A NOTE ON COMPLETE RESOLUTIONS

FOTINI DEMBEGIOTI AND OLYMPIA TALELLI

Abstract. It is shown that the Eckmann-Shapiro lemma holds for complete cohomology if and only if complete cohomology can be calculated using complete resolutions. It is also shown that for G an LH𝔉-group the kernels in a complete resolution of a ℤG-module coincide with Benson’s class of cofibrant modules.

1. Introduction

Let G be a group and ℤG its integral group ring. A ℤG-module M is said to admit a complete resolution (F, P, n) of coincidence index n, if there is an acyclic complex F = {(F_{i}, ϑ_{i})| i ∈ ℤ} of projective modules, and a projective resolution P = {(P_{i}, d_{i})| i ∈ ℤ, i ≥ 0} of M such that F and P coincide in dimensions greater than n

F :  ⋯ →F_{n+1}→ F_{n} → F_{n−1}→ ⋯ → F_{0} → F_{−1}→ F_{−2}→ ⋯ P :  ⋯ →P_{n+1}→ P_{n} → P_{n−1}→ ⋯ → P_{0} → M→ 0 ∥  ∥ ^{d_{n}}^{ϑ_{n}}

A ℤG-module module M is said to admit a complete resolution in the strong sense if there is a complete resolution (F, P, n) with Hom_{ℤG}(F, Q) acyclic for every ℤG-projective module Q.

It was shown by Cornick and Kropholler in [7] that if M admits a complete resolution (F, P, n) in the strong sense, then

Êxt_{ℤG}^{∗}(M, B) ≃ H^{∗}(Hom_{ℤG}(F, B))

where Êxt_{ℤG}^{∗}(M, ) is the P-completion of Ext_{ℤG}^{∗}(M, ) defined by Mislin for any group G [13], as

Êxt_{ℤG}^{k}(M, B) = lim_{r>k} S^{k−r}Ext_{ℤG}^{r}(M, B)

where S^{−m}T is the m-th left satellite of a functor T.

Alternative but equivalent definitions of the complete Êxt-groups were given by Benson and Carlson [5] and Vogel [9].

Complete cohomology Ĥ^{∗}(G, ) = Êxt_{ℤG}^{∗}(ℤ, ) generalizes the Tate cohomology defined for finite groups, and the Farrell-Tate cohomology defined for groups that have a finite-index subgroup of finite cohomological dimension.

It turns out that G admits a complete resolution in the strong sense if and only if the generalized cohomological dimension \underline{cd}G is finite [1], where

\underline{cd}G = sup\{n ∈ ℕ| ∃M ℤ-free, ∃F ℤG-free : Ext_{ℤG}^{n}(M, F) ≠ 0\}

was defined by Ikenaga in his study of generalized Farrell-Tate cohomology in [10].

Note that \underline{cd}G = Gcd_{ℤ}G [1], the Gorenstein projective dimension of G, which is defined via resolutions of the trivial ℤG-module ℤ by Gorenstein projective modules and is related to the G-dimension defined by Auslander in [2] (see also [3] and [6]).

A ℤG-module M is said to be Gorenstein projective if it admits a complete resolution in the strong sense of coincidence index 0, i.e. M is a kernel in a complete resolution in the strong sense.

Complete resolutions do not always exist, for example if G has a free abelian subgroup of infinite rank, then G does not admit a complete resolution [14].

If the complete cohomology can be calculated using complete resolutions, then one has certain advantages such as the Eckmann-Shapiro lemma and certain spectral sequences.

Here we show that the validity of the Eckmann-Shapiro lemma for complete cohomology actually implies that complete resolutions exist and that complete cohomology can be calculated using complete resolutions:

Theorem A. The following are equivalent for a group G.

(1) The Eckmann-Shapiro lemma holds for complete cohomology.

(2) G has a complete resolution and every complete resolution of G is a complete resolution in the strong sense.

(3) Complete cohomology can be calculated using complete resolu-

tions. We also show:

Theorem B. If a ℤG-module M is a kernel in a complete resolution and A is a ℤG-module that is ℤF -projective for every finite subgroup F of G, then M ⊗ A (with diagonal G-action) is projective as a ℤHmodule for every LH𝔉-subgroup H of G.

Theorem B in particular implies:

Corollary C. If G is an LH𝔉-group then the kernels of a complete resolution of a ℤG-module coincide with the cofibrant modules.

Corollary D. If G is an LH𝔉-group then every complete resolution of a ZG-module M is a complete resolution in the strong sense.

Cofibrant modules were introduced by Benson in [4] for his study of ℤG-modules that admit a projective resolution by finitely generated projective modules, when G is an LH𝔉-group. They are defined as follows: for any group G, a ℤG-module M is said to be cofibrant if M ⊗ B(G, ℤ) is a projective ℤG-module, where B(G, ℤ) is the set of bounded functions from G to ℤ.

The class LH𝔉 was defined by Kropholler in [11] as the smallest class of groups which contains the class of finite groups and whenever a group G admits a finite dimensional contractible G-CW-complex with stabilizers in H𝔉, then G is in H𝔉. LH𝔉 is the class of groups G such that every finitely generated subgroup of G is in H𝔉. The class LH𝔉 includes, for example, all soluble by finite groups, all groups with a faithful representation as endomorphisms of a Noetherian module over a commutative ring, and is extension closed, closed under ascending unions and closed under amalgamated free products and HNN extensions.

In [1] it was proved that if G is an LH𝔉-group and M a ZG-module that admits a projective resolution by finitely generated projective modules, then M is a Gorenstein projective ZG-module if and only if it is a cofibrant module.

Corollary C shows that for LH𝔉-groups the Gorenstein projective modules coincide with the cofibrant modules. We believe that

Conjecture A. For any group G the Gorenstein projective modules coincide with the cofibrant modules.

Conjecture B. (see also Conj. B in [15]) A ZG-module M admits a complete resolution if and only if it admits a complete resolution in the strong sense.

Conjecture B in particular implies that Gorenstein projectivity is a subgroup closed property. Here we show that this is so if the subgroup is in LH𝔉 (Corollary 2.2).

2. Proof of the results

Proof of Theorem A. (1) ⇒ (2) Assume that the Eckmann-Shapiro lemma holds for the complete cohomology of G, i.e.

Êxt_{ℤG}^{∗}(A, Hom_{ℤH}(ℤG, B)) ≃ Êxt_{ℤH}^{∗}(A|_{ℤH}, B)

for every ZG-module A, every ZH-module B and every subgroup H ≤ G . Applying the Eckmann-Shapiro lemma for the trivial subgroup H = 1, we have that Êxt_{ℤG}^{0}(A, Hom_{ℤ}(ℤG, B)) ≃ Êxt_{ℤ}^{0}(A|_{ℤ}, B) for every ZG-module A and every Z-module B. We have that Êxt_{ℤ}^{0}(A|_{ℤ}, B) = 0, since A, B have finite projective dimension over ℤ [11, 4.2].

It follows that Êxt_{ℤG}^{0}(A, Hom_{ℤ}(ℤG, A)) = 0 for any ZG-module A. For any ZG-module A there is a canonical injection A ↪ Hom_{ℤ}(ℤG, A). So, if I is injective then it is a direct summand of Hom_{ℤ}(ℤG, I). It follows that Êxt_{ℤG}^{0}(I, I) = 0, thus I has finite projective dimension [11, 4.2]. It is easy to see that the supremum of the projective dimensions of the injective ℤG-modules, spli ℤG, is finite hence (2) follows from [8, §4].

(2) ⇒ (3) by Theorem 1.2 of [7].

(3) ⇒ (1) It follows from (4) ⇒ (2) of Theorem 2.2 in [15] and [7] that every ℤG-module M admits a complete resolution in the strong sense (F, P, n) and Êxt_{ℤG}^{∗}(M, B) ≃ H^{n}(Hom_{ℤG}(F, B)). Clearly the computation of the complete cohomology via complete resolutions implies the validity of the Eckmann-Shapiro lemma.□

Lemma 2.1. Let G be a group.

(a) If M_{i}, i ∈ I, is a family of ℤG-modules that admit complete resolutions (in the strong sense) of coincidence index 0, then the direct sum ⊕_{i∈I}M_{i} admits a complete resolution (in the strong sense) of coincidence index 0.

(b) If M admits a complete resolution of coincidence index n and A is a ℤ-free ℤG-module, then M ⊗ A (with diagonal G-action) admits a complete resolution of coincidence index n.

(c) Let {(F_{i}, ϑ_{i}), i ∈ ℤ} be an acyclic complex of projective ℤGmodules with kernels M_{i}, i ∈ ℤ. If there is a bound on the projective dimensions of the M′_{i}s then M_{i} is projective for every i ∈ ℤ.

Proof. The proofs of (a) and (b) are straightforward.

(c) Let pd_{ℤG}M_{r} = k > 0 for some r ∈ ℤ. It follows from the short exact sequence 0 → M_{r} → F_{r} → M_{r−1} → 0 that pd_{ℤG}M_{r−1} = k + 1 and inductively pd_{ℤG}M_{r−s} = k + s for all s ∈ ℕ, a contradiction since there is a bound on the projective dimensions of the M′_{i}s. □

Proof of Theorem B. We first prove it for H𝔉-subgroups, using transfinite induction on the ordinal number α such that the subgroup H belongs to H_{α}𝔉. For α = 0, M ⊗ A is projective over any finite subgroup of G, because M is ℤ-free and A is projective over the finite subgroups of G. Assume the result is true for all H_{β}𝔉-subgroups of G for all β < α, and let H be an H_{α}𝔉-subgroup of G. There is an exact sequence of ℤH-modules

0→C_{r}→ ⋯ →C_{0}→ℤ→0

where each C_{i} is a direct sum of modules of the form ℤ[H/F], F ≤ H, F ∈ H_{β}𝔉, β < α. By the inductive hypothesis, M ⊗ A is projective over each ℤF, so each ℤ[H/F] ⊗ M ⊗ A is projective over ℤH, thus each C_{i} ⊗ M ⊗ A is projective over ℤH. Tensoring the above sequence with M ⊗ A gives us an exact sequence

0→C_{r} ⊗ M ⊗ A→ ⋯ →C_{0} ⊗ M ⊗ A→M ⊗ A→0.

It follows that pd_{ℤH}M ⊗ A < ∞ for any ℤG-module M that has a complete resolution of coincidence index 0. If there is not a bound on these projective dimensions, there is a family of ℤG-modules M_{n}, n ∈ ℕ that have complete resolutions of coincidence index 0, such that pd_{ℤH}M_{n} ⊗ A ≥ n for all n ∈ ℕ. Then the direct sum M = ⊕_{n∈ℕ}M_{n} has a complete resolution of coincidence index 0 by Lemma 2.1 (a), and pd_{ℤH}M ⊗ A ≥ pd_{ℤH}M_{n} ⊗ A ≥ n for all n ∈ ℕ, so pd_{ℤH}M ⊗ A is infinite, which is a contradiction. It follows that there is a bound on the projective dimensions pd_{ℤH}M ⊗ A where M is a ℤG-module that has a complete resolution of coincidence index 0. From Lemma 2.1 (b) and (c), we obtain that M ⊗ A is projective as a ℤH-module.

Next we prove the result for LH𝔉-subgroups, using induction on the cardinality of the subgroup. If H is a countable subgroup then it belongs to H𝔉, so the result has been proved. Assume that H is uncountable and the result is true for any subgroup with cardinality strictly smaller that |H|, and let M be a ℤG-module that has a complete resolution of coincidence index 0. Since H is uncountable it can be expressed as the union of an ascending chain of subgroups H_{α}, α < γ for some ordinal number γ such that each H_{α} has cardinality strictly smaller that |H|. By the inductive hypothesis M ⊗ A is projective over each ℤH_{α}, so by Lemma 5.6 in [4] pd_{ℤH}M ⊗ A ≤ 1. Thus we have proved that M ⊗ A has finite projective dimension over ℤH, for any ℤG-module M that has a complete resolution of coincidence index 0. It follows from the above argument that M ⊗ A is projective over H, for any ℤG-module M that has a complete resolution of coincidence index 0. □

Proof of Corollary C. Let P_{∗} be a complete resolution with kernels M_{i}, i ∈ ℤ. It is known [12] that B(G, ℤ) is a projective ℤF -module for any finite subgroup F of G. Hence it follows from Theorem B that each M_{i} is cofibrant. The converse was shown in [7].

Proof of Corollary D. Let P_{∗} be a complete resolution with kernels M_{i}, i ∈ ℤ. It follows from Corollary C that each M_{i} is a cofibrant module. There is a ℤ-split ℤG-monomorphism ℤ ↪ B(G, ℤ), hence if Q is a projective ℤG-module, then Q is a ℤG-direct summand of Hom_{ℤ}(B(G, ℤ), Q) (diagonal action), so the complex Hom_{ℤG}(P_{∗}, Q) is a direct summand of the complex Hom_{ℤG}(P_{∗}, Hom_{ℤ}(B(G, ℤ), Q)) ≃ Hom_{ℤG}(P_{∗} ⊗ B(G, ℤ), Q). This complex is exact because M_{i} ⊗ B(G, ℤ) are projective ℤG-modules, so P_{∗} ⊗ B(G, ℤ) splits.

It is not known whether Gorenstein projectivity is a subgroup closed property. Here we show that this is so if the subgroup is an LH𝔉-group.

Corollary 2.2. If G is a group and M is a Gorenstein projective module then M|_{H} is Gorenstein projective for any LH𝔉-subgroup H of G.

Proof. Since M is Gorenstein projective it admits a complete resolution in the strong sense of coincidence index 0 over ℤG, which is a complete resolution of coincidence index 0 over ℤH for any subgroup H of G. If the subgroup is in LH𝔉 then the resolution is a complete resolution in the strong sense by Corollary D. Hence M|_{H} is a Gorenstein projective module. □

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