ON THE GORENSTEIN DIMENSION OF SOLUBLE GROUPS

OLYMPIA TALELLI

1. Introduction

The Gorenstein projective dimension of an R-module M, Gpd_{R}M, is a refinement of the projective dimension of M over R, and was defined by Enochs and Jenda in [11]. This concept goes back to the concept of a relative homological dimension, called the G-dimension, which was defined by Auslander in [1] 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 this is finite, and is related to the invariants spliℤG, silpℤG and the generalized cohomological dimension of G, \underline{cd}G, where spliℤG is the supremum of the projective dimensions of the injective ℤGmodules, silpℤG is the supremum of the injective dimensions of the projective ℤG-modules, and

\underline{cd}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 [12] and Ikenaga [16] 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} 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.

Here we study the Gorenstein dimension of soluble groups and in particular how this is related to the Hirsch number of the group. Gedrich and Gruenberg proved the following Theorem which relates the Hirsch number, hG, of a soluble group G to spliℤG.

Theorem: [12] Let G be a soluble group and T the (unique) maximal normal torsion subgroup of G. Then hG ≤ spliℤG ≤ 2 + hG + spliℤT.

Our main result here relates the Hirsch number, hG, of a soluble group G to Gcd_{ℤ}G: Theorem: (Theorem 3.2) Let G be a soluble group and T the (unique) maximal normal torsion subgroup of G. Then

hG ≤ Gcd_{ℤ}G ≤ Gcd_{ℤ}T + hG + 1

Note that Gcd_{ℤ}G ≤ spliℤG ≤ Gcd_{ℤ}G + 1.

Both the above Theorems use the characterization of the Hirsch number of a soluble group as the homological dimension of the group over the rationals [21].

We believe that for soluble groups G, the Gorenstein dimension of G over ℤ coincides with the cohomological dimension of G over ℚ, i.e. cd_{ℚ}G = Gcd_{ℤ}G. This is not true in general, as there exist groups G with cd_{ℚ}G < Gcd_{ℤ}G.

2. Gorenstein dimension

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

An R-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 R-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 is not known whether a ℤG-module M that admits a complete resolution, also admits a complete resolution in the strong sense. However, this is so for a big class of groups which contains the class of soluble groups [6].

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

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

Examples:

Clearly every projective R-module is a Gorenstein projective Rmodule, hence Gpd_{R}M ≤ pd_{R}M. In particular Gcd_{ℤ}G ≤ cd_{ℤ}G.

If G is a finite group then every ℤG-module which is ℤ-free is a Gorenstein projective ℤG-module.

Indeed, if G is a finite group then there is a ℤ-split ℤG-exact sequence

(1)  \begin{aligned}0 → & ℤ → ℤG → K → 0 \\ & 1 ↦ ∑ g \end{aligned}

and if B is any ℤG-module, then tensoring (1) with B gives a ℤG-exact sequence

0 → B̀ → B̀ ⊗ ℤ̀G → B̀ ⊗ K̀ → 0

where B̀ ⊗ C̀ indicates diagonal action .

If B is ℤ-free then B̀ ⊗ ℤ̀G is a projective ℤG-module. So if

⋯ → P_{n} → P_{n−1} → ⋯ → P_{0} → ℤ → 0

is a projective resolution of G. i.e. a ℤG-projective resolution of the trivial ℤG-module ℤ, and A is a ℤG-module which is ℤ-free, then

(∗) ⋯ → P̀_{n} ⊗ À → ⋯ → P̀_{0} ⊗ À ⟶ ℤ̀G ⊗ À ⟶ ℤ̀G ⊗ (K̀ ⊗ À) → ⋯
↘  ↗ ↘  ↗
A  K̀ ⊗ À

is a complete resolution of A of coincidence index 0.

Now if

F : ⋯ ⟶ P̄_{n} ⟶^{ϑ_{n}} P̄_{n−1} ⟶ ⋯ ⟶ P̄_{0} ⟶^{ϑ_{0}} P_{−1} ⟶^{ϑ_{−1}} ⋯

is an acyclic complex of projective ℤG-modules then for any projective ℤG-module Q, the cochain complex Hom_{ℤG}(F, Q) is exact since Ext_{ℤG}^{1}(B, Q) = 0 for any ℤG-module B which is ℤ-free and

H^{i}(Hom_{ℤG}(F, Q)) ≃ Ext_{ℤG}^{1}(K_{i−1}, Q),

where K_{i−1} = imϑ_{i−1}, i ∈ ℤ. Hence (*) is a complete resolution of A in the strong sence of coincidence index 0, thus A is a Gorenstein projective ℤG-module.

It turns out that [2]

(a) If K is a subgroup of a group G and M a Gorenstein projective ℤK-module, then the induced module ℤ̀G ⊗_{ℤK} M is a Gorenstein projective ℤG-module.

(b) If 0→M′→M→M′′→0 is a short exact sequence of ℤG-modules and M′′ is Gorenstein projective, then M′ is Gorenstein projective if and only if M is Gorenstein projective. Moreover, direct sums and direct summands of Gorenstein projective modules are Gorenstein projective [15, Theorem 2.5].

It was also shown in [2] that the Gorenstein dimension of G enjoys the following properties:

Theorem 2.1. (1) If H ≤ G then Gcd_{ℤ}H ≤ Gcd_{ℤ}G.

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

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

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

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

In [15] the following characterization of finite Gorenstein projective dimension is given

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

(i) Gpd_{R}M ≤ n.

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

projective dimension. (iii) Ext_{R}^{i}(M, Q) = 0 for all i > n and all projective R-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_{R}M = sup{i ∈ ℕ| ∃L with pd_{R}L < ∞ : Ext_{R}^{i}(M, L) ≠ 0}

= sup{i ∈ ℕ| ∃Q R-projective : Ext_{R}^{i}(M, Q) ≠ 0}.

This in particular implies that if a group G has finite cohomological dimension over ℤ then this coincides with the Gorenstein dimension of G, i.e. if cd_{ℤ}G < ∞ then cd_{ℤ}G = Gcd_{ℤ}G.

More generally it was shown in [2] that for any group G the Gorenstein dimension of G, Gcd_{ℤ}G coincides with the generalized cohomological dimension, \underline{cd}G, defined by Ikenaga in [16] as

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

Remark: If a group G has torsion elements then its cohomological dimension over ℤ is infinite, however its Gorenstein dimension over ℤ could be finite. For example: If G is finite then we’ve seen that Gcd_{ℤ}G = 0, actually Gcd_{ℤ}G = 0 if and only if G is finite [9]. If G is a countable infinite locally finite group then Gcd_{ℤ}G = 1 [2]. More generally, it was shown in [2] that Gcd_{ℤ}G ≤ 1 if and only if G acts on a tree with finite vertex stabilizers. The proof of this result uses the Almost Stability Theorem of Dicks and Dunwoody [7]. If G is the free Burnside group G = B(n, e) of odd exponent e > 665 then by a result of Ivanov [17] it follows that Gcd_{ℤ}G ≤ 2 and by [2] it follows that Gcd_{ℤ}G = 2.

The finiteness of Gcd_{ℤ}G is proposed in [2] as an algebraic characterization of those groups G which admit a finite dimensional model for \underline{E}G, the classifying space for proper actions of G, namely:

Conjecture A: The following statements are equivalent for a group G:

(1) G admits a finite dimensional model for \underline{E}G.

(2) G admits a resolution of the form

0 → ⨁_{i_{n}∈I_{n}} ℤ(G/G_{i_{n}}) → ⋯ → ⨁_{i_{0}∈I_{0}} ℤ(G/G_{i_{0}}) → ℤ → 0

where G_{i_{j}} are finite groups.

(3) G is of type Φ.

(4) For any ℤG-module M, if H^{i}(F, M) = 0 for all i ≠ 0 and all finite subgroups F of G, then pd_{ℤG}M < ∞.

(5) Gcd_{ℤ}G < ∞. Note that a group G is said to be of type Φ if for every ℤG-module M, pd_{ℤG}M < ∞ if and only if pd_{ℤH}M < ∞ for every finite subgroup H of G.

The following theorem [22] generalizes a theorem of Kropholler and Mislin in [18]:

Theorem 2.3. If G is a group such that for every finite subgroup F of G the group N_{G}(F)/F is of type Φ and there is a bound on the orders of the finite subgroups, then G admits a finite dimensional model for \underline{E}G.

A consequence of this theorem is that Conjecture A holds for a big class of groups which contains in particular all soluble groups and all linear groups for which there is a bound on the orders of their finite subgroups.

A model for \underline{E}G is a G-CW-complex X in which all stabilizers are finite and is such that either of the following conditions hold:

(1) for every finite subgroup F of G, the fixed point subcomplex X^{F} is contractible.

(2) X is a terminal object in the homotopy category of G-CW-

complexes with finite stabilizers It is not hard to show that any two models of \underline{E}G are G-homotopy equivalent.

Recall that a G-CW-complex is a CW-complex X equipped with a G-action such that all elements of G map cells of X homeomorphically to cells, and such that whenever an element of G maps a cell to itself it does so via the identity map.

The space \underline{E}G plays a significant role in the Baum-Connes conjecture [3] which predicts that the K-homology of the reduces C^{∗}-algebra of a group G can be computed as the equivariant K-homology of the classifying space for proper actions of G, \underline{E}G. The space \underline{E}G plays also a role in the extension of the Atiyah-Segal Completion Theorem from finite to infinite groups [19].

3. Gorenstein dimension and Hirsch number

The following proposition was essentially proved by Gedrich and Gruenberg in [12] and its proof uses a result of Stammbach [21] which identifies the Hirsch number of a soluble group G as its homological dimension over the rationals, ℚ.

Proposition 3.1. Let G be a soluble group with Gcd_{ℚ}G < ∞. Then the Hirsch number of G, hG is finite.

Proof. Assume that hG is infinite and consider the first group in the sequence of derived factors 1, G^{(m−1)}, G^{(m−2)}/G^{(m−1)}, ⋯ having infinite Hirsch number. Let h(G^{(s−1)}/G^{(s)}) be infinite and h(G^{(s)}) finite. For each positive integer n we can find a subgroup H containing G^{(s)} so that H/G^{(s)} is free abelian of rank n. Hence

n ≤ h(H) < ∞.

By a result of Stammbach h(H) = hd_{ℚ}H. Let hd_{ℚ}H = λ. Then there is an injective ℚG-module I such that H_{λ}(G, I) ≠ 0. This in particular implies that the injective ℚG-module I cannot have a projective resolution of length less than λ, thus spliℚG ≥ λ ≥ n. But spliℚG = silpℚG = Gcd_{ℚ}G, hence Gcd_{ℚ}G ≥ n. Since n was arbitrary it follows that Gcd_{ℚ}G is infinite, a contradiction, thus hG is finite.

Theorem 3.2. Let G be a soluble group and T the unique maximal normal torsion subgroup of G. Then hG ≤ Gcd_{ℤ}G ≤ Gcd_{ℤ}T + hG + 1.

Proof. It follows from Proposition 3.1 that it is enough to consider the case where hG is finite. In that case Malcev’s structure theorem [20] says that there is a series

G_{0} < G_{1} < ⋯ < G_{n} < G_{n+1} = G

where G_{i} ◁ G, |G : G_{n}| < ∞, G_{0} is locally finite and G_{i}/G_{i−1}, 1 ≤ i ≤ n, are torsion free abelian of finite torsion free rank. Now by Theorem 2.1

Gcd_{ℤ}G = Gcd_{ℤ}G_{n} ≤ Gcd_{ℤ}G_{0} + Gcd_{ℤ}(G_{n}/G_{0}).

Since G_{n}/G_{0} is torsion free of finite Hirsch number it follows that it is countable and has finite cohomological dimension over ℤ, and thus Gcd_{ℤ}(G_{n}/G_{0}) = cd_{ℤ}(G_{n}/G_{0}) ≤ hG + 1 [5].

Hence by Theorem 2.1 we have that

(∗) Gcd_{ℤ}G ≤ Gcd_{ℤ}T + hG + 1.

Now Gcd_{ℚ}G ≤ Gcd_{ℤ}G and it follows essentially from Proposition 4.1 and Theorem 4.11 in [10] that if \underline{hd}_{ℚ}G is the supremum of those integers m for which there is an injective ℚG-module I with Tor_{m}^{ℚG}(I, M) ≠ 0 for some ℚG-module M then \underline{hd}_{ℚ}G = hd_{ℚ}G = hG and \underline{hd}_{ℚ}G ≤ Gcd_{ℚ}G.

Thus we have

hG = hd_{ℚ}G = \underline{hd}_{ℚ}G ≤ Gcd_{ℚ}G ≤ Gcd_{ℤ}G

and the result follows from (*).

If T is a locally finite group of cardinality less than ℵ_{ω} then it was shown by Dicks-Kropholler [8] that T admits a finite dimensional \underline{E}G which in particular implies that Gcd_{ℤ}T < ∞. This result together with Theorem 3.2 implies that

Corollary 3.3. Let G be a soluble group of cardinality < ℵ_{ω}. Then cd_{ℚ}G < ∞ if and only if Gcd_{ℤ}G < ∞ if and only if hG < ∞.

Hillman in [13] extended the notion of the Hirsch length of a soluble group to the class of elementary amenable groups. It was shown by Hillman and Linnell in [14] that Stammbach’s result extends to the class of elemantary amenable groups, i.e. the homological dimension over the rationals of an elemantary amenable group is its Hirsch length. Moreover Hillman-Linnell [14] and Wehrfitz [23] proved the corresponding of Malcev’s theorem for elementary amenable groups of finite Hirsch length.

Using the above we have the corresponding Theorem for elementary amenable groups, namely

Theorem 3.4. Let G be an elementary amenable group and T the unique maximal normal torsion subgroup of G. If hG is the Hirsch length of G then hG ≤ Gcd_{ℤ}G ≤ Gcd_{ℤ}T + hG + 1.

Remark: If cd_{ℚ}G < ∞ then cd_{ℚ}G ≤ Gcd_{ℤ}G. One does not expect cd_{ℚ}G = Gcd_{ℤ}G since there is a group G with cd_{ℚ}G = 2 and Gcd_{ℤ}G = cd_{ℤ}G = 3 [4]. However, we believe that if G is an elementary amenable group then Gcd_{ℤ}G = cd_{ℚ}G.

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