ON CERTAIN COHOMOLOGICAL INVARIANTS OF GROUPS

IOANNIS EMMANOUIL

Abstract. Let R be a left and right ℵ_{0}-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants silp ℤG and spli ℤG, which were introduced by Gedrich and Gruenberg in [13], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if \underline{cd} G = 0, where \underline{cd} is the generalized cohomological dimension of groups introduced by Ikenaga in [18].

0. Introduction

The classical Tate cohomology theory of finite groups [5, Chapter VI] was generalized by Farrell in [12] to the class of all groups with finite virtual cohomological dimension. In his attempt to extend the definition of Farrell cohomology to an even bigger class of groups, Ikenaga introduced in [18] the generalized cohomological dimension \underline{cd} G of a group G, by defining \underline{cd} G to be the supremum of all 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. Ikenaga showed that 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. We prove that the converse of the latter assertion is also true:

Theorem A. A group G is finite if and only if \underline{cd} G = 0.

This characterization of finiteness was conjectured in [19] by Ikenaga, who proved that the vanishing of \underline{cd} implies finiteness under the presence of an additional homological finiteness hypothesis. We are able to remove that hypothesis by translating the vanishing of the groups Ext_{ℤG}^{1}(M, P), where M and P are as above, into a Mittag-Leffler condition on certain inverse systems of Hom-groups. We note that this technique has been also used in [17], in order to establish a projectivity criterion, which was conjectured by Kaplansky, for modules over commutative domains.

The generalized cohomological dimension \underline{cd} G of G is closely related to the invariants spli ℤG and silp ℤG, which were introduced by Gedrich and Gruenberg in [13], 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, whereas silp ℤG is the supremum of the injective lengths of projective ℤG-modules. The relation between these two invariants has been studied by several authors in various contexts: Using the Hopf algebra structure of the group ring, Gedrich and Gruenberg have shown in [13, 1.6 and Corollary 5.9] that for any group G we have silp ℤG ≤ spli ℤG, with equality if spli ℤG is finite. The equality silp ℤG = spli ℤG has been established for groups in the the class H𝔉 of Kropholler (cf. [6, §6]) and, more generally, for groups in LH𝔉 (cf. [26, Corollary 1]). On the other hand, the equality between silp ℤG and spli ℤG is also known for groups G that have periodic cohomology after some steps (cf. [25, Theorem 3.2]); in that case, a result of Adem and Smith [1] shows that the finiteness of these invariants implies the existence of a free action of G on a CW-complex, which is homotopy equivalent to a sphere. We shall prove that the equality between silp ℤG and spli ℤG holds for any group G:

Theorem B. If G is any group, then silp ℤG = spli ℤG.

One may prove Theorem A using Theorem B and the main result of [7], where it is shown that the finiteness of G is equivalent to the equality spli ℤG = 1. As far as the proof of Theorem B is concerned, it suffices to establish the inequality spli ℤG ≤ silp ℤG; indeed, as we mentioned above, the reverse inequality has been proved by Gedrich and Gruenberg in [13]. We can easily reduce the proof to the case of a countable group. Then, the integral group ring ℤG is a countable ring and the proof of the inequality spli ℤG ≤ silp ℤG follows from a more general result, which we now describe.

Let R be any ring. Then, as in the special case where R is the integral group ring of a group, we may consider the injective lengths of projective left R-modules and the projective lengths of injective left R-modules, in order to define the left invariants l-silp R and l-spli R respectively. In the same way, we may consider right R-modules and define the right invariants r-silp R and r-spli R. We note that the distinction between left and right R-modules is superfluous if, for example, R is a commutative ring or the group algebra of a group. Gedrich and Gruenberg note in [13] that the relation between l-spli R and l-silp R is unclear for a general ring R and ask whether the finiteness of one implies that of the other. A positive answer to the latter question would imply the equality l-spli R = l-silp R; indeed, it is easily seen that if both invariants are finite, then they are equal. As shown by Faith and Walker in [10] and [11], the class of quasi-Frobenius rings is characterized by the vanishing of any one of the four invariants l-silp, r-silp, l-spli and r-spli. On the other hand, Jensen has proved in [21, 5.9] that if R is a commutative Noetherian ring, then we always have silp R = spli R. In the special case where R is an Artin algebra, the equality l-spli R = l-silp R is equivalent to a long-standing conjecture in representation theory, the so-called Gorenstein Symmetry Conjecture, which appears as conjecture 13 at the end of [2] (see also [3, §11] and [4, Chapter VII]). As shown by Happel in [16], the latter conjecture is closely related to the existence of Serre duality in the homotopy category of perfect complexes over R [23], which is itself a key hypothesis in Kontsevich’s formalism of Non-commutative Algebraic Geometry [22].

The ring R is called left ℵ_{0}-Noetherian if all left ideals of it are countably generated. In the same way, one defines the class of right ℵ_{0}-Noetherian rings. For example, any countable ring is both left and right ℵ_{0}-Noetherian. We prove the following result:

Theorem C. Let R be a ring which is both left and right ℵ_{0}-Noetherian. If both l-silp R and r-silp R are finite, then l-spli R = l-silp R and r-spli R = r-silp R.

As an immediate consequence of Theorem C, it follows that spli R ≤ silp R for any commutative ℵ_{0}-Noetherian ring R, with equality if silp R < ∞; this result may be viewed as a partial generalization of the result of Jensen mentioned above. As another consequence of Theorem C, we conclude that spli ℤG ≤ silp ℤG for any countable group G; as we noted above, the proof of Theorem B follows easily from this.

Instead of considering projective resolutions, we may consider flat resolutions and define the invariant l-sfli R (resp. r-sfli R) as the supremum of the weak dimensions of injective left (resp. right) R-modules. Since projective modules are flat, it is clear that l-sfli R ≤ l-spli R and r-sfli R ≤ r-spli R. Using a duality argument, we prove that we also have r-sfli R ≤ l-silp R if R is a left ℵ_{0}-Noetherian ring and l-sfli R ≤ r-silp R if R is a right ℵ_{0}-Noetherian ring. The use of flat modules and weak dimension in this setting enables us to use a surprising result of Jensen, which states that flat modules have finite projective dimension, provided that the finitistic dimension of the ring is finite (cf. [20, Proposition 6]). The proof of Theorem C is a consequence of these two results.

The contents of the paper are as follows: In Section 1, we present a few equivalent descriptions of the Mittag-Leffler condition on inverse systems of abelian groups and establish a relation between the vanishing of the Ext^{1}-group and the Mittag-Leffler condition on certain inverse systems of Hom-groups. In the following Section, we examine closely a natural transformation already considered by Ikenaga in [19] and use it in order to prove a key duality result. The latter result is then applied in Section 3, in order to relate the invariants silp and spli for ℵ_{0}-Noetherian rings. Finally, in Section 4, we establish the equality between silp and spli for integral group rings and prove that the finiteness of a group is equivalent to the vanishing of Ikenaga’s generalized cohomological dimension.

Acknowledgements. It is a pleasure to thank A. Beligiannis and O. Talelli, whose comments and suggestions were most helpful in writing this paper.

1. Preliminaries on the Mittag-Leffler condition

The goal of this preliminary Section is to record a few properties of inverse systems of abelian groups that are related to the Mittag-Leffler condition. All direct and inverse systems will be indexed by the ordered set ℕ of natural numbers.

Let (A_{n})_{n} be an inverse system of abelian groups with structural maps σ_{n,m} : A_{m} ⟶ A_{n}, n ≤ m, and consider the inverse limit A = lim_{⟵n}A_{n}, which is endowed with canonical maps s_{n} : A ⟶ A_{n}, n ≥ 0. The right derived functors of the inverse limit functor were introduced by Roos in [24]. In that paper, Roos showed that the higher inverse limits lim_{⟵}^{i} vanish for all i ≥ 2; the assumption that the inverse systems are indexed by ℕ is crucial here. For all n ≤ m we shall denote by A_{n,m} the image of σ_{n,m} : A_{m} ⟶ A_{n}; it is clear that im s_{n} ⊆ A_{n,m}. We consider for all n ≥ 0 the decreasing filtration

A_{n} = A_{n,n} ⊇ A_{n,n+1} ⊇ A_{n,n+2} ⊇ A_{n,n+3} ⊇ …

of A_{n} by the A_{n,m}’s. The inverse system (A_{n})_{n} is said to satisfy the Mittag-Leffler condition if these filtrations are eventually constant. In other words, (A_{n})_{n} satisfies the Mittag-Leffler condition if for all n ∈ ℕ there exists a suitable integer N = N(n) with N ≥ n, such that

A_{n,N} = A_{n,N+1} = A_{n,N+2} = A_{n,N+3} = …

Assuming that the inverse system (A_{n})_{n} satisfies the Mittag-Leffler condition, we shall refer to the subgroup A′_{n} = A_{n,N} ⊆ A_{n} (where N = N(n) as above) as the stable image. It is clear that the structural morphisms σ_{n,m} : A_{m} ⟶ A_{n} map the stable image A′_{m} ⊆ A_{m} onto the stable image A′_{n} ⊆ A_{n} for all n ≤ m. Therefore, we may consider the subsystem of stable images (A′_{n})_{n} ⊆ (A_{n})_{n}. It is easily seen that the inclusion (A′_{n})_{n} ↪ (A_{n})_{n} induces an isomorphism between the corresponding inverse limits lim_{⟵n}A′_{n} ≃ lim_{⟵n}A_{n}.

Lemma 1.1. Let (A_{n})_{n} be an inverse system of abelian groups with structural maps denoted by σ_{n,m} : A_{m} ⟶ A_{n}, n ≤ m, and consider the inverse limit A = lim_{⟵n}A_{n}. Then, the following conditions are equivalent:

(i) The inverse system (A_{n})_{n} satisfies the Mittag-Leffler condition.

(ii) For all n ∈ ℕ there exists an integer N = N(n) ≥ n, such that the subgroup A_{n,N} ⊆ A_{n} coincides with the image of the canonical map s_{n} : A ⟶ A_{n}.

(iii) For all n ∈ ℕ there exists an integer N = N(n) ≥ n, such that for any abelian group B the kernel of the additive map

(1)  Hom_{ℤ}(A_{n}, B) ⟶ Hom_{ℤ}(A, B),

which is induced by the canonical map s_{n} : A ⟶ A_{n}, coincides with the kernel of the additive map

(2)  Hom_{ℤ}(A_{n}, B) ⟶ Hom_{ℤ}(A_{N}, B),

which is induced by σ_{n,N} : A_{N} ⟶ A_{n}.

Proof. (i)→(ii): Assuming that the inverse system (A_{n})_{n} satisfies the Mittag-Leffler condition, let (A′_{n})_{n} be the subsystem of stable images of (A_{n})_{n} and consider the inverse limit A′ = lim_{⟵n}A′_{n}. As we have already noted, the inclusions A′_{n} ↪ A_{n} induce an isomorphism A′ ≃ A. The structural maps of the inverse system (A′_{n})_{n} being surjective, it is easily seen that the canonical maps s′_{n} : A′ ⟶ A′_{n} are surjective as well. Therefore, the commutativity of the diagram

A′ ⟶^{∼} A
s′_{n} ↓  ↓ s_{n}
A′_{n} ↪ A_{n}

shows that im s_{n} = A′_{n}. This completes the proof, since the stable image A′_{n} ⊆ A_{n} is the image of σ_{n,N} : A_{N} ⟶ A_{n} for some N ≥ n.

(ii)→(iii): Let B be an abelian group. Then, the kernel of (1) consists of those additive maps f : A_{n} ⟶ B which vanish when restricted to the image of the canonical map s_{n} : A ⟶ A_{n}. Similarly, the kernel of (2) consists of those additive maps f : A_{n} ⟶ B which vanish when restricted to the image of σ_{n,N} : A_{N} ⟶ A_{n}. Therefore, (iii) follows readily from (ii).

(iii)→(i): We fix n ∈ ℕ and choose an integer N = N(n) ≥ n as in (iii). We consider the cokernel of the canonical map s_{n} : A ⟶ A_{n} and note that the projection π : A_{n} ⟶ coker s_{n} is contained in the kernel of the additive map

Hom_{ℤ}(A_{n}, coker s_{n}) ⟶ Hom_{ℤ}(A, coker s_{n}).

Therefore, in view of our assumption, π is also contained in the kernel of the additive map

Hom_{ℤ}(A_{n}, coker s_{n}) ⟶ Hom_{ℤ}(A_{N}, coker s_{n}).

In other words, π vanishes in the image A_{n,N} of σ_{n,N} : A_{N} ⟶ A_{n}, i.e. we have A_{n,N} ⊆ im s_{n}. Then, the chain of inclusions

A_{n,N} ⊇ A_{n,N+1} ⊇ A_{n,N+2} ⊇ A_{n,N+3} ⊇ … ⊇ im s_{n} ⊇ A_{n,N}

shows that we actually have equalities

A_{n,N} = A_{n,N+1} = A_{n,N+2} = A_{n,N+3} = … = im s_{n}.

Since this is the case for all n ∈ ℕ, we conclude that the inverse system (A_{n})_{n} satisfies the Mittag-Leffler condition, as needed. □

The Mittag-Leffler condition was introduced by Grothendieck in [15], as a sufficient condition for the vanishing of lim_{⟵}^{1}. Even though this sufficient condition turns out to be necessary in the case of inverse systems of countable abelian groups (cf. [14]), in general, the vanishing of lim_{⟵}^{1} does not imply the Mittag-Leffler condition. In order to formulate a condition in terms

of the functor lim_{⟵}^{1}, which is equivalent to the Mittag-Leffler condition, one may proceed as follows: The category of inverse systems of abelian groups has arbitrary direct sums, which are computed pointwise. In particular, for any inverse system (A_{n})_{n} of abelian groups we may consider the direct sum of an infinite countable number of copies of itself. The latter is the inverse system (A_{n}^{(ℕ)})_{n}, whose structural maps A_{m}^{(ℕ)} ⟶ A_{n}^{(ℕ)} are those induced by the structural maps A_{m} ⟶ A_{n} of (A_{n})_{n} for all n ≤ m. In [9, Corollary 6], it is proved that the inverse system (A_{n})_{n} satisfies the Mittag-Leffler condition if and only if lim_{⟵n}^{1}A_{n}^{(ℕ)} = 0. We shall use that characterization of the Mittag-Leffler condition below.

We note that for any ring R and any direct system (M_{n})_{n} of left R-modules a contravariant functor 𝔉 from the category of left R-modules to that of abelian groups induces an inverse system of abelian groups (𝔉M_{n})_{n}, whose structural maps 𝔉M_{m} ⟶ 𝔉M_{n} are induced by the structural maps M_{n} ⟶ M_{m} of the direct system (M_{n})_{n} for all n ≤ m.

Proposition 1.2. (cf. [17, Example 2.4(4)]) Let R be a ring and (M_{n})_{n} a direct system of finitely generated left R-modules with direct limit M = lim_{⟶n}M_{n}. We consider a left R-module

P and assume that Ext_{R}^{1}(M, P^{(ℕ)}) = 0, where P^{(ℕ)} is the direct sum of an infinite countable number of copies of P. Then, the inverse system of abelian groups (Hom_{R}(M_{n}, P))_{n}, whose structural maps are induced by the structural maps of the direct system (M_{n})_{n}, satisfies the Mittag-Leffler condition.

Proof. It is well-known that one may express the Ext-groups of M in terms of the Ext-groups of the M_{n}’s, by means of short exact sequences

0 ⟶ lim_{⟵n}^{1} Ext_{R}^{∗−1}(M_{n}, ) ⟶ Ext_{R}^{∗}(M, ) ⟶ lim_{⟵n}Ext_{R}^{∗}(M_{n}, ) ⟶ 0.

In particular, there is a short exact sequence of abelian groups

0 ⟶ lim_{⟵n}^{1} Hom_{R}(M_{n}, P^{(ℕ)})⟶ Ext_{R}^{1}(M, P^{(ℕ)})⟶ lim_{⟵n}Ext_{R}^{1}(M_{n}, P^{(ℕ)})⟶ 0.

Our assumption about the vanishing of the group Ext_{R}^{1}(M, P^{(ℕ)}) therefore implies that

(3)  lim_{⟵n}^{1} Hom_{R}(M_{n}, P^{(ℕ)})= 0.

Since the left R-module M_{n} is finitely generated, the abelian group Hom_{R}(M_{n}, P^{(ℕ)}) can be identified with the direct sum Hom_{R}(M_{n}, P)^{(ℕ)} of an infinite countable number of copies of the group Hom_{R}(M_{n}, P). Then, the inverse system (Hom_{R}(M_{n}, P^{(ℕ)}))_{n} is identified with the direct sum of an infinite countable number of copies of the inverse system (Hom_{R}(M_{n}, P))_{n} and we may rewrite (3) into the form

lim_{⟵n}^{1} Hom_{R}(M_{n}, P)^{(ℕ)} = 0.

Using [9, Corollary 6], we conclude that the inverse system of abelian groups (Hom_{R}(M_{n}, P))_{n} satisfies the Mittag-Leffler condition, as needed. □

2. The natural transformation Φ

Let R be a ring. If P is a left R-module and D an abelian group, then the abelian group Hom_{ℤ}(P, D) of all additive maps from P to D can be endowed with the structure of a right R-module, by using the left R-module structure of P. If M is another left R-module, then we may consider the tensor product Hom_{ℤ}(P, D) ⊗_{R} M and define the map

(4)  Φ : Hom_{ℤ}(P, D) ⊗_{R} M ⟶ Hom_{ℤ}(Hom_{R}(M, P), D),

by letting Φ(f ⊗ m) be the operator which is given by g ↦ f(g(m)), g ∈ Hom_{R}(M, P), for all f ∈ Hom_{ℤ}(P, D) and m ∈ M. It is easily seen that Φ is a well-defined additive map, which is natural in P, D and M. As we are mainly interested in the dependence of Φ upon M, we shall denote the map (4) above by Φ_{M}.

In the special case where R is the group ring ℤG of a group G and P = ℤG, the natural transformation Φ was used in a crucial way by Ikenaga [19], in his attempt to prove that the vanishing of the generalized cohomological dimension \underline{cd} G implies the finiteness of G.

Our first goal is to examine some properties of Φ. In particular, we shall obtain conditions under which Φ is an isomorphism (or, at least, a monomorphism) of abelian groups.

Lemma 2.1. Let P, M be left R-modules and D an abelian group. Then, the additive map

Φ_{M} : Hom_{ℤ}(P, D) ⊗_{R} M ⟶ Hom_{ℤ}(Hom_{R}(M, P), D)

defined above is an isomorphism if one of the following two conditions is satisfied: (i) The R-module M is finitely generated and free. (ii) The abelian group D is divisible and the R-module M is finitely presented. Proof. (i) It suffices to consider the case where M = R. Then, the result is obvious since

both groups Hom_{ℤ}(P, D) ⊗_{R} R and Hom_{ℤ}(Hom_{R}(R, P), D) are identified with Hom_{ℤ}(P, D), in such a way that Φ corresponds to the identity map Hom_{ℤ}(P, D) ⟶ Hom_{ℤ}(P, D).

(ii) Let

F′ ⟶ F ⟶ M ⟶ 0

be an exact sequence of left R-modules, where F and F′ are finitely generated and free. Then, using the divisibility of D and the naturality of Φ, we obtain a commutative diagram of abelian groups with exact rows

Hom_{ℤ}(P, D) ⊗_{R} F′ → Hom_{ℤ}(P, D) ⊗_{R} F → Hom_{ℤ}(P, D) ⊗_{R} M → 0
Φ_{F′} ↓ Φ_{F} ↓ Φ_{M} ↓
Hom_{ℤ}(Hom_{R}(F′, P), D) → Hom_{ℤ}(Hom_{R}(F, P), D) → Hom_{ℤ}(Hom_{R}(M, P), D) → 0

In view of (i) above, the additive maps Φ_{F′} and Φ_{F} are bijective. Hence, using the 5-lemma, we conclude that the map Φ_{M} is bijective as well. □

Proposition 2.2. Let (M_{n})_{n} be a direct system of finitely presented left R-modules with direct limit M = lim_{⟶n}M_{n}. We consider a left R-module P and assume that Ext_{R}^{1}(M, P^{(ℕ)})= 0, where

P^{(ℕ)} is the direct sum of an infinite countable number of copies of P. Then, the additive map

Φ_{M} : Hom_{ℤ}(P, D) ⊗_{R} M ⟶ Hom_{ℤ}(Hom_{R}(M, P), D)

is injective for any divisible abelian group D.

Proof. Since the group Ext_{R}^{1}(M, P^{(ℕ)}) is trivial, we may apply Proposition 1.2 and conclude that the inverse system of abelian groups (Hom_{R}(M_{n}, P))_{n}, whose structural maps are induced by the structural maps of the direct system (M_{n})_{n}, satisfies the Mittag-Leffler condition. We consider a divisible abelian group D and let t_{n} : M_{n} ⟶ M be the canonical map for all n ∈ ℕ. In view of the naturality of Φ, the diagram

Hom_{ℤ}(P, D) ⊗_{R} M_{n} ⟶^{Φ_{Mn}} Hom_{ℤ}(Hom_{R}(M_{n}, P), D)
1⊗t_{n} ↓  ↓ t_{n∗}
Hom_{ℤ}(P, D) ⊗_{R} M ⟶^{Φ_{M}} Hom_{ℤ}(Hom_{R}(M, P), D)

is commutative for all n ∈ ℕ. We note that the abelian group Hom_{ℤ}(P, D) ⊗_{R} M is the direct limit of the direct system (Hom_{ℤ}(P, D) ⊗_{R} M_{n})_{n}. Therefore, in order to prove the injectivity of Φ_{M}, it suffices to show that

ker (t_{n∗} ∘ Φ_{M_{n}}) ⊆ ker (1 ⊗ t_{n})

for all n ∈ ℕ. To that end, we fix a non-negative integer n. The abelian group Hom_{R}(M, P) can be naturally identified with the inverse limit of the system (Hom_{R}(M_{n}, P))_{n}, in such a way that the canonical map from the inverse limit to Hom_{R}(M_{n}, P) is identified with the map

Hom_{R}(M, P) ⟶ Hom_{R}(M_{n}, P),

which is induced by t_{n} : M_{n} ⟶ M. Since the inverse system (Hom_{R}(M_{n}, P))_{n} satisfies the Mittag-Leffler condition, we may use Lemma 1.1 in order to find an integer N = N(n) ≥ n, such that the kernel of

t_{n∗} : Hom_{ℤ}(Hom_{R}(M_{n}, P), D) ⟶ Hom_{ℤ}(Hom_{R}(M, P), D)

coincides with the kernel of the additive map

τ_{n,N∗} : Hom_{ℤ}(Hom_{R}(M_{n}, P), D) ⟶ Hom_{ℤ}(Hom_{R}(M_{N}, P), D) ,

which is induced by the structural map τ_{n,N} : M_{n} ⟶ M_{N}. We now consider the commutative diagram

Hom_{ℤ}(P, D) ⊗_{R} M_{n} ⟶^{Φ_{Mn}} Hom_{ℤ}(Hom_{R}(M_{n}, P), D)
1⊗τ_{n,N} ↓  ↓ τ_{n,N∗}
Hom_{ℤ}(P, D) ⊗_{R} M_{N} ⟶^{Φ_{M_{N}}} Hom_{ℤ}(Hom_{R}(M_{N}, P), D)

The R-module M_{N} being finitely presented, the additive map Φ_{M_{N}} is bijective (Lemma 2.1(ii)) and hence

(5)  ker (τ_{n,N∗} ∘ Φ_{M_{n}}) = ker (1 ⊗ τ_{n,N}).

Since t_{N} ∘ τ_{n,N} = t_{n}, it follows that (1 ⊗ t_{N}) ∘ (1 ⊗ τ_{n,N}) = 1 ⊗ t_{n} and hence we conclude that

(6)  ker (1 ⊗ τ_{n,N}) ⊆ ker (1 ⊗ t_{n}).

On the other hand, in view of our choice of N ∈ ℕ, we have ker t_{n∗} = ker τ_{n,N∗} and hence

(7)  ker (t_{n∗} ∘ Φ_{M_{n}}) = Φ_{M_{n}}^{−1}(ker t_{n∗})= Φ_{M_{n}}^{−1}(ker τ_{n,N∗})= ker (τ_{n,N∗} ∘ Φ_{M_{n}}).

Combining (5), (6) and (7), it follows that ker (t_{n∗} ∘ Φ_{M_{n}}) ⊆ ker (1 ⊗ t_{n}), as needed. □

A left R-module is called countably presented if it is the cokernel of a linear map between countably generated free left R-modules. It is easily seen that the class of countably presented modules coincides with the class of those modules that may be expressed as direct limits of finitely presented modules. (We remind the reader of our convention that all direct systems be indexed by the ordered set of natural numbers.) We may therefore restate Proposition 2.2 as follows:

Proposition 2.3. Let P, M be left R-modules and assume that M is countably presented. If Ext_{R}^{1}(M, P^{(ℕ)}) = 0, where P^{(ℕ)} is the direct sum of an infinite countable number of copies of P, then the additive map

Φ_{M} : Hom_{ℤ}(P, D) ⊗_{R} M ⟶ Hom_{ℤ}(Hom_{R}(M, P), D)

is injective for any divisible abelian group D. □

Having fixed a left R-module P and a divisible abelian group D, we consider for any left R-module M a projective resolution

F_{∗} ⟶ M ⟶ 0.

Then, the natural transformation Φ induces a chain map

Φ_{F_{∗}} : Hom_{ℤ}(P, D) ⊗_{R} F_{∗} ⟶ Hom_{ℤ}(Hom_{R}(F_{∗}, P), D).

By applying homology, we obtain additive maps

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom_{ℤ}(P, D), M)⟶ Hom_{ℤ}(Ext_{R}^{n}(M, P), D),

n ≥ 0, which do not depend upon the particular choice of the projective resolution of M. It is clear that Φ_{M}^{(0)} can be identified with the map Φ_{M} studied before. Moreover, the Φ_{M}^{(n)}’s are natural in M and commute with the connecting homomorphisms, which are associated with any short exact sequence of left R-modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0.

In other words, Φ induces a morphism of homological exact ∂-functors

Φ^{(∗)} : Tor_{∗}^{R}(Hom_{ℤ}(P, D), )⟶ Hom_{ℤ}(Ext_{R}^{∗}( , P), D).

Let n be a non-negative integer and assume that the projective resolution F_{∗} ⟶ M ⟶ 0 is such that F_{i} is finitely generated for i = n, n+1. (Such a resolution exists if, for example, M is a module of type FP_{n+1}.) Then, the map Φ_{M}^{(n)} defined above is bijective; this is essentially shown by Ikenaga in [19, Proposition 1.7]. In the following result, we show that one may relax this homological finiteness condition on M and still be able to prove the injectivity of Φ_{M}^{(n)}, under the additional assumption that a certain Ext-group vanishes.

Proposition 2.4. Let n be a non-negative integer and consider two left R-modules P and M. We assume that:

(i) M admits a free resolution F_{∗} ⟶ M ⟶ 0, such that F_{i} is countably generated for i = n, n + 1 and

(ii) Ext_{R}^{n+1}(M, P^{(ℕ)}) = 0, where P^{(ℕ)} is the direct sum of an infinite countable number of copies of P. Then, the additive map

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom_{ℤ}(P, D), M) ⟶ Hom_{ℤ}(Ext_{R}^{n}(M, P), D)

is injective for any divisible abelian group D.

Proof. Let K be the cokernel of the map F_{n+1} ⟶ F_{n}. In view of our assumption (i), K is a countably presented left R-module which fits to an exact sequence

(8)  0 ⟶ K ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0.

Since Ext_{R}^{1}(K, P^{(ℕ)}) = Ext_{R}^{n+1}(M, P^{(ℕ)}) = 0, we may use Proposition 2.3 and conclude that the additive map

Φ_{K} : Hom_{ℤ}(P, D) ⊗_{R} K ⟶ Hom_{ℤ}(Hom_{R}(K, P), D)

is injective. By applying dimension shifting to the exact sequence (8), we obtain (composing connecting homomorphisms) injective additive maps

Tor_{n}^{R}(Hom_{ℤ}(P, D), M) ⟶ Hom_{ℤ}(P, D) ⊗_{R} K

and

Hom_{ℤ}(Ext_{R}^{n}(M, P), D) ⟶ Hom_{ℤ}(Hom_{R}(K, P), D).

Since Φ^{(∗)} commutes with connecting homomorphisms, we conclude that the following diagram of abelian groups (whose top raw is exact) is commutative

0 ⟶ Tor_{n}^{R}(Hom_{ℤ}(P, D), M) ⟶ Hom_{ℤ}(P, D) ⊗_{R} K
Φ_{M}^{(n)} ↓  ↓ Φ_{K}^{(0)}
Hom_{ℤ}(Ext_{R}^{n}(M, P), D) ⟶ Hom_{ℤ}(Hom_{R}(K, P), D)

Therefore, the injectivity of Φ_{M}^{(n)} follows from the injectivity of Φ_{K}^{(0)} = Φ_{K}. □

Corollary 2.5. Let n be a non-negative integer and consider a left R-module M. We assume that:

(i) M admits a free resolution F_{∗} ⟶ M ⟶ 0, such that F_{i} is countably generated for i = n, n + 1,

(ii) Ext_{R}^{n+1}(M, R^{(ℕ)}) = 0, where R^{(ℕ)} is the direct sum of an infinite countable number of copies of R, and

(iii) Ext_{R}^{n}(M, R) = 0. Then, Tor_{n}^{R}(I, M) = 0 for any injective right R-module I.

Proof. Since any injective right R-module is a direct summand of a cofree module, it suffices to consider the case where I = Hom_{ℤ}(R, D) for some divisible abelian group D. In that case, the vanishing of the group Tor_{n}^{R}(Hom_{ℤ}(R, D), M) follows from Proposition 2.4 (by letting P = R therein). □

3. The invariants silp and spli for ℵ_{0}-Noetherian rings

Let R be a ring. In connection with the existence of complete cohomological functors in the category of left R-modules, Gedrich and Gruenberg have defined in [13] the invariant l-silp R as the supremum of the injective lengths of projective left R-modules and the invariant l-spli R as the supremum of the projective lengths of injective left R-modules. In the same way, we may consider right R-modules and define the right invariants r-silp R and r-spli R. We note that, instead of considering projective resolutions, we may consider flat resolutions and define the invariants l-sfli R (resp. r-sfli R) as the supremum of the weak dimensions of injective left (resp. right) R-modules. Since projective modules are flat, it is clear that l-sfli R ≤ l-spli R and r-sfli R ≤ r-spli R.

Remark 3.1. Let R be a left ℵ_{0}-Noetherian ring, i.e. a ring all of whose left ideals are countably generated. Then, it is easy to show that any submodule of the left R-module R^{n} is countably generated for all n ≥ 1. Moreover, the same is true for all submodules of the direct sum R^{(ℕ)} of an infinite countable number of copies of the left R-module R. We conclude that a left R-module M is countably presented if and only if it is countably generated. Moreover, such an M possesses a resolution by countably generated free left R-modules.

Proposition 3.2. Let R be a left ℵ_{0}-Noetherian ring. Then, r-sfli R ≤ l-silp R.

Proof. We note that there is nothing to prove if l-silp R = ∞ and hence we may assume that l-silp R = n < ∞. We have to show that any injective right R-module has weak dimension ≤ n. Let I be an injective right R-module.

In view of the hypothesis made on R, we know that any countably generated left R-module M admits a resolution by countably generated free modules (cf. Remark 3.1). Our assumption about the value of l-silp R implies that the functors Ext_{R}^{n+1}( , P) and Ext_{R}^{n+2}( , P) are identically zero for any projective left R-module P. Therefore, we may apply Corollary 2.5 and conclude that Tor_{n+1}^{R}(I, M) = 0. Since this is the case for any countably generated left R-module M, the continuity of the Tor-functors implies that Tor_{n+1}^{R}(I, ) = 0. It follows that I has weak dimension ≤ n, as needed. □

Recall that the left finitistic dimension l-fin.dim R of a ring R is defined as the supremum of the projective dimensions of those left R-modules that have finite projective dimension. Therefore, the finiteness of l-fin.dim R is equivalent to the assertion that there is a uniform upper bound on the projective dimension of those left R-modules that have finite projective dimension. In the same way, we may consider right R-modules and define the right finitistic dimension r-fin.dim R.

Proposition 3.3. Let R be a ring, such that r-sfli R < ∞. Then, r-spli R ≤ r-fin.dim R.

Proof. Assume that r-sfli R = n < ∞. The inequality to be proved is obvious if r-fin.dim R = ∞ and hence we may assume that r-fin.dim R < ∞. Then, in order to show that r-spli R ≤ r-fin.dim R, it suffices to show that any injective right R-module has finite projective dimension. (Indeed, it would then follow that any injective right R-module has projective dimension bounded by the right finitistic dimension r-fin.dim R.)

Let I be an injective right R-module and consider an exact sequence of right R-modules

(9)  0 ⟶ M ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ I ⟶ 0,

with F_{i} projective for all i = 0, 1, …, n − 1. Since r-sfli R = n, the right R-module I has weak dimension ≤ n. The projective right R-module F_{i} is flat for all i = 0, 1, …, n − 1 and hence the right R-module M is also flat. As shown by Jensen in [20, Proposition 6], the finiteness of the right finitistic dimension r-fin.dim R of R implies that any flat right R-module has finite projective dimension; in particular, the right R-module M has finite projective dimension. In view of the exact sequence (9), we have Ext_{R}^{i+n}(I, ) = Ext_{R}^{i}(M, ) for all i ≥ 1 and hence the right R-module I has finite projective dimension as well. □

Corollary 3.4. Let R be a left ℵ_{0}-Noetherian ring, such that l-silp R < ∞. Then, r-spli R ≤ r-fin.dim R.

Proof. The assumption made on R implies that r-sfli R ≤ l-silp R (Proposition 3.2). Hence, r-sfli R is finite and the result follows from Proposition 3.3. □

By symmetry, i.e. by applying all of the above to the opposite R^{op} of a ring R, we obtain analogous results for right ℵ_{0}-Noetherian rings, i.e. for rings all of whose right ideals are countably generated. In particular, we have:

Corollary 3.4^{op}. Let R be a right ℵ_{0}-Noetherian ring, such that r-silp R < ∞. Then, l-spli R ≤ l-fin.dim R. □

Our next goal is to combine Corollaries 3.4 and 3.4^{op}, in order to obtain a left-right symmetric assertion. To that end, we shall need the following result, which is certainly well-known:

Lemma 3.5. If R is a ring, then l-fin.dim R ≤ l-silp R and r-fin.dim R ≤ r-silp R.

Proof. By symmetry, it suffices to prove the first one of the two inequalities. To that end, let M be a left R-module of finite projective dimension, say equal to m. Then, there is a projective left R-module P, such that Ext_{R}^{m}(M, P) ≠ 0. In particular, such a module P has injective dimension ≥ m and hence l-silp R ≥ m. The result follows since l-fin.dim R is the supremum of such m’s. □

We can now prove the following result (Theorem C of the Introduction):

Theorem 3.6. Let R be a ring which is both left and right ℵ_{0}-Noetherian. If both l-silp R and r-silp R are finite, then l-spli R = l-silp R and r-spli R = r-silp R.

Proof. It follows from Corollaries 3.4 and 3.4^{op} that r-spli R ≤ r-fin.dim R and l-spli R ≤ l-fin.dim R. Using Lemma 3.5, we conclude that r-spli R ≤ r-silp R and l-spli R ≤ l-silp R. In particular, r-spli R and l-spli R are also finite. As noted by Gedrich and Gruenberg in [13, 1.6], the finiteness of both l-silp R and l-spli R (resp. of both r-silp R and r-spli R) implies that l-silp R = l-spli R (resp. that r-silp R = r-spli R). □

If a ring R is isomorphic with its opposite R^{op}, then any left R-module M may be identified with a right R-module M′ and vice versa, in such a way that M and M′ have the same homological properties (in particular, the same projective and injective dimensions) as left and right modules respectively. In that case, we have l-spli R = r-spli R(= spli R) and l-silp R = r-silp R(= silp R). The following result is an immediate consequence of Theorem 3.6:

Corollary 3.7. Let R be a ring which is isomorphic with its opposite R^{op}. If R is left (and hence right) ℵ_{0}-Noetherian, then spli R ≤ silp R, with equality if silp R < ∞. □

Since commutative rings are obviously isomorphic with their opposites, we obtain the following partial generalization of a result of Jensen, who proved in [21, 5.9] that spli R = silp R if R is a commutative Noetherian ring.

Corollary 3.8. If R is a commutative ℵ_{0}-Noetherian ring, then spli R ≤ silp R, with equality if silp R < ∞. □

4. The case of group rings

Let k be a commutative ring, G a group and R = kG the corresponding group ring. Then, R is isomorphic with the opposite ring R^{op} and hence the distinction between left and right modules is redundant. Assume that the group G is countable and the commutative ring k is either countable or a field. Then, R is a left (and hence right) ℵ_{0}-Noetherian ring and hence we may apply Corollary 3.7, in order to conclude that spli kG ≤ silp kG. We shall prove that the latter inequality is valid for any group G.

Lemma 4.1. Let k be a commutative ring, G a group and M a kG-module. If M is flat as a kH-module for all countable subgroups H of G, then M is flat as a kG-module as well.

Proof. Let f : N′ ⟶ N be an injective kG-linear map. We have to show that the additive map

(10)  1_{M} ⊗ f : M ⊗_{kG} N′ ⟶ M ⊗_{kG} N

is also injective. Since the group G is the filtered colimit of its countable subgroups H, it is easily seen that the additive map above is the filtered colimit of the additive maps

1_{M} ⊗ f : M ⊗_{kH} N′ ⟶ M ⊗_{kH} N

over the countable subgroups H of G. The latter maps are injective, in view of our assumption about the flatness of M over kH, and hence the injectivity of (10) follows since filtered colimits preserve exactness. □

Proposition 4.2. Let G be a group and k a commutative ring, which is either countable or a field. Then, we have sfli kG ≤ silp kG.

Proof. There is nothing to prove if silp kG = ∞ and hence we may assume that silp kG = n < ∞. We have to show that any injective kG-module has weak dimension ≤ n. Let I be an injective kG-module and consider an exact sequence of kG-modules

(11)  0 ⟶ M ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ I ⟶ 0,

with F_{i} projective for all i = 0, 1, …, n − 1. If H is a countable subgroup of G, then kH is a left ℵ_{0}-Noetherian ring and hence we may apply Proposition 3.2 in order to conclude that sfli kH ≤ silp kH. Since silp kH ≤ silp kG = n (cf. [13, 5.1(iii)]), it follows that sfli kH ≤ n. Therefore, the kH-module I being injective, it has weak dimension ≤ n. Since the kH-module F_{i} is projective and hence flat for all i = 0, 1, …, n − 1, it follows that M is flat as a kHmodule. This is the case for any countable subgroup H of G and hence Lemma 4.1 implies that the kG-module M is flat. Then, the exact sequence (11) is a resolution of I by flat kG-modules of length n, as needed. □

Proposition 4.3. Let G be a group and k a commutative ring, which is either countable or a field. Then, we have spli kG ≤ silp kG.

Proof. The inequality is obvious if silp kG = ∞ and hence we may assume that silp kG < ∞. Then, we also have sfli kG < ∞ (Proposition 4.2) and hence spli kG ≤ fin.dim kG (Proposition 3.3). Since fin.dim kG ≤ silp kG (Lemma 3.5), it follows that spli kG ≤ silp kG. □

We recall that the self-injective dimension of a commutative ring k is the injective dimension of k as a k-module.

Theorem 4.4. Let G be a group and k a commutative ring, which is either countable Noetherian of finite self-injective dimension or a field. Then, we have silp kG = spli kG.

Proof. If silp kG is finite, then spli kG is also finite, in view of Proposition 4.3. On the other hand, Gedrich and Gruenberg have proved in [13, Theorem 2.4] that the finiteness of spli kG implies that silp kG < ∞. The equality in the statement of the Theorem therefore follows from [13, 1.6]. □

Since the self-injective dimension of ℤ is 1, the following result (which is precisely Theorem B of the Introduction) is an immediate consequence of Theorem 4.4.

Corollary 4.5. If G is a group, then we have silp ℤG = spli ℤG. □

Recall that Ikenaga has defined in [18] the generalized cohomological dimension \underline{cd} G of a group G to be the supremum of all 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. As an application of Corollary 4.5, we shall now prove the following finiteness criterion for G, which complements Theorem A of the Introduction.

Theorem 4.6. The following conditions are equivalent for a group G:

(i) \underline{cd} G = 0, (ii) silp ℤG = 1, (iii) spli ℤG = 1, (iv) G is finite. Proof. (i)→(ii): If \underline{cd} G = 0, then it is easily seen that projective ℤG-modules have injective

dimension ≤ 1 (cf. [19, Lemma 1.8(a)]).

(ii)→(iii): This follows from Corollary 4.5. (iii)→(iv): This is the main result of [7]. (iv)→(i): This is proved in [18, §II, Corollary 2]). □

Dembegioti and Talelli conjectured in [8, Conjecture A] that the invariants \underline{cd} G and spli ℤG of G are related by the equality spli ℤG = \underline{cd} G + 1. The results obtained above provide some evidence for the validity of that equality:

Corollary 4.7. Let G be a group. Then:

(i) We always have \underline{cd} G ≤ spli ℤG ≤ \underline{cd} G + 1, (ii) The equality spli ℤG = \underline{cd} G + 1 holds true if \underline{cd} G = 0. (iii) The equality spli ℤG = \underline{cd} G + 1 holds true if \underline{cd} G = 1. Proof. (i) As noted in [19, Lemma 1.8], we have \underline{cd} G ≤ silp ℤG ≤ \underline{cd} G + 1 and hence the

result follows from Corollary 4.5.

(ii) This follows from Theorem 4.6. (iii) If \underline{cd} G = 1, then 1 ≤ spli ℤG ≤ 2 (in view of (i) above). Therefore, if we assume that

spli ℤG ≠ 2, then it would follow that spli ℤG = 1. Using Theorem 4.6 once more, this would imply that \underline{cd} G = 0, a contradiction. □

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

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