ON THE GORENSTEIN AND COHOMOLOGICAL DIMENSION OF GROUPS

OLYMPIA TALELLI

Abstract. Here we relate the Gorenstein dimension of a group G, Gcd_{R}G, over ℤ and ℚ to the cohomological dimension of G, cd_{R}G, over ℤ and ℚ, and show that if G is in LH𝔉, a big class of groups defined by Kropholler in [12], then cd_{ℚ}G = Gcd_{ℚ}G and if G is torsion free then Gcd_{ℤ}G = cd_{ℤ}G. We also show that for any group G, Gcd_{ℚ}G ≤ cd_{ℤ}G.

1. Introduction

The Gorenstein projective dimension of an S-module M, Gpd_{S} M, is a refinement of the projective dimension of M over S, and was defined by Enochs and Jenda in [8]. This concept goes back to the concept of a relative homological dimension, called the G-dimension, which was defined by Auslander in [3] for finitely generated modules over commutative Noetherian rings and provided a characterization of the Gorenstein rings.

The Gorenstein dimension of a group G over ℤ, Gcd_{ℤ}G, is the Gorenstein projective dimension of ℤ, as a trivial ℤG-module. It turns out that Gcd_{ℤ}G coincides with the cohomological dimension of G over ℤ, cd_{ℤ}G, when the latter is finite, and is related ([1], [2]) to the invariants spliℤG, silpℤG and the generalized cohomological dimension of G, \underline{cd}_{ℤ}G, where spliℤG is the supremum of the projective dimensions of the injective ℤG-modules, silpℤG is the supremum of the injective dimensions of the projective ℤG-modules, and

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

These invariants were considered by Gedrich-Gruenberg [9] and Ikenaga

[11] in their study of extending the Farrell-Tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension, to a larger class of groups.

In [2] the finiteness of Gcd_{ℤ}G was proposed as an algebraic characterization of those groups G which admit a finite dimensional model for \underline{E}G, the classifying space for proper actions, equivalently as an algebraic characterization of those groups G which admit a resolution of finite length by direct sums of permutation modules induced by finite subgroups of G.

University of Athens, Department of Mathematics, Panepistimiopolis, GR-157 84, Athens, Greece. Email: otalelli@math.uoa.gr.

The Gorenstein dimension of a group G over ℚ, Gcd_{ℚ}G, is the Gorenstein projective dimension of ℚ as a trivial ℚG-module. The invariants spliℚG, silpℚG and the generalized cohomological dimension of G over ℚ, \underline{cd}_{ℚ}G are defined analogously. Although it is not known whether Gorenstein projectivity is a subgroup closed property, the Gorenstein dimension of a group G over ℤ and ℚ enjoys some nice properties (Theorem 3.1).

Here we study the Gorenstein dimension of a group G over ℤ and ℚ when G is in LH𝔉, a class of groups defined by Kropholler in [12]. The class H𝔉 was defined 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, or more generally, all elementary amenable groups, all linear groups, is extension closed and closed under ascending unions, amalgamated free products and HNN extensions.

We show that for G in LH𝔉, the Gorenstein dimension of G over ℚ coincides with the cohomological dimension of G over ℚ, Gcd_{ℚ}G = cd_{ℚ}G and when G is torsion free, then the Gorenstein dimension of G over ℤ coincides with the cohomological dimension of G over ℤ, Gcd_{ℤ}G = cd_{ℤ}G (Theorem 3.5 ).

We also show that for any group G, Gcd_{ℚ}G ≤ Gcd_{ℤ}G (Theorem 3.2).

One does not expect Gcd_{ℚ}G to be equal to Gcd_{ℤ}G since there are groups G with cd_{ℤ}G < ∞ and cd_{ℚ}G < cd_{ℤ}G [4], in which case Gcd_{ℚ}G = cd_{ℚ}G < cd_{ℤ}G = Gcd_{ℤ}G. However, we believe:

Conjecture 1.1. If G is an elementary amenable group then Gcd_{ℚ}G = Gcd_{ℤ}G.

Note that if G is an elementary amenable group, then since G is in LH𝔉, Gcd_{ℚ}G = cd_{ℚ}G.

2. Gorenstein dimension

Let S be an arbitrary ring with unit element 1 ≠ 0. All modules considered are left S-modules.

An S-module M is said to admit a complete resolution (F, P, 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}}

The number n is called the coincidence index of the complete resolution.

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

Definition 2.1. An S-module M is called Gorenstein projective if it admits a complete resolution in the strong sense of coincidence index 0.

If A is an S-module and n a non-negative integer, then the Gorenstein projective dimension of A is less than or equal to n, Gpd_{S}A ≤ n, if there is a resolution of A by Gorenstein projective S-modules of length n.

In [10] the following characterization of finite Gpd_{S}M is given:

Theorem 2.2. [10] Let M be an S-module with Gpd_{S}M < ∞, and n an integer. Then the following conditions are equivalent:

(i) Gpd_{S}M ≤ n.

(ii) Ext_{S}^{i}(M, L) = 0 for all i > n and all S-modules L with finite

projective dimension. (iii) Ext_{S}^{i}(M, Q) = 0 for all i > n and all projective S-modules Q. (iv) For every exact sequence 0→K_{n}→G_{n−1}→ ⋯ →G_{0}→M→0, where

G_{0}, …, G_{n−1} are Gorenstein projectives, then also K_{n} is Gorenstein projective.

Consequently, the Gorenstein projective dimension of M is determined by the formulas:

Gpd_{S}M = sup{i ∈ ℕ| ∃L with pd_{S}L < ∞ : Ext_{S}^{i}(M, L) ≠ 0}
= sup{i ∈ ℕ| ∃Q S-projective : Ext_{S}^{i}(M, Q) ≠ 0}.

Remark 2.3. If cd_{R}G < ∞ then it is well known that

cd_{R}G = sup{i ∈ ℕ| ∃F RG-projective : H^{i}(G, F) ≠ 0}.

It now follows from the characterization of the finite Gorenstein dimension obtained in Theorem 2.2 that if Gcd_{R}G < ∞ then cd_{R}G = Gcd_{R}G.

Following Ikenaga [11] we define the generalized cohomological dimension of a group G over ℚ, \underline{cd}_{ℚ}G, as

\underline{cd}_{ℚ}G = sup\{n ∈ ℕ| ∃ M, P : Ext_{ℚG}^{n}(M, P) ≠ 0, P ℚG-projective\}

and the generalized homological dimension of a group G over ℚ,

\underline{hd}_{ℚ}G = sup\{n ∈ ℕ| ∃ M, C : Tor_{n}^{ℚG}(M, C) ≠ 0, C ℚG-injective\}.

It is easy to see that if cd_{ℚ}G < ∞ then cd_{ℚ}G = \underline{cd}_{ℚ}G and if hd_{ℚ}G < ∞ then hd_{ℚ}G = \underline{hd}_{ℚ}G. Moreover, it is easy to see that \underline{cd}_{ℚ}G = silpℚG.

We obtain the following relations between spliRG, Gcd_{R}G and \underline{cd}_{R}G when R is the ring of integers ℤ or the ring of rationals ℚ.

Proposition 2.4. For any group G (1) Gcd_{R}G = \underline{cd}_{R}G

(2) \underline{hd}_{R}G ≤ \underline{cd}_{R}G

Proof. In [2] the statement (1) was proved for R = ℤ and its proof used the fact that spliℤG = silpℤG [7, Theorem 4.4]. However silpℚG = spliℚG [7, Theorem 4.4] and the proof for the case R = ℤ is valid also over ℚ. In [6, Theorem 4.11] the statement (2) was proved for R = ℤ. It is easy to see that the proof is also valid over ℚ. Note that the proof of (2) uses a certain characterization of a natural map which relates Tor_{∗}^{RG}( , ) to Ext_{RG}^{∗}( , ). □

Proposition 2.5. For any group G, spliRG < ∞ if and only if Gcd_{R}G < ∞.

Proof. If R = ℤ the result was shown in [1, Remark 2.10]. If R = ℚ then by [7, Theorem 4.4] silpℚG = spliℚG, hence by Proposition 2.4(1) silpℚG = spliℚG = \underline{cd}_{ℚ}G = Gcd_{ℚ}G. □

Remark 2.6. It is easy to see that the proof of Theorem 2.7 in [2] is valid for ℚG-modules, and thus we obtain that spliℚG < ∞ if and only if there is a ℚG-exact sequence 0 → ℚ → A with pd_{ℚG}A < ∞. Moreover pd_{ℚG}A = Gcd_{ℚ}G.

3. Gorenstein dimension and cohomological dimension

The Gorenstein dimension of a group G over R enjoys the following properties:

Theorem 3.1. (1) If H ≤ G then Gcd_{R}H ≤ Gcd_{R}G.

(2) If 1 → N → G → K → 1 is an extension of groups, then Gcd_{R}G ≤ Gcd_{R}N + Gcd_{R}K.

(3) If F is a finite subgroup of G, and N_{G}(F) its normalizer in G, then Gcd_{R}(N_{G}(F)/F) ≤ Gcd_{R}G.

(4) If 1 → N → G → K → 1 is an extension of groups with |N| < ∞ then Gcd_{R}G = Gcd_{R}K.

(5) If 1 → N → G → K → 1 is an extension of groups with |K| < ∞ then Gcd_{R}G = Gcd_{R}N.

Proof. The theorem was proved for R = ℤ in [2, Theorem 2.8]. The same proof goes through over ℚ . Note that we are using the characterization of the finiteness of spliℚG given in Remark 2.6. □

Theorem 3.2. For any group G, Gcd_{ℚ}G ≤ Gcd_{ℤ}G.

Proof. If Gcd_{ℤ}G < ∞ then by Proposition 2.5 spliℤG < ∞ hence by [9, Lemma 6.4] spliℚG < ∞ which implies that Gcd_{ℚ}G < ∞ by Proposition 2.5. If Gcd_{ℚ}G = n and Gcd_{ℤ}G = m then by Theorem 2.2 we obtain the following characterizations of n and m:

(i) n = sup{i ∈ ℕ| ∃F ℚG-projective : H^{i}(G, F) ≠ 0}
(ii) m = sup{i ∈ ℕ| ∃L with pd_{ℤG}L < ∞ : H^{i}(G, L) ≠ 0}.

But if F is a projective ℚG-module then clearly pd_{ℤG}F ≤ 1, hence it follows from (i) and (ii) that n ≤ m. □

Definition 3.3. [14] A group G is called of type Φ over R if it has the following property: for any RG-module M, pd_{RG}M < ∞ if and only if pd_{RH}M < ∞ for any finite subgroup H of G.

Proposition 3.4. If a group G is in LH𝔉 then

(1) findimRG = spliRG where findimRG is the supremum of the projective dimensions of the RG-modules of finite projective dimension.

(2) findimRG < ∞ if and only if G is of type Φ over R.

Proof. (1) For G in H𝔉 the result was proved in [5, Theorem C]. For R = ℤ and G in LH𝔉 the result was proved in [15, Corollary 2] and it is easy to see that the proof of Corollary 2 of [15] is valid over ℚ.

(2) Theorem 6 of [15] proves statement (2) for R = ℤ. It is easy to see that its proof is also valid over R = ℚ, and the result follows.

□

Theorem 3.5. If G is a group in LH𝔉 then (1) Gcd_{ℚ}G = cd_{ℚ}G.

(2) If G is torsion free then Gcd_{ℤ}G = cd_{ℤ}G.

Proof. (1) If Gcd_{ℚ}G < ∞ then by Proposition 2.5 spliℚG < ∞, hence by Proposition 3.4 (1) findimℚG < ∞, and thus by Proposition 3.4 (2), G is of type Φ. Since cd_{ℚ}H = 0 for every finite subgroup H of G it now follows that cd_{ℚ}G < ∞.

(2) If cd_{ℤ}G < ∞ then by Remark 2.3 cd_{ℤ}G = Gcd_{ℤ}G. If Gcd_{ℤ}G < ∞ then by Proposition 2.5 spliℤG < ∞ and since G is a torsion free group in LH𝔉 by Corollaries 2 and 4 of [15] it follows that cd_{ℤ}G < ∞. The result now follows.

□

In [13] the notion of jump cohomology was defined for a group G as follows:

Definition 3.6. [13] A group G has jump cohomology over R if there exists an integer k ≥ 0 such that if H is any subgroup of G with cd_{R}H < ∞, then cd_{R}H ≤ k. We allow k to be infinite.

The smallest such k is called the jump height of G over R.

Corollary 3.7. If G is a group in LH𝔉, then Gcd_{ℚ}G = cd_{ℚ}G =the jump height of G over ℚ.

Proof. In [13, Theorem 1.4] it was shown that if G is in H𝔉 then G has jump cohomology of height k over ℚ if and only if cd_{ℚ}G ≤ k. We obtain the same result for G in LH𝔉 through the proof of Theorem 7 in [15] which, it is easy to see that is valid over ℚ for any group G in LH𝔉. The result now follows from (1) of Theorem 3.5. □

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