: if S = R/(x) for an R–regular x, and N is a finite S–module with G-dim_{S} N < ∞, then G-dim_{R} N = G-dim_{S} N + 1 < ∞. Since the map R ⟶ R/(x) is Gorenstein, and thus quasi-Gorenstein by (7.7.1), the next theorem provides a broad generalization, as well as a converse. That it holds is a remarkable property of G-dimension with (almost) no counterpart for the familiar projective dimension: if x ∈ 𝔪 Ann_{R} N, then pd_{S} N = ∞, cf. [18; §3].

(7.11) Theorem. Let φ: (R, 𝔪) ⟶ (S, 𝔫) be a finite local homomorphism, which is quasi-Gorenstein at 𝔫. If N is a finite S–module, then

G-dim_{R} N = G-dim_{R} S + G-dim_{S} N .

In particular, G-dim_{R} N < ∞ if and only if G-dim_{S} N < ∞.

Proof. By the Auslander–Bridger equality (4.1.2), it suffices to prove the last assertion. By (4.1.4) and (4.2.3) we may assume that both R and S are complete. From (4.1.7) we see that the finiteness of G-dim_{R} N is equivalent to the inclusion N ∈ G(R), and that of G-dim_{S} N – to the inclusion N ∈ G(S). As φ is quasi-Gorenstein, (7.9) shows that these inclusions hold simultaneously. □

8. Quasi-Gorenstein homomorphisms

In this section φ: R ⟶ S denotes a homomorphism of noetherian rings.

Definition. The homomorphism φ is quasi-Gorenstein at a prime 𝔮 of S if the induced local homomorphism φ_{𝔮}: R_{𝔮∩R} ⟶ S_{𝔮} is quasi-Gorenstein at 𝔮S_{𝔮}; it is quasi-Gorenstein if it has this property at all 𝔮 ∈ Spec S.

We list a series of properties of quasi-Gorenstein homomorphisms, with indications of the results from which they are obtained by localization.

(8.1) Gorenstein homomorphisms. φ is Gorenstein (called “locally Gorenstein” in [4]) if and only if it is quasi-Gorenstein and locally of finite flat dimension. Cf. (7.7.1).

(8.2) Ascent and Descent. When R is Gorenstein, φ is quasi-Gorenstein if and only if S is Gorenstein. If S is Gorenstein and φ is locally of finite G-dimension, then φ is quasi-Gorenstein and R is Gorenstein at the prime ideals contracted from S. Cf. (7.7.2).

(8.3) Flat homomorphisms. A flat homomorphism φ is quasi-Gorenstein if and only if all the non-trivial fibers of φ are Gorenstein. Cf. (5.10) and (2.1).

(8.4) Essentially finite type. If φ is essentially of finite type and is quasi-Gorenstein at all maximal ideals of S, then it is quasi-Gorenstein. Cf. (4.5.1) and (5.8.1).

(8.5) Gorenstein formal fibers. If the formal fibers of R are Gorenstein and φ is quasi-Gorenstein at all maximal ideals of S, then φ is quasi-Gorenstein and all formal fibers of S are Gorenstein. Cf. (4.5.2), (5.8.2), and (5.9).

The proofs of the next two theorems follow those of [6; (6.10), (6.11)], and are omitted.

(8.6) Flat base change. Let φ be essentially of finite type, and let τ: R → T be a flat homomorphism. If φ is quasi-Gorenstein, then so is the induced homomorphism φ ⊗_{R} T: T → S ⊗_{R} T; when τ is faithfully flat the converse holds as well. □

(8.7) Completion. Let 𝔞 ⊂ R and 𝔟 ⊂ S be ideals such that φ(𝔞) ⊆ 𝔟 ≠ S, and let φ^{∗}: R^{∗} ⟶ S^{∗} be the induced homomorphism of the corresponding ideal-adic completions. If R has Gorenstein formal fibers and φ is quasi-Gorenstein, then so is φ^{∗}. □

The final results, involving a second homomorphism ψ: Q ⟶ R, have no analog for rings.

(8.8) Flat descent. If φ is faithfully flat and φψ is quasi-Gorenstein, then ψ is quasi-Gorenstein and φ is Gorenstein.

(8.9) Composition. If ψ and φ are quasi-Gorenstein, then so is φψ.

(8.10) Decomposition. Assume that φψ is quasi-Gorenstein.

(a) If ψ is quasi-Gorenstein, then so is φ.

(b) If ψ an φ are locally of finite G-dimension, then φ is quasi-Gorenstein, and ψ is quasi-Gorenstein at the prime ideals of R contracted from S.

Proof of Theorems (8.8) through (8.10). We may assume that ψ, φ, and hence φψ, are local. All three homomorphisms have finite G-dimension: this is the hypothesis in (8.10.b), follows from (4.7) in (8.8), and from (7.10) in the remaining two cases. From (7.1) we obtain the double inequalities

μ_{Q}^{i+depth Q} ≤ μ_{R}^{i+depth R} ≤ μ_{S}^{i+depth S} for i ∈ ℤ ,

which become equalities if and only if μ_{Q}^{i+depth Q} = μ_{S}^{i+depth S}. The assertions follow. □

Precisely as the concept introduced in this section extends that of Gorenstein homomorphism [4], one can generalize the Cohen–Macaulay homomorphisms of [6] as follows:

(8.11) Quasi-Cohen–Macaulay homomorphisms. A local homomorphism φ: (R, 𝔪) ⟶ (S, 𝔫) is quasi-Cohen–Macaulay at 𝔫 if G-dim φ < ∞, and a dualizing complex C′ for φ̂ has amp C′ = 0. It is easily seen that φ is quasi-Gorenstein at 𝔫 if and only it is quasi-Cohen– Macaulay at 𝔫 and has type 1.

Several properties of quasi-Gorenstein homomorphisms hold in the more general context of quasi-Cohen–Macaulay homomorphisms. However, if S is Cohen–Macaulay and G-dim φ < ∞, then φ is quasi-Cohen–Macaulay, but we do not know whether R is Cohen– Macaulay. This incomplete descent result is one reason not to pursue the venue. Another is that – at this time – we do not know whether the larger class is closed under composition, partly due to the unknown transitivity of the finiteness of G-dimension, cf. (4.8).

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U. S. A.

Matematisk Institut, Universitetsparken 5, DK–2100 København Ø, Denmark