ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION

LUCHEZAR L. AVRAMOV AND ALEX MARTSINKOVSKY

Contents

Introduction 1 1. Complexes. Minimality 3 2. Total reflexivity 6 3. Gorenstein dimension 8 4. Relative cohomology 12 5. Tate cohomology 18 6. Gorenstein perfection 24 7. Comparison morphisms 26 8. Minimal resolutions 29 9. Betti numbers 34 10. Bass numbers 36 References 40

Introduction

We study finite modules M over a noetherian (on both sides) ring R. Auslander and Bridger [2] singled out the class of R-modules of finite Goren-

stein dimension (or G-dimension) as a common generalization of modules of finite projective dimension over arbitrary rings, and arbitrary modules over commutative Gorenstein rings. Projective modules always have G-dimension 0. Over a Gorenstein ring the modules of G-dimension 0 are precisely the maximal Cohen-Macaulay modules; their central position in the category of finite R-modules is emphasized by the maximal Cohen-Macaulay approximations of Auslander and Buchweitz [3].

Classically, G-dimension is studied by means of functors Ext_{R}^{n}(M, ). Our approach is based on the interaction of this absolute cohomology theory with two other theories that are naturally defined on the category of modules of finite G-dimension.

The relative cohomology functors Ext_{G}^{n}(M, ) treat modules of G-dimension zero as projectives. Their vanishing defines a numerical invariant, the relative dimension rel dim_{G} M, which refines the classical projective dimension: rel dim_{G} M ≤ proj dim_{R} M, with equality when the projective dimension is finite.

Date: November 18, 2000, 11 h 33 min.

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

A.M. thanks Purdue University for hospitality and partial support.

At the other extreme, the Tate cohomology functors Ex̂t_{R}^{n}(M, ) capture only those properties of M that are shared by all its syzygy modules. This theory is rigid, in the sense that vanishing of any one of these functors implies the vanishing of all of them; vanishing characterizes modules of finite projective dimension.

Relative and Tate functors come equipped with natural transformations,

ε_{G}^{n}: Ext_{G}^{n} → Ext_{R}^{n}  and  ε_{R}^{n}: Ext_{R}^{n} → Ex̂t_{R}^{n} .

Our main result shows that, in all degrees, they provide remarkably tight connections between the three theories: There exists an exact sequence of functors

0 ⟶ Ext_{G}^{1} ⟶^{ε_{G}^{1}} Ext_{R}^{1} ⟶ ⋯ ⟶ Ext_{G}^{n} ⟶^{ε_{G}^{n}} Ext_{R}^{n} ⟶^{ε_{R}^{n}} Ex̂t_{R}^{n} ⟶^{δ_{R}^{n}} Ext_{G}^{n+1} ⟶ ⋯

This is new even for maximal Cohen-Macaulay modules over Gorenstein local rings.

In Section 1 we summarize some basic facts about complexes of modules, used throughout the paper, and introduce an intrinsic notion of minimality for complexes. Sections 2 and 3 provide a concise treatment of the classical aspects of finite G-dimension, using explicit constructions of complexes. This differs from the module-theoretic point of view in [2], where most results originate.

The next two sections contain elementary, self-contained constructions of the relative and Tate theories. Our presentation emphasizes the parallelism of their properties; unlike other treatments, we give essentially parallel arguments.

The relative cohomology in Section 4 is ‘relative homological algebra’ in the sense of Eilenberg and Moore [16], and MacLane [24], with respect to the class of modules of G-dimension 0. These sources are mostly concerned with existential aspects of the theory, while we focus on hands-on constructions of proper resolutions. As an application we recover properties of modules of finite G-dimension, due to Auslander and Bridger [6]. Also, we establish the equality of relative and Gorenstein dimensions, proved over commutative Gorenstein rings by Enochs and Jenda [19].

Tate cohomology, introduced through complete resolutions, has been the subject of several recent expositions, in particular by Buchweitz [11] and Cornick and Kropholler [13]. However, we could not find in the literature an adequate treatment of two essential points: functoriality in the contravariant argument and naturality of comparison maps. To tackle them, in Section 5 we incorporate into the notion of complete resolution a comparison morphism to a projective resolution. Such a device might prove useful in other contexts as well.

For the special class of G-perfect modules, defined in [2], relative and Tate cohomology groups have functorial comparisons with appropriate absolute homology groups: this is the topic of Section 6. In Section 7 we derive the exact sequence above, involving the comparison morphisms. In both sections the arguments heavily rely on explicit constructions developed earlier in the paper.

In Section 8 we produce and characterize minimal resolutions over commutative local rings. We prove that in all theories ‘minimal’ constructions yield minimal resolutions, and that our general concept of minimality specializes to a variety of notions, introduced earlier in the Gorenstein case in more or less ad hoc ways.

In the two final sections we test the techniques developed in the paper. For finite modules over a commutative local ring we consider relative and Tate versions, both of Betti numbers and of Bass numbers. Exploring the relation of these invariants with their absolute prototypes and with Auslander’s ‘delta invariants’, we uncover patterns that may be rather surprising from an absolutist perspective.

1. Complexes. Minimality

Let R be an associative ring and M = M(R) the category of left R-modules. Right R-modules are treated as left modules over the opposite ring, R^{o}. For

any left or right module M, the dual Hom_{R}(M, R) is equipped with the canonical action on the other side, and ( )^{∗} denotes either of the dualization functors

Hom_{R}( , R): M(R)^{op} ⟶ M(R^{o}) or Hom_{R^{o}}( , R): M(R^{o})^{op} ⟶ M(R) .

Complexes have differentials of degree −1, and are often displayed as sequences

B = ⋯ ⟶ B_{n+1} ⟶^{∂_{n+1}^{B}} B_{n} ⟶^{∂_{n}^{B}} B_{n−1} ⟶ ⋯

The dual complex of B is the complex of R^{o}-modules with (B_{−n})^{∗} in degree n:

B^{∗} = ⋯ ⟶ (B_{−n−1})^{∗} ⟶^{(∂_{−n}^{B})^{∗}} (B_{−n})^{∗} ⟶^{(∂_{−n+1}^{B})^{∗}} (B_{−n+1}^{∗})^{∗} ⟶ ⋯

We identify a module M and the complex with M in degree zero and 0 elsewhere.

Let i be an integer. The ith syzygy module of B is Ω^{i}B = Coker ∂_{i+1}^{B}. The ith shift of B is the complex Σ^{i}B with nth component equal to B_{n−i} and ∂_{n}^{Σ^{i}B} = (−1)^{i}∂_{n−i}^{B}. We denote B_{<i} or B_{⩽i−1} the subcomplex of B with nth component equal to B_{n} for n < i and to 0 for n ≥ i, and set B_{⩾i} = B_{>i−1} = B/B_{<i}.

A homomorphism β: B → C of degree i is a sequence of R-linear maps β_{n}: B_{n} → C_{n+i} for n ∈ ℤ. All homomorphisms of degree i form an abelian group, denoted Hom_{R}(B, C)_{i}, that we identify with ∏_{n∈ℤ} Hom_{R}(B_{n}, C_{n+i}). It appears as the ith component of a complex Hom_{R}(B, C) of abelian groups, with differential

∂(β_{n}) = (∂_{n+i}^{C} β_{n} − (−1)^{i}β_{n−1} ∂_{n}^{B}) for β = (β_{n}) ∈ Hom_{R}(B, C)_{i} .

For any i ∈ ℤ, the cycles in Hom_{R}(B, C)_{i} are the chain maps B → C of degree i. Chain maps β, β′: B → C are in the same homology class, denoted β ∼ β′, if and only if they are homotopic. As usual, we set H^{i} Hom_{R}(B, C) = H_{−i} Hom_{R}(B, C).

A morphism of complexes is a chain map of degree 0. A quasiisomorphism is a morphism β: B → C with H_{n}(β): H_{n}(B) → H_{n}(C) bijective for all n; if such a β is surjective, then each cycle of C is the image of some cycle of B. Since homology classes in H_{0} Hom_{R}(B, C) represent homotopy classes of morphisms, the Lifting Lemma and Extension Lemma below are obtained by unraveling definitions.

1.1. Let β: B → C be a morphism of complexes.

(1) Let P be a complex such that Hom_{R}(P , β): Hom_{R}(P , B) → Hom_{R}(P , C) is a quasiisomorphism. For each morphism γ: P → C there is a morphism α: P → B with γ ∼ β α (even γ = β α if Hom_{R}(P , β) is surjective). If γ′: P → C and α′: P → B satisfy γ′ ∼ γ and γ′ ∼ β α′, then α′ ∼ α.

(2) Let I be a complex such that Hom_{R}(β, I): Hom_{R}(C, I) → Hom_{R}(B, I) is a quasiisomorphism. For each morphism α: B → I there exists a morphism γ: P → B with γ β ∼ α (even γ β = α if Hom_{R}(β, I) is surjective). If α′: B → I and γ′: C → I satisfy α′ ∼ α and γ′ β ∼ α′, then γ′ ∼ γ.

This is often applied by means of a standard Comparison Lemma. A complex A is bounded below (respectively, above) if A_{i} = 0 for all i ≪ 0 (respectively, i ≫ 0).

1.2. Let β: B → C be a quasiisomorphism of complexes of R-modules.

(1) If P is a bounded below complex of projective R-modules, then Hom_{R}(P , β) is a quasiisomorphism; it is surjective when β is surjective.

(2) If I is a bounded above complex of injective R-modules, then Hom_{R}(β, I) is a quasiisomorphism; it is surjective when β is injective.

The fact that the induced maps are quasiisomorphisms can be proved along the lines of Lemma 4.3, using a simple Mittag-Leffler Criterion for acyclicity:

1.3. Let π^{(i)} : A^{(i)} → A^{(i−1)} be surjective morphisms of complexes, for i ≥ 1. If H_{n}(A^{(i)}) = 0 for all i ≥ 0 and all n ∈ ℤ, then H_{n}( \underleftarrow{lim} A^{(i)}) = 0 for all n ∈ ℤ.

Indeed, the definition of inverse limit yields a sequence of morphisms

0 ⟶ \underleftarrow{lim} A^{(i)} ⟶ ∏_{i⩾0} A^{(i)} ⟶^{λ} ∏_{i⩾0} A^{(i)}

where λ(a^{(i)}) = (a^{(i)} − π^{(i+1)}(a^{(i+1)})) .

The surjectivity of each π^{(i)} implies that λ is surjective. Since H_{n}( ∏_{i⩾0} A^{(i)}) = ∏_{i⩾0} H_{n}(A^{(i)}) = 0, the homology exact sequence yields H_{n}( \underleftarrow{lim} A^{(i)}) = 0.

A resolution (of length ≤ q) of a module M is a quasiisomorphism A → M with A_{n} = for all n < 0 (and A_{n} = 0 for n > q. The next result, sometimes called the Horseshoe Lemma, goes back to Cartan and Eilenberg [12, V §2].

1.4. Let 0 → M ⟶^{μ} M′ ⟶^{μ′} M′′ → 0 be an exact sequence of R-modules.

Given projective resolutions π: P → M and π′′: P′′ → M′′, there exists a commutative diagram with exact rows

[figure: 0; M; μ; M′; μ′; M′′; 0; 0; P; \overline{μ}; π; _{P_{′}}; _{\overline{μ}_{′}}; π′; P_{′′}; π′′; 0]

Given a commutative diagram of the form above with bounded below complexes of projective modules P′, P′, and P′′, a commutative diagram

[figure: 0; M; μ̃; γ; M′; μ′; γ′; M′′; γ′′; 0; 0; G; μ; _{G_{′}}; μ′; G_{′′}; 0]

with bottom row split exact in each degree and a quasiisomorphism γ, a morphism ϕ: P → G with γϕ = π, and a morphism ϕ′′: P′′ → G′′ with γ′′ϕ′′ = π′′, there exists a morphism ϕ′ with γ′ϕ′ = π′ making the following diagram commute

[figure: 0; G; μ̃; G′; μ′; G′′; 0; 0; P; \overline{μ}; ϕ; _{P_{′}}; _{\overline{μ}_{′}}; ϕ′; P_{′′}; ϕ′′; 0]

If γ: C → B and β: B → C are morphisms with β γ ∼ id_{C}, then γ is a right homotopy inverse of β, and β is a left homotopy inverse of γ. A homotopy equivalence is a morphism γ having a left homotopy inverse β, which itself has a left homotopy inverse γ′; in this case γ β = id_{B} γ β ∼ γ′β γ β ∼ γ′ id_{C} β = γ′β ∼ id_{B} so that γ and β are homotopy inverse morphisms. The same conclusion holds if γ has a right homotopy inverse β′ which itself has a right homotopy inverse γ′′.

Well known sources of homotopy equivalences come from 1.2 and 1.1:

1.5. A quasiisomorphism β: B → C is a homotopy equivalence if

(1) B_{n} and C_{n} are projective for all n, and vanish for all n ≪ 0; or

(2) B_{n} and C_{n} are injective for all n, and vanish for all n ≫ 0.

A complex A contractible if id_{A} ∼ 0_{A}. A subcomplex A of a complex B is irrelevant if A is contractible and A_{n} is a direct summand of B_{n} for each n ∈ ℤ.

1.6. Lemma. If A is an irrelevant subcomplex of a complex B and C = B/A, then the canonical morphism β: B → C is a homotopy equivalence.

Proof. The exact sequence 0 → A → B ⟶^{β} C → 0 induces exact sequences

0 ⟶ Hom_{R}(C, A) ⟶ Hom_{R}(C, B) ⟶^{Hom_{R}(C,β)} Hom_{R}(C, C) ⟶ 0

0 ⟶ Hom_{R}(C, B) ⟶^{Hom_{R}(β,B)} Hom_{R}(B, B) ⟶ Hom_{R}(A, B) ⟶ 0

As Hom_{R}(C, A) and Hom_{R}(A, B) are contractible along with A, their homology is trivial, so the maps induced by β are quasiisomorphisms. From 1.1 we get morphisms γ , γ′: C → B with βγ = id_{C} and γ′β = id_{B}.

We introduce a concept of minimality, applicable to any complex: A complex B is minimal if each homotopy equivalence β: B → B is bijective.

1.7. Proposition. Let B be a complex of R-modules.

(1) The following conditions are equivalent.

(i) The complex B is minimal.

(ii) Each morphism β: B → B homotopic to id_{B} is an isomorphism. (iii) If β: B → C and γ: C → B are homotopy inverses, then β is injective,

γ is surjective, Ker γ is contractible, and C = Im β ⊕ Ker γ.

(2) Every homotopy equivalence between minimal complexes is an isomorphism.

(3) If B is minimal and A is an irrelevant subcomplex, then A = 0.

Proof. (1) (i) ⟹ (ii). If β ∼ id_{B}, then β^{2} ∼ β ∼ id_{B}, so β is its own homotopy inverse. In particular, β is a homotopy equivalence, and so an isomorphism.

(ii) ⟹ (iii). The morphism α = γ β: B → B is homotopic to id_{B}, and so is an isomorphism. Thus, γ′ = α^{−1}γ: C → B is a homotopy equivalence with γ′β = id_{B}, hence C = Im β ⊕ K for K = Ker γ′ = Ker γ. In the exact sequence

0 ⟶ Hom_{R}(K, K) ⟶ Hom_{R}(K, C) ⟶^{Hom_{R}(K,γ′)} Hom_{R}(K, B) ⟶ 0

the map Hom_{R}(K, γ′) is a homotopy equivalence, and so a quasiisomorphism. It follows that H Hom_{R}(K, K) = 0, so the cycle id_{K} is a boundary, that is, id_{K} ∼ 0_{K}.

(iii) ⟹ (i) is clear.

(2) follows immediately from property (1.iii) of minimal complexes.

(3) If B is minimal and A is an irrelevant subcomplex, then β: B → B/A is a homotopy equivalence by Lemma 1.6, so A = Ker β = 0 by property (1.iii).

1.8. Example. Let B be a complex with B_{n} is injective for all n, B_{n} = 0 for n > 0, and H_{n}(B) = 0 for n ≠ 0. Using Proposition 1.7.1 and well known properties of injective envelopes, one sees that B is minimal if and only if B_{n} is an injective envelope of Ker(∂_{n}) for all n ≤ 0. Thus, for ‘injective resolutions’ our concept of minimality agrees with the classical notion.

2. Total reflexivity

For the rest of the paper R denotes a left and right noetherian ring.

Let G be a finite R-module. Recall that G is said to be reflexive if the canonical map ζ^{G}: G → G^{∗∗} is bijective. We say that G is totally reflexive^{1} if, in addition

(∗) Ext_{R}^{n}(G, R) = 0 = Ext_{R^{o}}^{n}(G^{∗}, R) for all n > 0 .

Problem. Are the conditions defining total reflexivity independent?

2.1. Examples. Let R be a commutative ring and G a finite R-module.

(1) If R is local Gorenstein, then the following are equivalent: (i) G is totally reflexive; (ii) G is maximal Cohen-Macaulay; (iii) Ext_{R}^{n}(G, R) = 0 for all n > 0: cf. [28, (4.15)] for (iii) ⟹ (ii) and [10, (3.3.10.d)] for (ii) ⟹ (i).

(2) If R = ⨁_{i=0}^{∞} R_{i} is graded with R_{0} a field, G is graded, and (∗) holds, then the following are equivalent: (i) G is reflexive; (ii) ζ^{G} is injective; (iii) ζ^{G} is surjective.

Indeed, by the Hilbert-Serre Theorem, the Laurent series ∑_{i∈ℤ} rank_{R_{0}}(G_{i})t^{i} is the Laurent expansion around t = 0 of a rational function H_{G}(t). If N is a graded R-module such that Ext_{R}^{n}(G, N) = 0 for n ≫ 0, then [7, Theorem 1] yields

∑_{n}(−1)^{n}H_{Ext_{R}^{n}(G,N)}(t) = \frac{H_{G}(t^{−1}) · H_{N}(t)}{H_{R}(t^{−1})} .

Applying the formula to the pairs of modules (G, R) and (G^{∗}, R), we get

H_{G^{∗}}(t) = \frac{H_{G}(t^{−1}) · H_{R}(t)}{H_{R}(t^{−1})}  and  H_{G^{∗∗}}(t) = \frac{H_{G^{∗}}(t^{−1}) · H_{R}(t)}{H_{R}(t^{−1})} .

hence H_{G^{∗∗}}(t) = H_{G}(t). Thus, rank_{k} G_{n} = rank_{k} G_{n}^{∗∗} for each n ∈ ℤ. Since ζ^{G} is a degree 0 homomorphisms of graded vector spaces, our assertion follows.

We recall from [2] some basic properties of totally reflexive modules. Proofs are included for three reasons: not all statements appear in [2] in the form or generality needed here; some of the original arguments do not carry over to the non-commutative case; we are able to offer some substantial shortcuts.

The following full subcategories of M(R) are of special interest in this paper:

• F = F(R) whose objects are the finite R-modules.

• G = G(R) whose objects are the totally reflexive R-modules.

• P = P(R) whose objects are the finite projective R-modules.

For a full subcategory C ⊆ M, we say that a complex C with C_{n} ∈ C for all n is a C-complex. A totally acyclic complex is a P-complex T with H_{n}(T) = 0 = H_{n}(T^{∗}) for all n ∈ ℤ. A C-resolution of an R-module M is a quasiisomorphism γ: G → M where G is a C-complex with G_{n} = 0 for n < 0; it has an associated exact sequence

G^{+} = ⋯ ⟶ G_{n} ⟶^{∂_{n}} G_{n−1} ⟶ ⋯ ⟶ G_{1} ⟶^{∂_{1}} G_{0} ⟶^{γ} M ⟶ 0 .

A complete resolution of M is a diagram T ⟶^{ϑ} P ⟶^{π} M where π is a P-resolution, T is a totally acyclic complex, ϑ is a morphism, and ϑ_{n} is bijective for all n ≫ 0.

1Auslander and Bridger [2] introduced these modules and called them modules of G-dimension zero, in anticipation of their role in the construction of a homological dimension. The difficulty of using that name is quite obvious (the reader should try substituting ‘modules of projective dimension zero’ for ‘projective modules’). Another name, systematically used by Enochs and collaborators, cf. e.g. [19], is Gorenstein projective modules, referring to their role in a relative cohomology theory. We have opted for a name reflecting their intrinsic module-theoretic properties.

The three lemmas that follow reproduce, and partly generalize, portions of [2, (4.12), (4.9), (3.11), (4.11)], sometimes with considerably shorter proofs.

2.2. Lemma. Let G be a totally reflexive R-module.

(1) Ext_{R}^{n}(G, N) = 0 for all n > 0 and all N ∈ M with proj dim_{R} N < ∞.

(2) The R^{o}-module G^{∗} is totally reflexive.

Proof. (1) As R is noetherian and M is finite, the functor Ext_{R}^{n}(M, ) commutes with arbitrary direct sums, so Ext_{R}^{n}(M, Q) = 0 for all n > 0 when Q is projective. When 1 ≤ proj dim_{R} N = p < ∞, choose an exact sequence 0 → N′ → Q → N → 0 with Q projective. By induction we have Ext_{R}^{n}(M, Q) = 0 and Ext_{R}^{n+1}(M, N′) = 0 for all n > 0, so the cohomology exact sequence yields Ext_{R}^{n}(M, N) = 0.

(2) The condition Ext_{R^{o}}^{n}(G^{∗}, R) = 0 for n > 0 holds by hypothesis. The R^{o}-module G^{∗} is reflexive because G is in F. Thus, we get Ext_{R^{o}^{o}}^{n}((G^{∗})^{∗}, R) = Ext_{R}^{n}((G^{∗})^{∗}, R) ≅ Ext_{R}^{n}(G, R) = 0 for n > 0, so G^{∗} is totally reflexive over R^{o}.

2.3. Lemma. The category G is closed under extensions and kernels of epimorphisms, and G(R) contains P(R).

Proof. The last assertion is elementary. Let E = 0 → E → F → G → 0 be an exact sequence with G ∈ G; we assume that one of E or F is in G, and show that so is the other. The sequence E^{∗} = 0 → G^{∗} → F^{∗} → E^{∗} → 0 is exact because Ext_{R}^{1}(G, R) = 0. Thus, we obtain a commutative diagram with exact rows

\xymatrix{0 \ar[r] & E \ar[r] \ar[d]^{ζ^{E}} & F \ar[r] \ar[d]^{ζ^{F}} & G \ar[r] \ar[d]^{ζ^{G}}_{≅} & 0 \\ 0 \ar[r] & E^{∗∗} \ar[r] & F^{∗∗} \ar[r] & G^{∗∗}}

By assumption ζ^{G} and one of ζ^{E}, ζ^{F} is bijective. The Snake Lemma shows all three are bijective, so E^{∗∗} is exact. Since Ext_{R}^{n}(G, R) = 0 for n > 0, the exact sequence E yields Ext_{R}^{n}(F, R) ≅ Ext_{R}^{n}(E, R) for each n > 0; one of these groups vanishes by assumption, hence both do. As Ext_{R^{o}}^{n}(G^{∗}, R) = 0 for n > 0, the exact sequence E^{∗} shows that Ext_{R^{o}}^{n}(E^{∗}, R) → Ext_{R^{o}}^{n}(F^{∗}, R) is bijective for n > 1 and surjective for n = 1; it is injective also for n = 1 because E^{∗∗} is exact. By assumption, at least one of Ext_{R^{o}}^{n}(E^{∗}, R) or Ext_{R^{o}}^{n}(F^{∗}, R) vanishes, hence both do.

2.4. Lemma. Let T be a P-complex with H_{n}(T) = 0 for all n ∈ ℤ.

The following conditions are equivalent.

(i) H_{n}(T^{∗}) = 0 for all n ∈ ℤ.

(ii) Ω^{n}T is totally reflexive for each n ∈ ℤ.

(iii) H_{n} Hom_{R}(T , N) = 0 for all n ∈ ℤ and each N ∈ M with proj dim_{R} N < ∞.

Proof. (i) ⟹ (ii). Fix n ∈ ℤ and set G = Ω^{n}T. As Σ^{−n}T_{⩾n} is a P-resolution of G, we have Ext_{R}^{i}(G, R) ≅ H_{−i−n}(T^{∗}) = 0 for i > 0 and G^{∗} ≅ Ker(∂_{n+1}^{T})^{∗} = Ker(∂_{−n}^{T^{∗}}). As T^{∗} is exact, Ker(∂_{−n}^{T^{∗}}) = Ω^{−n+1}T^{∗} and Σ^{n−1}T _{⩾−n+1}^{∗} is a Presolution of G^{∗}. Since T^{∗∗} = T is exact, we get Ext_{R}^{i}(G^{∗}, R) ≅ H_{i−n+1}(T^{∗∗}) = 0 for i > 0 and G^{∗∗} ≅ Ker(∂_{−n+2}^{T^{∗}})^{∗} = Ker(∂_{n−1}^{T^{∗∗}}) = Ker(∂_{n−1}^{T}) = G.

(ii) ⟹ (iii). For each n the complex Σ^{−n+1}T_{⩾n−1} is a P-resolution of Ω^{n−1}T, so H_{−n} Hom_{R}(T , N) ≅ Ext_{R}^{1}(Ω^{n−1}T , N); this module vanishes by Lemma 2.2.1.

(iii) ⟹ (i) is clear.

3. Gorenstein dimension

In this section R is a noetherian (on both sides) ring.

A G-resolution G → M is said to be strict if G_{n} is projective for all n ≥ 1.

3.1. Theorem. Let M be a finite module over a noetherian ring R.

For each integer g ≥ 0 the following conditions are equivalent.

(i) M has a G-resolution of length ≤ g.

(ii) M has a strict G-resolution of length ≤ g.

(iii) M has a P-resolution P with Ω^{g}P ∈ G.

(iv) M has a complete resolution σ: S → P with σ_{n} bijective for all n ≥ g.

(v) M has a complete resolution ϑ: T → P with ϑ_{n} bijective for all n ≥ g and surjective for all n ∈ ℤ

Proofs of the results in this section are collected at the end of the section.

Conditions (i) and (iii) reproduce conditions (b) and (c) of [2, (3.13.)]; here as there, their equivalence is obtained as an immediate consequence of Lemmas 2.3 and 3.9. The equivalence of (i) and (iv) appears in [6, (4.4.4)].

In [2, (3.1)] Auslander and Bridger assign to each R-module M a number G-dim_{R} M, called the Gorenstein dimension, or G-dimension, of M. If M ≠ 0 is finite, then the equivalence of conditions (a) and (c) in [2, (3.13)] shows that

G-dim_{R} M = inf \{g ∈ ℕ | \begin{matrix}there exists a G-resolution \\ G → M of length ≤ g \end{matrix}\} .

We take this equality as the definition of G-dimension for a finite module M, and observe the following consequences: G-dim_{R} M = −∞ if and only if M = 0, and G-dim_{R} M = 0 if and only if M ≠ 0 and M is totally reflexive.

Recall that R is regular (of dimension at most d) if each finite module, left or right, has finite projective dimension (at most d). We say that R is Gorenstein (of dimension at most d) if each finite module, left or right, has finite G-dimension (at most d). By Auslander and Bridger [2, (4.20)] (for d < ∞) and Goto [22] this definition yields the classical notion: for each maximal ideal 𝔪 the injective dimension inj dim_{R_{𝔪}}R_{𝔪} is finite (at most d). The following result might be ‘well known’; for completeness, we incclude a proof later in this section.

3.2. Theorem. For a ring R and an integer d ≥ 0 the following are equivalent. (i) R is Gorenstein of dimension at most d. (ii) R is noetherian with inj dim_{R} R ≤ d and inj dim_{R^{o}} R ≤ d.

If condition (ii) holds, then inj dim_{R} R = inj dim_{R^{o}} R by Zaks [31, p. 84].

3.3. Examples. Let R be a noetherian ring.

(1) If d = 0, then by Eilenberg and Nakayama [17, Theorem 18] condition (ii) can be weakened to (ii′) R is self-injective on one side, or strengthened to (ii′′) is quasi-Frobenius, that is, left and right artinian and left and right self-injective.

(2) Let 𝕜 be a commutative ring, R a 𝕜-algebra with a 𝕜-linear involution x ↦ \overline{x}, and let ( )^{∨} be the functor Hom_{𝕜}( , 𝕜): M(R)^{op} → M(R), where for M ∈ M(R) the ring R acts on M^{∨} by (xα)(y) = α(\overline{x} y). If R is in P(𝕜) and the R-module R^{∨} is isomorphic to R, then G-dim_{R} M ≤ proj dim_{𝕜} M for each M ∈ F(R).

Indeed, each K ∈ F(R)∩P(𝕜) satisfies K^{∨} ∈ F(R)∩P(𝕜) and K^{∨∨} ≅ K. Thus, if 0 → J → L → K^{∨} → 0 is an 𝕜-split exact sequence in F(R) with middle term in P(R), then the dual sequence 0 → K → L^{∨} → J^{∨} → 0 has the same properties.

Each P(R)-resolution P → M is a P(𝕜)-resolution. If proj dim_{𝕜} M = p < ∞, then K = Ω^{p}P is in F(R) ∩ P(𝕜), so as above we obtain a 𝕜-split exact sequence 0 → Ω^{p}P → T′ → K′ → 0 in F(R), with T′ ∈ P(R) and K′ ∈ F(R) ∩ P(𝕜). Iterating, we extend T_{⩾p} = P_{⩾p} to a 𝕜-split exact complex T with Ω^{p}T = Ω^{p}P. Finally, the isomorphisms T^{∗} ≅ Hom_{R}(T , R^{∨}) ≅ Hom_{𝕜}(T , 𝕜) yield H(T^{∗}) = 0.

We give names to two more full subcategories of F:

• G̃ = G̃(R) whose objects are the finite R-modules with G-dim_{R} M < ∞.

• P̃ = P̃(R) whose objects are the finite R-modules with proj dim_{R} M < ∞. We summarize basic properties of these categories, extending [2, (3.14), (3.16)].

3.4. Proposition. The category G̃(R) is closed under extensions, kernels of epimorphisms, cokernels of monomorphisms, and G̃(R) contains P̃(R).

Problem. Describe the noetherian rings R for which G̃(R) = P̃(R).

3.5. Examples. Let R be a noetherian ring.

(1) If R is Gorenstein, then R regular if and only if G̃(R) = P̃(R).

(2) If R is local, commutative, Golod, and not a hypersurface, then G̃(R) = P̃(R). Indeed, by Lescot [23, (6.5)], cf. also [5, (5.3.3.5)], if P → M is a minimal free

resolution and proj dim_{R} M = ∞, then rank_{R} P_{n+1} > rank_{R} P_{n} for all n > d = depth R. Assuming G̃(R) ≠ P̃(R), find M ∈ G(R) \ P̃(R) by condition 3.1.iii, set M′ = Ω^{−2d−3}(P^{∗}), and choose a minimal free resolution P′ → M′. As H_{−n}(P^{∗}) = Ext_{R}^{n}(M, R) = 0 for n > 0, we have (Σ^{2d+3}(P^{∗}))_{⩽2d+2} ≅ P ′_{⩽2d+2}, so rank_{R} P′_{d+2} = rank_{R} P_{d+1}^{∗} < rank_{R} P_{d+2}^{∗} = rank_{R} P′_{d+1}, contradicting Lescot’s Theorem.

(3) If R is commutative, 𝔞 is a proper ideal, and f is a non-zero-divisor contained in 𝔞^{2}, then G̃(R/(f)) ≠ P̃(R/(f)) by [8, (3.2)].

Our treatment of G-dimension is based on the following constructions.

3.6. Construction. Let π: P → M be a P-resolution.

Fix an integer g ≥ 0, set G = Ω^{g}P. and choose a P-resolution λ: L → G^{∗} over R^{o}. Noting that Σ^{g−1}((P_{<g})^{∗}) is a P-complex which is trivial in negative degrees and has Ω^{1−g}(P^{∗}) as degree 0 homology, choose a lifting of the canonical map

Ω^{1−g}(P^{∗}) = Coker((∂_{g−1}^{P})^{∗}) ↠ Im((∂_{g}^{P})^{∗}) ↪ Ker((∂_{g+1}^{P})^{∗}) = G^{∗}

to a morphism κ: Σ^{g−1}((P_{<g})^{∗}) → L. Define a graded module S and homomorphisms ∂^{S}: S → S and σ: S → P of degree −1 and 0, respectively, by

S_{n} = \begin{cases}P_{n} \\ P_{g} \\ L_{g−n−1}^{∗} \end{cases}  ∂_{n}^{S} = \begin{cases}∂_{n}^{P} \\ λ_{0}^{∗} ζ^{G} ω_{g}^{P} \\ (∂_{g−n}^{L})^{∗} \end{cases}  σ_{n} = \begin{cases}id & P_{n} \\ id & P_{g} \\ κ_{g−n−1}^{∗} \end{cases} for n > g; ; for n = g; ; for n < g ,

where ω_{g}^{P} : P_{g} → G is the canonical surjection.

Fact. (S, ∂^{S}) is a P-complex, σ is morphism with σ_{n} bijective for all n ≥ g, and

H_{n}(S) = \begin{cases}0 & for n ≥ g + 1 ; \\ Ker(ζ^{G}) & for n = g; \\ Coker(ζ^{G}) & for n = g − 1 ; \\ Ext_{R}^{g−1−n}(G^{∗}, R) & for n ≤ g − 2 ; \end{cases}

H_{n}(S^{∗}) = \begin{cases}0 & for n ≥ −g; \\ Ext_{R^{o}}^{−n−g}(G, R) & for n ≤ −g − 1 , \end{cases}

If G is totally reflexive, then S ⟶^{σ} P ⟶^{π} M is a complete resolution with Ω^{g}S = G.

Proof. Easy verifications show that (S, ∂^{S}) is a complex and σ is a morphism, bijective in degrees ≥ g. We get an exact sequence 0 → Σ^{g−1}(L^{∗}) → S → P_{⩾g} → 0 of complexes. The associated homology exact sequence breaks down to produce two series of isomorphisms, H_{n}(S) ≅ H_{n}(P) = 0 for n ≥ g + 1 and Ext_{R^{o}}^{g−1−n}(G^{∗}, R) = H_{n}(L^{∗}) ≅ H_{n}(S) for n ≤ g − 2, and an exact sequence

0 ⟶ H_{g}(S) ⟶ G ⟶^{ζ^{G}} G^{∗∗} ⟶ H_{g−1}(S) ⟶ 0 .

A similar computation with the exact sequence 0 → (P_{⩾g})^{∗} → S^{∗} → Σ^{1−g}L → 0 yields the expressions for H_{n}(S^{∗}). When G is totally reflexive these computations imply H(S) = 0 = H(S^{∗}); each S_{n} is in P, so σ is a complete resolution.

3.7. Construction. Let σ: S → P be a morphism of complexes.

Fix a number g ≤ ∞, set T_{n} = (S ⊕ P_{<g} ⊕ Σ^{−1}P_{<g})_{n} for all n ∈ ℤ, and for x ∈ S_{n}, y ∈ P_{n}, y′ ∈ P_{n+1} define maps ∂^{T} of degree −1, and ϑ, β, α of degree 0:

∂^{T} : T → T  by  ∂_{n}^{T} (x, y, y′) = (∂_{n}^{S}(x) , ∂_{n}^{P} (y) , y − ∂_{n+1}^{P}(y′));
ϑ: T → P  by  ϑ_{n}(x, y, y′) = σ(x) + y;
β: T → S  by  β_{n}(x, y, y′) = x;
α: S → T  by  α_{n}(x) = (x, 0, 0) .

Fact. (T , ∂^{T}) is a complex, ϑ is a morphism with σ = ϑ α and ϑ_{n} surjective for n < g, the maps α and β are inverse homotopy equivalences with βα = id_{S}.

If S ⟶^{σ} P ⟶^{π} M is a complete resolution with σ_{n} bijective for n ≥ g, then T ⟶^{ϑ} P ⟶^{π} M is a surjective complete resolution with ϑ_{n} bijective for n ≥ g.

Proof. The properties of ∂^{T}, ϑ, and βα follow directly from the formulas. To get a homomorphism ξ: T → T with α β = id_{T} +∂^{T} ξ + ξ∂^{T}, set ξ_{n}(x, y, y′) = (0, y′, 0).

When σ is a complete resolution each T_{n} is in P by definition. As α is a homotopy equivalence, so is α^{∗}, hence H(T) ≅ H(S) = 0 and H(T^{∗}) ≅ H(S^{∗}) = 0.

3.8. Construction. Let T ⟶^{ϑ} P ⟶^{π} M be a surjective complete resolution, with ϑ_{n} bijective for n ≥ g. Set Q = Ker ϑ, denote κ the inclusion Q ⊆ T, and set

G_{n} = \begin{cases}Q_{n−1} & for n ≥ 1 ; \\ Ω^{0}T & for n = 0 ; \\ 0 & for ≤ −1 ; \end{cases}  ∂_{n}^{G} = \begin{cases}−∂_{n−1}^{Q} & for n ≥ 2 ; \\ −(Ω^{0}κ) ∘ ω_{0}^{Q} & for n = 1 ; \end{cases}

T_{n}^{♭} = \begin{cases}T_{n} & for n ≥ 0 ; \\ Ω^{0}T & for n = −1 ; \\ 0 & for n ≤ −2 ; \end{cases}  ∂_{n}^{T^{♭}} = \begin{cases}∂_{n}^{T} & for n ≥ 1 ; \\ ω_{0}^{T} & for n = 0 , \end{cases}

κ_{n}^{♭} = \begin{cases}κ_{n} & for n ≥ 0 ; \\ id_{Ω^{0}T} & for n = −1 ; \\ 0 & for ≤ −2 ; \end{cases}  ϑ_{n}^{♭} = \begin{cases}ϑ_{n} & for n ≥ 0 ; \\ Ω^{0}ϑ & for n = −1 ; \\ 0 & for n ≤ −2 , \end{cases}

where ω_{0}^{Q}: Q_{0} → Ω^{0}Q and ω_{0}^{T} : T_{0} → Ω^{0}T are the canonical maps.

Fact. (G, ∂^{G}) and (T^{♭}, ∂^{T^{♭}}) are complexes, κ^{♭}: Σ^{−1}G → T^{♭} and ϑ^{♭}: T^{♭} → P are

morphisms, the sequence 0 → Σ^{−1}G ⟶^{κ^{♭}} T^{♭} ⟶^{ϑ^{♭}} P → 0 is exact, and γ: G → M

with γ_{0} : G_{0} = Ω^{0}T ⟶^{Ω^{0}ϑ} Ω^{0}P = M is a strict G-resolution of length g.

Proof. All but the last statement result from direct verifications. As H(T^{♭}) = 0, from the exact sequence of complexes we see that H_{n}(G) = 0 for n ≠ 0 and H_{0}(γ): H_{0}(G) → M is bijective. By construction, G_{n} = 0 for n < 0 or n > g. The module G_{0} = Ω^{0}T is totally reflexive by Lemma 2.4. If n ≥ 1, then G_{n} is the kernel of ϑ_{n−1}, an epimorphism of finite projective modules, so G_{n} is in P.

The next result is from [6, (3.13)], with a new proof we learned from Idun Reiten.

3.9. Lemma. If M has a G-resolution G → M of length ≤ g, then in each Presolution P → M the module Ω^{n}P is totally reflexive for all n ≥ g.

Proof. Choose a comparison morphism P → G. Its mapping cone is exact, so for each n ≥ g we have an exact sequence of R-modules

0 → Ω^{n}P → P_{n−1} → ⋯ → P_{g} → P_{g−1} ⊕ G_{g} → ⋯ → P_{0} ⊕ G_{1} → G_{0} → 0

Using Lemma 2.3, we get first P_{i−1} ⊕ G_{i} ∈ G for i = 1, …, g, then Ω^{n}P ∈ G.

Proof of Theorem 3.1. (i) ⟹ (iii) by Lemma 3.9.

(iii) ⟹ (iv) by Construction 3.6.

(iv) ⟹ (v) by Construction 3.7.

(v) ⟹ (ii) by Construction 3.8.

(ii) ⟹ (i) holds by definition.

Proof of Proposition 3.4. Any P-resolution is a G-resolution, so G̃ ⊇ P̃.

Let M = 0 → M → M′ → M′′ → 0 be an exact sequence of R-modules, and let 0 → P → P′ → P′′ → 0 be an exact sequence of complexes provided by the Horseshoe Lemma 1.4. For each n ∈ ℤ we then have an exact sequence 0 → Ω^{n}P → Ω^{n}P′ → Ω^{n}P′′ → 0. Applying to it Lemma 2.3, and using Lemma 3.9 and condition 3.1.iii to check membership in G̃, we see that G̃ is closed under extensions and kernels of epimorphisms.

If M and M′ are in G, choose P-resolutions P → M and P′ → M′, lift M → M′ to a morphism P → P′, and form its mapping cone P′′. The exact sequence 0 → P′ → P′′ → ΣP → 0 shows that P′′ is a P-resolution of M′′ and yields exact sequences 0 → Ω^{n}P′ → Ω^{n}P′′ → Ω^{n−1}P → 0. As above, we get M′′ ∈ G.

Proof of Theorem 3.2. (i) ⟹ (ii). By assumption, for each left ideal 𝔞 there is a P-resolution P → R/𝔞 with Ω^{d}P ∈ G. Thus, Ext_{R}^{d+1}(R/𝔞, R) ≅ Ext_{R}^{1}(Ω^{d}P , R) = 0, so inj dim_{R} R ≤ d by Baer’s Criterion; inj dim_{R^{o}} R ≤ d follows by symmetry.

(ii) ⟹ (i). For M ∈ F choose P-resolutions P → M and L → (Ω^{d}P)^{∗} and form a complex S as in Construction 3.6. It satisfies H_{n}(S^{∗}) = 0 for n ≥ −d and

H_{n}(S^{∗}) = Ext_{R}^{−n−d}(Ω^{d}P , R) ≅ Ext_{R}^{−n}(M, R)

for n ≤ −d − 1. Since inj dim_{R} R ≤ d, the last module vanishes, hence S^{∗} is exact. For each integer n, we then get

H_{n}(S) = H_{n} Hom_{R}(S^{∗}, R) = Ext_{R^{o}}^{d+1}(Ω^{−n−d−1}(S^{∗}), R) .

The last module vanishes because inj dim_{R^{o}} R ≤ d, so S is exact as well. By Lemma 2.4, we conclude that Ω^{d}P = Ω^{d}S is totally reflexive.

4. Relative cohomology

All relative notions below are introduced with respect to the class G of totally reflexive module, references to which are dropped when no ambiguity arises.

A sequence E of homomorphisms in M is proper exact if the induced sequence Hom_{R}(G, E) is exact for all G ∈ G; since R is in G, proper exact sequences are exact. A proper resolution G → M is a G-resolution whose associated exact sequence G^{+} is proper. We denote \overline{G} the full subcategory of the category F of finite R-modules whose objects are the modules admitting some proper resolution.

There are handy sufficient conditions for properness of sequences and resolutions.

4.1. Lemma. (1) Each split exact sequence is proper.

(2) Each exact sequence 0 → N → N′ → N′′ → 0 with proj dim_{R} N < ∞ is proper.

(3) Each strict G-resolution G → M of finite length is proper.

Proof. (1) is clear. Lemma 2.2.1 shows that in the induced exact sequence

0 ⟶ Hom_{R}(G, N) ⟶ Hom_{R}(G, N′) ⟶ Hom_{R}(G, N′′) ⟶ Ext_{R}^{1}(G, N)

the Ext vanishes, so (2) holds. (2) implies (3) by induction on the length of G.

By Theorem 3.1, Part (3) of the lemma yields an inclusion G̃(R) ⊆ \overline{G}(R).

Problem. Is there a ring R for which G̃(R) ≠ \overline{G}(R)?

Choosing for each M ∈ \overline{G} a proper resolution G → M, we define for each n ∈ ℤ and each N ∈ M a relative cohomology group

Ext_{G}^{n}(M, N) = H^{n} Hom_{R}(G, N) .

Choosing a projective resolution π: P → M and a morphism of complexes ϕ: P → G that lifts the identity map of M, we define a comparison homomorphism

ε_{G}^{n}(M, N): Ext_{G}^{n}(M, N) ⟶ Ext_{R}^{n}(M, N)

by setting ε_{G}^{n}(M, N) = H^{n} Hom_{R}(ϕ, N): H^{n} Hom_{R}(G, N) → H^{n} Hom_{R}(P , N).

4.2. Theorem. The assignment (M, N) ↦ Ext_{G}^{n}(M, N) defines a functor

Ext_{G}^{n} : \overline{G}(R)^{op} × M(R) ⟶ M(ℤ)

and the maps ε_{G}^{n}(M, N) yield a morphism of functors ε_{G}^{n}: Ext_{G}^{n} → Ext_{R}^{n} such that:

(1) Ext_{G}^{n} and ε_{G}^{n} are independent of the choices of resolutions and lifting. (2.a) For any integer g ≥ 0 the following conditions are equivalent.

(i) G-dim_{G} M ≤ g.

(ii) Ext_{G}^{n}(M, ) = 0 for all n > g.

(iii) Ext_{G}^{g+1}(M, ) = 0 .

(iv) Each proper resolution F → M has Ω^{n}F ∈ G for all n ≥ g. (2.b) ε_{G}^{0} : Ext_{G}^{0} → Hom_{R} is an isomorphism and Ext_{G}^{n}(M, ) = 0 for n < 0.

(3) If proj dim_{R} M < ∞, then ε_{G}^{n}(M, N) is bijective for each n ∈ ℤ.

(4) If G-dim_{R} M < ∞ and N is projective, then ε_{G}^{n}(M, N) is bijective for all n ∈ ℤ.

Remark. Converses to Parts (3) and (4) are established in Corollary 7.2.

4.3. Lemma. Given proper resolutions γ: G → M and γ′: G′ → M′, and Presolutions π: P → M and π′: P′ → M′, there exist unique up to homotopy morphisms of complexes ϕ: P → G and ϕ′: P′ → G′ with π = γ ϕ and π′ = γ′ ϕ′.

For each homomorphism of modules μ: M → M′ there exists a unique up to homotopy morphism μ̃, making the right hand square of the diagram

\xymatrix{P \ar[d]^{\overline{μ}} \ar[r]^{ϕ} & G \ar[d]^{μ̃} \ar[r]^{γ} & M \ar[d]^{μ} \\ _{P_{′}} \ar[r]^{ϕ′} & _{G_{′}} \ar[r]^{γ′} & M′}

commutes; for each choice of μ̃ there exists a unique up to homotopy morphism \overline{μ}, making the left hand square commute up to homotopy.

If μ = id_{M}, then μ̃ and \overline{μ} are homotopy equivalences.

Proof. The sequence G′^{+} is proper exact, hence H Hom_{R}(G_{i}, G′^{+}) = 0 for all i ∈ ℤ. Note that G = \underrightarrow{lim} G_{⩽i}, and consider the exact sequences of complexes

0 ⟶ G_{⩽i−1} ⟶ G_{⩽i} ⟶ Σ^{i}G_{i} ⟶ 0 .

As G_{⩽−1} = 0, using the induced exact sequence of complexes of abelian groups

0 ⟶ Hom_{R} (Σ^{i}G_{i}, G′^{+}) ⟶ Hom_{R}(G_{⩽i}, G′^{+}) ⟶ Hom_{R}(G_{⩽i−1}, G′^{+}) ⟶ 0

and induction on i we get H Hom_{R}(G_{⩽i}, G′^{+}) = 0 for all i ∈ ℤ. Thus, we have

H Hom_{R}(G, G′^{+}) = H Hom_{R} ( \underrightarrow{lim} G_{⩽i}, G′^{+}) = H ( \underleftarrow{lim} Hom_{R} (G_{⩽i}, G′^{+})) = 0

with the last equality coming from the Mittag-Leffler Criterion 1.3.

The exact sequence 0 → Σ^{−1}M′ → G′^{+} → G′ → 0 induces an exact sequence

0 ⟶ Hom_{R}(G, Σ^{−1}M′) ⟶ Hom_{R}(G, G′^{+}) ⟶ Hom_{R}(G, G′) ⟶ 0

For each n, the connecting map in the homology exact sequence is an isomorphism

H_{n} Hom_{R}(G, G′) ≅ H_{n−1} Hom_{R}(G, Σ^{−1}M′) .

Composing it with H_{n−1} Hom_{R}(G, Σ^{−1}M′) = H_{n} Hom_{R}(G, M′) we obtain the map H_{n} Hom_{R}(G, γ′), so Hom_{R}(G, γ′) is a quasiisomorphism. The Lifting Lemma 1.1.1 yields a unique up to homotopy morphism μ̃ with γ′ μ̃ = μ γ.

If μ = id_{M}, then reversing the roles of M and M′ we get a morphism μ̃: G′ → G inducing id_{M}. Thus, μ̃ μ′: G′ → G′ induces id_{M}, and hence is homotopic to id_{G}′. By symmetry, μ′ μ̃ ∼ id_{G}, so μ̃ is a homotopy equivalence.

By 1.2.1 and 1.1.1 we get morphisms ϕ, ϕ′ with π = γ ϕ, π′ = γ′ ϕ′, then a morphism μ′ with ϕ \overline{μ}′ ∼ μ′ ϕ′, all unique up to homotopy. If μ = id_{M} all these maps are quasiisomorphism, so \overline{μ} is a homotopy equivalence by loc. cit.

Proof of Theorem 4.2. The first assertion of Lemma 4.3, applied to the chosen resolutions of M, M′, yields the naturality of Ext_{G}^{n} and ε_{G}^{n}.

(1) is due to the last assertion of the same lemma.

(2.a) is proved after Proposition 4.4.

(2.b) results from the left exactness of Hom_{R}( , N).

(3) Note that if M has a P-resolution P of finite length, then this is a proper G-resolution by Lemma 4.1.3, so we can take G = P and ϕ = id_{P}.

(4) is proved after Proposition 4.7.

Exact sequences of relative Ext groups exist in certain cases.

4.4. Proposition. For each R-module M ∈ \overline{G} and each proper exact sequence N = 0 → N ⟶^{ν} N′ ⟶^{ν′} N′′ → 0 of R-modules there exist natural in M and N homomorphisms ð_{G}^{n}(M, N), such that the sequence below is exact

[figure: ⋯; Ext_{G}^{n}(M, N); Ext_{G}^{n}(M,ν); Ext_{G}^{n}(M, N′); Ext_{G}^{n}(M,ν′); Ext_{G}^{n}(M, N′′); ð_{G}^{n}(M,N); Ext_{G}^{n+1}(M, N); ⋯]

and the connecting maps ð^{n}(M, N): Ext_{R}^{n}(M, N′′) → Ext_{R}^{n+1}(M, N) satisfy

ð^{n}(M, N) ∘ ε_{G}^{n}(M, N′′) = ε_{G}^{n+1}(M, N) ∘ ð_{G}^{n}(M, N) for all n ∈ ℤ .

Proof. Let ϕ: P → G be a morphism from a projective resolution of M to a proper resolution of M, with H_{0}(ϕ) = id_{M}. It induces a commutative diagram

\xymatrix{0 \ar[r] & Hom_{R}(G, N) \ar[r] \ar[d]^{Hom_{R}(ϕ,N)} & Hom_{R}(G, N′) \ar[r] \ar[d]^{Hom_{R}(ϕ,N′)} & Hom_{R}(G, N′′) \ar[r] \ar[d]^{Hom_{R}(ϕ,N′′)} & 0 \\ 0 \ar[r] & Hom_{R}(P , N) \ar[r] & Hom_{R}(P , N′) \ar[r] & Hom_{R}(P , N′′) \ar[r] & 0}

of complexes of abelian groups; the bottom (respectively, top) row is exact because each P_{n} is in P (respectively each G_{n} is in G) and N is (respectively, proper) exact. The homology exact sequence of the top row is the desired long exact sequence. The commutativity of the diagram gives the formula for the connecting maps. Naturality in N is clear; naturality in M follows from Lemma 4.3.

Proof of Theorem 4.2.2.b. (ii) ⟹ (iii) and (iv) ⟹ (i) are clear.

(i) ⟹ (ii). In view of Theorems 4.2.1 and 3.1 we can compute Ext_{G}^{n}(M, ) from a proper resolution γ: G → M of length ≤ g.

(iii) ⟹ (iv) Let F → M be a proper resolution and set G = Ω^{g}F. Since Σ^{−g}F_{⩾g} → G is a proper resolution, we get Ext_{G}^{1}(G, ) ≅ Ext_{G}^{g+1}(M, ) = 0. Proposition 4.4 applied to the proper exact sequence 0 → Ω^{g+1}F → F_{g+1} ⟶^{ω} G → 0 shows that Hom_{R}(G, ω): Hom_{R}(G, F_{g+1}) → Hom_{R}(G, G) is surjective. Thus, G is isomorphic to a direct summand σ(G) of F_{g+1}, so G is in G by Lemma 2.3.

A version of the Horseshoe Lemma for proper resolutions reads as follows.

4.5. Lemma. Let M = 0 → M ⟶^{μ} M′ ⟶^{μ′} M′′ → 0 be a proper exact sequence of finite R-modules with M and M′′ in \overline{G}.

If γ: G → M and γ′′: G′′ → M′′ are proper resolutions, while π: P → M and π′′: P′′ → M′′ are P-resolutions, then there exists a commutative diagram

[figure: 0; M; μ; M′; μ′; M′′; 0; 0; G; μ̃; γ; _{G_{′}}; μ′; γ′; G_{′′}; γ′′; 0; 0; P; \overline{μ}; ϕ; _{P_{′}}; _{\overline{μ}_{′}}; ϕ′; P_{′′}; ϕ′′; 0]

of morphisms of complexes where the middle and bottom rows are split exact as sequences of graded modules, γ′ is a proper resolution, π = γ′ϕ′ is a P-resolution, and there are equalities γϕ = π and γ′′ϕ′′ = π′′.

Proof. As M is proper, the map Hom_{R}(G′′_{0}, M′) → Hom_{R}(G′′_{0}, M′′) is surjective, so there exists a homomorphism ν′: G′′_{0} → M′ with μ′ν′ = γ′_{0}. In the diagram

\xymatrix{ & 0 & 0 & 0 \\ 0 \ar[r] & M \ar[r]^{μ} \ar[u] & M′ \ar[r]^{μ′} \ar[u] & M′′ \ar[r] \ar[u] & 0 \\ 0 \ar[r] & G_{0} \ar[r]^{μ̃_{0}} \ar[u]_{γ_{0}} & G_{0} ⊕ G′′_{0} \ar[r]^{μ′_{0}} \ar[u]_{γ′_{0}} & G′′_{0} \ar[r] \ar[u]_{γ′′_{0}} & 0 \\ 0 \ar[r] & Ω^{1}G \ar[r] \ar[u] & Ker γ′_{0} \ar[r] \ar[u] & Ω^{1}G′′ \ar[r] \ar[u] & 0 \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

where γ′_{0} = [μγ_{0} ν′], while μ̃_{0} and μ′_{0} are the canonical maps, the external columns and the top row are proper exact by hypothesis, and the middle row is proper exact by Lemma 4.1.1. Thus, for each G ∈ G the induced diagram

\xymatrix{ & 0 & 0 & 0 \\ 0 \ar[r] & Hom_{R}(G, M) \ar[r] \ar[u] & Hom_{R}(G, M′) \ar[r] \ar[u] & Hom_{R}(G, M′′) \ar[r] \ar[u] & 0 \\ 0 \ar[r] & Hom_{R}(G, G_{0}) \ar[r] \ar[u] & Hom_{R}(G, G′_{0} ⊕ G′′_{0}) \ar[r] \ar[u] & Hom_{R}(G, G′′_{0}) \ar[r] \ar[u] & 0 \\ 0 \ar[r] & Hom_{R}(G, Ω^{1}G) \ar[r] \ar[u] & Hom_{R}(G, Ker γ′_{0}) \ar[r] \ar[u] & Hom_{R}(G, Ω^{1}G′′) \ar[r] \ar[u] & 0 \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

is commutative with exact external columns, top row, and middle row. Applying the Snake Lemma to these rows, one sees that the middle column and bottom row are exact. Thus, all rows and columns in the original diagram are proper exact.

Iterating the procedure we obtain the upper tier of the desired diagram of complexes. The lower tier is produced by applying the Horseshoe Lemma 1.4.

4.6. Proposition. For each proper exact sequence M = 0 → M ⟶^{μ} M′ ⟶^{μ′} M′′ → 0 of R-modules of finite G-dimension and each R-module N there are natural in M and N homomorphisms ð_{G}^{n}(M, N), such that the sequence below is exact

[figure: ⋯; Ext_{G}^{n}(M′′, N); Ext_{G}^{n}(μ′,N); Ext_{G}^{n}(M′, N); Ext_{G}^{n}(μ,N); Ext_{G}^{n}(M, N); ð̂_{G}^{n}(M,N); Ext_{G}^{n+1}(M′′, N); ⋯]

and the connecting maps ð^{n}(M, N): Ext_{R}^{n}(M, N) → Ext_{R}^{n+1}(M′′, N) satisfy

ð^{n}(M, N) ∘ ε_{G}^{n}(M, N) = ε_{G}^{n+1}(M′′, N) ∘ ð_{G}^{n}(M, N) for all n ∈ ℤ .

Proof. The lower tier of the diagram in Lemma 4.5 yields a commutative diagram

\xymatrix{0 \ar[r] & Hom_{R}(G′′, N) \ar[r] \ar[d]^{Hom_{R}(ϕ′′,N)} & Hom_{R}(G′, N) \ar[r] \ar[d]^{Hom_{R}(ϕ′,N)} & Hom_{R}(G, N) \ar[r] \ar[d]^{Hom_{R}(ϕ,N)} & 0 \\ 0 \ar[r] & Hom_{R}(P′′, N) \ar[r] & Hom_{R}(P′, N) \ar[r] & Hom_{R}(P , N) \ar[r] & 0}

with exact rows. The homology exact sequence of the top row is the desired long exact sequence. Commutativity of the diagram gives the formula for the connecting maps. Naturality in N is clear. To get naturality in M, take a morphism M → Ṁ and apply Hom_{R}( , N) to the commuting up to homotopy diagram

[figure: 0; G; μ̃; G′; μ′; G′′; 0; 0; Ġ; ˜μ̇; _{Ġ_{′}}; _{˜μ̇_{′}}; Ġ_{′′}; 0]

with rows provided by Construction 4.5 and columns by Lemma 4.3.

Finite direct sums of proper resolutions are proper resolutions, and for each finite R-module G the functor Hom_{R}(G, ) commutes with arbitrary direct sums, hence

4.7. Proposition. For each finite set \{M_{i} ∈ \overline{G} | i ∈ I\} the module ∐_{i∈I} M_{i} is in \overline{G}, and for any family \{N_{j} ∈ M | j ∈ J\} there is a natural isomorphism

Ext_{G}^{n} ( ∐_{i∈I} M_{i}, ∐_{j∈J} N_{j}) ≅ ∐_{(i,j)∈I×J} Ext_{G}^{n}(M_{i}, N_{j}) .  □

Proof of Theorem 4.2.4. By assumption, G-dim_{R} M = g < ∞ and N is projective. The map ε_{G}^{n}(M, N): Ext_{G}^{n}(M, N) → Ext_{R}^{n}(M, N) is bijective for n ≤ 0 by Theorem 4.2.2.b. By induction on g we show that it is bijective for all n. If g = 0 and n > 0, then Ext_{G}^{n}(M, R) = 0 by Theorem 4.2.2a, and Ext_{R}^{n}(M, R) = 0 by definition. If g > 0 Theorem 3.1 yields a G-resolution G → M of length g, so G-dim_{R} Ω^{1}G < g. Thus, each ε_{G}^{n}(G_{0}, R) and ε_{G}^{n}(Ω^{1}G, R) is bijective by the induction hypothesis. The Five-Lemma, used on the commutative diagram given by Proposition 4.6 and the exact sequence 0 → Ω^{1}G → G_{0} → M → 0, shows that ε_{G}^{n}(M, R) is bijective.

For each finite module M ≠ 0 a relative G-dimension is defined by

rel dim_{G} M = inf \{g ∈ ℕ | \begin{matrix}there exists a proper G-resolution \\ G → M of length ≤ g \end{matrix}\}

and set rel dim_{G} 0 = −∞; thus, rel dim_{G} M = ∞ for M ∉ \overline{G}. Over commutative Gorenstein local rings the following is proved by Enochs and Jenda [19, (5.2)].

4.8. Proposition. If M is a finite R-module M, then G-dim_{R} M = rel dim_{G} M.

Proof. Any proper resolution is a G-resolution, so G-dim_{R} M ≤ rel dim_{G} M. Equality holds trivially when M = 0 or G-dim_{R} M = ∞, so we assume that G-dim_{R} M = g is finite. Theorem 3.1 yields a strict G-resolution of length ≤ g; such a resolution is proper by Lemma 4.1.3. Thus, rel dim_{G} M ≤ g, hence rel dim_{G} M = g = G-dim_{R} M.

The next theorem combines several results of Auslander and Bridger, cf. [2, (3.14), (4.13.a), (4.15)] (where some are proved under a commutative hypotheses). Our proofs, based on relative cohomology, are far from the original arguments.

4.9. Theorem. For any finite R-module M the following hold.

(1) There are inequalities

sup{n ∈ ℕ | Ext_{R}^{n}(M, R) ≠ 0} ≤ G-dim_{R} M ≤ proj dim_{R} M

and equalities hold to the left of any finite dimension.

(2) If 0 → M → M′ → M′′ → 0 is an exact sequence, then

G-dim_{R} M′ ≤ max{G-dim_{R} M , G-dim_{R} M′′}

with strict inequality possible only if G-dim_{R} M = G-dim_{R} M′′ + 1 .

(3) If M = M′ ⊕ M′′, then G-dim_{R}(M) = max{G-dim_{R} M , G-dim_{R} M′}.

(4) Each G-resolution G → M has Ω^{n}G ∈ G for all n ≥ G-dim_{R} M.

Proof. (1) The inequality on the right hand side follows directly from Lemma 2.3, and Theorem 4.2.3 implies that equality holds when proj dim_{R} M is finite.

The inequality on the left hand side is clear, so assume G-dim_{R} M = g < ∞ and Ext_{R}^{g}(M, R) = 0. For g = 0 this means that M^{∗} = 0; as M is reflexive, we get M = 0, contradicting our hypothesis. When g ≥ 1, Proposition 4.8 yields a proper resolution G → M of length g, and shows each proper resolution has length at least g. Our assumption means that the map ∂_{g}^{∗}: G_{g−1}^{∗} → G_{g}^{∗} is surjective. The R^{o}-module G_{g}^{∗} is then projective, so ∂_{g}^{∗} is split by a homomorphism σ: G_{g}^{∗} → G_{g−1}^{∗}. Thus, σ^{∗} ∂_{g} = id_{G_{g}}, hence E = 0 → G_{g} → ∂_{g}(G_{g}) → 0 is an irrelevant subcomplex of G. We then get a proper resolution G/E → M of length < g, which is impossible.

(2) The assertion holds trivially if any one of the modules is equal to 0, so we assume that all G-dimensions involved are non-negative. If both G-dim_{R} M and G-dim_{R} M′′ are infinite, then there is nothing to prove. If one is finite and the other is not, then Proposition 3.4 yields G-dim_{R} M′ = ∞, so the desired equality holds. If G-dim_{R}(M) and G-dim_{R} M′′ are finite, then Proposition 3.4 shows that G-dim_{R} M′ is finite as well. To see the desired (in)equality it suffices to bear in mind (1) while staring at the cohomology exact sequence

[figure: Ext_{R}^{n−1}(M, R); Ext_{R}^{n}(M′′, R); Ext_{R}^{n}(M′, R); Ext_{R}^{n}(M, R); Ext_{R}^{n+1}(M′′, R)]

(3) In view of (2) we may assume that M′, M′′, and their direct sum have finite G-dimension. Now apply Proposition 4.7 with N = R, and use (1).

(4) is an easy consequence of (2).

4.10. Remark. Yoneda congruence relations involving only proper short exact sequences define equivalence classes of proper exact sequences of length n, starting at N and ending at M. The result is an abelian group of proper extensions that depends functorially on M and N, cf. MacLane [24, XII §4]. The map sending the class of each proper sequence to its congruence class with respect to Yoneda equivalence using all short exact sequences is a morphism of extension functors.

If M has a proper G-resolution, then there exists a canonical isomorphism of the group of proper extensions with Ext_{G}^{n}(M, N); it transforms the morphism of extension functors into the morphism ε^{n}: Ext_{G}^{n} → Ext_{R}^{n}, cf. [24, XII §9].

5. Tate cohomology

For each module M of finite G-dimension over a noetherian ring R choose a complete resolution T ⟶^{ϑ} P ⟶^{π} M by Theorem 3.1, then for each R-module N and for each n ∈ ℤ define a Tate cohomology group by the equality

Ex̂t_{R}^{n}(M, N) = H^{n} Hom_{R}(T , N) .

These groups come equipped with comparison homomorphisms

ε_{R}^{n}(M, N): Ext_{R}^{n}(M, N) ⟶ Ex̂t_{R}^{n}(M, N)

given by H^{n} Hom_{R}(ϑ, N): H^{n} Hom_{R}(P , N) → H^{n} Hom_{R}(T , N).

5.1. Example. Let Π be a finite group, R = ℤ[Π] its group ring, and turn ℤ into an R-module by gm = m for g ∈ Π and m ∈ ℤ. The map ∑_{g∈Π} a_{g}g ↦ ∑_{g∈Π} a_{g}g^{−1} is an involution, and the map h ↦ (∑_{g∈Π} a_{g}g ↦ a_{h}) defines an R-linear isomorphism R ≅ R^{∨}, so G-dim_{R} ℤ = 0 by Example 3.3.2, hence Ex̂t_{R}^{n}(ℤ, N) is defined. This is the original nth Tate cohomology group Ĥ^{n}(Π, N), cf. [12, XII §3].

5.2. Theorem. The assignment (M, N) ↦ Ex̂t_{R}^{n}(M, N) defines a functor

Ex̂t_{R}^{n}: G̃(R)^{op} × M(R) ⟶ M(ℤ)

and the maps ε_{R}^{n}(M, N) yield a morphism of functors ε_{R}^{n}: Ext_{R}^{n} → Ex̂t_{R}^{n} such that:

(1) Ex̂t_{R}^{n} and ε_{R}^{n} are independent of the choices of resolutions and lifting.

(2) For any integer g the following conditions are equivalent.

(i) G-dim_{R} M ≤ g.

(ii) ε_{R}^{n}(M, N): Ext_{R}^{n}(M, N) → Ex̂t_{R}^{n}(M, N) is bijective for all n > g.

(3) If proj dim_{R} M < ∞, then Ex̂t_{R}^{n}(M, ) = 0 for all n ∈ ℤ.

(4) If proj dim_{R} N < ∞, then Ex̂t_{R}^{n}( , N) = 0 for all n ∈ ℤ.

Remark. Converses to Parts (3) and (4) are proved in Theorem 5.9.

5.3. Lemma. Let T ⟶^{ϑ} P ⟶^{π} M and T′ ⟶^{ϑ′} P′ ⟶^{π′} M′ be complete resolutions.

For each homomorphism of modules μ: M → M′ there exists a unique up to homotopy morphism \overline{μ}, making the right hand square of the diagram

\xymatrix{T \ar[d]^{μ̂} \ar[r]^{ϑ} & P \ar[d]^{\overline{μ}} \ar[r]^{π} & M \ar[d]^{μ} \\ _{T_{′}} \ar[r]^{ϑ′} & _{P_{′}} \ar[r]^{π′} & M′}

commute, and for each choice of \overline{μ} there exists a unique up to homotopy morphism μ̂, making the left hand square commute up to homotopy.

If μ = id_{M}, then \overline{μ} and μ̂ are homotopy equivalences.

Proof. The existence of \overline{μ} and its uniqueness up to homotopy are classical, cf. 1.2.1 and 1.5; thus, we only have to deal with μ̂.

Assuming first that ϑ′ is surjective, we set Q′ = Ker ϑ′ and Q′^{(i)} = Q′_{>−i}. Thus, for each i ∈ ℤ we get an exact sequence of complexes

0 ⟶ Σ^{−i}(Q′_{−i}) ⟶ Q′^{(i)} ⟶^{β′^{(i)}} Q′^{(i−1)} ⟶ 0

with β′^{(i)} the canonical epimorphism. It induces an exact sequence of complexes

0 ⟶ Hom_{R} (T , Σ^{−i}(Q′_{−i})) ⟶ Hom_{R} (T , Q′^{(i)}) ⟶ Hom_{R} (T , Q′^{(i−1)}) ⟶ 0

As Q′^{(i)} = 0 for i ≪ 0, we assume by induction on i that H_{n} Hom_{R} (T , Q′^{(i−1)}) vanishes for some i and all n. Lemma 2.2.1 yields H_{n} Hom_{R} (T , Σ^{−i}Q′_{−i}) = 0, hence H_{n} Hom_{R} (T , Q′^{(i)}) = 0. Since Q′ = \underleftarrow{lim} Q′^{(i)}, we get

Hom_{R}(T , Q′) = Hom_{R} (T , \underleftarrow{lim} Q′^{(i)}) = \underleftarrow{lim} Hom_{R} (T , Q′^{(i)})

so H_{n} Hom_{R}(T , Q′) = 0 for all n ∈ ℤ by the Mittag-Leffler Criterion 1.3. The exact

sequence of complexes 0 → Q′ → T′ ⟶^{ϑ′} P′ → 0 induces an exact sequence

0 ⟶ Hom_{R}(T , Q′) ⟶ Hom_{R}(T , T′) ⟶^{Hom_{R}(T ,ϑ′)} Hom_{R}(T , P′) ⟶ 0

of complexes of abelian groups, which shows Hom_{R}(T , ϑ′) is a surjective quasiisomorphism. By 1.1.1 we get a morphism μ̂ satisfying ϑ′ μ̂ = \overline{μ} ϑ.

In general, factor ϑ′ as T′ ⟶^{α} T′′ ⟶^{ϑ′′} P′, with a homotopy equivalence α and surjective morphism ϑ′′, cf. Construction 3.7. Thus, Hom_{R}(T , α) is a quasiisomorphism. The map Hom_{R}(T , ϑ′′) is one by the special case above, hence so is Hom_{R}(T , ϑ′) = Hom_{R}(T , ϑ′′) ∘ Hom_{R}(T , α). Now 1.1.1 yields a unique up to homotopy morphism μ̂ such that ϑ′ μ̂ ∼ \overline{μ} ϑ.

If μ = id_{M}, then reversing the roles of M and M′ we get a morphism μ′: T′ → T inducing id_{M}. Thus, μ̃ μ′: T′ → T′ induces id_{M}, and hence is homotopic to id_{T}′. By symmetry, μ′ μ̂ ∼ id_{T}, so μ̃ is a homotopy equivalence.

Proof of Theorem 5.2. The naturality of Ex̂t_{R}^{n} and ε_{R}^{n} is obtained by applying the first assertion of Lemma 5.3 to the chosen complete resolutions of M and M′.

(1) is a consequence of the last assertion of that lemma.

(3) If proj dim_{R} M = p < ∞, then in Construction 3.6 choose a P-resolution P → M of length p, set g = p + 1, and resolve Ω^{p}G = 0 with K = 0; the resulting complete resolution is 0 → P → M, hence Ex̂t_{R}^{n}(M, ) = 0.

(4) results directly from Lemma 2.4.

(2) (i) ⟹ (ii) By (1) and Theorem 3.1 the morphism ε_{R}^{n}(M, ) can be computed using a Tate resolution ϑ: T → P with ϑ_{n} bijective for n ≥ G-dim_{R} M, so ε_{R}^{n}(M, ) is an isomorphism for n > G-dim_{R} M.

(ii) ⟹ (i) If ε_{R}^{n}(M, ) is bijective, then in particular Ext_{R}^{n}(M, R) = Ex̂t_{R}^{n}(M, R). The last group is trivial by (4), so Ext_{R}^{n}(M, R) = 0 for n > g. Theorem 4.9.1 now implies g ≤ G-dim_{R} M.

With a fixed first argument, Ex̂t_{R}^{∗}(M, ) is a cohomology functor on M.

5.4. Proposition. For each R-module M of finite G-dimension and each exact sequence N = 0 → N ⟶^{ν} N′ ⟶^{ν′} N′′ → 0 of R-modules there exist natural in M and N homomorphisms ð̂^{n}(M, N), such that the sequence below is exact

[figure: ⋯; Ex̂t_{R}^{n}(M, N); Ex̂t_{R}^{n}(M,ν); Ex̂t_{R}^{n}(M, N′); Ex̂t_{R}^{n}(M,ν′); Ex̂t_{R}^{n}(M, N′′); ð̂^{n}(M,N); Ex̂t_{R}^{n+1}(M, N); ⋯]

and the connecting maps ð^{n}(M, N): Ext_{R}^{n}(M, N′′) → Ext_{R}^{n+1}(M, N) satisfy

ð̂^{n}(M, N) ∘ ε_{R}^{n}(M, N′′) = ε_{R}^{n+1}(M, N) ∘ ð^{n}(M, N) for all n ∈ ℤ .

Proof. A complete resolution T → P → M induces a commutative diagram

\xymatrix{0 \ar[r] & Hom_{R}(P , N) \ar[r] \ar[d]^{Hom_{R}(ϑ,N)} & Hom_{R}(P , N′) \ar[r] \ar[d]^{Hom_{R}(ϑ,N′)} & Hom_{R}(P , N′′) \ar[r] \ar[d]^{Hom_{R}(ϑ,N′′)} & 0 \\ 0 \ar[r] & Hom_{R}(T , N) \ar[r] & Hom_{R}(T , N′) \ar[r] & Hom_{R}(T , N′′) \ar[r] & 0}

of complexes, whose rows are exact because P_{n} and T_{n} are projective for all n. The homology exact sequence of the bottom row is the desired long exact sequence. The commutativity of the diagram gives the formula for the connecting maps. Naturality in N is clear; naturality in M follows from Lemma 5.3.

We prove a version of the Horseshoe Lemma for complete resolutions.

5.5. Lemma. Let M = 0 → M ⟶^{μ} M′ ⟶^{μ′} M′′ → 0 be an exact sequence of finite R-modules with max{G-dim_{R} M, G-dim_{R} M′′} ≤ g < ∞.

If π: P → M and π′′: P′′ → M′′ are P-resolutions, then there exists a commutative diagram with exact rows

[figure: 0; M; μ; M′; μ′; M′′; 0; 0; P; \overline{μ}; π; _{P_{′}}; _{\overline{μ}_{′}}; π′; P_{′′}; π′′; 0; 0; T; μ̂; ϑ; _{T_{′}}; μ′; ϑ′; T_{′′}; ϑ′′; 0]

whose columns are surjective complete resolutions that are bijective in degrees ≥ g.

Proof. The classical Horseshoe Lemma 1.4 provides the upper tier of the diagram. In the exact sequence 0 → Ω^{g}P → Ω^{g}P′ → Ω^{g}P′′ → 0 all modules are totally reflexive by Lemmas 3.9 and 2.3. Lemma 2.2 then guarantees the exactness of the top row in the following commutative diagram:

[figure: 0; (Ω^{g}P′′)^{∗}; (Ω^{g}P′)^{∗}; (Ω^{g}P)^{∗}; 0; 0; L_{′′}; ψ′′; λ′′; _{L_{′}}; ψ′; λ′; L; λ; 0]

The rest of the diagram is obtained by choosing P-resolutions λ and λ′′, then using 1.4 again; thus, the bottom row is split exact, considered as a sequence of graded R-modules. Lift ω_{g}^{P} to a morphism κ: Σ^{g−1}((P_{<g})^{∗}) → L, do likewise for ω_{g}^{P′′}, then use 1.4 for a third time to get a commutative diagram with exact rows

[figure: 0; L′′; ψ′′; L′; ψ′; L; 0; 0; Σ^{g−1}((P ′′_{<g})^{∗}); κ′′; Σ^{g−1}((P ′′_{<g})^{∗}); κ′; Σ^{g−1}((P_{<g})^{∗}); κ; 0]

where the maps in the bottom row are induced by \overline{μ} and \overline{μ}′.

Feeding these data to Construction 3.6, form the columns of the diagram

[figure: 0; M; μ; M′; μ′; M′′; 0; 0; P; \overline{μ}; π; _{P_{′}}; _{\overline{μ}_{′}}; π′; P_{′′}; π′′; 0; 0; S; ξ; σ; _{S_{′}}; ξ′; σ′; S_{′′}; σ′′; 0]

where ξ_{n} = π_{n} for n ≥ g, ξ_{n} = ψ_{g−n−1}^{′∗} for n < g, and ξ′ is defined similarly. The explicit descriptions of its maps show that the diagram commutes. Processing its columns through Construction 3.7, we obtain the columns of the desired diagram, then complete its construction by setting μ̂(x, y, y′) = (ξ(x), \overline{μ}(y), \overline{μ}(y′)), and defining μ̂′ is in a similar manner. Once again, the commutativity of the diagram results from the explicit formulas used in its construction.

The next rsult reflects the fact that Tate cohomology is the restriction to G̃ of a cohomology functor on F, cf. 5.11.

5.6. Proposition. For each exact sequence M = 0 → M ⟶^{μ} M′ ⟶^{μ′} M′′ → 0 of R-modules of finite G-dimension and each R-module N there exist natural in M and N homomorphisms ð̂^{n}(M, N), such that the sequence below is exact

[figure: ⋯; Ex̂t_{R}^{n}(M′′, N); Ex̂t_{R}^{n}(μ′,N); Ex̂t_{R}^{n}(M′, N); Ex̂t_{R}^{n}(μ,N); Ex̂t_{R}^{n}(M, N); ð̂^{n}(M,N); Ex̂t_{R}^{n+1}(M′′, N); ⋯]

and the connecting maps ð^{n}(M, N): Ext_{R}^{n}(M, N) → Ext_{R}^{n+1}(M′′, N) satisfy

ð̂^{n}(M, N) ∘ ε_{R}^{n}(M, N) = ε_{R}^{n+1}(M′′, N) ∘ ð^{n}(M, N) for all n ∈ ℤ .

Proof. Since all modules P_{n} and T_{n} are projective, applying Hom_{R}( , N) to the lower tier of the diagram in Construction 5.5, we get a commutative diagram

\xymatrix{0 \ar[r] & Hom_{R}(P′′, N) \ar[r] \ar[d]^{Hom_{R}(ϑ′′,N)} & Hom_{R}(P′, N) \ar[r] \ar[d]^{Hom_{R}(ϑ′,N)} & Hom_{R}(P , N) \ar[r] \ar[d]^{Hom_{R}(ϑ,N)} & 0 \\ 0 \ar[r] & Hom_{R}(T′′, N) \ar[r] & Hom_{R}(T′, N) \ar[r] & Hom_{R}(T , N) \ar[r] & 0}

with exact rows. The homology exact sequence of the bottom row is the desired long exact sequence. Commutativity of the diagram gives the formula for the connecting maps. Naturality in N is clear. To get naturality in M, take a morphism M → Ṁ and apply Hom_{R}( , N) to the commuting up to homotopy diagram

[figure: 0; T; μ̂; T′; μ′; T ′′; 0; 0; Ṫ; ˆμ̇; _{Ṫ_{′}}; _{ˆμ̇_{′}}; Ṫ_{′′}; 0]

with rows provided by Construction 5.5 and columns by Lemma 5.3.

Finite direct sums of complete resolutions are complete resolutions, and for each finite R-module T the functor Hom_{R}(T, ) commutes with all direct sums, hence 5.7. Proposition. For each finite set {M_{i} ∈ G̃| i ∈ I} the module ∐_{i∈I} M_{i} is in G̃, and for any family {N_{j} ∈ M | j ∈ J} there is a natural isomorphism

Ex̂t_{R}^{n}( ∐_{i∈I} M_{i}, ∐_{j∈J} N_{j}) ≅ ∐_{(i,j)∈I×J} Ex̂t_{R}^{n}(M_{i}, N_{j}) .  □

Before going further, we describe a type of resolutions of length ≤ 1.

A G-approximation of an R-module M is a resolution χ: B → M, with B_{0} ∈ G, B_{1} ∈ P̃, and B_{n} = 0 for n ≥ 2; complying with tradition, we use the same name for the associated exact sequence B^{+} = 0 → B_{1} → B_{0} → M → 0. By Example 2.1, when R is a Gorenstein local ring this is a maximal Cohen-Macaulay approximation in the sense of Auslander and Buchweitz [3].

5.8. Lemma. Let M be a finite R-module.

(1) M has a G-approximation with proj dim_{R} B_{1} < g if and only if G-dim_{R} M ≤ g.

If 0 → Y ⟶^{υ} X ⟶^{χ} M → 0 is a G-approximation, then for all R-modules L ∈ G̃ and N ∈ M the following hold.

(2) If χ′: G → M is a homomorphism with G ∈ G, then there exists a homomorphism ψ: G → X with χψ = χ′.

(3) For all n ∈ ℤ the following homomorphisms are bijective:

Ex̂t_{R}^{n}(L, χ): Ex̂t_{R}^{n}(L, M) ⟶^{≅} Ex̂t_{R}^{n}(L, X) .
Ex̂t_{R}^{n}(χ, N): Ex̂t_{R}^{n}(M, N) ⟶^{≅} Ex̂t_{R}^{n}(X, N) .

(4) There is a natural exact sequence where ν(α ⊗ y)(x) = α(x)y:

0 ⟶ Ex̂t_{R}^{−1}(M, N) ⟶ X^{∗} ⊗_{R} N ⟶^{ν} Hom_{R}(X, N) ⟶ Ex̂t_{R}^{0}(M, N) ⟶ 0

(5) If M = 0 → K → P → M → 0 is an exact sequence with P ∈ P, then for all n ∈ ℤ Propositions 5.4 and 5.6 yield isomorphisms

ð^{n}(L, M): Ex̂t_{R}^{n}(L, M) ⟶^{≅} Ex̂t_{R}^{n+1}(L, K) ;
ð^{n}(M, N): Ex̂t_{R}^{n}(K, N) ⟶^{≅} Ex̂t_{R}^{n+1}(M, N) .

Proof. (1) If χ: B → M is a G-approximation with proj dim_{R} B_{1} = p < g, then splice it with a P-resolution of B_{1} of length p to get a strict G-resolution of M of length p + 1 ≤ g. Conversely, if G-dim_{R} M ≤ g, then choose a strict G-resolution G → M of length ≤ g by Theorem 3.1 and set B_{0} = G_{0}, B_{1} = Ω^{1}G, υ = ∂_{1}.

(2) is a consequence of Lemma 4.1.2.

For (3) and (5) use Propositions 5.4 and 5.6 with Theorems 5.2.3 and 5.2.4.

(4) By (3) we may assume that M = X. Construction 3.6 yields a complete resolution S → S_{⩾0} → X. The exactness of S^{∗} implies that Ω^{1}S^{∗} is isomorphic to X^{∗} and that Σ^{−1}((S_{<0})^{∗}) → X^{∗} is a projective resolution. In view of the canonical isomorphism Hom_{R}(S_{<0}, N) ≅ (S_{<0})^{∗} ⊗_{R} N we have an exact sequence

0 ⟶ Hom_{R}(S_{⩾0}, N) ⟶ Hom_{R}(S, N) ⟶ (S_{<0})^{∗} ⊗_{R} N ⟶ 0

A direct computation with the expression for ∂_{0}^{S} from Construction 3.6 shows that in degrees 1, 0 the associated homology sequence is the desired exact sequence.

Unlike absolute or relative cohomology, no numerical invariant is defined by vanishing of Tate cohomology functors. The reason is that this cohomolgy theory is rigid, a fact established in the next result.

5.9. Theorem. For a module M of finite G-dimension the following are equivalent. (i) proj dim_{R} M < ∞. (ii) Ex̂t_{R}^{n}(M, ) = 0 for some n ∈ ℤ. (ii′) Ex̂t_{R}^{n}(M, ) = 0 for all n ∈ ℤ. (iii) Ex̂t_{R}^{n}( , M) = 0 for some n ∈ ℤ. (iii′) Ex̂t_{R}^{n}( , M) = 0 for all n ∈ ℤ. (iv) Ex̂t_{R}^{0}(M, M) = 0 .

Proof. Theorems 5.2.3 and 5.2.4 show that (i) implies (ii′) and (iii′), from where (ii) and (iii) clearly follow. We choose a G-approximation 0 → Y → X → M → 0 and use Lemma 5.8.3 to replace M by X in conditions (ii), (iii), and (iv). Choosing a surjective complete resolution T → P → X, from Lemma 5.8.5 we get isomorphisms Ex̂t_{R}^{n}(X, Ω^{−n}T) ≅ Ex̂t_{R}^{0}(X, X) ≅ Ex̂t_{R}^{n}(Ω^{n}T , X); we see that (ii) implies (iv), and so does (iii). If (iv) holds, then ν: X^{∗} ⊗_{R} X → Hom_{R}(X, X) is surjective by Lemma 5.8.4, so X is projective by [12, (VII.3.1)]; thus, (i) holds.

The next construction goes back to Buchweitz [11] and Auslander-Buchweitz [3].

5.10. Construction. Let S ⟶^{σ} P ⟶^{π} M be a complete resolution.

Set U = Ω^{0}S and M′ = Coker Ω^{0}(πσ). Choose an epimorphism φ′: V → M′ with V ∈ P, denote μ: M → M′ the canonical map, then choose a homomorphism φ: V → M with μφ = φ′. Set X = U ⊕ V, define a map χ: X → M by χ(u, v) = Ω^{0}(σ)(u) + φ(v). Finally, set Y = Ker χ and let υ: Y → X denote the inclusion.

Fact. A = ⋯ → 0 → Y ⟶^{υ} U ⊕ V → 0 → ⋯ is a G-approximation.

Proof. It is easy to see that χ is surjective, so the sequence A^{+} is exact. By Lemma 2.4 the module U is totally reflexive, hence so is U ⊕V by Proposition 2.3. Choosing a morphism τ: V → P with π_{0}τ_{0} = φ, we get a commutative diagram

[figure: S; σ′; S_{⩾0} ⊕ V; π′; χ′; U ⊕ V; χ; S; σ; P; π; M]

where σ′_{<0} = 0, σ′_{n}(x) = (x, 0) for n ≥ 0, π′ = (σ′_{⩾0}, id_{V}), χ′(s, v) = σ_{⩾0}(s) + τ(v). The top row is a complete resolution of U ⊕ V. Thus, H^{n} Hom_{R}(id_{S}, Y) represents for each n ∈ ℤ the map Ex̂t_{R}^{n}(χ, Y), which is therefore bijective. The exact sequence of Proposition 5.6 yields Ex̂t_{R}^{0}(Y, Y) = 0, so proj dim_{R} Y < ∞ by Theorem 5.9.

5.11. Remark. During the 1980ies, Pierre Vogel developed, but never published, a cohomology theory that associates to each pair (M, N) of modules over an arbitrary ring R a sequence of abelian groups Ex^{∨}t_{R}^{n}(M, N) for n ∈ ℤ, and comes equipped with a natural transformation Ext_{R}^{n}(M, N) → Ex^{∨}t_{R}^{n}(M, N) of cohomology functors. He also proved that if G is a finite group, then there is a natural isomorphism Ex^{∨}t_{ℤ[G]}^{n}(ℤ, N) ≅ Ĥ^{n}(G, N) of his theory and Tate cohomology.

Vogel’s argument extends, essentially verbatim, to establish that if R is noetherian and M is a finite module of finite G-dimension, then there is a natural isomorphism of cohomology functors Ex^{∨}t_{R}^{n}(M, N) ≅ Ex̂t_{R}^{n}(M, N), compatible with the natural maps from Ext_{R}^{n}(M, N). The construction of Vogel cohomology and the proof of the comparison theorem are presented in [21, I], and in [25, §2].

6. Gorenstein perfection

In this section we study a class of modules of finite G-dimension whose relative and Tate cohomology can be bounded by means of absolute (co)homology groups.

To each R-module M ∈ G̃ we associate an R^{o}-module M^{†}, setting 0^{†} = 0 and

M^{†} = Ext_{R}^{g}(M, R) when M ≠ 0 and g = G-dim_{R} M.

An R-module M is said to be G-perfect^{2} if it is finite, has finite G-dimension, and there is an integer g such that Ext_{R}^{n}(M, R) = 0 for n ≠ g. In view of Theorems 4.2.2 and 4.9.1, this notion can be introduced alternatively by requiring the existence of an integer g such that G-dim_{R} M = g and Ext_{R}^{n}(M, R) = 0 for n < g.

6.1. Examples. Let M be a finite R-module.

(1) M is G-perfect with G-dim_{R} M = 0 if and only if M ∈ G \ 0.

(2) If proj dim_{R} M ≠ ±∞ and Ext_{R}^{n}(M, R) = 0 for n < proj dim_{R} M, then M is G-perfect with G-dim_{R} M = proj dim_{R} M, cf. Theorem 4.9.

(3) If R is commutative and Gorenstein, then M is G-perfect with G-dim_{R} M = g if and only if it is Cohen-Macaulay of codimension g, that is, depth_{R_{𝔪}} M_{𝔪} = dim_{R_{𝔪}} M_{𝔪} = dim R_{𝔪} − g for all 𝔪 ∈ Max(R) ∪ Supp(M), cf. [10, (3.3.10.c)].

6.2. Construction. Let M be a G-perfect R-module with G-dim_{R} M = g.

Let π: P → M a P-resolution, set G = Ω^{g}P, denote ω: P_{g} → G the canonical surjection and ι: G → P_{g−1} the canonical injection, then define a complex (G, ∂^{G}) and a map ϕ: P → G of degree 0 by

G_{n} = \begin{cases}0 \\ G \\ P_{n} \end{cases}  ∂_{n}^{G} = \begin{cases}0 \\ ι \\ ∂_{n}^{P} \end{cases}  ϕ_{n} = \begin{cases}0 \\ ω \\ id & P_{n} \end{cases}  for n > g; ; for n = g; ; for n < g .

Fact. The map ϕ: P → G is a surjective quasiisomorphism, the complex Σ^{g}G^{∗} is a strict G-resolution of M^{†} and the map ϕ^{∗}: G^{∗} → P^{∗} is a quasiisomorphism.

If T ⟶^{ϑ} P ⟶^{π} M is a surjective complete resolution with ϑ_{n} bijective for n ≥ g, then the inclusion ι of Q = Ker ϑ into T defines a surjective complete resolution

Σ^{g−1}T ⟶^{ϑ′} Σ^{g−1}(Q^{∗}) → M^{†} with ϑ′ = Σ^{g−1}(ι^{∗}) and ϑ′_{n} bijective for n ≥ g.

Proof. The map ϕ is surjective by definition and a quasiisomorphism by inspection.

The module G is totally reflexive by Lemma 3.9, so G^{∗} is totally reflexive by Lemma 2.2.2. We have an exact sequence 0 → F → P → G → 0 of complexes of R-modules with F = Ker ϕ. The dual sequence of complexes of R^{o}-modules 0 → G^{∗} → P^{∗} → F^{∗} → 0 is exact: there could be a problem only in degree −g, but the sequence 0 → G^{∗} → P_{g}^{∗} → F_{g}^{∗} → 0 is exact By Lemma 2.2.1. For n > −g we have H_{n}(F^{∗}) = 0 because F_{n} = 0. Left exactness of Hom_{R}( , R) yields H_{−g}(F^{∗}) = 0. For n < −g we have H_{n}(F^{∗}) = H_{n}(P^{∗}) = Ext_{R}^{−n}(M, R) = 0. Thus, H_{n}(F^{∗}) = 0 for all n, so ϕ^{∗} is a quasiisomorphism. In particular, Ω^{−g}G^{∗} = H_{−g}(G^{∗}) ≅ Ext_{R}^{−g}(M, R) = M^{†}, so Σ^{g}G^{∗} is a strict G-resolution of M^{†}.

As each P_{n} is projective, the sequence of complexes 0 → P^{∗} → T^{∗} → Q^{∗} → 0 is exact. Because H(T^{∗}) = 0, the homology exact sequence yields H_{−n}(Q^{∗}) ≅ H_{−n−1}(P^{∗}) = Ext_{R}^{n+1}(M, R) for all n ∈ ℤ. Since M is G-perfect with G-dim_{R} M = g, it follows that Σ^{g−1}(Q^{∗}) is a P-resolution of M^{†} over R^{o}.

2Auslander and Bridger, who first considered such modules over commutative rings in [2, (4.34)], call them perfect. In view of Example 6.1.2 the notion of G-perfection extends that of perfection, as defined by Rees [29]. We prefer to have distinct names for different concepts.

Under condition 3.2.ii, Part (3) of the next theorem is contained in [11, (6.3.4)].

6.3. Theorem. Let M be a G-perfect R-module with G-dim_{R} M = g.

(1) The R^{o}-module M^{†} is G-perfect with G-dim_{R^{o}} M^{†} = g.

(2) There is a natural isomorphism of R-modules M ≅ M^{††}.

(3) For each R-module N there exist natural in M and N homomorphisms

Tor_{g−n}^{R}(M^{†}, N) ⟶ Ext_{G}^{n}(M, N)

which are bijective for n ≥ 2 and surjective for n = 1.

(4) For each R-module N there exist natural in M and N homomorphisms

Ex̂t_{R}^{n}(M, N) ⟶ Tor_{g−n−1}^{R}(M^{†}, N)

which are bijective for n ≤ −2 and make the following sequence exact

[figure: 0; Ex̂t_{R}^{−1}(M, N); Tor_{g}^{R}(M^{†}, N); Ext_{R}^{0}(M, N); ⋯; Tor_{g−n}^{R}(M^{†}, N); Ext_{R}^{n}(M, N); ε_{R}^{n}(M,N); Ex̂t_{R}^{n}(M, N); Tor_{g−n−1}^{R}(M^{†}, N); ⋯; Ext_{R}^{g}(M, N); Ex̂t_{R}^{g}(M, N); 0]

Proof. We adopt the notation used in the preceding construction.

(1) The proper resolution Σ^{g}G^{∗} of M^{†} yields G-dim_{R^{o}} M^{†} ≤ g, as well as the second isomorphism of R-modules below:

Ext_{R^{o}}^{n}(M^{†}, R) ≅ Ext_{G}^{n}(M^{†}, R) ≅ H_{−n} ((Σ^{g}(G^{∗}))^{∗}) ≅ H_{g−n}(G) = \begin{cases}0 & if n ≠ g; \\ M & if n = g; \end{cases}

the first isomorphism comes from Theorem 4.2.4, the third from the isomorphisms of complexes (Σ^{g}(G^{∗}))^{∗} ≅ Σ^{−g}(G^{∗∗}) ≅ Σ^{−g}G. As a consequence, G-dim_{R^{o}} M^{†} = g.

(2) Due to (1), for n = g the formula displayed above becomes M^{††} ≅ M.

(3) In view of (1), we can apply Construction 6.2 to the R^{o}-module M^{†}. It yields a surjective quasiisomorphism ϕ′: P′ → G′ from a P-resolution P′ → M^{†} over R^{o}, such that G^{′∗} is a proper G-resolution of M^{††} ≅ M over R, and ϕ^{′∗}: G^{′∗} → P^{′∗} is a quasiisomorphism. We have a natural commutative diagram with exact row

[figure: (Σ^{−g}P′) ⊗_{R} N; (Σ^{−g}ϕ′)⊗_{R}N; ξ; (Σ^{−g}G′) ⊗_{R} N; ξ; Hom_{R}(Σ^{g}(P^{′∗}), N); Hom_{R}(Σ^{g}(ϕ^{′∗}),N); Hom_{R}(Σ^{g}(G^{′∗}), N); 0]

where ξ denotes the morphism of complexes given by (ξ(x ⊗ y))(α) = α(x)y. The composition of the maps in the square induces maps Tor_{g−n}^{R}(M^{†}, N) → Ext_{G}^{n}(M, N) for all n ∈ ℤ. They are bijective for n ≥ 2 and surjective for n = 1 because the maps ξ are isomorphisms, and (Σ^{−g}ϕ′) ⊗_{R} N is bijective in negative degrees.

(4) In view of the canonical isomorphism ξ: Q^{∗} ⊗_{R} M → Hom_{R}(Q, N), the exact sequence of complexes of Construction 6.2 yields an exact sequence

0 ⟶ Σ^{g−1}(Q^{∗}) ⊗_{R} M ⟶ Hom_{R}(Σ^{g−1}T , N) ⟶ Hom_{R}(Σ^{g−1}P , N) ⟶ 0

The associated homology exact sequence yields the desired assertions.

7. Comparison morphisms

By Theorem 3.1 and Lemma 4.1.3, three cohomology theories—absolute, relative, and Tate—are defined on the category G̃ of modules of finite G-dimension. Our main result demonstrates that the morphisms of functors ε_{G}^{n} of Theorem 4.2 and ε_{R}^{n} of Theorem 5.2 yield a very tight relation between these theories.

7.1. Theorem. Let M be a finite R-module with G-dim_{R} M = g < ∞.

For each R-module N there exist natural in M and N homomorphisms δ_{R}^{n}(M, N), such that the following sequence is exact:

[figure: 0; Ext_{G}^{1}(M, N); ε_{G}^{1}(M,N); Ext_{R}^{1}(M, N); ⋯; Ext_{G}^{n}(M, N); ε_{G}^{n}(M,N); Ext_{R}^{n}(M, N); ε_{R}^{n}(M,N); Ex̂t_{R}^{n}(M, N); δ_{R}^{n}(M,N); Ext_{G}^{n+1}(M, N); ⋯; Ex̂t_{R}^{g}(M, N); 0]

Comparing the theorem with Theorems 5.2.3, and 5.2.4. we get:

7.2. Corollary. For M ∈ G̃ the following conditions are equivalent.

(i) ε_{G}^{n}(M, ): Ext_{G}^{n}(M, ) → Ext_{R}^{n}(M, ) is an isomorphism for all n ∈ ℤ.

(ii) ε_{G}^{n}( , M): Ext_{G}^{n}( , M) → Ext_{R}^{n}( , M) is an isomorphism for all n ∈ ℤ.

(iii) proj dim_{R} M < ∞.

Proof of Theorem 7.1. The exact sequences of complexes of R-modules in Construction 3.8 is split-exact in each degree, so it induces an exact sequence

0 → Hom_{R}(P , N) ⟶^{Hom_{R}(ϑ^{♭},N)} Hom_{R}(T^{♭}, N) ⟶^{Hom_{R}(κ^{♭},N)} Hom_{R}(Σ^{−1}G, N) → 0

of complexes of abelian groups. Its cohomology exact sequence has the form

[figure: ⋯; ð^{n}; H^{n} Hom_{R}(T^{♭}, N); H^{n} Hom_{R}(κ^{♭},N); H^{n} Hom_{R}(Σ^{−1}G, N); H^{n+1} Hom_{R}(P , N); H^{n+1} Hom_{R}(ϑ^{♭},N); H^{n+1} Hom_{R}(T^{♭}, N); ⋯]

We set out to identify the modules and maps appearing in this sequence. Since P → M is a projective resolution, we have

H^{n+1} Hom_{R}(P , N) = Ext_{R}^{n+1}(M, N) for n ∈ ℤ .

The left exactness of Hom_{R}( , N) yields H^{n} Hom_{R}(T^{♭}, N) = 0 for n ≤ 0, and

H^{n+1} Hom_{R}(T^{♭}, N) = H^{n+1} Hom_{R}(T , N) = Ex̂t_{R}^{n+1}(M, N) for n ≥ 0 .

Since H^{n+1} Hom_{R}(ϑ^{♭}, N) = H^{n+1} Hom_{R}(ϑ, N) for n ≥ 0, Theorem 5.2.1 gives

H^{n} Hom_{R}(ϑ^{♭}, N) = ε_{R}^{n}(M, N): Ext_{R}^{n}(M, N) ⟶ Ex̂t_{R}^{n}(M, N) for n ≥ 1 .

As G → M is a proper resolution, from Theorem 4.2.1 we see that

H^{n} Hom_{R}(Σ^{−1}G, N) = H^{n+1} Hom_{R}(G, N) = Ext_{G}^{n+1}(M, N) for n ∈ ℤ .

Next we prove that ð^{n} = ε_{G}^{n+1}(M, N) for n ≥ 0. To this end we form, following [9, §2.6], the fundamental commutative diagram with exact rows

[figure: 0; Σ^{−1}G; κ^{♭}; T^{♭}; ϑ^{♭}; P; 0; 0; Σ^{−1}G; κ̇; D; ϑ̇; φ̇; C; φ; 0]

where C is the mapping cone of κ^{♭} and D its mapping cylinder, that is:

C_{n} = G_{n} ⊕ T_{n}^{♭}  ∂_{n}^{C}(y, x) = (∂_{n}^{G}(y) , κ_{n−1}^{♭}(y) + ∂_{n}^{T^{♭}}(x))
D_{n} = G_{n+1} ⊕ G_{n} ⊕ T_{n}^{♭}  ∂_{n}^{D}(y′, y, x) = (y − ∂_{n+1}^{G}(y′), ∂_{n}^{G}(y) , κ_{n−1}^{♭}(y) + ∂_{n}^{T^{♭}}(x))

and the morphisms are described as follows:

κ̇(y′) = (y′, 0, 0) ,  ϑ̇(y′, y, x) = (y, x) ;
φ̇(y′, y, x) = κ^{♭}(y′) + ϑ^{♭}(x) ,  φ(y, x) = ϑ^{♭}(x) .

It induces a commutative diagram with exact rows

[figure: 0 → Hom_{R}(P , N); Hom_{R}(ϑ^{♭},N); Hom_{R}(φ,N); Hom_{R}(T^{♭}, N); Hom_{R}(κ^{♭},N); Hom_{R}(φ̇,N); Hom_{R}(Σ^{−1}G, N) → 0; 0 → Hom_{R}(C, N); Hom_{R}(ϑ̇,N); Hom_{R}(D, N); Hom_{R}(κ̇,N); Hom_{R}(Σ^{−1}G, N) → 0]

The connecting maps of the cohomology exact sequences yield a commutative square

[figure: H^{n} Hom_{R}(Σ^{−1}G, N); ð^{n}; H^{n+1} Hom_{R}(P , N); H^{n+1} Hom_{R}(φ,N); H^{n+1} Hom_{R}(G, N); H^{n+1} Hom_{R}(θ,N); H^{n+1} Hom_{R}(C, N)]

where θ: C → G is the morphism given by θ(y′, x) = y′; note that H_{0}(θ) = id_{M}.

The morphism φ is clearly surjective; it is split as a homomorphism of graded R-modules because each P_{n} is projective. We have an exact sequence of complexes

0 ⟶ E ⟶^{ξ} C ⟶^{φ} P ⟶ 0 where

E_{n} = G_{n+1} ⊕ G_{n} , ∂^{E}(y′, y) = (y, 0) , ξ(y′, y) = (y′, κ(y)) .

The map (y′, y) ↦ (0, y′) is a homotopy between id_{E} and 0_{E}, so E is an irrelevant subcomplex of C, hence φ is a homotopy equivalence by Lemma 1.6. We choose a homotopy inverse ψ: P → C of ϕ and note that H_{0}(ψ) = H_{0}(φ)^{−1} = id_{M}^{−1} = id_{M}.

As Hom_{R}(φ, N) and Hom_{R}(ψ, N) are inverse homotopy equivalences, we have H^{n+1} Hom_{R}(φ, N)^{−1} = H^{n+1} Hom_{R}(ψ, N). Thus, the commutative square yields

ð^{n} = H^{n+1} Hom_{R}(φ, N)^{−1} ∘ H^{n+1} Hom_{R}(θ, N)
= H^{n+1} Hom_{R}(ψ, N) ∘ H^{n+1} Hom_{R}(θ, N)
= H^{n+1} Hom_{R}(θ ψ, N)

Furthermore, H_{0}(θ ψ) = H_{0}(θ) H_{0}(ψ) = id_{M}, so the morphism θ ψ: P → G lifts the identity map of M, hence H^{n+1} Hom_{R}(θ ψ, N) = ε_{G}^{n+1}(M, N) by Theorem 4.2.1.

To finish the construction of the exact sequence in the theorem, we set

δ^{n}(M, N) = H^{n+1} Hom_{R}(κ^{♭}, N): Ex̂t_{R}^{n}(M, N) → Ext_{G}^{n+1}(M, N) for n ≥ 1 .

It is clear from the construction that the exact sequence is functorial in N. For any homomorphism μ: M → M′, form a commutative diagram

[figure: 0; Σ^{−1}G; κ^{♭}; T^{♭}; ϑ^{♭}; P; 0; 0; Σ^{−1}G′; κ^{♭}; μ̂|_{Σ^{−1}G}; _{T_{′}_{♭}}; ϑ^{♭}; μ̂; _{P_{′}}; \overline{μ}; 0]

by first choosing μ̂ from Lemma 5.3 so that the right hand square commutes, then taking α to be restriction of μ̂. Since H(T^{♭}) and H(T′^{♭}), are both trivial, the induced commutative diagram in homology shows that H_{−1}(α) = μ. Thus, for all n ∈ ℤ we get H^{n} Hom_{R}(\overline{μ}, N) = Ext_{R}^{n}(μ, N), H^{n} Hom_{R}(μ̂, N) = Ex̂t_{R}^{n}(μ, N), and H^{n} Hom_{R}(α, N) = Ext_{G}^{n+1}(μ, N). Applying Hom_{R}( , N) to the diagram and taking homology, we obtain the desired naturality in M.

Some extra structure comes for free with the various cohomology functors.

7.3. Remark. Let R be a noetherian ring which is an algebra over a commutative ring 𝕜. For a finite R-module M of finite G-dimension and an R-module N, let F^{n}(M, N) stand for Ext_{R}^{n}(M, N), its relative or Tate variants, or Tor_{n}^{R}(M, N).

(1) For each n the group F^{n}(M, N) has a structure of 𝕜-module, which is natural for maps induced by homomorphisms of either module argument, for comparison maps between various functors, and for connecting maps of long exact sequences.

Indeed, in each case there is an R-linear resolution P and a functor F on M(R) such that F^{n}(M, N) = H^{n} F(P , N) for all n ∈ ℤ. The action of 𝕜 on either P or N induces the same action on the complex F(P , N), and passes to homology. All maps listed above are either induced by 𝕜-linear morphisms of complexes or are connecting maps in exact sequences of 𝕜-linear morphisms of such complexes.

(2) The 𝕜-module F^{n}(M, N) is annihilated by the ideal ann_{𝕜}(M) + ann_{𝕜}(N). Indeed, denote λ_{x} any R-linear map given by left multiplication with x ∈ 𝕜. If

xN = 0, then λ_{x}^{N} = 0, hence F(P , λ_{x}^{N}) = 0. If xM = 0, then λ_{x}^{P} is a morphism with H_{0}(λ_{x}^{P}) = λ_{x}^{M} = 0. By the lifting property for resolutions of the corresponding type, λ_{x}^{P} is the homotopic to 0^{P}, so H(F(λ_{x}^{P} , N)) = H(F(0^{P} , N)) = 0.

(3) If R is finite over 𝕜 and N is finite over R, then F^{n}(M, N) is finite over 𝕜. Indeed, the image \overline{𝕜} of 𝕜 in R is noetherian by the theorem of Eakin-Nagata-

Eisenbud [18]. Each P_{n} is finite over R, hence also over \overline{𝕜}. Being the homology of a complex of finite \overline{𝕜}-modules, F^{n}(M, N) is finite over \overline{𝕜}, hence over 𝕜.

We close off the discussion of relative and Tate functors with a note on homology.

7.4. Remark. If M is a right R-module of finite G-dimension, G → M is a proper resolution and T → P → M is a complete resolution, one sets

Tor_{n}^{G}(M, N) = H_{n}(G ⊗_{R} N)  and  Tôr_{n}^{R}(M, N) = H_{n}(T ⊗_{R} N) .

for each R-module N and each n ∈ ℤ. The resulting homology groups come equipped with natural in both arguments Tôr_{n}^{G}(M, N) → Tor_{n}^{R}(M, N) → Tor_{n}^{G}(M, N). The arguments in Sections 2 through 6 dualize perfectly, so all the results in Sections 2 through 6 have valid analogs in homology.

8. Minimal resolutions

In this section R is a commutative noetherian local ring with unique maximal ideal 𝔪 and residue field k = R/𝔪, and M is a finite R-module.

We denote f-rank_{R} M the maximal rank of a free direct summand of M. Recall that M has a free cover, that is, a surjective homomorphism φ: F → M with F free and φ ⊗_{R} k bijective. Classically, a free resolution P → M is said to be minimal if P_{n} → Ω^{n}P is a free cover for each n ∈ ℤ, that is, if ∂(P) ⊆ 𝔪P. This is a special case of the general concept of minimality defined in Section 1.

8.1. Proposition. Over a local ring R a complex P of finite free modules is minimal if and only if ∂(P) ⊆ 𝔪P.

Proof. Assume first that ∂(P) ⊆ 𝔪P, and let β: P → P be a morphism such that β = id_{P} +θ∂ + ∂θ. Por each n we then have β_{n} ⊗_{R} k = id_{P_{n}⊗_{R}k}. As P_{n} is free of finite rank, β_{n} is an isomorphism, so P is minimal by Proposition 1.7.1.

Assume next that ∂(P) ⊈ 𝔪P, choose y ∈ P_{n} with ∂(y) = x ∉ 𝔪P_{n−1}. Setting E_{n} = Ry, E_{n−1} = Rx, and E_{i} = 0 for i ≠ n, n−1 we get a contractible subcomplex E ⊆ P. This subcomplex is irrelevant because y (respectively, x) is part of a basis of P_{n} (respectively, P_{n−1}). Thus, P is not minimal by Proposition 1.7.3.

We need some properties of syzygy modules in minimal free resolutions.

8.2. Lemma. Let R be a local ring, ρ: R → k the canonical surjection, M a finite R-module, and P → M a minimal free resolution.

(1) f-rank_{R}(Ω^{n}P) ≥ rank_{k} Ext_{R}^{n}(M, ρ) for all n ∈ ℤ.

(2) f-rank_{R}(Ω^{n}P) = 0 for n > max{0, depth R − depth_{R} M}.

Proof. (1) Choose homomorphisms φ_{1}, …, φ_{r}: P_{n} → R with φ_{j}∂_{n+1}^{P} = 0, such that the images of cls(φ_{1}), …, cls(φ_{r}) ∈ H^{n} Hom_{R}(P , R) = Ext_{R}^{n}(M, R) form a basis of Im H^{n} Hom_{R}(P , ρ). The map x ↦ (ρ φ_{1}(x), …, ρ φ_{r}(x)) is then a surjective homomorphism P_{n} → k^{r}. It follows that φ: P_{n} → R^{r}, given by x ↦ (φ_{1}(x), …, φ_{r}(x)), is a surjective homomorphism. Since φ∂_{n+1}^{P} = 0, the map φ factors through a homomorphism Ω^{n}P → R^{r}, which is necessarily surjective; thus, f-rank_{R}(Ω^{n}P) ≥ r.

(2) is known, cf. e.g. [5, (1.2.5)] for a proof.

The following results are due to Auslander and Bridger [2, (4.30), (4.13.b)]. We give proofs based on Auslander’s earliest exposition of G-dimension in [1, §3.2].

8.3. Theorem. Let R be a local ring, M a finite module, and G-dim_{R} M = g < ∞.

(1) If x ∈ 𝔪 is R-regular and M-regular, then G-dim_{R/Rx}(M/xM) = g.

(2) G-dim_{R} M = depth R − depth_{R} M.

Proof. Set ( ) = ( ⊗_{R} (R/Rx)), and let G → M be a G-resolution of length g.

(1) Let first g = 0 and choose a totally acyclic complex S with Ω^{0}S = M, cf. Construction 3.6. We obtain Ω^{0}(\overline{S}) ≅ \overline{M} and H(\overline{S}) = 0 from the exact sequence 0 → S ⟶^{x} S → \overline{S} → 0, and H Hom_{R}(S, \overline{R}) = 0 from the exact sequence 0 → Hom_{R}(S, R) ⟶^{x} Hom_{R}(S, R) → Hom_{R}(S, \overline{R}) → 0. Thus, H Hom\overline{R}(\overline{S}, \overline{R}) ≅ H Hom_{R}(S, \overline{R}) = 0, so G-dim\overline{R} \overline{M} = 0 by Lemma 2.4.

For any g, the sequence 0 → G ⟶^{x} G → \overline{G} → 0 is exact because each G_{i} is contained in a free module. Thus, \overline{G} is a resolution of \overline{M}. We have just shown that each \overline{G}_{i} is in G(\overline{R}), so G-dim\overline{R} \overline{M} ≤ g. By Theorem 4.9.1, in the exact sequence

Ext_{R}^{g}(M, R) ⟶^{x} Ext_{R}^{g}(M, R) → Ext_{R}^{g}(M, \overline{R}) → Ext_{R}^{g+1}(M, R)

the last module vanishes and the first one does not. We obtain G-dim\overline{R} \overline{M} = g from the following relations, the first of which is a standard isomorphism:

Extg ; \overline{R}(\overline{M}, \overline{R}) ≅ Ext_{R}^{g}(M, \overline{R}) ≅ Ext_{R}^{g}(M, R)/x Ext_{R}^{g}(M, R) ≠ 0 .

(2) Set t = depth_{R} M and d = depth R. Let first d = 0 and pick a monomorphism ι: k → R. For each L ∈ F(R) let L(ι) denote the image of Hom_{R}(L, ι) in L^{∗}; we remark that 𝔪L(ι) = 0, and that L(ι) = 0 if and only if L = 0. If g = 0, then we have M^{∗}(ι) ⊆ M^{∗∗} ≅ M, hence t = 0. Assuming g > 0, we have an exact sequence 0 → G_{g} ⟶^{∂} G_{g−1} → Ω^{g−1}G → 0. Since Ext_{R}^{1}(G_{g−1}, R) = 0, it induces an exact

sequence G_{g−1}^{∗} ⟶^{∂^{∗}} G_{g}^{∗} → E → 0 with E = Ext_{R}^{1}(Ω^{g−1}G, R). Dualization yields E^{∗} ≅ Ker(∂) = 0. It follows that E(ι) = 0, hence E = 0, and thus Ω^{g−1}G is in G by Theorem 4.9.1. This implies G-dim_{R} M < g, contradicting our assumption.

Let now d ≥ 1. For t = 0 the exact sequence 0 → Ω^{1}G → G_{0} → M → 0 yields G-dim_{R} Ω^{1}G = g − 1 and depth_{R}(Ω^{1}G) = 1, so we may assume t ≥ 1. Pick an R⊕M-regular element x ∈ 𝔪. We get G-dim\overline{R}(\overline{M}) = g from (1), and G-dim\overline{R}(\overline{M}) = depth \overline{R} − depth\overline{R}(\overline{M}) by induction on d; the difference equals d − t.

We say that a complete resolution T → P → M (respectively, a proper resolution G → M, a G-approximation χ: B → M) is minimal if the complexes T and P are minimal (respectively, the complex G is minimal, the complex B is minimal).

The results that follow establish the existence of minimal resolutions of each kind, and their uniqueness up to isomorphism.

8.4. Theorem. Let R be a local ring and M a finite module of finite G-dimension.

(1) M has a minimal complete resolution; more precisely: (1′) A minimal complete resolution S ⟶^{σ} P ⟶^{π} M is produced by Construction

3.6, using a minimal free resolution π: P → M, a module G = Ω^{g}P with G-dim_{R} G = 0 and f-rank_{R} G = 0 (by Theorems 8.3.2 and 3.1, and Lemma 8.2.2, any g > G-dim_{R} M would do) and a minimal free resolution λ: L → G^{∗}.

(2) A complete resolution T ⟶ P → M is minimal if and only if

∂(T) ⊆ 𝔪T  and  ∂(P) ⊆ 𝔪P .

(3) Let T → P → M and T′ → P′ → M be complete resolutions, and let μ̂: T →

T′ and \overline{μ}: P → P′ be morphisms over id_{R} M, given by Lemma 5.3.

If T → P → M is minimal, then μ̂ and \overline{μ} are isomorphisms onto direct summands; if both resolutions are minimal, then μ̂ and \overline{μ} are isomorphisms.

(4) In every minimal complete resolution T ⟶^{ϑ} P ⟶^{π} M the homomorphism ϑ_{n} is bijective for all n > G-dim_{R} M.

Proof. (2) follows from Proposition 8.1, and (3) from Proposition 1.7.

(1′) As ∂(P) ⊆ 𝔪P and ∂(L) ⊆ 𝔪L, Construction 3.6 yields a complete resolution S → P → M with ∂_{n}^{S}(S_{n}) ⊆ 𝔪S_{n−1} for n ≠ g. It also shows that ∂_{g}^{S} factors through G = Ω^{g}S, so G ⊈ 𝔪S_{g−1} implies f-rank_{R} G > 0, contradicting our hypothesis. Thus, S → P → M is minimal by (2).

(4) By Theorem 5.2.2 the map ε_{R}^{n}(M, k): Ext_{R}^{n}(M, k) → Ex̂t_{R}^{n}(M, k) is bijective for all n > G-dim_{R} M. Furthermore, by Theorem 5.2.1 this homomorphism can be computed as H^{n} Hom_{R}(ϑ, k): Hom_{R}(P , k) → Hom_{R}(T , k). From Part (2) above we obtain H^{n} Hom_{R}(ϑ, k) = Hom_{R}(ϑ_{n}, k). Referring to Nakayama’s Lemma, we conclude that ϑ_{n} is bijective for all n > G-dim_{R} M.

8.5. Theorem. Let R be a local ring and M a finite module of finite G-dimension.

(1) M has a minimal proper resolution; more precisely:

(1′) Let A^{+} = 0 → Y ⟶^{υ} U ⊕ V ⟶^{χ} M → 0 be the G-approximation produced by Construction 5.10 from the minimal complete resolution in Theorem 8.4.1′ and a free cover φ′: V → Coker Ω^{0}(πσ); the resolution F → M obtained by splicing A with a minimal free resolution κ: K → Y is minimal.

(2) A proper resolution G → M is minimal if and only if the following hold: (a) G_{n} is free of finite rank for n ≥ 1 .

(b) ∂_{n}(G_{n}) ⊆ 𝔪G_{n−1} for n ≥ 2 .

(c) ∂_{1}(G_{1}) contains no non-zero free direct summand of G_{0} .

(3) Let G → M and G′ → M be proper resolutions, and let μ̃: G → G′ be a

morphism over id_{R} M, given by Lemma 4.3.

If the first resolution is minimal, then μ̃ is an isomorphism onto a direct summand; if both resolutions are minimal, then μ̃ is an isomorphisms.

(4) If G → M is a minimal proper resolution, then

G-dim_{R} M = sup{n ∈ ℤ | G_{n} ≠ 0} .

Proof. (3) results from Proposition 1.7, and (2) is proved in two installments.

(2′) Here we show that if G → M is a proper resolution satisfying the conditions in (2), then the complex G is minimal. By Proposition 1.7.1 it suffices to show that if β: G → G is a morphism and there are maps θ_{n}: G_{n} → G_{n+1} with

β_{n} − id_{G_{n}} = θ_{n−1}∂_{n} + ∂_{n+1}θ_{n} for n ≥ 0 ,

then each β_{n} is bijective. By (a) the module G_{n} is free of finite rank for n ≥ 1, so for these n it suffices to show that the right hand side is contained in 𝔪G_{n}. Condition (b) guarantees this for n ≥ 2, and reduces it to Im(θ_{0}∂_{1}) ⊆ 𝔪G_{1} for n = 1. Assuming the inclusion fails, we can find x ∈ Im ∂_{1} with θ_{0}(x) ∉ 𝔪G_{1}; as G_{1} is free, Rx is a free direct summand of G_{0}, contradicting (c). Thus, β_{n} is an isomorphism for n ≥ 1. It follows that in the commutative diagram

\xymatrix{0 \ar[r] & Ω^{1}G \ar[r] \ar[d]^{Ω^{1}(β)} & G_{0} \ar[r] \ar[d]^{β_{0}} & H_{0}(G) \ar[r] \ar[d]^{H_{0}(β)} & 0 \\ 0 \ar[r] & Ω^{1}G \ar[r] & G_{0} \ar[r] & H_{0}(G) \ar[r] & 0}

the homomorphism Ω^{1}(β) is bijective. On the other hand, H_{0}(β) = H_{0}(id_{G}) is the identity map. Thus, β_{n} is an isomorphism also for n = 0.

(1′) The resolution F → M satisfies conditions (2.a) and (2.b) by construction. Assume that (2.c) fails, and choose y ∈ Y so that ∂_{1}(y) = (u, v) ∈ U ⊕ V = F_{0} generates a non-zero free direct summand. We then get a surjective homomorphism λ: U ⊕ V → R with λ(u) + λ(v) = 1, so λ(u) or λ(v) is invertible. In the first case Ru ≠ 0 is a free direct summand of U, contradicting Lemma 8.2.2. In the second case Rv ≠ 0 is a direct summand of V; since φ′(v) = μ χ(u, v) = μ χ∂_{1}(y) = 0, we have Ker(φ′ ⊗_{R} k) ∋ v ⊗ 1 ≠ 0, contradicting the choice of φ′: V → M′ as a free cover. Thus, F → M satisfies condition (2.c) as well, hence is minimal by (2′).

(2′′) Let G → M be a minimal proper resolution. By (3) it is isomorphic to the resolution in (1′), for which we have just checked the conditions in (2).

(4) Let G → M be a minimal proper resolution. By definition of G-dimension, G-dim_{R} M ≤ sup{n ∈ ℤ | G_{n} ≠ 0}. The opposite inequality comes from (3).

By Part (3) of the next result, if R is Gorenstein then minimal G-approximations are minimal Cohen-Macaulay approximations in the sense of Auslander. Thus, Part (1) contains his existence theorem for such approximations, and its extension to Cohen-Macaulay rings with dualizing module by Enochs, Jenda, and Xu [20, (5.1)].

8.6. Theorem. Let R be a local ring and M a finite module of finite G-dimension.

(1) M has a minimal G-approximation; more precisely: (1′) The G-approximation A → M described in Theorem 8.5.1′ is minimal.

(2) For a G-approximation χ: B → M the following conditions are equivalent. (i) B is minimal. (ii) υ(B_{1}) contains no non-zero direct summand of B_{0}. (iii) υ(B_{1}) contains no non-zero free direct summand of B_{0}. (iv) Each homomorphism β_{0} : B_{0} → B_{0} with χβ_{0} = χ is bijective.

(v) The complex G obtained by splicing B with a minimal free resolution

κ: K → B_{1} is minimal.

(3) Let χ: B → M and χ′: B′ → M be G-approximations, and let β: B → B′ be

a morphism with β_{0} : B_{0} → B′_{0} given by Lemma 5.8.2 and β_{1} = β_{0}|_{B_{1}}.

If the first approximation is minimal, then β is an isomorphism onto a direct summand; if both approximations are minimal, then β is an isomorphism.

(4) If 0 → Y → X → M → 0 is a minimal G-approximation and M ≠ 0, then

G-dim_{R} M = sup{0, proj dim_{R} Y + 1} .

Proof. We start by proving that the conditions in (2) are equivalent.

(i) ⟹ (ii). If C is a direct summand of B_{0} contained in υ(B_{1}), then setting E_{0} = C, E_{1} = υ^{−1}(C), and E_{n} = 0 for n ≠ 0, 1 we get an irrelevant subcomplex of B. Since B is minimal, we have E = 0 by Proposition 1.7.2.

(ii) ⟹ (iii) is clear.

(iii) ⟹ (v). We have G_{⩾1} ≅ ΣK by definition, so G_{n} is free for n ≥ 1 and ∂_{n}(G_{n}) ⊆ 𝔪G_{n−1} for n ≥ 2. Since Im ∂_{1} = Im υ, this module contains no non-zero free direct summand of G_{0} by hypothesis. Thus, G is minimal by Theorem 8.5.2.

(v) ⟹ (iv). Let β_{0} : B_{0} → B_{0} be a homomorphism with χβ_{0} = χ, and choose a morphism β̃: K → K lifting the induced map B_{1} → B_{1}. Setting β_{n} = β̃_{n−1} for n ≥ 1, we obtain a morphism β: G → G with H_{0}(β) = id_{M}. The resolution G → M being proper by Lemma 4.1.3, the morphism β is a homotopy equivalence by Lemma 4.3. By hypothesis, G is a minimal complex, so β is an isomorphism; in particular, β_{0} is bijective.

(iv) ⟹ (i). Let β: B → B be a morphism with β ∼ id_{B}. As H_{0}(β) = H_{0}(id_{B}) = id_{M}, we have χβ_{0} = χ, so β_{0} is an isomorphism; as υ is injective, β_{1} is an isomorphism as well. Thus, B is minimal by Proposition 1.7.1.

(1′) and (4) follow from the corresponding parts of Theorem 8.5, using (2.v).

(3) Follows from Proposition 1.7.

For a module M of finite G-dimension choose a minimal free resolution P → M, minimal G-approximations 0 → Y_{n} → X_{n} → Ω^{n}P → 0. The uniqueness of minimal free resolutions and Theorem 8.6.3 imply that all choices yield isomorphic modules X_{n}. We introduce the nth G-delta invariant of M by the formula

δ_{G}^{n}(M) = f-rank_{R}(X_{n}) for n ∈ ℤ .

When R is Gorenstein this is Auslander’s definition of δ^{n}(M), cf. [4, p. 314].

8.7. Theorem. If (R, 𝔪, k) is a local ring, M is a finite R-module, G-dim_{R} M is finite, and ε_{R}^{n}(M, k): Ext_{R}^{n}(M, k) → Ex̂t_{R}^{n}(M, k) is the comparison map, then

δ_{G}^{n}(M) = rank_{k} Ker ε_{R}^{n}(M, k) for all n ∈ ℤ .

Furthermore, δ_{G}^{n}(M) = 0 for all n < 0 and all n > G-dim_{R} M.

Proof. In view of Theorem 5.2.2 the second assertion follows from the first.

Let P → M be a minimal free resolution. By Proposition 5.6 and Theorem 5.9, the exact sequence 0 → Ω^{1}P → P_{0} → M → 0 induces a commutative diagram

\xymatrix{Ext_{R}^{n}(Ω^{1}P , k) \ar[r]^{≅} \ar[d]^{ε_{R}^{n}(Ω^{1}P ,k)} & Ext_{R}^{n+1}(M, k) \ar[d]^{ε_{R}^{n+1}(M,k)} \\ Ex̂t_{R}^{n}(Ω^{1}P) \ar[r]^{≅} & Ex̂t_{R}^{n+1}(M, k)}

Thus, it suffices to prove the equality δ_{G}(M) = rank_{k} ( Ker ε_{R}^{0}(M, k)).

Let B = 0 → Y ⟶^{υ} X ⟶^{χ} M → 0 be a minimal G-approximation. Decompose X as U ⊕ V where V is free with rank_{R} V = δ_{G}(M), and let ι: U → X and π: X → V be the canonical maps. Since f-rank_{R} U = 0, Theorem 8.4.1 yields a complete resolution S ⟶^{ϑ} P → U with ∂(S) ⊆ 𝔪S and ϑ_{n} = id_{S_{n}} for n ≥ 0. Thus, ε_{R}^{n}(U, k): Ext_{R}^{n}(U, k) → Ex̂t_{R}^{n}(U, k) is bijective for n ≥ 0, so in the diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] \\  & Ker ε_{R}^{0}(M, k) \ar@{-->}[r] \ar[d] & Hom_{R}(V, k) \ar[d]^{Hom_{R}(π,k)} \\ 0 \ar[r] & Hom_{R}(M, k) \ar[r]^{Hom_{R}(χ,k)} \ar[d]_{ε_{R}^{0}(M,k)} & Hom_{R}(X, k) \ar[r]^{Hom_{R}(υ,k)} \ar[d]^{ε_{R}^{0}(U,k)∘Hom_{R}(ι,k)} & Hom_{R}(Y, k) \\  & Ex̂t_{R}^{0}(M, k) \ar[r]^{Ex̂t_{R}^{0}(ι χ,k)} & Ex̂t_{R}^{0}(U, k)}

the right hand column is exact. The left hand column is exact by definition and the bottom square commutes by Proposition 5.6. This defines the hyphenated arrow, which is necessarily injective. It is surjective if Im Hom_{R}(π, k) ⊆ Im Hom_{R}(χ, k). In view of the exactness of the middle row, this condition can be rewritten as Hom_{R}(υ, k) ∘ Hom_{R}(π, k) = 0. The composition is equal to Hom_{R}(πυ, k), and the last map is trivial because πυ(Y) is contained in 𝔪V by the minimality of B.

As illustration and for later use, we record some computations of Auslander.

8.8. Examples. Let R be a non-regular Gorenstein local ring.

(1) The residue field has trivial delta invariants: δ_{G}^{n}(k) = 0 for all n ∈ ℤ; this is announced in [4, (5.7)], and follows also from the more general [26, Theorem 6].

(2) If ℓℓ(R) denotes the Loewy length of R, that is, the smallest integer r such that 𝔪^{r} is contained in an ideal generated by some system of parameters of R, then δ_{G}^{0}(R/𝔪^{s}) = 1 for all s ≥ ℓℓ(R): a proof may be found in [15, p. 273].

9. Betti numbers

In this section R is a commutative noetherian local ring with residue field k. The rank of the nth free module in a minimal free resolution of a finite R-module M is called the nth Betti number of M; it is denoted β_{n}^{R}(M), and is equal to rank_{k} Ext_{R}^{n}(M, k). If G-dim_{R} M < ∞, then by 7.3 the numbers

β_{n}^{G}(M) = rank_{k} Ext_{G}^{n}(M, k)  and  β̂_{n}^{R}(M) = rank_{k} Ex̂t_{R}^{n}(M, k) .

are non-negative integers, that we call the nth relative Betti number and nth stable Betti number of M, respectively. We relate them to their absolute counterparts.

9.1. Theorem. Let M be a finite R-module with G-dim_{R} M = g < ∞. Let T → P → M be a complete resolution and 0 → Y → X → M → 0 a G-approximation.

(1) β̂_{n}^{R}(M) = β̂_{n}^{R}(X) = β̂_{n−i}^{R}(Ω^{i}T) = β̂_{−n−1}^{R}(X^{∗}) for all n ∈ ℤ and all i ∈ ℤ. If the complete resolution is minimal, then β̂_{n}^{R}(M) = rank_{R} T_{n} for all n ∈ ℤ. If M is G-perfect, then β̂_{n}^{R}(M) = β̂_{g−1−n}^{R}(M^{†}) for all n ∈ ℤ.

(2) If proj dim_{R} M < ∞, then β_{n}^{G}(M) = β_{n}^{R}(M) and β̂_{n}^{R}(M) = 0 for all n ∈ ℤ.

(3) If proj dim_{R} M = ∞, then the following tables express the relative Betti numbers and the stable Betti numbers of M.

n ⋯  0  1  ⋯  g + 1 ⋯
β_{n}^{G}(M) 0 β_{0}^{R}(M) β_{0}^{R}(M) − β_{0}^{R}(X) + β_{0}^{R}(Y) = δ_{G}^{1}(M) ≤ β_{1}^{R}(M) β_{n−1}^{R}(Y)  0

\begin{matrix}n & ⋯ & −1 & 0 & ⋯ & g & ⋯ \\ β̂_{n}^{R}(M) & β_{−n−1}^{R}(X^{∗}) & β_{0}^{R}(X^{∗}) − δ_{G}^{0}(X) & β_{0}^{R}(X) − δ_{G}^{0}(X) & β_{n}^{R}(X) & β_{g}^{R}(X) − δ_{G}^{g}(X) & β_{n}^{R}(M) \end{matrix}

Furthermore, β_{n}^{G}(M) > 0 if n = 0 or 2 ≤ n ≤ g, and β̂_{n}^{R}(M) > 0 for all n ∈ ℤ.

Proof. (1) Lemmas 5.8.3 and 5.8.5 yield equalities β̂_{n}^{R}(X) = β̂_{n}^{R}(M) = β̂_{n−i}^{R}(Ω^{i}T). If T is minimal, then β̂_{n}^{R}(M) = rank_{R} T_{n} by Theorem 8.4.2. If M is G-perfect, then β̂_{n}^{R}(M) = β̂_{g−1−n}^{R}(M^{†}) for all n by the last computation and Construction 6.2. As X is G-perfect with X^{†} = X^{∗}, we get β̂_{n}^{R}(X) = β̂_{−n−1}^{R}(X^{∗}) for all n ∈ ℤ.

(2) results from Theorems 4.2.3 and 5.2.3.

(3) Theorem 4.2.2 yields the expressions for β_{n}^{G}(M) when n ≤ 0 or n > g. G-approximations are proper exact sequences and Ext_{G}^{n}(X, k) = 0 for n ≠ 0 by Theorem 4.2.2, so Proposition 4.6 yields isomorphisms and an exact sequence:

Ext_{G}^{n−1}(Y, k) ≅ Ext_{G}^{n}(M, k) for all n ≥ 2 ;
0 ⟶ Ext_{G}^{0}(M, k) ⟶ Ext_{G}^{0}(X, k) ⟶ Ext_{G}^{0}(Y, k) ⟶ Ext_{G}^{1}(M, k) ⟶ 0 .

By Theorems 4.2.2.b and 4.2.3 all relative Ext modules above, except for Ext_{G}^{n}(M, k) with n ≥ 1, are isomorphic to the corresponding absolute Ext modules. This gives the expressions for β_{n}^{G}(M) for n ≥ 2, and the first expression for β_{1}^{G}(M). The second expression and the inequality for β_{1}^{G}(M) come from Theorem 7.1.

Observe that the expression for β̂_{n}^{R}(M) is given by Theorem 5.2.2 for n > g and by Theorem 7.1 for n = g. Due to (1), this observation applied to X and to X^{∗} gives the tabulated values of β̂_{n}^{R}(M) for 0 ≤ n ≤ g and for n ≤ −1, along with an equality β̂_{0}^{R}(M) = β_{0}^{R}(X^{∗}) − δ_{G}^{0}(X^{∗}); it remains to note that δ_{G}^{0}(X^{∗}) = δ_{G}^{0}(X).

Assume β̂^{n}(M) = 0 for some n ∈ ℤ. In view of (1), this yields T_{n} = 0 in a minimal complete resolution T → P → M, hence Ex̂t_{R}^{n}(M, N) = 0 for all Rmodules N. By Theorem 5.9 this implies proj dim_{R} M < ∞, a contradiction.

The theorem suggests how to find a Betti sequence with a hole at n = 1.

9.2. Example. If R is a non-regular Gorenstein local ring with dim R = d, then

β_{n}^{G}(k) = \begin{cases}0 & for n < 0 , n = 1 , and n > d; \\ 1 & for n = 0 ; \\ β_{d−n}^{R}(k) & for 2 ≤ n ≤ d; \end{cases}

β̂_{n}^{R}(k) = β_{n}^{R}(k) + β_{d−1−n}^{R}(k) for all n ∈ ℤ .

Indeed, the value of β_{n}^{G}(k) is given by Theorem 4.2.2.b for n ≤ 0, and by Theorem 6.3.3 for n ≥ 2. Theorems 6.3.4 and 5.2.2 compute β̂_{n}^{R}(k) for n < −1 and for n > d. Theorem 8.7 and Remark 8.8.1 imply that the exact sequences of Theorems 7.1 and 6.3.4 split, yielding β_{1}^{G}(k) = 0, and β̂_{n}^{R}(k) = β_{n}^{R}(k) + β_{d−1−n}^{R}(k) for −1 ≤ n ≤ d.

In order to compare the asymptotes of a complete resolution we raise the

Problem. If G-dim_{R} M < proj dim_{R} M, does lim_{n→∞} \frac{β̂_{n}^{R}(M)}{β̂_{i−n}^{R}(M)} exist for some i ∈ ℤ?

9.3. Remarks. (1) By Theorem 9.1 it suffices to treat the case when M is totally reflexive with f-rank_{R} M = 0. For such modules Theorems 6.3.4 and 5.2.2 yield equalities β̂_{n}^{R}(M)/β̂_{i−n}^{R}(M) = β_{n}^{R}(M)/β_{n+g−i−1}^{R}(M^{∗}) for all n ≥ 0.

(2) For R Gorenstein and M = k the limit is known to exist if i = d − 1, cf. Example 9.2, but not in general, cf. [5, §4.3]. Also, limits for various i may differ: if R = k[x, y, z]/(x^{2} − y^{2}, y^{2} − z^{2}, xy, yz, xz), then β_{n}^{R}(k) = (b^{n+1} − b^{−n−1})/\sqrt{5} for all n ≥ 0 and b = (3 + \sqrt{5})/2, cf. [30, Satz 9], so lim_{n→∞} β̂_{n}^{R}(k)/β̂_{i−n}^{R}(k) = b^{i+1}.

For more examples, consider the complete intersection dimension, defined by

CI-dim_{R} M = inf{proj dim_{Q}(M ⊗_{R} R′) − proj dim_{Q} R′}

with infimum taken over all diagrams of local homomorphisms R → R′ ← Q where the first map is flat and the second is surjective with kernel generated by a regular sequence. There are inequalities G-dim_{R} M ≤ CI-dim_{R} M ≤ proj dim_{R} M, with equalities to the left of any finite dimension, cf. [8, (1.4)]. The ring R is a complete intersection if and only if CI-dim_{R} M < ∞ for each M ∈ F(R), cf. [8, (1.3)].

9.4. Example. If CI-dim_{R} M < proj dim_{R} M, then lim_{n→∞} β̂_{n}^{R}(M)/β̂_{i−n}^{R}(M) exists for each i ∈ ℤ, does not depend on i, and is a rational number; when the sequence (β_{n}^{R}(M))_{n⩾0} is bounded the sequence (β̂_{n}^{R}(M))_{n∈ℤ} is constant.

Indeed, by [8, (8.1)] there exist polynomials b_{±}^{M}(t) ∈ ℚ[t], with equal leading terms, such that β_{n}^{R}(M) = b_{+}^{M}(n) for all even n ≫ 0 and β_{n}^{R}(M) = b_{−}^{M}(n) for all odd n ≫ 0. We may assume M ∈ G, cf. Remark 9.3.1, and then by [6, (2.6.2), (3.3)] the polynomials for M and M^{∗} have the same degree, say c. If c = 0, then β_{n}^{R}(M) = b^{M} and β_{n}^{R}(M^{∗}) = b^{M^{∗}} for n ≥ 0 by [8, (7.3.2)]. If T is a complete resolution of M, then each Ω^{i}T is in G by Lemma 2.4, so Theorem 9.1 yields

β_{0}^{R}(M^{∗}) = β̂_{0}^{R}(M^{∗}) = β̂_{0}^{R}(Ω^{−1}T) = β_{0}^{R}(Ω^{−1}T) = β_{1}^{R}(Ω^{−1}T) = β_{0}^{R}(M) .

10. Bass numbers

Let R be a local ring with residue field k. The nth Bass number of a finite R-module N is the number of copies of the injective envelope of k in the nth module of a minimal injective resolution of N; it is denoted μ_{n}^{R}(N) and is equal to rank_{k} Ext_{R}^{n}(k, N). To use a similar expression to define stable and relative analogs of Bass numbers, we need G-dim_{R} k to be finite. This imposes restrictions on R.

10.1. Remark. For a local ring R with residue field k the following are equivalent. (i) R is Gorenstein. (ii) G-dim_{R} N ≤ dim R for each finite R-module N . (iii) G-dim_{R} N < ∞ for each finite R-module N . (iv) k is a G-perfect R-module with G-dim_{R} k = dim R and k^{†} ≅ k.

(v) G-dim_{R} k < ∞.

Indeed, the equivalence of (i) and (ii) comes from Theorem 3.2. By Theorem 4.2.2.a condition (iv) can be restated as μ_{R}^{n}(R) = 0 for n ≠ dim R and μ_{R}^{dim R}(R) = 1, and condition (v) as μ_{R}^{n}(R) = 0 for n ≫ 0. Thus, (i), (iv), and (v) are equivalent by the classical characterization of Gorenstein local rings by Bass, cf. [10, §3.3]

For the rest of this section R is a commutative noetherian local Gorenstein ring and set d = dim R. For each finite R-module N we introduce an nth relative Bass number and an nth stable Bass number by

μ_{G}^{n}(N) = rank_{k} Ext_{G}^{n}(k, N)  and  μ_{R}^{n}(N) = rank_{k} Ex̂t_{R}^{n}(k, N) .

10.2. Remark. The Bass numbers of the R-module k are equal to its Betti numbers, so Example 9.2 expresses μ_{G}^{n}(k) and μ_{R}^{n}(k) in terms of μ_{R}^{n}(k).

10.3. Theorem. Let N be a finite R-module with G-dim_{R} N = g < ∞. Let T → P → N be a complete resolution and 0 → Y → X → N → 0 a G-approximation.

(1) μ_{R}^{n}(N) = μ_{R}^{n}(X) = μ_{R}^{n+i}(Ω^{i}T) = β̂_{d−1−n−i}^{R}(Ω^{i}T) for all n ∈ ℤ and all i ∈ ℤ.

(2) If proj dim_{R} N < ∞, then μ_{G}^{n}(N) = μ_{R}^{n}(N) and μ_{R}^{n}(N) = 0 for all n ∈ ℤ.

(3) If proj dim_{R} N = ∞ then the following tables, where t = depth_{R} N, express the relative Bass numbers and the stable Bass numbers of N.

n ⋯  0  1  ⋯  d  ⋯
μ_{G}^{n}(N)  0  μ_{R}^{0}(N) ≤ μ_{R}^{1}(N) β_{d−n}^{R}(N)  0

n ⋯  t − 2  ⋯  d  d + 1  ⋯
μ_{R}^{n}(N) β_{d−n−1}^{R}(N) μ_{R}^{d}(Ω^{d−n}T) μ_{R}^{d}(Ω^{0}T) ≤ μ_{R}^{d}(N)  μ_{R}^{n}(N)

Furthermore, μ_{n}^{R}(N) > 0 for all n, and μ_{n}^{G}(N) > 0 for 2 ≤ n ≤ d.

Proof. (1) The first two equalities come from Lemmas 5.8.3 and 5.8.5. We obtain the third equality directly from Theorem 6.3.4 when n + i ≤ −2, then shifting degrees by means of Lemma 5.8.5 we extend it to all values of n and i.

(2) results from Theorems 4.2.4 and 5.2.4.

(3) The first table comes from Theorem 4.2.2.b for n ≤ 0 and from Theorem 7.1 for n = 1. Since k is G-perfect with G-dim_{R} k = d, cf. Remark 10.1, the tabulated values of μ_{G}^{n}(N) for n ≥ 2 are given by Theorem 6.3.3.

Recalling that Ext_{R}^{n}(k, N) = 0 for n < t, we fill in the second table using Theorem 6.3.4 for n ≤ t − 2, Part (1) for t − 2 < n < d, Part (1) with Theorem 7.1 for n = d, and Theorem 5.2.2 for n ≥ d + 1.

If μ_{R}^{n}(N) = 0 for some n, then (1) yields β̂_{d−1−n}^{R}(Ω^{0}T) = 0. Using alternatively parts (3) and (1) of Theorem 9.1, for i > G-dim_{R} N we get equalities 0 = β̂_{i}^{R}(Ω^{0}T) = β̂_{i}^{R}(N) = β_{i}^{R}(N) contradicting the hypothesis proj dim_{R} N = ∞.

The number depth_{G} N = inf{n ∈ ℤ | μ_{n}^{G}(N) ≠ 0} was introduced in [27, §2.3], in search of an equality of Auslander-Buchsbaum type for relative invariants. We obtain an amusing catalog of all cases where such an equality holds.

10.4. Corollary. An equality rel dim_{G} N = depth_{G} R−depth_{G} N holds if and only if N satisfies one of the following disjoint conditions: depth_{R} N = 0, or depth_{G} N = 1, or depth_{R} N = 2, or proj dim_{R} N ≤ depth R − 3.

Proof. Applying Proposition 4.8, Theorem 8.3, and Theorem 4.2.4, one obtains

rel dim_{G} N = G-dim_{R} N = depth R − depth_{R} N = depth_{G} R − depth_{R} N .

Thus, N satisfies a ‘relative Auslander-Buchsbaum Equality’ if and only if depth_{G} N = depth_{R} N. From Theorem 10.3 one reads off the implications:

depth_{G} N = 0 ⟺ depth_{R} N = 0 .
depth_{G} N = 1 ⟹ depth_{R} N = 1 .
depth_{G} N = 2 ⟸ depth_{R} N = 2 .
If n ≥ 3, then depth_{G} N = n ⟺ depth_{R} N = n and proj dim_{R} N < ∞ .

The list above is just another format of the desired assertion.

The balance of this section is devoted to showing that all admissible configurations of vanishing Betti numbers and Bass numbers do occur, by constructing relevant examples and analyzing them through the results of the preceding sections. It may come as a surprise that relative Bass sequences and relative Betti sequences can have a gap at n = 1, or that the relative Bass sequence of a nontrivial module can be identically equal to zero: no such pattern occurs in absolute cohomology.

10.5. Examples. Let R be a non-regular Gorenstein ring of dimension d.

The following table lists examples of all possible patterns of vanishing in pairs (μ_{G}^{0}(N), μ_{G}^{1}(N)) for an indecomposable R-module N with proj dim_{R} N = ∞.

μ_{G}^{0} = 0 = μ_{G}^{1}  μ_{G}^{0} > 0 = μ_{G}^{1} μ_{G}^{0} = 0 < μ_{G}^{1}  μ_{G}^{0} > 0 < μ_{G}^{1}
d = 0  No such N  Each N
d = 1 Each torsion-free N  No such N R/𝔪^{s} for s ≥ ℓℓ(R)
d ≥ 2  𝔪  k  N_{d−1}  N_{d}

The assertions on k, 𝔪, and N_{i} come from Remark 10.2, and Propositions 10.6, 10.8, respectively; the rest follows from Proposition 10.7 and Example 8.8.2.

10.6. Proposition. If R is a non-regular Gorenstein local ring with dim R = d ≥ 1, then the following hold:

μ_{G}^{n}(𝔪) = \begin{cases}0 & for n ≤ 1 and n ≥ d + 1 ; \\ β_{n−d}^{R}(𝔪) & for 2 ≤ n ≤ d; \end{cases}

μ̂_{R}^{n}(𝔪) = \begin{cases}μ_{R}^{n}(𝔪) + μ_{R}^{d−n+1}(𝔪) & for n ≠ 1, d; \\ μ_{R}^{d}(𝔪) & for n = 1, d . \end{cases}

Proof. Since R is Gorenstein, we have Ext_{R}^{n}(k, R) = 0 for n ≠ d and Ext_{R}^{d}(k, R) ≅ k. Let ρ: R → k be the canonical surjection, let P → k be a minimal free resolution, and let X → Ω^{n}P be a G-approximation. We then have

rank_{k} Ext_{R}^{d}(k, ρ) ≤ f-rank_{R} Ω^{n}P ≤ f-rank_{R} X = δ_{G}^{d}(k) = 0 .

where the first inequality comes from Lemma 8.2.1 and the equality from Remark 8.8.1. Thus, the cohomology exact sequence induced by the short exact sequence 0 → 𝔪 → R ⟶^{ρ} k → 0 yields equalities

μ_{R}^{n}(𝔪) = μ_{R}^{n−1}(k) for all n ∈ ℤ; μ_{R}^{n}(𝔪) = \begin{cases}μ_{R}^{n−1}(k) & for n ≠ d; \\ μ_{R}^{d−1}(k) + 1 & for n = d . \end{cases}

The expressions for μ_{R}^{n}(𝔪) now follow from those for μ_{R}^{n}(k), cf. Remark 10.2.

For n ≠ 1 the values of μ_{G}^{n}(𝔪) are given by Theorem 10.3. To prove that μ_{G}^{1}(𝔪) vanishes we use an exact sequence given by Theorem 7.1:

0 ⟶ Ext_{G}^{1}(k, 𝔪) ⟶ Ext_{R}^{1}(k, 𝔪) ⟶^{ε_{R}^{1}(k,𝔪)} Ex̂t_{R}^{1}(k, 𝔪) ⟶ Ext_{G}^{2}(k, 𝔪) .

If d = 1, then Ext_{G}^{2}(k, 𝔪) = 0 by Theorem 4.2.2, so ε_{R}^{1} is surjective, hence μ_{G}^{1}(𝔪) = μ_{R}^{1}(𝔪) − μ_{R}^{1}(𝔪); the difference vanishes by the computation above.

If d ≥ 2, then Proposition 5.4 yields a commutative diagram

\xymatrix{Hom_{R}(k, R) \ar[r] & Hom_{R}(k, k) \ar[r] \ar[d]_{ε_{R}^{0}(k,k)} & Ext_{R}^{1}(k, 𝔪) \ar[d]^{ε_{R}^{1}(k,𝔪)} \ar[r] & Ext_{R}^{1}(k, R) \\ Ex̂t_{R}^{0}(k, R) \ar[r] & Ex̂t_{R}^{0}(k, k) \ar[r] & Ex̂t_{R}^{1}(k, 𝔪) \ar[r] & Ex̂t_{R}^{1}(k, R)}

The upper ends vanish by the depth hypothesis, the lower ones by Theorem 5.2.4. As ε_{R}^{0}(k, 𝔪) is injective by Theorem 8.7 and Example 8.8.1, so is ε_{R}^{1}(k, 𝔪).

Next we take a look at dimension one.

10.7. Proposition. If R is a non-regular Gorenstein local ring with dim R = 1 and N is a finite R-module, then μ_{G}^{1}(N) = δ_{G}^{0}(N).

Remark. The module N is torsion-free if and only if it is maximal Cohen-Macaulay.

Proof. For convenience of notation, we set r = δ_{G}^{0}(N).

If depth N = 1, then N is maximal Cohen-Macaulay, so N ≅ R^{r} ⊕G with G ∈ G and f-rank_{R} G = 0. From Proposition 4.7 and Theorem 4.2.4 we obtain

Ext_{G}^{1}(k, N) ≅ Ext_{G}^{1}(k, R^{r}) ⊕ Ext_{G}^{1}(k, G)
≅ Ext_{R}^{1}(k, R^{r}) ⊕ Ext_{G}^{1}(k, G)
≅ k^{r} ⊕ Ext_{G}^{1}(k, G) .

Since 0 → 𝔪 ⟶^{ι} R → 0 is a G-resolution of k, from Construction 6.2 we obtain a proper resolution 0 → R^{∗} ⟶^{ι^{∗}} 𝔪^{∗} → 0 of k^{†} = k, and hence an exact sequence

Hom_{R}(𝔪^{∗}, G) ⟶^{Hom_{R}(ι^{∗},G)} Hom_{R}(R^{∗}, G) ⟶ Ext_{G}^{1}(k, G) ⟶ 0 .

Take any α ∈ Hom_{R}(R^{∗}, G). If α^{∗}: G^{∗} → R^{∗∗} = R is surjective, then R splits off as a direct summand of G^{∗}, so G ≅ G^{∗∗} has a direct summand isomorphic to R, a contradiction. Thus, α^{∗}(G^{∗}) ⊆ 𝔪, hence α^{∗} = ιβ for some β: G^{∗} → 𝔪. Dualizing, we get α = α^{∗∗} = Hom_{R}(ι^{∗}, G)(β^{∗}), hence Ext_{G}^{1}(k, G) = 0, and so μ_{G}^{1}(N) = r.

If depth N = 0, then Theorem 8.5 yields a proper exact sequence

0 ⟶ R^{s} ⟶^{∂} R^{r} ⊕ G ⟶ N ⟶ 0

where G is maximal Cohen-Macaulay with f-rank_{R} G = 0 and ∂(R^{s}) contains no nontrivial free direct summand of R^{r}. Thus, the composition σ: R^{s} → R^{r}⊕G → R^{r} satisfies σ(R^{s}) ⊆ 𝔪R^{r}. We have Ext_{G}^{1}(k, G) = 0 by the case treated above, and Ext_{G}^{2}(k, R^{s}) = 0 by Proposition 4.2.2.a, so by Proposition 4.6 the sequence

Ext_{G}^{1}(k, R^{s}) ⟶^{Ext_{G}^{1}(k,σ)} Ext_{G}^{1}(k, R^{r}) ⟶ Ext_{G}^{1}(k, N) ⟶ 0

is exact. Now Ext_{G}^{1}(k, σ) = 0 by the minimality of σ, so μ_{G}^{1}(N) = r, as desired.

The missing patterns in dimensions ≥ 2 are provided by

10.8. Proposition. Let R be a non-regular Gorenstein local ring with dim R = d ≥ 2, let x_{1}, …, x_{d} be a maximal regular sequence, and set R_{i} = R/(x_{1}, …, x_{i}).

For i = d − 1 and i = d there exist exact sequences

0 ⟶ R_{i} ⟶ N_{i} ⟶ 𝔪 ⟶ 0

where N_{i} is indecomposable, proj dim_{R} N_{i} = ∞, and

μ_{G}^{n}(N_{i}) = \binom{i}{d − n}  for  n = 0, 1 .

Proof. The exact sequences 0 → R_{i} ⟶^{x_{i+1}} R_{i} → R_{i+1} → 0 induce exact sequences

⋯ ⟶^{x_{i+1}} Ext_{R}^{n}(k, R_{i}) ⟶ Ext_{R}^{n}(k, R_{i+1}) ⟶ Ext_{R}^{n+1}(k, R_{i}) ⟶^{x_{i+1}} ⋯

where multiplication by x_{i+1} is the zero map. As R_{0} = R and R is Gorenstein, we have μ_{R}^{n}(R_{0}) = \binom{0}{d−n}. Induction on i yields μ_{R}^{n}(R_{i}) = \binom{i}{d−n} for all 0 ≤ i ≤ d and n ∈ ℤ. If i ≥ d − 1 then μ_{R}^{2}(R_{i}) ≠ 0, so Ext_{R}^{1}(𝔪, R_{i}) ≅ Ext_{R}^{2}(k, R_{i}) ≠ 0, hence for i ≥ d − 1 there exists a non-split exact sequence 0 → R_{i} → N_{i} → 𝔪 → 0. It is proper by Lemma 4.1.2, so Proposition 4.4 and Corollary 7.2 give an exact sequence

Ext_{G}^{0}(k, 𝔪) ⟶ Ext_{R}^{1}(k, R_{i}) ⟶ Ext_{G}^{1}(k, N_{i}) ⟶ Ext_{G}^{1}(k, 𝔪)

Both ends vanish by Proposition 10.6, hence μ_{G}^{1}(N_{i}) = μ_{R}^{1}(R_{i}) = \binom{i}{d−n} for n = 0, 1.

Denote T_{i} the torsion submodule of N_{i}. The defining exact sequence shows that proj dim N_{i} is infinite, that N_{i} has rank 1, and that T_{i} is isomorphic to R_{i}; in particular, T_{i} is indecomposable. If N_{i} = V_{i} ⊕ W_{i} with V_{i} ≠ 0 and W_{i} ≠ 0, we may assume that rank V_{i} = 0, and hence V_{i} ⊆ T_{i}. As V_{i} is a direct summand of N_{i}, it is also a direct summand of T_{i}, hence V_{i} = T_{i}. This implies that the defining sequence splits, contrary to our choice. We conclude that N_{i} is indecomposable.

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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

Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

   E-mail address: alexmart@neu.edu