FINITENESS CRITERIA IN GORENSTEIN HOMOLOGICAL ALGEBRA

IOANNIS EMMANOUIL AND OLYMPIA TALELLI

Abstract. In this paper, we examine the class of modules of finite Gorenstein projective dimension and study approximations of modules in that class by modules, which are either Gorenstein projective or else have finite projective dimension. We also examine the relevance of complete cohomology in the study of modules of finite Gorenstein projective dimension and obtain, over group rings, such a finiteness criterion that involves only complete cohomology.

Contents

0. Introduction 1 1. Modules of finite Gorenstein projective dimension 4 2. Stabilizing with respect to modules of finite projective dimension 8 3. Stabilizing with respect to Gorenstein projective modules 11 4. Finiteness of Gorenstein projective dimension and complete cohomology 13 5. Modules of finite Gorenstein injective dimension 16 6. The special case of group rings 17 References 20

0. Introduction

Auslander and Bridger defined in [1], [2] the G-dimension G-dim_{R}M of any finitely generated module M over a commutative Noetherian local ring R and showed that R is Gorenstein if and only if the G-dimension of its residue field is finite. The G-dimension generalizes the projective dimension, in the sense that if M is a finitely generated module of finite projective dimension, then G-dim_{R}M = pd_{R}M. This definition was extended to all modules over any ring R by Enochs and Jenda in [14]: A module is said to be Gorenstein projective if it is a syzygy of a doubly infinite acyclic complex of projective modules

⋯ ⟶ P_{n+1} ⟶ P_{n} ⟶ P_{n−1} ⟶ ⋯,

which remains acyclic when applying the functor Hom_{R}( , P) for any projective module P. Then, modules of finite Gorenstein projective dimension are defined in the standard way, by using resolutions by Gorenstein projective modules. As shown in [10, Theorem 4.2.6], the Gorenstein projective dimension of a finitely generated module over a commutative Noetherian ring agrees with its G-dimension. The relative cohomology theory obtained by resolving the first argument of the Hom functor by Gorenstein projective modules has been studied in detail in [4] for modules of finite Gorenstein projective dimension; for such modules, the relative theory is closely related to both of the ordinary and the complete cohomology.

The concept of a maximal Cohen-Macaulay approximation, which was introduced and studied by Auslander and Buchweitz in [3] for finitely generated modules over a commutative Gorenstein local ring, has been generalized by Holm who showed in [18, Theorem 2.10] that if R is any ring and M is an R-module of finite Gorenstein projective dimension, then there exists a short exact sequence of R-modules

Research supported by a GSRT/Greece excellence grant, cofunded by the ESF/EU and National Resources.

(1)  0 ⟶ K ⟶ G ⟶ M ⟶ 0,

where G is Gorenstein projective and pd_{R}K < ∞. Such an exact sequence satisfies a weak form of uniqueness, as in the classical Schanuel’s lemma. We shall describe in section 2 the precise sense in which G may be regarded as an approximation of M by a Gorenstein projective module: We consider the stabilization 𝔉𝔓-R-Mod of the category of R-modules with respect to the class of modules of finite projective dimension and denote by 𝔉𝔓-FGP(R) (resp. by 𝔉𝔓-GP(R)) the full subcategory of 𝔉𝔓-R-Mod consisting of the modules of finite Gorenstein projective dimension (resp. of the Gorenstein projective modules). Then, the assignment M ↦ G extends to a functor μ : 𝔉𝔓-FGP(R) ⟶ 𝔉𝔓-GP(R), which is right adjoint to the inclusion functor 𝔉𝔓-GP(R) ↪ 𝔉𝔓-FGP(R). Vanishing of the counit of this adjunction characterizes the modules of finite projective dimension among the modules of finite Gorenstein projective dimension.

As shown in [11, Lemma 2.17], one may use the infinite-loop-space-like structure of the class of Gorenstein projective modules, in order to dualize (1) and obtain for any module M of finite Gorenstein projective dimension an exact sequence of R-modules

(2)  0 ⟶ M ⟶ A ⟶ G′ ⟶ 0,

where G′ is Gorenstein projective and pd_{R}A < ∞. As above, such an exact sequence satisfies a weak form of uniqueness (à la Schanuel). We shall explain in section 3 the precise sense in which A may be regarded as an approximation of M by a module of finite projective dimension: We consider the stabilization 𝔊𝔓-R-Mod of the category of R-modules with respect to the class of Gorenstein projective modules and denote by 𝔊𝔓-FGP(R) (resp. by 𝔊𝔓-FP(R)) the full subcategory of 𝔊𝔓-R-Mod consisting of the modules of finite Gorenstein projective dimension (resp. of finite projective dimension). Then, the assignment M ↦ A extends to a functor ν : 𝔊𝔓-FGP(R) ⟶ 𝔊𝔓-FP(R), which is left adjoint to the inclusion functor 𝔊𝔓-FP(R) ↪ 𝔊𝔓-FGP(R). Vanishing of the unit of this adjunction characterizes the Gorenstein projective modules among the modules of finite Gorenstein projective dimension.

The two reformulations of the finiteness of the Gorenstein projective dimension of a module M, which are described by the existence of short exact sequences as in (1) and (2) above, are somehow analogous to the interpretations of an element α ∈ Ext_{R}^{n}(X, Y) (where X, Y are two modules and n is a positive integer) in classical homological algebra. On one hand, α may be described by using a projective resolution P_{∗} ⟶ X ⟶ 0 as the cohomology class of a cocycle f : Ω^{n}X ⟶ Y. This interpretation is analogous to the derivation of the exact sequence (1) out of the assumption that the module M has finite Gorenstein projective dimension; cf. the proof of [18, Theorem 2.10] and [4, Construction 5.10]. Alternatively, we may use Yoneda’s approach and describe α as the equivalence class of an n-fold extension

0 ⟶ Y ⟶ Z ⟶ P_{n−2} ⟶ ⋯ ⟶ P_{0} ⟶ X ⟶ 0,

where the modules P_{0}, …, P_{n−2} are projective. One may construct such an extension by starting from a projective resolution P_{∗} ⟶ X ⟶ 0 and a cocycle f : Ω^{n}X ⟶ Y that represents α and then define Z to be the pushout of f and the inclusion Ω^{n}X ↪ P_{n−1}. This approach is analogous to the derivation of the exact sequence (2) out of the exact sequence (1), which is presented in the proof of [11, Lemma 2.17] (and reproduced in the proof of Proposition 1.2 below).

The conditions characterizing the finiteness of the Gorenstein projective dimension of a module M that are described by the existence of short exact sequences as in (1) and (2) above may be relaxed, by removing the assumption that the modules K and A therein have finite projective dimension and assuming instead that certain elements of complete cohomology groups vanish. Using the complete cohomology functors Êxt_{R}^{∗}( , ), which were defined by Benson, Carlson, Mislin and Vogel in [7], [17] and [21], we prove that a module M has finite Gorenstein projective dimension if and only if one of the following two conditions is satisfied:

(i) There exists a short exact sequence of R-modules 0 ⟶ K ⟶ G ⟶ M ⟶ 0, where G is Gorenstein projective, such that the image of the element ξ ∈ Ext_{R}^{1}(M, K) that classifies the extension vanishes in the group Êxt_{R}^{1}(M, K).

(ii) There exists a short exact sequence of R-modules 0 ⟶ M ⟶^{ι} A ⟶ G′ ⟶ 0, where G′ is Gorenstein projective, such that the image of ι ∈ Hom_{R}(M, A) vanishes in the group Êxt_{R}^{0}(M, A). It is interesting to note that conditions (i) and (ii) above involve no a priori assumption about the finiteness of the (Gorenstein) projective dimension of the modules K and A therein.

We now consider the special case where R is the integral group ring ℤG of a group G and examine the Gorenstein cohomological dimension Gcd_{ℤ}G of G over ℤ, which is defined to be the Gorenstein projective dimension of the trivial ℤG-module ℤ. It follows from the discussion above that Gcd_{ℤ}G < ∞ if and only if there exists a short exact sequence of ℤG-modules

0 ⟶ ℤ ⟶ A ⟶ \overline{A} ⟶ 0,

where pd_{ℤG}A < ∞ and \overline{A} is Gorenstein projective. A stronger version of that characterization was obtained in [6, Theorem 2.7], where the finiteness of the Gorenstein cohomological dimension of G over ℤ was shown to be equivalent to the existence of a ℤ-split monomorphism of ℤG-modules ℤ ⟶ A, where A is ℤ-free and pd_{ℤG}A < ∞. This result shows that the finiteness of the Gorenstein cohomological dimension of G over ℤ may be described solely in terms of ordinary group cohomology. Using in an essential way the Hopficity of the group ring ℤG, we shall obtain in section 6, an even stronger version of the latter characterization. We prove that the finiteness of Gcd_{ℤ}G is equivalent to the existence of a ℤ-split monomorphism of ℤGmodules ι : ℤ ⟶ A, where A is ℤ-free, such that the image of ι ∈ Hom_{ℤG}(ℤ, A) = H^{0}(G, A) vanishes in the group Ĥ^{0}(G, A). Hence, the finiteness of the Gorenstein cohomological dimension of G over ℤ may be also described solely in terms of complete cohomology. Note that this result is stronger than the characterization of finiteness described in condition (ii) above, in the case of the trivial ℤG-module ℤ, since there is no additional assumption about the module \overline{A} = coker ι being Gorenstein projective.

The Gorenstein cohomological dimension Gcd_{ℤ}G is a generalization of the ordinary cohomological dimension cd_{ℤ}G of G over ℤ: If cd_{ℤ}G is finite, then we have an equality Gcd_{ℤ}G = cd_{ℤ}G. On the other hand, it is easily seen that the vanishing of Gcd_{ℤ}G characterizes the finite groups, i.e. Gcd_{ℤ}G = 0 if and only if the group G is finite. We note that the cohomological dimension cd_{ℤ}G of any non-trivial finite group G is infinite. C.T.C. Wall has proved in [24] that cd_{ℤ}G < ∞ if and only if G acts freely and cellularly on a contractible finite dimensional CWcomplex. We expect that the finiteness of the Gorenstein cohomological dimension Gcd_{ℤ}G of G over ℤ should play a central role in any attempt to generalize Wall’s theorem, in a way that includes actions with finite stabilizers. Indeed, if G acts cellularly with finite stabilizers on a contractible finite dimensional CW-complex X, then the cellular chain complex of X is a Gorenstein projective resolution of G, i.e. a resolution of the trivial ℤG-module ℤ by Gorenstein projective modules.

Notations and terminology. Unless otherwise specified, all modules considered in this paper will be left modules over a fixed ring R. For any module M we shall denote by ΩM the kernel of a surjective R-linear map P ⟶ M, where P is a projective module. Even though ΩM is not uniquely determined by M, Schanuel’s lemma implies uniqueness up to addition of a projective module. We also consider the iterates Ω^{n}M, n ≥ 1, which are defined inductively by letting Ω^{1}M = ΩM and Ω^{n}M = ΩΩ^{n−1}M for all n > 1. If P_{∗} ⟶ M ⟶ 0 is a projective resolution of M, then Ω^{n}M is identified with the image of the map P_{n} ⟶ P_{n−1} for all n ≥ 1. A doubly infinite acyclic complex of projective modules is called totally acyclic if the complex remains exact after applying the functor Hom_{R}( , P) for any projective module P. In this way, the Gorenstein projective modules are precisely the syzygies of the totally acyclic complexes of projective modules.

Acknowledgements. It is a pleasure to thank A. Beligiannis and the anonymous referee for their very helpful comments and suggestions that improved the paper.

1. Modules of finite Gorenstein projective dimension

In this preliminary section, we shall review a few equivalent formulations of the finiteness of the Gorenstein projective dimension of a module and record some consequences concerning the relation between modules of finite Gorenstein projective dimension, modules of finite projective dimension and Gorenstein projective modules.

Lemma 1.1. A module M is Gorenstein projective if and only if there exists a short exact sequence of R-modules 0 ⟶ M ⟶ P ⟶ G ⟶ 0, where P is projective and G is Gorenstein projective.

Proof. If M is Gorenstein projective, then we may obtain a short exact sequence as in the statement by considering a totally acyclic complex of projective modules that has M as a syzygy. Conversely, the existence of such a short exact sequence implies that the module M is Gorenstein projective, in view of [18, Theorem 2.5]. □

The next result (more precisely, the equivalence between assertions (i) and (iii) therein) is a higher dimensional analogue of the above lemma.

Proposition 1.2. The following conditions are equivalent for a module M and a non-negative integer n:

(i) Gpd_{R}M ≤ n,

(ii) there exists a short exact sequence of R-modules 0 ⟶ K ⟶ G ⟶ M ⟶ 0, where G is Gorenstein projective and pd_{R}K ≤ n − 1 (if n = 0, this is understood to mean K = 0) and

(iii) there exists a short exact sequence of R-modules 0 ⟶ M ⟶ A ⟶ G′ ⟶ 0, where G′ is Gorenstein projective and pd_{R}A ≤ n.

Proof. The equivalence between (i) and (ii) has been shown in [18, Theorem 2.10]. The implication (ii)→(iii) has been proved in [11, Lemma 2.17], as follows: Assume that there exists a short exact sequence of R-modules as in (ii). Then, Lemma 1.1 implies that G fits into a short exact sequence of R-modules

0 ⟶ G ⟶ P ⟶ G′ ⟶ 0,

where P is projective and G′ is Gorenstein projective. We consider the pushout of the diagram

\begin{matrix}G & ⟶ & M \\ ↓ \\ P \end{matrix}

which fits into a commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ K ⟶ G ⟶ M ⟶ 0
∥ ↓ ↓
0 ⟶ K ⟶ P ⟶ A ⟶ 0
↓  ↓
G′ = G′
↓  ↓
0  0

Since pd_{R}K ≤ n − 1, the horizontal exact sequence in the middle of the diagram shows that pd_{R}A ≤ n. Hence, the rightmost vertical exact sequence has all of the desired properties.

(iii)→(ii): Assume that there exists a short exact sequence of R-modules as in (iii). Since pd_{R}A ≤ n, there exists a short exact sequence of R-modules

0 ⟶ K ⟶ P ⟶ A ⟶ 0,

where P is projective and pd_{R}K ≤ n − 1. We now consider the pullback of the diagram

\begin{aligned}& & M \\ & & ↓ \\ P & ⟶ & A \end{aligned}

which fits into a commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ K ⟶ G ⟶ M ⟶ 0
∥ ↓ ↓
0 ⟶ K ⟶ P ⟶ A ⟶ 0
↓  ↓
G′ = G′
↓  ↓
0  0

Since G′ is Gorenstein projective, we may apply Lemma 1.1 to the vertical exact sequence in the middle of the diagram, in order to conclude that G is Gorenstein projective as well. Hence, the top horizontal exact sequence has all of the desired properties. □

Remarks 1.3. (a) Let M be a module of finite Gorenstein projective dimension. Then, the two constructions described in the proof of the equivalence between conditions (ii) and (iii) in Proposition 1.2 above are inverse to each other, provided that we make the obvious choices in the second step in each direction. (What we mean by this is that having performed the construction described in the proof of the implication (ii)→(iii) in the short exact sequence 0 ⟶ K ⟶ G ⟶ M ⟶ 0, there is an obvious choice for a description of A as a quotient of the form P/K. Similarly, having performed the construction described in the proof of the implication (iii)→(ii) in the short exact sequence 0 ⟶ M ⟶ A ⟶ G′ ⟶ 0, there is an obvious choice for a description of G as the kernel of a surjective map P ⟶ G′.) This assertion follows from the following two elementary observations:

(a1) If the commutative square

\begin{matrix}G & ⟶ & M \\ ↓ & & ↓ \\ P & ⟶ & A \end{matrix}

is a pushout diagram and the morphism G ⟶ P is injective, then the square is also a pullback diagram.

(a2) If the commutative square

\begin{matrix}G & ⟶ & M \\ ↓ & & ↓ \\ P & ⟶ & A \end{matrix}

is a pullback diagram and the morphism P ⟶ A is surjective, then the square is also a pushout diagram.

(b) We may restate the equivalence between conditions (ii) and (iii) in Proposition 1.2 as follows: Any short exact sequence of R-modules 0 ⟶ M ⟶ A ⟶ G′ ⟶ 0 as in Proposition 1.2(iii) induces a short exact sequence 0 ⟶ ΩA ⟶ ΩG′ ⟶ M ⟶ 0, which is of the form required in Proposition 1.2(ii), since the operation Ω maps modules of finite projective dimension (resp. Gorenstein projective modules) onto modules of finite projective dimension (resp. onto Gorenstein projective modules). Conversely, for any short exact sequence of Rmodules 0 ⟶ K ⟶ G ⟶ M ⟶ 0 as in Proposition 1.2(ii), we may express G as ΩG′ for some Gorenstein projective module G′. Then, there is an induced description of K as ΩA for some module A, which has necessarily finite projective dimension. Hence, the given exact sequence is of the form 0 ⟶ ΩA ⟶ ΩG′ ⟶ M ⟶ 0 and thus induces a short exact sequence 0 ⟶ M ⟶ A ⟶ G′ ⟶ 0, which is of the form required in Proposition 1.2(iii).

Even though the exact sequences described in Proposition 1.2(ii) and (iii) are by no means unique, they do satisfy a weak form of uniqueness (à la Schanuel), as shown below.

Lemma 1.4. Let M be a module of finite Gorenstein projective dimension.

(i) Assume that 0 ⟶ K ⟶ G ⟶ M ⟶ 0 and 0 ⟶ \overline{K} ⟶ \overline{G} ⟶ M ⟶ 0 are two short exact sequences of R-modules with K, \overline{K} of finite projective dimension and G, \overline{G} Gorenstein projective. Then, there is an isomorphism G ⊕ \overline{K} ≃ \overline{G} ⊕ K.

(ii) Assume that 0 ⟶ M ⟶ A ⟶ G′ ⟶ 0 and 0 ⟶ M ⟶ \overline{A} ⟶ \overline{G′} ⟶ 0 are two short exact sequences of R-modules with A, \overline{A} of finite projective dimension and G′, \overline{G′} Gorenstein projective. Then, there is an isomorphism G′ ⊕ \overline{A} ≃ \overline{G′} ⊕ A.

Proof. (i) Since G is Gorenstein projective and pd_{R}\overline{K} < ∞, the group Ext_{R}^{1}(G, \overline{K}) is trivial and hence there exists an R-linear map λ : G ⟶ \overline{G} making the right square in the following diagram with exact rows commutative

0 ⟶ K ⟶ G ⟶ M ⟶ 0
l ↓ λ ↓ ∥
0 ⟶ \overline{K} ⟶ \overline{G} ⟶ M ⟶ 0

Then, there exists a unique R-linear map l : K ⟶ \overline{K}, which makes the left square of the diagram commutative as well. We may view the pair (l, λ) as a quasi-isomorphism between the complexes 0 ⟶ K ⟶ G ⟶ 0 and 0 ⟶ \overline{K} ⟶ \overline{G} ⟶ 0 and consider the corresponding mapping cone (which is an acyclic complex). We therefore obtain a short exact sequence

0 ⟶ K ⟶ G ⊕ \overline{K} ⟶ \overline{G} ⟶ 0.

Since \overline{G} is Gorenstein projective and pd_{R}K < ∞, the group Ext_{R}^{1}(\overline{G}, K) is trivial and hence the above exact sequence splits. It follows that there exists an isomorphism G ⊕ \overline{K} ≃ \overline{G} ⊕ K, as needed.

(ii) We use an argument, which is similar to that used in the proof of (i) above: Since G′ is Gorenstein projective and pd_{R}\overline{A} < ∞, the group Ext_{R}^{1}(G′, \overline{A}) is trivial and hence there exists an R-linear map ϱ : A ⟶ \overline{A} making the left square in the following diagram with exact rows commutative

0 ⟶ M ⟶ A ⟶ G′ ⟶ 0
∥ ϱ ↓ r ↓
0 ⟶ M ⟶ \overline{A} ⟶ \overline{G′} ⟶ 0

Then, there exists a unique R-linear map r : G′ ⟶ \overline{G′}, which makes the right square of the diagram commutative as well. We may view the pair (ϱ, r) as a quasi-isomorphism between the complexes 0 ⟶ A ⟶ G′ ⟶ 0 and 0 ⟶ \overline{A} ⟶ \overline{G′} ⟶ 0 and consider the corresponding mapping cone (which is an acyclic complex). We therefore obtain a short exact sequence

0 ⟶ A ⟶ G′ ⊕ \overline{A} ⟶ \overline{G′} ⟶ 0.

Since \overline{G′} is Gorenstein projective and pd_{R}A < ∞, the group Ext_{R}^{1}(\overline{G′}, A) is trivial and hence the above exact sequence splits. It follows that there exists an isomorphism G′ ⊕ \overline{A} ≃ \overline{G′} ⊕ A, as needed. □

As an immediate consequence of Proposition 1.2, we may obtain a simple characterization of those modules of finite Gorenstein projective dimension which are Gorenstein projective (or have finite projective dimension), in terms of the functor Ext_{R}^{1}.

Corollary 1.5. Let M be a module of finite Gorenstein projective dimension. Then:

(i) M is Gorenstein projective if and only if the functor Ext_{R}^{1}(M, ) vanishes on all modules of finite projective dimension and

(ii) M has finite projective dimension if and only if the functor Ext_{R}^{1}( , M) vanishes on all Gorenstein projective modules.

Proof. (i) If M is Gorenstein projective then Ext_{R}^{1}(M, A) = 0 for all modules A of finite projective dimension. Conversely, assume that this condition is satisfied and consider a short exact sequence of R-modules

0 ⟶ K ⟶ G ⟶ M ⟶ 0,

where G is Gorenstein projective and pd_{R}K < ∞. Since the group Ext_{R}^{1}(M, K) is trivial, the above exact sequence splits. Then, M is a direct summand of the Gorenstein projective module G and hence it is itself Gorenstein projective (cf. [18, Theorem 2.5]).

(ii) If M has finite projective dimension, then Ext_{R}^{1}(G, M) = 0 for all Gorenstein projective modules G. Conversely, assume that this condition is satisfied and consider a short exact sequence of R-modules

0 ⟶ M ⟶ A ⟶ G′ ⟶ 0,

where G′ is Gorenstein projective and pd_{R}A < ∞. Since the group Ext_{R}^{1}(G′, M) is trivial, the above exact sequence splits. Then, M is a direct summand of the module A and hence it has finite projective dimension; in fact, pd_{R}M ≤ pd_{R}A. □

If 𝔛 is a class of modules, then the left (resp. right) Ext^{1}-orthogonal of 𝔛 is the class ^{⊥}𝔛 (resp. 𝔛^{⊥}), consisting of those modules Y for which the group Ext_{R}^{1}(Y, X) (resp. Ext_{R}^{1}(X, Y)) is trivial for all X ∈ 𝔛. In this way, we may rephrase Corollary 1.5 by saying that ^{⊥}FP(R) ∩ FGP(R) = GP(R) and GP(R)^{⊥} ∩ FGP(R) = FP(R), where FP(R) (resp. FGP(R), resp. GP(R)) is the class of all modules of finite projective dimension (resp. of all modules of finite Gorenstein projective dimension, resp. of all Gorenstein projective modules). In particular, we may state the following corollary, dealing with rings R of finite left Gorenstein global dimension; these are precisely those rings R, over which the class FGP(R) comprises of all modules.

Corollary 1.6. If R is a ring of finite left Gorenstein global dimension, then ^{⊥}FP(R) = GP(R) and GP(R)^{⊥} = FP(R). □

Corollary 1.7. Let M be a module of finite Gorenstein projective dimension and consider a non-negative integer n. Then, Gpd_{R}M ≤ n if and only if the functor Ext_{R}^{n+1}(M, ) vanishes on all modules of finite projective dimension.

Proof. Since M has finite Gorenstein projective dimension, the n-th syzygy module Ω^{n}M has finite Gorenstein projective dimension as well (cf. [18, Theorem 2.24]). Moreover, we have Gpd_{R}M ≤ n if and only if Ω^{n}M is Gorenstein projective (cf. [18, Proposition 2.7]). Since the functors Ext_{R}^{1}(Ω^{n}M, ) and Ext_{R}^{n+1}(M, ) are naturally isomorphic, the result is an immediate consequence of Corollary 1.5. □

2. Stabilizing with respect to modules of finite projective dimension

Let M, N be two modules. Then, the set consisting of those R-linear maps f : M ⟶ N that factor through a module of finite projective dimension is a subgroup of the abelian group Hom_{R}(M, N). (This follows since the direct sum of two modules of finite projective dimension has finite projective dimension as well.) We shall denote by 𝔉𝔓-Hom_{R}(M, N) the corresponding quotient group and let [f] = [f]_{𝔉𝔓} be the class of any R-linear map f ∈ Hom_{R}(M, N) therein. We note that for any three modules M, N and L the composition of R-linear maps induces a well-defined biadditive map

𝔉𝔓-Hom_{R}(N, L) × 𝔉𝔓-Hom_{R}(M, N) ⟶ 𝔉𝔓-Hom_{R}(M, L).

We shall denote by 𝔉𝔓-R-Mod the category whose objects are all modules and whose morphism sets are given by the abelian groups 𝔉𝔓-Hom_{R}(M, N) (with composition of morphisms induced by the composition of R-linear maps). Then, Lemma 1.4 implies that the Gorenstein projective modules G and G′, which are defined for any module M of finite Gorenstein projective dimension in Proposition 1.2(ii) and (iii), are uniquely determined up to isomorphism as objects of the category 𝔉𝔓-R-Mod.

Lemma 2.1. Let f : M ⟶ N be an R-linear map, where M, N are two modules of finite Gorenstein projective dimension. We also consider two exact sequences of R-modules

0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0 and 0 ⟶ L ⟶^{ȷ} H ⟶^{q} N ⟶ 0,

where K, L have finite projective dimension and G, H are Gorenstein projective. Then: (i) There exists an R-linear map g : G ⟶ H, such that qg = fp. (ii) If g, g′ : G ⟶ H are two R-linear maps with qg = fp and qg′ = fp, then [g] = [g′] ∈

𝔉𝔓-Hom_{R}(G, H).

(iii) If [f] = [0] ∈ 𝔉𝔓-Hom_{R}(M, N) and g : G ⟶ H is an R-linear map such that qg = fp, then [g] = [0] ∈ 𝔉𝔓-Hom_{R}(G, H).

Proof. (i) Since G is Gorenstein projective and pd_{R}L < ∞, the group Ext_{R}^{1}(G, L) is trivial and hence the additive map q_{∗} : Hom_{R}(G, H) ⟶ Hom_{R}(G, N) is surjective. Therefore, there exists an R-linear map g : G ⟶ H such that fp = q_{∗}(g), as needed.

(ii) Let g, g′ : G ⟶ H be two R-linear maps with qg = fp and qg′ = fp.

0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0
g ↓↓ g′  f ↓
0 ⟶ L ⟶^{ȷ} H ⟶^{q} N ⟶ 0

Then, q(g′ − g) = qg′ − qg = fp − fp = 0 and hence there exists an R-linear map h : G ⟶ L, such that g′ − g = ȷl. Since L has finite projective dimension, we conclude that [g] = [g′] ∈ 𝔉𝔓-Hom_{R}(G, H).

(iii) Assume that f factors as the composition of two R-linear maps M ⟶^{a} Λ ⟶^{b} N, where the module Λ has finite projective dimension. Let

0 ⟶ Λ′ ⟶ P ⟶^{π} Λ ⟶ 0

be a short exact sequence of R-modules with P projective (and Λ′ of finite projective dimension). We may construct such an exact sequence by considering the beginning of a projective resolution of Λ. Then, invoking (i) above, we may find R-linear maps α : G ⟶ P and β : P ⟶ H, such that πα = ap and qβ = bπ.

0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0
α ↓  a ↓
0 ⟶ Λ′ ⟶ P ⟶^{π} Λ ⟶ 0
β ↓  b ↓
0 ⟶ L ⟶^{ȷ} H ⟶^{q} N ⟶ 0

We conclude that q(βα) = (ba)p = fp and hence for any R-linear map g : G ⟶ H with qg = fp we have [g] = [βα] ∈ 𝔉𝔓-Hom_{R}(G, H) (cf. (ii) above). This finishes the proof, since we obviously have [βα] = [0] ∈ 𝔉𝔓-Hom_{R}(G, H). □

Let 𝔉𝔓-GP(R) and 𝔉𝔓-FGP(R) be the full subcategories of 𝔉𝔓-R-Mod, whose objects are the Gorenstein projective modules and the modules of finite Gorenstein projective dimension respectively; then, 𝔉𝔓-GP(R) is a full subcategory of 𝔉𝔓-FGP(R). We note that if G, A are two modules with G Gorenstein projective and A of finite projective dimension, then the group Ext_{R}^{1}(G, ΩA) is trivial, in view of the finiteness of the projective dimension of ΩA; cf. Corollary 1.5(i). It follows that any R-linear map f : G ⟶ A factors as the composition of R-linear maps G ⟶ P ⟶^{p} A, where P is a projective module and p is surjective.^{1} Therefore, we conclude that an R-linear map between Gorenstein projective modules factors through a module of finite projective dimension if and only if it factors through a projective module. Hence, the category 𝔉𝔓-GP(R) defined above is precisely the stable category of Gorenstein projective modules.

We observe that Lemmas 1.4(i) and 2.1 imply that there is a well-defined additive functor

μ : 𝔉𝔓-FGP(R) ⟶ 𝔉𝔓-GP(R),

which maps a module M of finite Gorenstein projective dimension to the Gorenstein projective module G that appears in the short exact sequence of Proposition 1.2(ii) and the class [f] ∈ 𝔉𝔓-Hom_{R}(M, N) of an R-linear map f : M ⟶ N to the class [g] ∈ 𝔉𝔓-Hom_{R}(G, H) of any R-linear map g : G ⟶ H that satisfies the condition described in Lemma 2.1(i). We can now state the main result of this section.

1Dually, we may express G as ΩG′, for some Gorenstein projective module G′, and use the vanishing of the abelian group Ext_{R}^{1}(G′, A), in order to factor the R-linear map f : G ⟶ A as the composition of R-linear maps G ⟶^{ι} P′ ⟶ A, where P′ is a projective module and ι is injective.

Theorem 2.2. The additive functor μ : 𝔉𝔓-FGP(R) ⟶ 𝔉𝔓-GP(R) defined above is right adjoint to the inclusion functor 𝔉𝔓-GP(R) ↪ 𝔉𝔓-FGP(R).

Proof. We fix a Gorenstein projective module G and let N be a module of finite Gorenstein projective dimension. We also consider a short exact sequence of R-modules

0 ⟶ L ⟶^{ȷ} H ⟶^{q} N ⟶ 0,

where pd_{R}L < ∞ and H is Gorenstein projective (cf. Proposition 1.2(ii)). We note that the additive map

(3)  [q]_{∗} : 𝔉𝔓-Hom_{R}(G, H) ⟶ 𝔉𝔓-Hom_{R}(G, N)

is natural in both G (this is straightforward) and N (this follows from Lemma 2.1). We shall establish the adjunction in the statement of the theorem, by proving that the additive map (3) above is bijective. Indeed, since the module G is Gorenstein projective and pd_{R}L < ∞, the group Ext_{R}^{1}(G, L) is trivial. Hence, the additive map

q_{∗} : Hom_{R}(G, H) ⟶ Hom_{R}(G, N)

is surjective, whence the surjectivity of (3). As far as the injectivity of (3) is concerned, assume that g : G ⟶ H is an R-linear map, such that [qg] = [q][g] = [q]_{∗}[g] = [0] ∈ 𝔉𝔓-Hom_{R}(G, N). Then, we may consider the commutative diagram

0 ⟶ 0 ⟶ G = G ⟶ 0
g ↓  qg ↓
0 ⟶ L ⟶^{ȷ} H ⟶^{q} N ⟶ 0

and invoke Lemma 2.1(iii), in order to conclude that [g] = [0] ∈ 𝔉𝔓-Hom_{R}(G, H). □

We shall now prove that triviality of the counit of the adjunction described in Theorem 2.2 characterizes the modules of finite projective dimension among the modules of finite Gorenstein projective dimension.

Corollary 2.3. The following conditions are equivalent for a module N of finite Gorenstein projective dimension:

(i) pd_{R}N < ∞,

(ii) 𝔉𝔓-Hom_{R}(G, N) = 0 for all Gorenstein projective modules G and

(iii) there exists a short exact sequence of R-modules 0 ⟶ L ⟶ H ⟶^{q} N ⟶ 0, where pd_{R}L < ∞ and H is Gorenstein projective, such that [q] = [0] ∈ 𝔉𝔓-Hom_{R}(H, N).

Proof. The implications (i)→(ii) and (ii)→(iii) are obvious and hence it only remains to show that (iii)→(i). To that end, we note that the adjunction isomorphism (3) established in the proof of Theorem 2.2 implies, in particular, that the additive map

[q]_{∗} : 𝔉𝔓-Hom_{R}(H, H) ⟶ 𝔉𝔓-Hom_{R}(H, N)

is bijective. Since [q]_{∗}[1_{H}] = [q][1_{H}] = [q1_{H}] = [q] = [0] ∈ 𝔉𝔓-Hom_{R}(H, N), we conclude that [1_{H}] = [0] ∈ 𝔉𝔓-Hom_{R}(H, H); hence, the identity map of H factors through a module A of finite projective dimension. It follows that H is a direct summand of A and hence H has finite projective dimension as well. Being a Gorenstein projective module of finite projective dimension, the module H is necessarily projective (cf. [18, Proposition 2.27]). It follows readily that pd_{R}N ≤ pd_{R}L + 1 < ∞, as needed. □

3. Stabilizing with respect to Gorenstein projective modules

Let M, N be two modules. Then, the set consisting of those R-linear maps f : M ⟶ N that factor through a Gorenstein projective module is a subgroup of the abelian group Hom_{R}(M, N). (This follows since the direct sum of two Gorenstein projective modules is Gorenstein projective as well.) We shall denote by 𝔊𝔓-Hom_{R}(M, N) the corresponding quotient group and let [f] = [f]_{𝔊𝔓} be the class of any R-linear map f ∈ Hom_{R}(M, N) therein. We note that for any three modules M, N and L the composition of R-linear maps induces a well-defined biadditive map

𝔊𝔓-Hom_{R}(N, L) × 𝔊𝔓-Hom_{R}(M, N) ⟶ 𝔊𝔓-Hom_{R}(M, L).

We shall denote by 𝔊𝔓-R-Mod the category whose objects are all modules and whose morphism sets are given by the abelian groups 𝔊𝔓-Hom_{R}(M, N) (with composition of morphisms induced by the composition of R-linear maps). Then, Lemma 1.4 implies that the modules K and A, which are defined for any module M of finite Gorenstein projective dimension in Proposition 1.2(ii) and (iii), are uniquely determined up to isomorphism as objects of the category 𝔊𝔓-R-Mod.

Lemma 3.1. Let f : M ⟶ N be an R-linear map, where M, N are two modules of finite Gorenstein projective dimension. We also consider two exact sequences of R-modules

0 ⟶ M ⟶^{ι} A ⟶^{p} G′ ⟶ 0 and 0 ⟶ N ⟶^{ȷ} B ⟶^{q} H′ ⟶ 0,

where A, B have finite projective dimension and G′, H′ are Gorenstein projective. Then: (i) There exists an R-linear map g : A ⟶ B, such that gι = ȷf. (ii) If g, g′ : A ⟶ B are two R-linear maps with gι = ȷf and g′ι = ȷf, then [g] = [g′] ∈

𝔊𝔓-Hom_{R}(A, B).

(iii) If [f] = [0] ∈ 𝔊𝔓-Hom_{R}(M, N) and g : A ⟶ B is an R-linear map such that gι = ȷf, then [g] = [0] ∈ 𝔊𝔓-Hom_{R}(A, B).

Proof. (i) Since G′ is Gorenstein projective and pd_{R}B < ∞, the group Ext_{R}^{1}(G′, B) is trivial and hence the additive map ι^{∗} : Hom_{R}(A, B) ⟶ Hom_{R}(M, B) is surjective. Therefore, there exists an R-linear map g : A ⟶ B such that ȷf = ι^{∗}(g), as needed.

(ii) Let g, g′ : A ⟶ B be two R-linear maps with gι = ȷf and g′ι = ȷf.

0 ⟶ M ⟶^{ι} A ⟶^{p} G′ ⟶ 0
f ↓  g ↓↓ g′
0 ⟶ N ⟶^{ȷ} B ⟶^{q} H′ ⟶ 0

Then, (g′ − g)ι = g′ι − gι = ȷf − ȷf = 0 and hence there exists an R-linear map h : G′ ⟶ B, such that g′ − g = hp. Since G′ is Gorenstein projective, we conclude that [g] = [g′] ∈ 𝔊𝔓-Hom_{R}(A, B).

(iii) Assume that f factors as the composition of two R-linear maps M ⟶^{a} Γ ⟶^{b} N, where the module Γ is Gorenstein projective. Let

0 ⟶ Γ ⟶^{k} P ⟶ Γ′ ⟶ 0

be a short exact sequence of R-modules with P projective and Γ′ Gorenstein projective. We may construct such an exact sequence by considering a totally acyclic complex of projective modules with Γ as a syzygy (cf. Lemma 1.1). Then, invoking (i) above, we may find R-linear maps α : A ⟶ P and β : P ⟶ B, such that αι = ka and βk = ȷb.

0 ⟶ M ⟶^{ι} A ⟶^{p} G′ ⟶ 0
a ↓  α ↓
0 ⟶ Γ ⟶^{k} P ⟶ Γ′ ⟶ 0
b ↓  β ↓
0 ⟶ N ⟶^{ȷ} B ⟶^{q} H′ ⟶ 0

We conclude that (βα)ι = ȷ(ba) = ȷf and hence for any R-linear map g : A ⟶ B with gι = ȷf we have [g] = [βα] ∈ 𝔊𝔓-Hom_{R}(A, B) (cf. (ii) above). This finishes the proof, since we obviously have [βα] = [0] ∈ 𝔊𝔓-Hom_{R}(A, B). □

Let 𝔊𝔓-FP(R) and 𝔊𝔓-FGP(R) be the full subcategories of 𝔊𝔓-R-Mod, whose objects are the modules of finite projective and finite Gorenstein projective dimension respectively; then, 𝔊𝔓-FP(R) is a full subcategory of 𝔊𝔓-FGP(R). As we have noted in the discussion following Lemma 2.1, any R-linear map f : G ⟶ A, where G is Gorenstein projective and A has finite projective dimension, factors through a projective module. It follows that an R-linear map between modules of finite projective dimension factors through a Gorenstein projective module if and only if it factors through a projective module. Hence, the category 𝔊𝔓-FP(R) defined above is precisely the stable category of modules of finite projective dimension.

We observe that Lemmas 1.4(ii) and 3.1 imply that there is a well-defined additive functor

ν : 𝔊𝔓-FGP(R) ⟶ 𝔊𝔓-FP(R),

which maps a module M of finite Gorenstein projective dimension to the module A of finite projective dimension that appears in the short exact sequence of Proposition 1.2(iii) and the class [f] ∈ 𝔊𝔓-Hom_{R}(M, N) of an R-linear map f : M ⟶ N to the class [g] ∈ 𝔊𝔓-Hom_{R}(A, B) of any R-linear map g : A ⟶ B that satisfies the condition described in Lemma 3.1(i). We can now state the main result of this section.

Theorem 3.2. The additive functor ν : 𝔊𝔓-FGP(R) ⟶ 𝔊𝔓-FP(R) defined above is left adjoint to the inclusion functor 𝔊𝔓-FP(R) ↪ 𝔊𝔓-FGP(R).

Proof. We fix a module B of finite projective dimension and let M be a module of finite Gorenstein projective dimension. We also consider a short exact sequence of R-modules

0 ⟶ M ⟶^{ι} A ⟶^{p} G′ ⟶ 0,

where pd_{R}A < ∞ and G′ is Gorenstein projective (cf. Proposition 1.2(iii)). We note that the additive map

(4)  [ι]^{∗} : 𝔊𝔓-Hom_{R}(A, B) ⟶ 𝔊𝔓-Hom_{R}(M, B)

is natural in both B (this is straightforward) and M (this follows from Lemma 3.1). We shall establish the adjunction in the statement of the theorem, by proving that the additive map (4) above is bijective. Indeed, since the module G′ is Gorenstein projective and pd_{R}B < ∞, the group Ext_{R}^{1}(G′, B) is trivial. Hence, the additive map

ι^{∗} : Hom_{R}(A, B) ⟶ Hom_{R}(M, B)

is surjective, whence the surjectivity of (4). As far as the injectivity of (4) is concerned, assume that g : A ⟶ B is an R-linear map, such that [gι] = [g][ι] = [ι]^{∗}[g] = [0] ∈ 𝔊𝔓-Hom_{R}(M, B).

Then, we may consider the commutative diagram

0 ⟶ M ⟶^{ι} A ⟶^{p} G′ ⟶ 0
gι ↓  g ↓
0 ⟶ B = B ⟶ 0 ⟶ 0

and invoke Lemma 3.1(iii), in order to conclude that [g] = [0] ∈ 𝔊𝔓-Hom_{R}(A, B). □

We shall now prove that triviality of the unit of the adjunction described in Theorem 3.2 characterizes the Gorenstein projective modules among the modules of finite Gorenstein projective dimension.

Corollary 3.3. The following conditions are equivalent for a module M of finite Gorenstein projective dimension:

(i) M is Gorenstein projective,

(ii) 𝔊𝔓-Hom_{R}(M, B) = 0 for all modules B with pd_{R}B < ∞ and

(iii) there exists a short exact sequence of R-modules 0 ⟶ M ⟶^{ι} A ⟶ G′ ⟶ 0, where pd_{R}A < ∞ and G′ is Gorenstein projective, such that [ι] = [0] ∈ 𝔊𝔓-Hom_{R}(M, A).

Proof. The implications (i)→(ii) and (ii)→(iii) are obvious and hence it only remains to show that (iii)→(i). To that end, we note that the adjunction isomorphism (4) established in the proof of Theorem 3.2 implies, in particular, that the additive map

[ι]^{∗} : 𝔊𝔓-Hom_{R}(A, A) ⟶ 𝔊𝔓-Hom_{R}(M, A)

is bijective. Since [ι]^{∗}[1_{A}] = [1_{A}][ι] = [1_{A}ι] = [ι] = [0] ∈ 𝔊𝔓-Hom_{R}(M, A), we conclude that [1_{A}] = [0] ∈ 𝔊𝔓-Hom_{R}(A, A); hence, the identity map of A factors through a Gorenstein projective module G. It follows that A is a direct summand of G and hence A is Gorenstein projective (cf. [18, Theorem 2.5]). Being a Gorenstein projective module of finite projective dimension, the module A is necessarily projective (cf. [18, Proposition 2.27]). Then, Lemma 1.1 implies that M is Gorenstein projective, as needed. □

4. Finiteness of Gorenstein projective dimension and complete cohomology

Farrell has generalized in [15] the classical Tate cohomology theory of finite groups (cf. [9, Chapter XII] and [8, Chapter VI]) to the class of groups with finite virtual cohomological dimension. More generally, the existence of terminal completions of the ordinary Ext functors has been studied by Gedrich and Gruenberg in [16]. Mislin developed in [21] a generalization of the Tate-Farrell cohomology, that may be defined over any ring R. Using an approach that involves satellites, he defined for any module M complete cohomology functors Êxt_{R}^{∗}(M, ) and a natural transformation Ext_{R}^{∗}(M, ) ⟶ Êxt_{R}^{∗}(M, ) as the projective completion of the ordinary Ext functors Ext_{R}^{∗}(M, ); Mislin’s approach is heavily influenced by Gedrich and Gruenberg’s theory of terminal completions. As an immediate consequence of the definition, we note that the complete cohomology groups Êxt_{R}^{∗}(M, N) vanish if pd_{R}N < ∞. Equivalent definitions of the complete cohomology functors have been independently formulated by Vogel in [17] (using a hypercohomology approach) and by Benson and Carlson in [7] (using projective resolutions). Using the approach by Benson and Carlson, it follows that the elements in the kernel of the canonical map Hom_{R}(M, N) ⟶ Êxt_{R}^{0}(M, N) are those R-linear maps f : M ⟶ N, which are such that the R-linear map Ω^{n}f : Ω^{n}M ⟶ Ω^{n}N induced by f between the n-th syzygy modules of M and N factors through a projective module for n ≫ 0. For a thorough discussion of complete cohomology, the reader may also consult [5].

Assume that M is a module of finite Gorenstein projective dimension. Then, M admits a strong complete projective resolution, i.e. there exists a totally acyclic complex of projective modules P_{∗}, which coincides with an ordinary projective resolution Q_{∗} of M in sufficiently large degrees. If τ_{∗} : P_{∗} ⟶ Q_{∗} is a chain map which is the identity in sufficiently large degrees, then the complete cohomology functors Êxt_{R}^{∗}(M, ) may be computed as the cohomology groups of the complex Hom_{R}(P_{∗}, ), whereas the canonical map Ext_{R}^{∗}(M, ) ⟶ Êxt_{R}^{∗}(M, ) is that induced by the chain map τ_{∗}. This is an immediate consequence of Mislin’s satellite approach (cf. [21, Lemma 2.4]). In particular, since the strong complete projective resolution of M may be chosen to coincide with an ordinary projective resolution of M in degrees ≥ Gpd_{R}M, it follows that the canonical map Ext_{R}^{n}(M, N) ⟶ Êxt_{R}^{n}(M, N) is bijective if n > Gpd_{R}M and surjective if n = Gpd_{R}M.

Remark 4.1. Let M be a module of finite injective dimension. Then, as shown by Holm in [19, Theorem 2.2], we have an equality Gpd_{R}M = pd_{R}M. The key point in his proof consists in showing that the finiteness of Gpd_{R}M implies that pd_{R}M is finite as well. An alternative proof of the latter implication may be obtained by using complete cohomology, as follows: In view of the finiteness of Gpd_{R}M, the module M admits a strong complete projective resolution P_{∗} and the complete cohomology functors Êxt_{R}^{∗}(M, ) may be computed as the cohomology groups of the complex Hom_{R}(P_{∗}, ). Since P_{∗} is an acyclic complex of projective modules, the complex of abelian groups Hom_{R}(P_{∗}, L) is acyclic for any module L of finite injective dimension. In particular, the complex Hom_{R}(P_{∗}, M) is acyclic and hence Êxt_{R}^{∗}(M, M) = 0. As shown by Kropholler in [20, §4.2], one may use the Benson-Carlson approach to complete cohomology, in order to show that the vanishing of the group Êxt_{R}^{0}(M, M) implies that the projective dimension of M is finite, as needed.

The next result shows that the conditions on the finiteness of the projective dimension of the modules K and A that appear in the short exact sequences in Proposition 1.2(ii) and (iii) may be relaxed to the assertion that certain elements of complete cohomology groups vanish.

Theorem 4.2. The following conditions are equivalent for a module M:

(i) Gpd_{R}M < ∞,

(ii) there exists a short exact sequence of R-modules 0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0, where G is Gorenstein projective, such that the image of the classifying element ξ ∈ Ext_{R}^{1}(M, K) vanishes in the group Êxt_{R}^{1}(M, K) and

(iii) there exists a short exact sequence of R-modules 0 ⟶ M ⟶^{ȷ} A ⟶^{q} G′ ⟶ 0, where G′ is Gorenstein projective, such that the image of ȷ ∈ Hom_{R}(M, A) vanishes in the group Êxt_{R}^{0}(M, A).

Proof. (i)→(ii): Our hypothesis implies the existence of a short exact sequence of R-modules

0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0,

where G is Gorenstein projective and pd_{R}K < ∞ (cf. Proposition 1.2(ii)). Then, Êxt_{R}^{1}(M, K) is the trivial group and hence the image of the classifying element ξ vanishes therein.

(ii)→(i): We consider the commutative diagram with exact rows

Hom_{R}(G, K) ⟶^{ι^{∗}} Hom_{R}(K, K) ⟶^{∂} Ext_{R}^{1}(M, K)
↓ ↓ ↓
Êxt_{R}^{0}(G, K) ⟶^{ι^{∗}}  Êxt_{R}^{0}(K, K) ⟶^{∂} Êxt_{R}^{1}(M, K)

Since the module G is Gorenstein projective, the additive map Hom_{R}(G, K) ⟶ Êxt_{R}^{0}(G, K) is surjective. Then, our assumption about the vanishing of the image [ξ] of ξ = ∂(1_{K}) in the group Êxt_{R}^{1}(M, K) is easily seen to imply the existence of an R-linear map f ∈ Hom_{R}(G, K), such that [fι] = [1_{K}] ∈ Êxt_{R}^{0}(K, K). Therefore, if we denote by g the endomorphism 1_{K} − fι of K, then the induced endomorphism Ω^{n}g of Ω^{n}K factors through a projective module for n ≫ 0. On the other hand, it is clear that the endomorphism Ω^{n}(fι) = (Ω^{n}f)(Ω^{n}ι) of Ω^{n}K factors through Ω^{n}G for all n. Since G is Gorenstein projective, this is also the case for all of the Ω^{n}G’s (cf. Lemma 1.1). Any projective module being Gorenstein projective, it follows that Ω^{n}1_{K} = Ω^{n}(g + fι) = Ω^{n}g + Ω^{n}(fι) factors through a Gorenstein projective module for n ≫ 0. Since Ω^{n}1_{K} is the identity map of Ω^{n}K, we conclude that Ω^{n}K is a direct summand of a Gorenstein projective module and hence it is itself Gorenstein projective for n ≫ 0 (cf. [18, Theorem 2.5]). It follows that Gpd_{R}K < ∞ and the short exact sequence

0 ⟶ K ⟶^{ι} G ⟶^{p} M ⟶ 0

then implies that Gpd_{R}M ≤ Gpd_{R}K + 1 < ∞.

(i)→(iii): Our hypothesis implies the existence of a short exact sequence of R-modules

0 ⟶ M ⟶^{ȷ} A ⟶^{q} G′ ⟶ 0,

where G′ is Gorenstein projective and pd_{R}A < ∞ (cf. Proposition 1.2(iii)). Then, Êxt_{R}^{0}(M, A) is the trivial group and hence the image of ȷ vanishes therein.

(iii)→(i): We consider the commutative diagram with exact rows

Hom_{R}(G′, A) ⟶^{q^{∗}} Hom_{R}(A, A) ⟶^{ȷ^{∗}} Hom_{R}(M, A)
↓  ↓  ↓
Êxt_{R}^{0}(G′, A) ⟶^{q^{∗}}  Êxt_{R}^{0}(A, A) ⟶^{ȷ^{∗}}  Êxt_{R}^{0}(M, A)

Since the module G′ is Gorenstein projective, the additive map Hom_{R}(G′, A) ⟶ Êxt_{R}^{0}(G′, A) is surjective. Then, our assumption about the vanishing of the image [ȷ] of ȷ = ȷ^{∗}(1_{A}) in the group Êxt_{R}^{0}(M, A) is easily seen to imply the existence of an R-linear map f ∈ Hom_{R}(G′, A), such that [fq] = [1_{A}] ∈ Êxt_{R}^{0}(A, A). Therefore, if we denote by g the endomorphism 1_{A} − fq of A, then the induced endomorphism Ω^{n}g of Ω^{n}A factors through a projective module for n ≫ 0. On the other hand, it is clear that the endomorphism Ω^{n}(fq) = (Ω^{n}f)(Ω^{n}q) of Ω^{n}A factors through Ω^{n}G′ for all n. Since G′ is Gorenstein projective, this is also the case for all of the Ω^{n}G′’s (cf. Lemma 1.1). Any projective module being Gorenstein projective, it follows that Ω^{n}1_{A} = Ω^{n}(g + fq) = Ω^{n}g + Ω^{n}(fq) factors through a Gorenstein projective module for n ≫ 0. Since Ω^{n}1_{A} is the identity map of Ω^{n}A, we conclude that Ω^{n}A is a direct summand of a Gorenstein projective module and hence it is itself Gorenstein projective for n ≫ 0 (cf. [18, Theorem 2.5]). It follows that Gpd_{R}A < ∞ and the short exact sequence

0 ⟶ M ⟶^{ȷ} A ⟶^{q} G′ ⟶ 0

then implies that Gpd_{R}M < ∞ (cf. [18, Theorem 2.24]). □

Remark 4.3. A short exact sequence of R-modules 0 ⟶ K ⟶ G ⟶ M ⟶ 0 induces an R-linear map f : ΩM ⟶ K, whose image under the connecting homomorphism

∂ : Hom_{R}(ΩM, K) ⟶ Ext_{R}^{1}(M, K)

is the classifying element ξ ∈ Ext_{R}^{1}(M, K). The R-linear map f is unique up to linear maps that factor through projective modules and hence its image [f] in the complete cohomology group Êxt_{R}^{0}(ΩM, K) depends only upon the given short exact sequence. Moreover, the image of [f] under the connecting homomorphism

(5)  ∂ : Êxt_{R}^{0}(ΩM, K) ⟶ Êxt_{R}^{1}(M, K)

is the image [ξ] of the classifying element ξ in the group Êxt_{R}^{1}(M, K); this follows from the commutativity of the diagram

Hom_{R}(ΩM, K) ⟶^{∂} Ext_{R}^{1}(M, K)
↓  ↓
Êxt_{R}^{0}(ΩM, K) ⟶^{∂} Êxt_{R}^{1}(M, K)

The connecting homomorphism (5) is evidently bijective; this follows since the complete cohomology groups are trivial in the case where the first argument is a projective module. Hence, the assumption about the vanishing of the element [ξ] ∈ Êxt_{R}^{1}(M, K), which appears in Theorem 4.2(ii), is equivalent to the vanishing of the element [f] ∈ Êxt_{R}^{0}(ΩM, K).

5. Modules of finite Gorenstein injective dimension

The class of Gorenstein injective modules was defined by Enochs and Jenda in [14], by using an approach dual to that defining Gorenstein projective modules. Modules of finite Gorenstein injective dimension are then defined using resolutions by Gorenstein injective modules. The arguments that were employed in order to establish the properties of modules of finite Gorenstein projective dimension in the previous sections can be easily dualized and lead to analogous results, concerning modules of finite Gorenstein injective dimension. In fact, as shown in [18, Theorem 2.15] and [11, Lemma 2.18], the following conditions are equivalent for a module M and a non-negative integer n:

(i) Gid_{R}M ≤ n,

(ii) there exists a short exact sequence of R-modules 0 ⟶ M ⟶ Q ⟶ L ⟶ 0, where Q is Gorenstein injective and id_{R}L ≤ n − 1 (if n = 0, this is understood to mean L = 0) and

(iii) there exists a short exact sequence of R-modules 0 ⟶ Q′ ⟶ B ⟶ M ⟶ 0, where Q′ is Gorenstein injective and id_{R}B ≤ n. As with Gorenstein projective dimension, there is a weak form of uniqueness (à la Schanuel) for the short exact sequences in (ii) and (iii) above. This uniqueness may be used in order to turn the assignments M ↦ Q and M ↦ B into functors between subcategories of appropriate stabilizations of the category of R-modules. These functors are adjoints of suitable inclusion functors and hence the exact sequences above may be considered as approximations of M, in the sense of Auslander and Buchweitz. We leave it to the reader to complete the details, by providing the dual arguments to those used in sections 1, 2 and 3 above.

In analogy with Theorem 4.2, we may characterize the finiteness of the Gorenstein injective dimension of M, by relaxing the hypotheses on the finiteness of the injective dimension of the modules L and B which appear in the exact sequences of (ii) and (iii) above, in a way that involves Nucinkis’ injective complete cohomology Ẽxt (cf. [22]). This cohomology theory may be defined by the duals of the three approaches that define the (projective) complete cohomology Êxt. The complete cohomology groups Ẽxt_{R}^{∗}(M, N) vanish if M is a module of finite injective dimension. It is interesting to note that the complete cohomology functors Êxt and Ẽxt are naturally isomorphic if and only if the ring R has finite left Gorenstein global dimension (cf. [22, Theorem 5.2]). The proof of the following result proceeds along the same lines as that of Theorem 4.2, by using the dual arguments; the details are left to the reader.

Theorem 5.1. The following conditions are equivalent for a module M:

(i) Gid_{R}M < ∞,

(ii) there exists a short exact sequence of R-modules 0 ⟶ M ⟶ Q ⟶ L ⟶ 0, where Q is Gorenstein injective, such that the image of the classifying element ξ ∈ Ext_{R}^{1}(L, M) vanishes in the group Ẽxt_{R}^{1}(L, M) and

(iii) there exists a short exact sequence of R-modules 0 ⟶ Q′ ⟶ B ⟶^{q} M ⟶ 0, where Q′ is Gorenstein injective, such that the image of q ∈ Hom_{R}(B, M) vanishes in the group Ẽxt_{R}^{0}(B, M). □

6. The special case of group rings

In this final section, we consider a commutative ring k, a group G and let R = kG be the associated group algebra. Then, using the diagonal action of the group G, the tensor product^{2} M ⊗N of two kG-modules is also a kG-module. Moreover, if M is a projective kG-module and N is a kG-module which is k-projective, then the kG-module M ⊗ N is projective as well (cf. [8, Corollary 5.7]). Therefore, if P_{∗} ⟶ M ⟶ 0 is a projective resolution of a kG-module M and N is a k-projective kG-module, then P_{∗} ⊗ N ⟶ M ⊗ N ⟶ 0 is a projective resolution of the kG-module M ⊗ N; it follows that we then have an equality Ω^{n}(M ⊗ N) = Ω^{n}M ⊗ N for all n ≥ 0, whereas pd_{kG}(M ⊗ N) ≤ pd_{kG}M.

Lemma 6.1. Let k be a commutative ring and consider a direct system of k-modules and linear maps

P_{0} ⟶^{f_{0}} P_{1} ⟶^{f_{1}} ⋯ ⟶^{f_{n−1}} P_{n} ⟶^{f_{n}} ⋯

We assume that f_{n} is a split monomorphism for all n ≥ 0 and define the k-module P to be the direct limit of the system. Then:

(i) the canonical map P_{0} ⟶ P is a split monomorphism and

(ii) if P_{n} is projective for all n, then P is projective as well.

Proof. In view of our hypothesis about f_{n}, we can assume that there exists a k-module Q_{n+1}, such that P_{n+1} = P_{n} ⊕ Q_{n+1} and the structural map f_{n} : P_{n} ⟶ P_{n+1} is the map x ↦ (x, 0), x ∈ P_{n}, for all n ≥ 0. Then, we have P_{n} = P_{0} ⊕(⨁_{i=1}^{n} Q_{i}) for all n ≥ 0 and hence we conclude that P ≃ P_{0} ⊕(⨁_{i=1}^{∞} Q_{i}). The validity of (i) is then clear. As far as (ii) is concerned, we note that the projectivity of the P_{n}’s implies that all of the Q_{n}’s are projective modules; hence, P is projective as well. □

Proposition 6.2. Let k be a commutative ring and consider a k-module A and a split monomorphism of k-modules ι : k ⟶ A. We also consider the tensor powers A^{⊗n}, n ≥ 1, and define the direct system of k-modules and homomorphisms

k ⟶ A ⟶ A^{⊗2} ⟶ ⋯ ⟶ A^{⊗n} ⟶ ⋯,

whose structural maps A^{⊗n} = k ⊗ A^{⊗n} ⟶ A ⊗ A^{⊗n} = A^{⊗n+1} are the linear maps ι ⊗ 1_{A^{⊗n}} for all n ≥ 0. Let A′ be the direct limit of this direct system. Then:

(i) the canonical map ι′ : k ⟶ A′ is a split monomorphism and

(ii) if the module A is projective, then A′ is projective as well.

2All tensor products in this section will be understood to be over k.

Proof. This is an immediate consequence of Lemma 6.1, since our assumption that ι is a split monomorphism implies that the structural maps A^{⊗n} ⟶ A^{⊗n+1} are split monomorphisms, whereas the projectivity of A implies that the tensor powers A^{⊗n} are projective modules for all n ≥ 0. □

Gedrich and Gruenberg introduced in [16] the invariants spli kG and silp kG, as the supremum of the projective lengths of injective kG-modules and the supremum of the injective lengths of projective kG-modules respectively. The finiteness of these invariants is closely related to the existence of strong complete (projective or injective) resolutions of kG-modules, as shown in [16, §4]. If k is a Noetherian ring of finite self-injective dimension, then we always have an equality spli kG = silp kG (cf. [13, Theorem 4.4]). In the proof of the next result, we shall use the following simple observation.

Observation 6.3. Let k be a commutative ring of finite global dimension and consider a group G. Then, all Gorenstein projective kG-modules are k-projective. Indeed, let M be a Gorenstein projective kG-module. Then, considering a totally acyclic complex of projective kG-modules that has M as a syzygy, we conclude that for any non-negative integer n there exists an exact sequence of kG-modules

0 ⟶ M ⟶ P_{1} ⟶ ⋯ ⟶ P_{n} ⟶ N ⟶ 0,

where P_{i} is projective for all i = 1, …, n (and N is Gorenstein projective). Since any projective kG-module is a projective k-module, we may let n be the global dimension of the ring k and conclude that M = Ω^{n}N is k-projective, as needed.

Theorem 6.4. Let k be a commutative Noetherian ring of finite global dimension. Then, the following conditions are equivalent for a group G:

(i) Gpd_{kG}M < ∞ for any kG-module M,

(ii) Gcd_{k}G < ∞,

(iii) there exists a short exact sequence of kG-modules 0 ⟶ k ⟶ A ⟶ \overline{A} ⟶ 0, where pd_{kG}A < ∞ and \overline{A} is Gorenstein projective,

(iv) there exists a k-split monomorphism of kG-modules k ⟶ A, where A is k-projective and pd_{kG}A < ∞,

(v) there exists a k-split monomorphism of kG-modules ι : k ⟶ A, where A is k-projective, such that the image of ι ∈ Hom_{kG}(k, A) = H^{0}(G, A) vanishes in the group Êxt_{kG}^{0}(k, A) = Ĥ^{0}(G, A) and

(vi) spli kG = silp kG < ∞.

Proof. The implication (i)→(ii) is obvious, since Gcd_{k}G = Gpd_{kG}k, whereas the implication (ii)→(iii) follows from Proposition 1.2.

(iii)→(iv): As we noted in Observation 6.3 above, any Gorenstein projective kG-module is k-projective. In particular, a short exact sequence of kG-modules as in (iii) is necessarily k-split and the kG-module A therein is k-projective.

(iv)→(v): This is clear since the assumption that the kG-module A has finite projective dimension implies that the complete cohomology group Êxt_{kG}^{0}(k, A) = Ĥ^{0}(G, A) is trivial.

(v)→(iv): Let P_{∗} ⟶ k ⟶ 0 be a projective resolution of the trivial kG-module k. For notational simplicity, we shall denote by K_{i}, i ≥ 0, the corresponding syzygy modules of k (i.e. we let K_{i} = Ω^{i}k, i ≥ 0). Then, P_{∗} ⊗ A ⟶ k ⊗ A ⟶ 0 is a projective resolution of k ⊗ A = A and K_{i} ⊗ A, i ≥ 0, are the corresponding syzygy modules of A. A lifting of the kG-linear map ι is provided by the chain map 1 ⊗ ι : P_{∗} ⟶ P_{∗} ⊗ A. Using the Benson-Carlson approach to complete cohomology, we conclude that the vanishing of the image of ι under the canonical map H^{0}(G, A) ⟶ Ĥ^{0}(G, A) implies the existence of a non-negative integer s, which is such that the map 1 ⊗ ι : K_{s} ⟶ K_{s} ⊗ A factors through a projective kG-module P. We note that for any kG-module B the map 1 ⊗ ι ⊗ 1 : K_{s} ⊗ B ⟶ K_{s} ⊗ A ⊗ B factors through the kG-module P ⊗ B.

We now consider the direct system of kG-modules (A^{⊗n})_{n}, with structural maps A^{⊗n} ⟶ A^{⊗n+1} given by ι ⊗ 1_{A^{⊗n}} for all n ≥ 0, and let A′ be its direct limit. Then, Proposition 6.2 implies that the kG-module A′ is k-projective and the natural kG-linear map ι′ : k ⟶ A′ is a k-split monomorphism. In view of the very definition of A′, it follows that there exists an isomorphism of kG-modules σ : A ⊗ A′ ⟶ A′, which is such that the composition A′ ⟶^{ι⊗1} A ⊗ A′ ⟶^{σ} A′ is the identity map of A′. Since the map 1 ⊗ ι ⊗ 1 : K_{s} ⊗ A′ ⟶ K_{s} ⊗ A ⊗ A′ factors through the kG-module P ⊗ A′, we conclude that the identity map of K_{s} ⊗ A′ factors through P ⊗ A′ as well. Since A′ is k-projective, the kG-module P ⊗ A′ is projective; hence, being a direct summand of it, the kG-module K_{s} ⊗ A′ is projective as well. Then, the exact sequence

0 ⟶ K_{s} ⊗ A′ ⟶ P_{s−1} ⊗ A′ ⟶ ⋯ ⟶ P_{0} ⊗ A′ ⟶ k ⊗ A′ ⟶ 0

provides us with a projective resolution of the kG-module k ⊗ A′ = A′ of length s and hence pd_{kG}A′ ≤ s < ∞.

(iv)→(vi): The assumptions made on k imply that we have an equality spli kG = silp kG (cf. [13, Theorem 4.4]); therefore, the result will follow if we show that any injective kG-module has finite projective dimension. To that end, let I be an injective kG-module. Then, the k-split monomorphism of kG-modules ι : k ⟶ A induces a monomorphism of kG-modules 1 ⊗ ι : I ⟶ I ⊗ A, which is split in view of the injectivity of I. Hence, we are reduced to showing that the kG-module I ⊗ A has finite projective dimension. Let P_{∗} ⟶ I ⟶ 0 be a projective resolution of I and assume that gl.dim k = n. Then, in the exact sequence

0 ⟶ Ω^{n}I ⟶ P_{n−1} ⟶ ⋯ ⟶ P_{0} ⟶ I ⟶ 0

the kG-modules Ω^{n}I, P_{n−1}, …, P_{0} are k-projective. Since A is k-projective (and hence k-flat), there is an induced exact sequence of kG-modules

0 ⟶ Ω^{n}I ⊗ A ⟶ P_{n−1} ⊗ A ⟶ ⋯ ⟶ P_{0} ⊗ A ⟶ I ⊗ A ⟶ 0.

Hence, the finiteness of pd_{kG}(I ⊗ A) will follow if we show that pd_{kG}(L ⊗ A) < ∞ for any kprojective kG-module L. But this is clear since for such a kG-module L we have an inequality pd_{kG}(L ⊗ A) ≤ pd_{kG}A < ∞.^{3}

(vi)→(i): Let M be an arbitrary kG-module. Then, as shown by Gedrich and Gruenberg in [16, §4], the finiteness of spli kG implies the existence of an acyclic complex (P_{∗}, ∂_{∗}) of projective kG-modules, which coincides with a projective resolution of M in sufficiently large degrees. In view of the finiteness of silp kG, any projective kG-module has finite injective dimension; hence, the complex (P_{∗}, ∂_{∗}) is totally acyclic, i.e. it is a strong complete projective resolution of M. The existence of such a resolution shows that the kG-module M has finite Gorenstein projective dimension, as needed. □

Remarks 6.5. (i) Let k be a commutative ring, G a group and M a k-projective kG-module. We assume that there exists a k-split monomorphism of kG-modules ι : k ⟶ A, where A is k-projective, such that the image of 1⊗ι ∈ Hom_{kG}(M, M ⊗A) vanishes in Êxt_{kG}^{0}(M, M ⊗A). Then, the same arguments that were used in the proof of the implication (v)→(iv) in Theorem 6.4 may be employed in order to show that there exists a k-split monomorphism of kG-modules k ⟶ A′, where A′ is k-projective, such that pd_{kG}(M ⊗A′) < ∞. This result is closely related to the finiteness criterion proved by Cornick and Kropholler in [12, Lemma 2.2 and Remark 2.4]; we avoid any assumption about the existence of a multiplicative structure on A, at the expense of replacing A with A′ (which is still a k-projective kG-module that contains k as a k-split kG-submodule).

3The above argument shows that spli kG ≤ gl.dim k + pd_{kG}A.

(ii) In the case where k is the ring of integers, the search for a module A with the properties stated in Theorem 6.4(iv) was motivated by certain considerations in the study of free group actions on finite dimensional homotopy spheres (cf. [23], [24]).

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Department of Mathematics, University of Athens, Athens 15784, Greece

  E-mail address: emmanoui@math.uoa.gr

Department of Mathematics, University of Athens, Athens 15784, Greece

  E-mail address: otalelli@math.uoa.gr