GORENSTEIN DIMENSION AND PROPER ACTIONS

ABDOLNASER BAHLEKEH, FOTINI DEMBEGIOTI, AND OLYMPIA TALELLI

Abstract. We conjecture that a group G admits a finite dimensional classifying space for proper actions if and only if the Gorenstein projective dimension of G is finite. We verify the 1dimensional case of this conjecture. Some evidence is given for the hypothesis that the Gorenstein projective ℤG-modules are precisely D. Benson’s class of cofibrant modules.

1. Introduction

In 1967 Auslander [2] introduced a new invariant for finitely generated modules over commutative Noetherian rings: a relative homological dimension called the G-dimension. The “G” is for Gorenstein, since through this, a characterization of the Gorenstein rings was given, namely:

For a local ring R, R is Gorenstein if and only if the residue field R/m has finite G-dimension, where m is the maximal ideal of R.

If R is a commutative Noetherian ring, a finitely generated R-module M is said to have finite G-dimension G−dim_{R}M < ∞ if it has a resolution of finite length by G-modules. A finitely generated R-module E is called a G-module if Ext_{R}^{m}(E, R) = 0 for m > 0, Ext_{R}^{m}(Hom_{R}(E, R), R) = 0 for m > 0 and the canonical map δ_{E} : E → Hom_{R}(Hom_{R}(E, R), R) given by δ_{E}(x)(f) = f(x) for f ∈ Hom_{R}(E, R) and x ∈ E, is an isomorphism.

Auslander and Bridger in [3] proved that G-dim_{R}M ≤ pd_{R}M for every finitely generated R-module M over a commutative Noetherian ring R, with G-dim_{R}M = pd_{R}M when pd_{R}M is finite.

Over a general ring R, Enochs and Jenda defined in [11] a homological dimension, the Gorenstein projective dimension, Gpd_{R}M, for any R-module M. It is defined via resolutions by Gorenstein projective modules. Avramov, Buchweitz, Martsinkovsky and Reiten proved [5, Theorem 4.2.6.] that if R is a commutative Noetherian ring and M is a finitely generated R-module, then M is Gorenstein projective if and only if G-dim_{R}M = 0.

It turns out that an R-module M is Gorenstein projective if it admits a complete resolution in the strong sense of coincidence index 0.

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 sufficiently high dimensions

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 turns out that the Gorenstein projective dimension of ℤ as a trivial ℤG-module, where ℤG is the integral group ring of a group G, Gcd_{ℤ}G, is closely related to spli ℤG, the supremum of the projective dimensions of the injective ℤG-modules, namely spli ℤG < ∞ if and only if Gcd_{ℤ}G < ∞ [1, Remark 2.10].

In [25] it was conjectured: Conjecture A. The following statements are equivalent for a group G.

(1) G admits a finite dimensional model for \underline{E}G, the classifying space for proper actions.

(2) G is of type Φ.

(3) spli ℤG < ∞

(4) findim ℤG < ∞.

where spli ℤG is the supremum of the projective lengths of the injective ℤG-modules and findim ℤG is the supremum of the projective lengths of the ℤG-modules of finite projective dimension. A group G is 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.

For G a discrete group, a proper G-CW-complex is one in which all cell stabilizers are finite. A classifying space for proper actions of G, denoted by \underline{E}G, is a proper G-CW-complex such that for each finite subgroup K of G the fixed point subcomplex \underline{E}G^{K} is contractible. Equivalently, \underline{E}G is a terminal object in the homotopy category of proper G-CW-complexes, which implies that \underline{E}G is unique up to Gequivariant homotopy. Associated to any group G there is a model for \underline{E}G. So Conjecture A proposes algebraic characterizations of those groups G which admit a finite dimensional model for \underline{E}G.

Note that for any group G

findim ℤG ≤ spli ℤG

and if spli ℤG < ∞ then findim ℤG = spli ℤG.

Spli ℤG < ∞ if and only if Gcd_{ℤ}G < ∞, however Gcd_{ℤ}G is a more natural algebraic invariant for the above conjecture, since if G admits a contractible G-CW-complex X with finite cell stabilizers, then the cellular chain complex of X is a Gorenstein projective resolution of G (Prop. 3.1).

In the frame of Conjecture A we show:

Theorem 3.6. A group G acts on a tree with finite vertex stabilizers if and only if Gcd_{ℤ}G ≤ 1.

We also give a characterization of finite Gcd_{ℤ}G:

Theorem 2.7. For any group G, Gcd_{ℤ}G < ∞ if and only if there is a ℤ-split ℤG-exact sequence 0 → ℤ → A where A is a ℤ-free ℤG-module of finite projective dimension. Moreover, Gcd_{ℤ}G = pd_{ℤG}A.

From C. Martinez-Perez’s main theorem in [20] and our characterization of Gcd_{ℤ}G follows

Theorem 3.4. Suppose that a group G admits a finite dimensional \underline{E}G and sup\{m| H_{0} ≨ H_{1} ≨ ⋯ ≨ H_{m}, H_{i} non-trivial finite subgroups of G} = λ is finite.

Then the minimal dimension for a model of \underline{E}G ≤ λ + Gcd_{ℤ}G. We use the characterization of finite Gcd_{ℤ}G of Theorem 2.7 to show

that Gcd_{ℤ}G behaves well with closure properties.

Theorem 2.8.

(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.

We also show (Theorem 2.5) that Gcd_{ℤ}G coincides with the generalized cohomological dimension of G, \underline{cd}G, defined by Ikenaga in his study of generalized Farrell-Tate cohomology in [15]. \underline{Cd}G is defined as follows:

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

In [4], Benson introduced the cofibrant modules for his study of ℤGmodules of type FP_{∞} when G is in the class LH𝔉, a large class of groups defined by Kropholler in [16].

For any group G, a ℤG-module M is said to be cofibrant if M ⊗ B(G, ℤ) with diagonal G-action is a projective ℤG-module, where B(G, ℤ) is the set of bounded functions from G to ℤ.

A ℤG-module A is said to be of type FP_{∞} if A admits a projective resolution by finitely generated projective modules.

The class H𝔉 was defined by Kropholler in [16] 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.

It not difficult to see that every cofibrant ℤG-module is Gorenstein projective. We believe that in essence the two notions, cofibrant and Gorenstein projective, are the same. Here we obtain some results pointing to this direction. The following result is an analogue of Lemma 4.5 in [4] for Gorenstein projectives.

Theorem 4.2 ((iv), (v)). M is a Gorenstein projective ℤG-module if and only if Gpd_{ℤG}M < ∞ and every extension of M by a ℤG-module of finite projective dimension splits.

The following result is an analogue of Theorem 5.7 in [4] for Gorenstein projectives.

Theorem 4.3. Let G be a group in H𝔉, and M a ℤG-module with Gpd_{ℤG}M < ∞. Then pd_{ℤG}M < ∞ if and only if pd_{ℤF}M < ∞ for every finite subgroup F of G.

The following Theorem gives information about the structure of Gorenstein projectives of type FP_{∞} if G is in LH𝔉, and is an analogue of Theorem A´ in [17].

Theorem 4.5. Let G be in LH𝔉. Then the following statements are equivalent for a ℤG-module M:

(1) M is Gorenstein projective and of type FP_{∞}.

(2) M is a direct summand of a module M_{E} which admits a series

0 = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{n} = M_{E}

in which the sections M_{i}/M_{i−1} are of the form ℤ̀G⊗_{ℤE} V where E is a finite elementary abelian subgroup of G and V a finitely generated ℤE-module which is ℤ-free.

(3) M is a direct summand of a module M_{F} which admits a series

0 = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{n} = M_{F}

in which the sections M_{i}/M_{i−1} are of the form ℤ̀G⊗_{ℤF} N where F is a finite subgroup of G and N a finitely generated ℤF - module which is ℤ-free.

Combining Theorem A´ in [17] with our Theorem 4.5 we obtain: Corollary 4.6. Suppose that G is an LH𝔉-group and A a ZG-

module. Then A is a Gorenstein projective ZG-module of type FP_{∞} if and only if it is a cofibrant module of type FP_{∞}.

We propose two conjectures: Conjecture A´ . The following statements are equivalent for a

group G.

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

(2) G is of type Φ.

(3) Gcd_{ℤ}G < ∞.

(4) findim ℤG < ∞. It is known that (1) ⇒ (2) ⇒ (3) ⇒ (4) (cf. [25]) and (4) ⇒ (1) was

shown for certain classes of groups ([18], [25], [26]).

Conjecture B. For any group G, Gcd_{ℤ}G + 1 = findim ℤG. Conjecture B was proved for some classes of groups in [7].

2. Gorenstein dimension

For G a group, K a subgroup, M a ℤK-module, and A, B ℤGmodules, we will use the notation ℤ̀G ⊗_{ℤK} M and À ⊗ B̀ to indicate the G-action, induced in the first and diagonal in the second case.

Examples 2.1. Examples of Gorenstein projective modules: (a) Every projective module is Gorenstein projective.

(b) If F is a finite group then every ℤF -module which is ℤ-free is Gorenstein projective.

(c) 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.

(d) If G has periodic cohomology after some steps and G is in LH𝔉, then G admits a complete resolution in the strong sense. Clearly, every kernel of this resolution is Gorenstein projective.

(e) 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 [14, Theorem 2.5].

Proof. (a) is clear from the definition.

(b) It is well known that if F is a finite group then the trivial ℤF - module ℤ admits a complete resolution of coincidence index 0

[figure: P :; ⋯; ⟶; P_{1}; ⟶; P_{0}; ⟶; P_{−1}; ⟶; P_{−2}; ⟶; ⋯; ℤ]

and since if Q is a projective ℤF -module

(1)  Ext_{ℤF}^{i}(A, Q) = 0 for all i > 1 and any ℤF -module A

it follows that P is a complete resolution in the strong sense.

If M is a ℤF -module which is ℤ-free then P̀⊗M̀ is a complete resolution of coincidence index 0, which is a complete resolution of M in the strong sense by (1).

(c) [figure: If P :; ⋯; ⟶; P_{1}; ⟶; P_{0}; ⟶; P_{−1}; ⟶; P_{−2}; ⟶; ⋯; M]

is a complete ℤK-resolution of M in the strong sense of coincidence index 0, then ℤ̀G ⊗_{ℤK} P is a complete ℤG-resolution of M in the strong sense of coincidence index 0, since induction takes projectives to projectives and (ind, res) is an adjoint pair.

(d) If G has periodic cohomology after some steps then findim ℤG < ∞ [22] and it was shown in [26] that if G is in LH𝔉 then findim ℤG < ∞ if and only if spli ℤG < ∞, hence by [24, Theorem 2.2] G admits a complete resolution in the strong sense.

□

Definition 2.2. [14] Let A be a ℤG-module, and n a non-negative integer. Then the Gorenstein projective dimension of A is less than or equal to n, Gpd_{ℤG}A ≤ n, if there is a resolution of A by Gorenstein projective modules, of length n.

In [14] the following characterization of finite Gorenstein projective dimension is given:

Theorem 2.3. [14, Theorem 2.20] 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}.

If G admits a complete resolution in the strong sense of coincidence index n, (F, P, n), then Gcd_{ℤ}G ≤ n since

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

is a resolution of ℤ of length n by Gorenstein projective ℤG-modules. Conversely, if Gcd_{ℤ}G ≤ n, then by Schanuel’s Lemma for Gorenstein projectives [14, 2.7] there is an exact sequence

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

with P_{i} projectives and R_{n} a Gorenstein projective ℤG-module. This clearly gives rise to a complete resolution of G in the strong sense of coincidense index n. Thus Gcd_{ℤ}G ≤ n if and only if G admits a complete resolution of G in the strong sense of coincidense index n.

Proposition 2.4.

(a) If cd_{ℤ} G < ∞, then cd_{ℤ} G = Gcd_{ℤ}G.

(b) Gcd_{ℤ}G < ∞ if and only if spli ℤG < ∞ [1, Remark 2.10].

(c) For every ℤG-module M, Gpd_{ℤG}M ≤ Gcd_{ℤ}G + 1. If M is ℤ-free then Gpd_{ℤG}M ≤ Gcd_{ℤ}G.

Proof. (a) is an immediate consequence of Theorem 2.3. The proof of (b) is based on the fact that spli ℤG < ∞ if and only if G admits a complete resolution in the strong sense [24, Theorem 2.2].

(c) Assume that Gcd_{ℤ}G < ∞, so spli ℤG < ∞ by (b) and silp ℤG < ∞ by [12], where silp ℤG is the supremum of the injective lengths of the projective ℤG-modules, . Let M be a ℤ-free ℤG-module and P a Gorenstein projective ℤG-module and F a complete resolution of P of coincidence index 0. Then F̀ ⊗ M̀ is a complete resolution of P̀ ⊗ M̀ of coincidence index 0, and it is a complete resolution in the strong sense because silp ℤG < ∞. It follows that P̀ ⊗ M̀ is Gorenstein projective.

Now, if Gcd_{ℤ}G = n and 0→P_{n}→ ⋯ → ⋯ →P_{0}→ℤ→0 is a Gorenstein projective resolution of ℤ, then

0→P̀_{n} ⊗ M̀→ ⋯ → ⋯ →P̀_{0} ⊗ M̀→M→0

is a Gorenstein projective resolution of M, so Gpd_{ℤG}M ≤ n. If M is not ℤ-free, a dimension shifting argument shows that Gpd_{ℤG}M ≤ n+1. □

Theorem 2.5. For any group G, Gcd_{ℤ}G = \underline{cd}G. Consequently, if \underline{cd}G < ∞, then \underline{cd}G = sup\{n ∈ ℕ| ∃P ℤG-projective : H^{n}(G, P) ≠ 0\}.

Proof. We have that \underline{cd}G = sup\{n ∈ ℕ| ∃M ℤ-free, ∃P ℤG-free : Ext_{ℤG}^{n}(M, F) ≠ 0\} [15], and it is easy to see that \underline{cd}G ≤ silp ℤG ≤ \underline{cd}G + 1. Since silp ℤG = spli ℤG [10], we have by Prop. 2.4 (b) that Gcd_{ℤ}G < ∞ if and only if \underline{cd}G < ∞.

So we only have to prove that Gcd_{ℤ}G = \underline{cd}G when Gcd_{ℤ}G is finite. By the characterization of the finite Gpd_{ℤG}M of Theorem 2.3 it follows that Gcd_{ℤ}G ≤ \underline{cd}G and that \underline{cd}G = sup\{Gpd_{ℤG}M| M a ℤ-free ℤG-module\}, hence by Prop. 2.4(c) \underline{cd}G ≤ Gcd_{ℤ}G. □

Theorem 2.6. Let 0→A→B→C→0 be a short exact sequence of Rmodules.

(a) If any two of the modules A, B, C, have finite Gorenstein projective dimension, then so has the third [14, Theorem 2.24 ].

(b) (i) If Gpd_{R}A > Gpd_{R}B then Gpd_{R}C = Gpd_{R}A + 1. (ii) If Gpd_{R}A ≤ r and Gpd_{R}B ≤ r then Gpd_{R}C ≤ r + 1.

Proof. (b) is an immediate consequence of (a) and the long exact Extsequence with projective coefficients. □

The following result gives another characterization of finite Gcd_{ℤ}.

Theorem 2.7. Let G be a group. Then Gcd_{ℤ}G < ∞ if and only if there is a ℤ-split ℤG-exact sequence 0 → ℤ → A where A is a ℤ-free ℤG-module of finite projective dimension. Moreover, pd_{ℤG}A = Gcd_{ℤ}G.

Proof. Gcd_{ℤ}G < ∞ if and only if spli ℤG < ∞ (Prop. 2.4(b)), and spli ℤG < ∞ if and only if there is a ℤ-split ℤG-exact sequence 0 → ℤ → A where A is a ℤ-free ℤG-module of finite projective dimension [24, Theorem 2.2].

Assume that Gcd_{ℤ}G = n. Then Gpd_{ℤG}A ≤ n from Prop. 2.4(c) and it follows from Theorem 2.3 that Ext_{ℤG}^{k}(A, P) = 0 for k > n and all projective modules P, so pd_{ℤG}A ≤ n. Since Gcd_{ℤ}G = n, it follows from Theorem 2.3 that H^{n}(G, Q) ≠ 0 for some projective ℤG-module Q. If Ā is the cokernel of the map 0 → ℤ → A, then Ā is ℤ-free, so Gpd_{ℤG}Ā ≤ n and Ext_{ℤG}^{n+1}(Ā, Q) = 0. From the long exact sequence

⋯ → Ext_{ℤG}^{n}(A, Q) → H^{n}(G, Q) → Ext_{ℤG}^{n+1}(Ā, Q) = 0

it follows that Ext_{ℤG}^{n}(A, Q) ≠ 0, so pd_{ℤG}A ≥ n. □

Using this characterization we obtain the following properties of Gcd_{ℤ}G.

Theorem 2.8.

(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.

Proof. (1) Immediate from Theorem 2.7.

(2) If Gcd_{ℤ}K and Gcd_{ℤ}N are finite then spli ℤK and spli ℤN are finite, so spli ℤG is finite [12]. It follows that Gcd_{ℤ}G is finite. Assume Gcd_{ℤ}K = k and Gcd_{ℤ}N = n, so there is a Gorenstein projective resolution of N of length n.

(2)  0→M_{n}→ ⋯ →M_{1}→M_{0}→ℤ→0

Applying ℤG ⊗_{ℤN} to (2) we obtain a Gorenstein projective resolution of the ℤG-module ℤG ⊗_{ℤN} ℤ ≃ ℤK of length n, so Gpd_{ℤG}ℤK ≤ n. Every ℤK-module is a ℤG-module via π : G→K. It follows that Gpd_{ℤG}P ≤ n for any projective ℤKmodule P.

We claim that any Gorenstein projective ℤK-module L has Gpd_{ℤG}L ≤ n. Indeed L has finite Gorenstein projective dimension over ℤG by Prop. 2.4(c), because Gcd_{ℤ}G is finite. Assume that Gpd_{ℤG}L > n, and let 0→L→P_{0}→P_{1} ⋯ be an exact sequence of ℤK-modules with P_{i} projective over ℤK for all i, and L_{i} = ker(P_{i}→P_{i+1}). Since Gpd_{ℤG}L > n ≥ Gpd_{ℤG}P_{0}, it follows from Theorem 2.6(b)(i) that Gpd_{ℤG}L_{1} = Gpd_{ℤG}Q + 1. Inductively, Gpd_{ℤG}L_{i} = Gpd_{ℤG}L + i, so ⊕_{i}L_{i} has infinite Gorenstein projective dimension over ℤG, which is a contradiction by Prop. 2.4(c).

Consider a Gorenstein projective resolution

0→Q_{k}→ ⋯ →Q_{0}→ℤ→0

of K. Since Gpd_{ℤG}Q_{i} ≤ n, it follows inductively from Theorem 2.6(b)(ii) that Gpd_{ℤG}ℤ ≤ n+k, so Gcd_{ℤ}G ≤ Gcd_{ℤ}N +Gcd_{ℤ}K.

(3) Using (1) we may assume that N_{G}(F) = G. Assume also that Gcd_{ℤ}G is finite. Since spli ℤ(G/F) ≤ spli ℤG [25, Prop. 3.5], it follows from Prop. 2.4(b) that Gcd_{ℤ}(G/F) is finite.

Let 0 → ℤ → A be a ℤ-split ℤG-exact sequence with pd_{ℤG}A = Gcd_{ℤ}G (Prop. 2.7). As in the proof of Proposition 3.5 [25] we can prove that the ℤ(G/F)-module A^{F} has finite projective dimension pd_{ℤ(G/F)}A^{F} ≤ pd_{ℤG}A. Since 0 → ℤ → A^{F} is a ℤsplit ℤ(G/F)-exact sequence, it follows from Theorem 2.7 that Gcd_{ℤ}(G/F) = pd_{ℤ(G/F)}A^{F} ≤ pd_{ℤG}A = Gcd_{ℤ}G.

(4) This follows from (2) and (3) using the fact that the Gorenstein dimension of a finite group is 0.

(5) This follows from (1) and (2) using the fact that the Gorenstein

dimension of a finite group is 0.

□

Remarks 2.9.

(1) If G is a virtually torsion free group with vcdG < ∞, then by Prop. 2.4(a) and Theorem 2.8(5) vcdG = Gcd_{ℤ}G. There are groups G which fit into an extension of the form 1→F →G→K→1, with F finite, and cd_{ℤ}K < ∞ and do not have a torsion free subgroup of finite index. By Theorem 2.8(4), however, Gcd_{ℤ}G = Gcd_{ℤ}K.

An example of such a group is the following: G = A∗_{H,φ}B, where A = ⟨a_{1}, a_{2}, a_{3}, a, d/[a_{i}, a_{j}] = [a_{i}, d] = [a, d] = d^{p} = 1, a_{1}^{a} = a_{2}, a_{2}^{a} = a_{3}, a_{3}^{a} = a_{1}a_{2}^{−3}a_{3}^{2}⟩,

B = ⟨b_{1}, b_{2}, b_{3}, b, e/[b_{i}, b_{j}] = [b_{i}, e] = [b, e] = e^{p} = 1, b_{1}^{b} = b_{2}, b_{2}^{b} = b_{3}, b_{3}^{b} = b_{1}b_{2}^{−3}b_{3}^{2}⟩,

H = ⟨a_{1}, a_{2}^{p}, a_{3}, d⟩ ≤ A and φ : H→B with φ(a_{1}) = b_{1}^{p}e, φ(a_{2}^{p}) = b_{2}, φ(a_{3}) = b_{3}^{p}, φ(d) = e.

Note that A ≃ B ≃ (ℤ×ℤ×ℤ×ℤ_{p})◁ℤ, and vcdA = vcdB = 4.

It follows that ⟨d⟩ ≤ ∩{N ≤ G| |G : N| < ∞}, hence G does not have a torsion free subgroup of finite index. Moreover, there is a group extension 1→⟨d⟩→G→K→1 with cdK < ∞.

This group was constructed by Dyer in [9] as a counterexample to a conjecture related to residual finiteness.

(2) If G is the fundamental group π(G, X, T) of a graph of groups and

0→ ⨁_{i∈I} ℤ(G/G_{e_{i}})→ ⨁_{j∈J} ℤ(G/G_{v_{j}})→ℤ→0

the corresponding cellular chain complex (cf. [8]) then one obtains from Theorem 2.6 relations between the Gorenstein projective dimension of G and the Gorenstein projective dimensions of the vertex and edge groups. In particular

(a) Suppose that G = G_{1}∗_{H}G_{2} and m = max{Gcd_{ℤ}G_{1}, Gcd_{ℤ}G_{2}}. Then m ≤ Gcd_{ℤ}G ≤ m + 1.

(b) Suppose that G = A∗_{H}. Then Gcd_{ℤ}A ≤ Gcd_{ℤ}G ≤ Gcd_{ℤ}A+ 1.

3. Gorenstein dimension and proper actions

Proposition 3.1. If G admits a contractible G-CW-complex X with finite cell stabilizers then the cellular chain complex of X is a Gorenstein projective resolution of G.

Proof. X is a G-CW-complex equipped with a G-action such that all elements of G map cells of X homeomorphically to cells is such a way that the setwise stabilizer of each cell is equal to the pointwise stabilizer. Hence the cellular chain complex of X is an exact sequence of ℤG-modules of the form ⨁_{i∈H_{i}} Z(G/H_{i}) where H_{i}, i ∈ I, are finite subgroups of G. Since ℤ[G/H_{i}] ≃ ℤG⊗_{ℤH_{i}} ℤ, it follows from Examples 2.1(b) and (c) that ℤ(G/H_{i}) is Gorenstein projective for i ∈ I. The result follows from the fact that direct sums of Gorenstein projective modules are Gorenstein projective (Example 2.1(e)). □

An immediate corollary of Proposition 3.1 is:

Corollary 3.2. If G admits a finite dimensional model for \underline{E}G and dim\underline{E}G = n, then Gcd_{ℤ}G ≤ n.

Remark 3.3. One does not expect Gcd_{ℤ}G to be the minimal dimension of a model for \underline{E}G, since there is a group G with vcdG = 3, hence Gcd_{ℤ}G = 3, with any model for its classifying space for proper actions having dimension at least 4 [19].

It was proved in [20, Theorem 3.10] that if a group G admits a finite dimensional model for \underline{E}G and sup\{m| H_{0} ≨ H_{1} ≨ ⋯ ≨ H_{m}, H_{i} non-trivial finite subgroups of G} = λ is finite, then the minimal dimension for a model of \underline{E}G is ≤ λ + pd_{ℤG}B(G, ℤ).

Combining this with Theorem 2.7 we obtain:

Theorem 3.4. If G is as above then the minimal dimension for a model of \underline{E}G ≤ λ + Gcd_{ℤ}G.

A reformulation of Conjecture A in [25] is the following:

Conjecture 3.5. The following statements are equivalent for a group G.

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

(2) G is of type Φ.

(3) Gcd_{ℤ}G < ∞.

(4) findim ℤG < ∞

In the frame of Conjecture 3.5 we show:

Theorem 3.6. Let G be a group. Then Gcd_{ℤ}G ≤ 1 if and only if G acts on a tree with finite vertex stabilizers.

Proof. Assume that Gcd_{ℤ}G ≤ 1. It follows from Theorem 2.7 that there is ℤ-free module A with pd_{ℤG}A ≤ 1 and A^{G} ≠ 0. Then G is the fundamental group of a graph of finite groups, by the Corollary of [23].

Conversely, assume that G acts on a tree with finite vertex stabilizers. Then Gcd_{ℤ}G ≤ 1 from Corollary 3.2. □

4. Gorenstein projectives and cofibrant modules

Proposition 4.1. Every cofibrant module is Gorenstein projective.

Proof. This follows from Theorem 3.5 of [6]. □

Theorem 4.2. Some properties of Gorenstein projectives.

(i) Let M, N be R-modules. If M is Gorenstein projective then the natural map \underline{Hom}_{R}(M, N)→Êxt_{R}^{0}(M, N) is an isomorphism.

(ii) If M is Gorenstein projective and it has finite projective dimen-

sion then M is projective. (iii) Let M be an R-module with finite Gorenstein projective dimen-

sion n. Then there is a short exact sequence 0→K→N→M→0 where N is Gorenstein projective and pd_{ℤG}K = n − 1.

(iv) If M is Gorenstein projective then every extension of M by an

R-module of finite projective dimension splits.

(v) If M has finite Gorenstein projective dimension and every extension of M by an R-module of finite projective dimension splits, then M is Gorenstein projective.

(vi) If M is of type FP_{∞} and has finite Gorenstein projective dimension then there is a complete resolution in the strong sense of M of finite type.

Proof. (i) \underline{Hom}_{R}(M, N) is the quotient of Hom_{R}(M, N) by the subgroup of maps that factor through projectives modules. Êxt_{R}^{∗}( , ) denotes the complete cohomology (cf. [13], [21]). When complete resolutions in the strong sense exist, they can be used to compute complete cohomology [6]. If M is Gorenstein projective then a complete resolution in the strong sense exists for M:

[figure: ⋯; ⟶; P_{1}; ⟶; d_{1}; P_{0}; ⟶; d_{0}; P_{−1}; ⟶; P_{−2}; ⟶; ⋯; π; M]

and Êxt_{R}^{0}(M, N) = kerd_{1}^{∗}/imd_{0}^{∗}:

Hom_{R}(P_{−1}, N) ⟶^{d_{0}^{∗}} Hom_{R}(P_{0}, N) ⟶^{d_{1}^{∗}} Hom_{R}(P_{1}, N).

Since kerd_{1}^{∗} = imπ^{∗} ≃ Hom_{R}(M, N) we have that Êxt_{R}^{0}(M, N) ≃ Hom_{R}(M, N)/imd_{0}^{∗}, so it remains to show that every map M→N that factors through a projective module Q, factors through P_{−1}. For this it suffices to show that every map f : M→Q factors through P_{−1}. This follows from the fact that P is a complete resolution in the strong sense, so Hom_{R}(P_{−1}, Q) ⟶^{d_{0}^{∗}}

Hom_{R}(P_{0}, Q) ⟶^{d_{1}^{∗}} Hom_{R}(P_{1}, Q) is exact. (ii) This is Proposition 2.27 of [14] and it follows from Theorem 2.3. (iii) This is proved in [14, Theorem 2.10]. (iv) Let 0→N→A→M→0 be a short exact sequence where M is

Gorenstein projective and N has finite projective dimension. Then the following sequence is exact

Hom_{R}(A, N)→Hom_{R}(N, N)→Ext_{R}^{1}(M, N)

where Ext_{R}^{1}(M, N) = 0 from Theorem 2.3. It follows that Hom_{R}(A, N)→Hom_{R}(N, N) is an epimorphism, so 0→N→A is a split monomorphism.

(v) Since Gpd_{R}M is finite, it follows from (iii) that there is an exact sequence 0→K→N→M→0 where N is Gorenstein projective and pd_{R}K is finite. By hypothesis this sequence splits, so M is a direct summand of N and thus Gorenstein projective by Example 2.1(e).

(vi) We may assume that M is Gorenstein projective and of type FP_{∞}. Then there is a short exact sequence

0→M→P →L→0

where P is a projective module, and Hom_{R}(P, Q)→Hom_{R}(M, Q) is an epimorphism for every projective module Q, equivalently every map M→Q factors through P. There is a free module F such that P is a direct summand of F, and since M is finitely generated, it is contained in a finitely generated free submodule F_{−1} ≤ F. Then it is easy to see that every map M→Q factors through F_{−1}, equivalently Hom_{R}(F_{−1}, Q)→Hom_{R}(M, Q) is an epimorphism for every projective module Q. From the short exact sequence 0→M→F_{−1}→F_{−1}/M→0 and the corresponding long exact sequence

Hom_{R}(F_{−1}, Q)→Hom_{R}(M, Q)→Ext_{ℤG}^{1}(F_{−1}/M, Q)→0

it follows that Ext_{R}^{1}(F_{−1}/M, ) vanishes on projectives. Since Gpd_{R}(F_{−1}/M) ≤ 1 it follows from Theorem 2.3 that F_{−1}/M is Gorenstein projective. Since F_{−1} is finitely generated it follows that F_{−1}/M is finitely generated. Inductively we build an exact sequence

(3)  0→M→F_{−1}→F_{−2}→ ⋯

where F_{i} is a finitely generated R-module for all i ≤ −1, and the sequence

(4)  ⋯ →Hom(F_{−2}, Q)→Hom(F_{−1}, Q)→Hom(M, Q)→0

is exact for every projective module Q.

Splicing (3) with a projective resolution of M by finitely generated modules, we obtain a complete resolution of M of finite type, which is a complete resolution in the strong sense because of (4) and the fact that Ext_{R}^{i}(M, ) vanishes on projectives for i > 0.

□

Theorem 4.3. Let G be a group in H𝔉, and M a ℤG-module with Gpd_{ℤG}M < ∞. Then pd_{ℤG}M < ∞ if and only if pd_{ℤF}M < ∞ for every finite subgroup F of G.

For the proof of the Theorem we need the following Lemma.

Lemma 4.4. Let G be a group and M a Gorenstein projective ℤGmodule. Then M is a Gorenstein projective ℤK-module for every H𝔉subgroup K of G.

Proof. K ∈ H_{α}𝔉 for some ordinal α. We use induction on α. If α = 0 then K is a finite group, so the result follows from Example 2.1(b). Assume the result is true for all subgroups in H_{β}𝔉 with β < α and let K ∈ H_{α}𝔉. Then there is a finite dimensional contractible K-CW-complex X such that each cell stabilizer belongs to H_{β}𝔉 for some β < α. Tensoring the cellular chain complex of X by M we obtain an exact sequence of ℤK-modules

0→C̀_{n} ⊗ M̀→ ⋯ →C̀_{0} ⊗ M̀→M→0

where each C̀_{i} ⊗ M̀ is a direct sum of modules of the form ℤ̀K ⊗_{ℤL} M for L ∈ H_{β}𝔉 with β < α. By the inductive hypothesis M is Gorenstein projective as a ℤL-module, so each C̀_{i}⊗M̀ is Gorenstein projective as a ℤK-module (Example 2.1(c), (e)). It follows that Gpd_{ℤK}M < ∞. We claim that Gpd_{ℤK}M = 0. Assume that Gpd_{ℤK}M = n > 0. Since M is Gorenstein projective as a ℤG-module there is a ℤG-exact sequence 0→M→P_{0}→P_{1}→ ⋯ such that P_{i} is a projective ℤG-module for all i ≥ 0. Let M_{i} = ker(P_{i}→P_{i+1}) and N = ⨁_{i≥0} M_{i}. As in the proof of Theorem 2.8 (2) it follows that N has infinite Gorenstein projective dimension over K, which is a contradiction since we have shown that if N is a Gorenstein projective ℤG-module then Gpd_{ℤK}N < ∞.

□

Proof of Theorem 4.3: The proof is analogous to the proof of Theorem 5.7 in [4], using the previous Lemma and the fact that if M is a Gorenstein projective module of finite projective dimension then M is projective (Theorem 4.2(ii)).

Theorem 4.5. Let G be in LH𝔉. Then the following statements are equivalent for a ℤG-module M:

(1) M is Gorenstein projective and of type FP_{∞}.

(2) M is a direct summand of a module M_{E} which admits a series

0 = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{n} = M_{E}

in which the sections M_{i}/M_{i−1} are of the form ℤ̀G⊗_{ℤE} V where E is a finite elementary abelian subgroup of G and V a finitely generated ℤE-module which is ℤ-free.

(3) M is a direct summand of a module M_{F} which admits a series

0 = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{n} = M_{F}

in which the sections M_{i}/M_{i−1} are of the form ℤ̀G⊗_{ℤF} N where F is a finite subgroup of G and N a finitely generated ℤEmodule which is ℤ-free.

Proof. (1)⇒(2) The proof of this follows essentially the proof of Theorem A´ (1)⇒(2) of Kropholler in [17]. One needs:

(a) A direct sum of Gorenstein projectives is Gorenstein projective. This holds by Example 2.1(e).

(b) If C is Gorenstein projective then there is a projective module P and an injection C→P. This holds by definition.

(c) If M is Gorenstein projective then the natural map

\underline{Hom}_{R}(M, N)→Êxt_{R}^{0}(M, N)

is an isomorphism for every module N. This holds by Theorem 4.2(i).

(2)⇒(3) is clear.

(3)⇒(1) If F is a finite subgroup of G and N is a finitely generated ℤF -module which is ℤ-free then ℤ̀G⊗_{ℤF} N is Gorenstein projective by Example 2.1(b)(c), and of type FP_{∞}, since N is of type FP_{∞} over ℤF. Since being Gorenstein projective and being of type FP_{∞} are extension and direct summand closed properties, it follows that M is Gorenstein projective and of type FP_{∞}.

□

Corollary 4.6. Let G be in LH𝔉. Then the following statements are equivalent for a ℤG-module M:

(1) M is Gorenstein projective and of type FP_{∞}.

(2) M is cofibrant and of type FP_{∞}.

Proof. It follows from Theorem A´ of [17] and Theorem 4.5. □

Acknowledgements. The first author would like to thank the Department of Mathematics of the University of Athens, for its hospitality during his visit in the spring semester of 2008 and the Center of Excellence for Mathematics (University of Isfahan) for partial support. The third author was partially supported by ELKE (grant #70/4/6411).

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