A HOMOLOGICAL CHARACTERIZATION OF LOCALLY FINITE GROUPS

IOANNIS EMMANOUIL

Abstract. We show that a group G is locally finite if and only if Tor_{1}^{ℤG}(M, I) = 0 for all ℤG-modules M and I, with M ℤ-free and I injective. To that end, we relate the finiteness of a finitely generated group G to the finiteness of the injective dimension of induced modules. We also examine the relation between the finiteness of a group G and the finiteness of the projective dimension of coinduced modules.

0. Introduction

If G is a finite group, then the class of induced ℤG-modules coincides with the class of coinduced modules (cf. [3, Chapter III, Proposition 5.9]). Conversely, if there exist two nonzero abelian groups A and B, such that the induced module ℤG ⊗ A is isomorphic with the coinduced module Hom(ℤG, B), then we may apply the invariance functor H^{0}(G, ) to both sides and conclude that the group G is finite. In this note, we shall examine certain criteria for a group G to be finite (or locally finite), by requiring that induced modules share certain homological properties with coinduced modules and vice versa.

Farrell has generalized in [8] the classical Tate cohomology theory of finite groups to the class of those groups that have finite virtual cohomological dimension. In his attempt to extend Farrell’s definition to an even bigger class of groups, Ikenaga introduced in [10, §II] the generalized cohomological dimension \underline{cd} G of a group G, by defining \underline{cd} G to be the supremum of those integers n, for which there exist a ℤ-free ℤG-module M and a projective ℤG-module P, such that Ext_{ℤG}^{n}(M, P) ≠ 0. It is clear that \underline{cd} G ≤ n if and only if Ext_{ℤG}^{n+1}(M, P) = 0 for any two ℤG-modules M and P, with M ℤ-free and P projective. If the group G has finite virtual cohomological dimension, then \underline{cd} G = vcd G. In particular, if G is a finite group, then \underline{cd} G = 0. The converse of the latter assertion, which was conjectured by Ikenaga in [11], was established in [6, Theorem 4.6]. Thus, a group G is finite if and only if \underline{cd} G = 0, i.e. if and only if Ext_{ℤG}^{1}(M, P) = 0 for all ℤG-modules M and P, with M ℤ-free and P projective. We may express this latter condition by saying that a projective ℤG-module P behaves like coinduced modules, as far as the functor Ext_{ℤG}^{1} is concerned, in the sense that P annihilates the functor Ext_{ℤG}^{1}(M, ), whenever M is a ℤG-module, which annihilates the functors Ext_{ℤG}^{1}( , C) for all coinduced modules C. This relation between projective and coinduced ℤG-modules is obvious if G is a finite group; in that case, any projective ℤG-module is a direct summand of a coinduced module.

Ikenaga has also defined in [10, §III] the generalized homological dimension \underline{hd} G of a group G as the supremum of those integers n, for which there exist a ℤ-torsion-free ℤG-module M and an injective ℤG-module I, such that Tor_{n}^{ℤG}(M, I) ≠ 0. This definition was slightly modified in [7, §4], where the generalized homological dimension \underline{hd}′G of G was defined as the supremum of those integers n, for which there exist a ℤ-free ℤG-module M and an injective ℤG-module I, such that Tor_{n}^{ℤG}(M, I) ≠ 0. It is clear that \underline{hd}′G ≤ \underline{hd} G. Moreover, we have \underline{hd}′G ≤ n if and only if Tor_{n+1}^{ℤG}(M, I) = 0 for any two ℤG-modules M and I, with M ℤ-free and I injective. Even though it is not a straightforward consequence of the definitions, we always have \underline{hd}′G ≤ \underline{cd} G (cf. [7, Theorem 4.11]). As shown by Ikenaga in [10, §III], if G is a locally finite group, then \underline{hd} G = 0. We shall prove that the vanishing of the generalized homological dimension is actually equivalent to the local finiteness of the group.

Theorem. The following conditions are equivalent for a group G:

(i) G is locally finite,

(ii) \underline{hd} G = 0,

(iii) \underline{hd}′G = 0.

In other words, we shall prove that a group G is locally finite if and only if Tor_{1}^{ℤG}(M, I) = 0 for all ℤG-modules M and I, with M ℤ-torsion-free (or ℤ-free) and I injective. We may express this latter condition by saying that an injective ℤG-module I behaves like induced modules, as far as the functor Tor_{1}^{ℤG} is concerned, in the sense that I annihilates the functor Tor_{1}^{ℤG}(M, ), whenever M is a ℤG-module, which annihilates the functors Tor_{1}^{ℤG}( , C) for all induced modules C. This relation between injective and induced ℤG-modules is obvious if G is a locally finite group; in that case, any injective ℤG-module is a direct summand of a filtered colimit of induced modules.

We note that the above characterization of local finiteness of G, in terms of the vanishing of \underline{hd} G, is proved for solvable groups in [10, §III] and for groups that satisfy one of a number of additional hypotheses in [12].

Our proof uses in a crucial way the invariants spli ℤG and sfli ℤG that are associated with the projective (resp. flat) length of injective ℤG-modules (cf. [1], [7] and [9]). In particular, we relate the finiteness of a finitely generated group G to the finiteness of the injective dimension of the ℤG-module ℤG (and more generally to the finiteness of the injective dimension of induced modules). Dually, we examine the relation between the finiteness of a group G and the finiteness of the projective dimension of the ℤG-module Hom(ℤG, ℚ/ℤ) (and more generally to the finiteness of the projective dimension of coinduced modules).

Notations and terminology. For any two abelian groups A, B we shall denote by Hom(A, B) the group Hom_{ℤ}(A, B) of all additive maps from A to B. In the same way, we shall denote the tensor product A ⊗_{ℤ} B simply by A ⊗ B. If G is a group, then an induced (resp. coinduced) module is a ℤG-module of the form ℤG⊗A (resp. Hom(ℤG, A)), where A is an abelian group.

1. Preliminaries

In this preliminary Section, we shall collect certain properties of the generalized homological dimension of groups and related invariants, that will be used in the sequel. The generalized homological dimensions \underline{hd} G and \underline{hd}′G of G are closely related to the invariant sfli ℤG, which is defined as the supremum of the flat lengths of injective ℤG-modules (cf. [1], [7]).

The following result is essentially due to Ikenaga (cf. [10, §III, Proposition 8] and [11, Lemma 1.8]). The properties that are described below for the invariant \underline{hd}′ are also satisfied by Ikenaga’s invariant \underline{hd} and his arguments work without any change in our setting (see also [1, Propositions 3.10 and 3.12]).

Proposition 1.1. Let G be a group.

(i) If H ⊆ G is a subgroup, then \underline{hd}′H ≤ \underline{hd}′G.

(ii) We have \underline{hd} G ≤ sfli ℤG ≤ \underline{hd}′G + 1. □

The invariant sfli ℤG is closely related to the invariant spli ℤG, which was defined by Gedrich and Gruenberg in [9], in connection with the existence of complete cohomological functors in the category of ℤG-modules. Here, spli ℤG is the supremum of the projective lengths of injective ℤG-modules. The following result is precisely [7, Theorem 4.8(i)].

Proposition 1.2. If G is a countable group, then sfli ℤG ≤ spli ℤG ≤ sfli ℤG + 1. □

We note that for any group G we have spli ℤG = silp ℤG, where silp ℤG is the supremum of the injective lengths of projective ℤG-modules [6, Corollary 4.5]. The inequality silp ℤG ≤ spli ℤG was proved by Gedrich and Gruenberg in [9, 1.6 and Corollary 5.9].

Corollary 1.3. Let G be a countable group, such that \underline{hd}′G = n < ∞. Then, the ℤG-module ℤG has injective dimension ≤ n + 2.

Proof. Since \underline{hd}′G = n, Proposition 1.1(ii) implies that sfli ℤG ≤ n + 1. Then, invoking Proposition 1.2, we conclude that spli ℤG ≤ n + 2. Therefore, we also have silp ℤG ≤ n + 2; in particular, the projective ℤG-module ℤG has injective dimension ≤ n + 2. □

2. The proof of the Theorem

As we have already mentioned, \underline{hd} G = 0 if the group G is locally finite. Conversely, we shall prove that the vanishing of \underline{hd}′G implies that G is locally finite. In view of Proposition 1.1(i), this will follow if we can show that a finitely generated group G with \underline{hd}′G = 0 is necessarily finite. Hence, it suffices to prove the next result, which relates the finiteness of a finitely generated group G to the finiteness of the injective dimension of induced modules.

Proposition 2.1. The following conditions are equivalent for a finitely generated group G: (i) G is finite, (ii) \underline{hd}′G = 0, (iii) the homology functors H_{i}(G, ) vanish on injective modules for all i ≥ 1 and the ℤG-

module ℤG has injective dimension ≤ 2,

(iv) the homology functors H_{i}(G, ) vanish on injective modules for all i ≥ 1 and the ℤGmodule ℤG has finite injective dimension,

(v) the homology functors H_{i}(G, ) vanish on injective modules for all i ≥ 1 and there is a non-zero induced ℤG-module of finite injective dimension.

Proof. The implication (i)→(ii) is clear, whereas the implication (ii)→(iii) follows from the equality H_{i}(G, ) = Tor_{i}^{ℤG}(ℤ, ) and Corollary 1.3. Since (iv) is a weak version of (iii) and the ℤG-module ℤG is obviously induced, it only remains to show that (v)→(i). To that end, we consider an induced module M = ℤG ⊗ A of finite injective dimension, where A is a non-zero abelian group, and fix an injective resolution of finite length

0 ⟶ M ⟶ I_{0} ⟶ I_{1} ⟶ ⋯ ⟶ I_{n} ⟶ 0.

Let K_{i} be the corresponding i-th cosyzygy module of M; in particular, K_{n} = I_{n}. Using our assumption that injective modules are acyclic for group homology, an inductive argument shows that H_{n−i}(G, K_{n−i}) = 0 for all i = 0, 1, …, n−1. In particular, we have H_{1}(G, K_{1}) = 0. Then, the short exact sequence of ℤG-modules

0 ⟶ M ⟶ I_{0} ⟶ K_{1} ⟶ 0

induces the exact sequence

0 = H_{1}(G, K_{1}) ⟶ H_{0}(G, M) ⟶ H_{0}(G, I_{0}).

Since H_{0}(G, M) = H_{0}(G, ℤG ⊗ A) = A ≠ 0, it follows that the group H_{0}(G, I_{0}) is non-trivial. Then, one may use a result of Strebel (cf. [3, §III.5, Exercise 4(b)]) as in [11, Corollary 1.6], in order to conclude that the group G is finite, as needed. □

Remarks 2.2. (i) It is shown in [7, Theorem 4.11] that \underline{hd}′G ≤ \underline{cd} G for any group G. Hence, if G is a finitely generated group, then the implication

\underline{hd}′G = 0 ⟹ G is finite,

established in Proposition 2.1, is stronger than the implication

\underline{cd} G = 0 ⟹ G is finite,

which was proved in [6, Theorem 4.6].

(ii) Let R be a ring and consider a class 𝔛 of left R-modules. Then, the left Tor_{1}-orthogonal of 𝔛 is the class ^{⊤}𝔛, consisting of those right R-modules M for which Tor_{1}^{R}(M, N) = 0 for all N ∈ 𝔛. In the same way, if 𝔜 is a class of right R-modules, then we may define the right Tor_{1}-orthogonal of 𝔜, as the class 𝔜^{⊤}, consisting of those left R-modules N for which Tor_{1}^{R}(M, N) = 0 for all M ∈ 𝔜. It is clear that we always have 𝔛 ⊆ (^{⊤}𝔛)^{⊤}.

In particular, let G be a group and consider the class ℑ consisting of all induced ℤGmodules. Then, ^{⊤}ℑ is easily seen to be the class consisting of those ℤG-modules M which are ℤ-torsion-free. It follows that the vanishing of \underline{hd} G is equivalent to the assertion that the class of injective ℤG-modules is contained in (^{⊤}ℑ)^{⊤} and may be interpreted by saying that injective ℤG-modules have a similar homological behavior as induced modules.

On the other hand, if G is a locally finite group, then injective ℤG-modules are obviously contained in (^{⊤}ℑ)^{⊤}. Indeed, any injective ℤG-module is a direct summand of a coinduced module and coinduced modules are, in this case, filtered colimits of induced modules. (If A is any abelian group, then we may express the coinduced module Hom(ℤG, A) as the filtered colimit of the ℤG-modules ℤG ⊗_{ℤH} Hom(ℤG, A), where H runs through the collection of finite subgroups of G. The claim follows since for any finite subgroup H of G the ℤH-module Hom(ℤG, A) is coinduced and hence induced; cf. [3, Chapter III, Proposition 5.9].)

Proposition 2.1 relates the finiteness of a finitely generated group G to the injective dimension of the ℤG-module ℤG. In order to put in perspective that relation, the following result may be useful.

Proposition 2.3. A group G is finite if and only if the ℤG-module ℤG has injective dimension ≤ 1.

Proof. If G is finite, then, as shown by Gedrich and Gruenberg in [9, Corollary 5.3], we have silp ℤG = spli ℤG ≤ 1. In particular, the ℤG-module ℤG has injective dimension ≤ 1.

Conversely, assume that the ℤG-module ℤG has injective dimension ≤ 1. Then, for any subgroup H ⊆ G the ℤH-module ℤH has injective dimension ≤ 1 as well. (This follows since ℤH is a direct summand of ℤG as a ℤH-module, whereas any injective ℤG-module is injective as a ℤH-module.) In particular, if the subgroup H ⊆ G is countable, then [2, Corollary 1.4] implies that spli ℤH = silp ℤH ≤ 1. Hence, invoking the main result of [4], we conclude that any countable subgroup H of G is necessarily finite. This obviously implies that the group G is itself finite, as needed. □

Remark 2.4. In view of the result above, Proposition 2.1 (and more specifically conditions (iii) and (iv) therein) may be expressed as follows: If G is a finitely generated group, then we may characterize its finiteness by relaxing the condition on the injective dimension id_{ℤG}ℤG of the ℤG-module ℤG that appears in Proposition 2.3 (namely, that id_{ℤG}ℤG ≤ 1) and assume that id_{ℤG}ℤG ≤ 2 or else that id_{ℤG}ℤG is merely finite, provided that injective ℤG-modules are known to be acyclic for group homology.

3. On the vanishing of \underline{cd}

Let G be a group. The proof of the implication

\underline{cd} G = 0 ⟹ G is finite,

which was presented in [6, Theorem 4.6], is based upon the proof of the implication

spli ℤG = 1 ⟹ G is finite,

which is itself proved in [4], by using the Almost Stability Theorem of Dicks and Dunwoody [5]. We shall obtain an alternative proof of the former implication, by dualizing the arguments in the proof of Proposition 2.1. In particular, we shall relate the finiteness of a group G to the finiteness of the projective dimension of coinduced modules. To that end, we note that for any group G we have silp ℤG ≤ \underline{cd} G + 1, as shown in [11, Lemma 1.8]. Since we also have spli ℤG = silp ℤG (cf. [6, Corollary 4.5]), it follows that spli ℤG ≤ \underline{cd} G + 1. In particular, if \underline{cd} G = n < ∞, then spli ℤG ≤ n + 1 and hence all injective ℤG-modules have projective dimension ≤ n + 1.

Proposition 3.1. The following conditions are equivalent for a group G:

(i) G is finite,

(ii) \underline{cd} G = 0,

(iii) the cohomology functors H^{i}(G, ) vanish on projective modules for all i ≥ 1 and the ℤG-module Hom(ℤG, ℚ/ℤ) has projective dimension ≤ 1,

(iv) the cohomology functors H^{i}(G, ) vanish on projective modules for all i ≥ 1 and the ℤG-module Hom(ℤG, ℚ/ℤ) has finite projective dimension,

(v) the cohomology functors H^{i}(G, ) vanish on projective modules for all i ≥ 1 and there is a non-zero coinduced module of finite projective dimension.

Proof. The implication (i)→(ii) is proved in [10, §2, Corollary 2], whereas the implication (ii)→(iii) follows from the equality H^{i}(G, ) = Ext_{ℤG}^{i}(ℤ, ) and the discussion above. Since (iv) is a weak version of (iii) and the ℤG-module Hom(ℤG, ℚ/ℤ) is obviously coinduced, it only remains to show that (v)→(i). To that end, we consider a coinduced module M = Hom(ℤG, A) of finite projective dimension, where A ia a non-zero abelian group, and fix a projective resolution of finite length

0 ⟶ P_{n} ⟶ ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ 0.

Let K_{i} be the corresponding i-th syzygy module of M; in particular, K_{n} = P_{n}. Using our assumption that projective modules are acyclic for group cohomology, an inductive argument shows that H^{n−i}(G, K_{n−i}) = 0 for all i = 0, 1, …, n−1. In particular, we have H^{1}(G, K_{1}) = 0. Then, the short exact sequence of ℤG-modules

0 ⟶ K_{1} ⟶ P_{0} ⟶ M ⟶ 0

induces the exact sequence

H^{0}(G, P_{0}) ⟶ H^{0}(G, M) ⟶ H^{1}(G, K_{1}) = 0.

Since H^{0}(G, M) = H^{0}(G, Hom(ℤG, A)) = A ≠ 0, it follows that the group H^{0}(G, P_{0}) is nontrivial. Then, the ℤG-module P_{0} being projective, we conclude that the group G is finite, as needed. □

Remark 3.2. Let R be a ring and consider a class 𝔛 of left R-modules. Then, the left Ext^{1}- orthogonal of 𝔛 is the class ^{⊥}𝔛, consisting of those left R-modules M for which Ext_{1}^{R}(M, N) = 0 for all N ∈ 𝔛. In the same way, if 𝔜 is a class of left R-modules, then we may define the right Ext^{1}-orthogonal of 𝔜, as the class 𝔜^{⊥}, consisting of those left R-modules N for which Ext_{1}^{R}(M, N) = 0 for all M ∈ 𝔜. It is clear that we always have 𝔛 ⊆ (^{⊥}𝔛)^{⊥}.

In particular, let G be a group and consider the class ℭ consisting of all coinduced ℤGmodules. Then, ^{⊥}ℭ is easily seen to be the class consisting of those ℤG-modules M which are ℤ-free. It follows that the vanishing of \underline{cd} G is equivalent to the assertion that the class of projective ℤG-modules is contained in (^{⊥}ℭ)^{⊥} and may be interpreted by saying that projective ℤG-modules have a similar cohomological behavior as coinduced modules.

On the other hand, if G is a finite group, then projective ℤG-modules are obviously contained in (^{⊥}ℭ)^{⊥}. Indeed, any projective ℤG-module is a direct summand of an induced module and induced modules are, in this case, coinduced; cf. [3, Chapter III, Proposition 5.9].

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

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