ON THE FLAT LENGTH OF INJECTIVE MODULES

IOANNIS EMMANOUIL AND OLYMPIA TALELLI

Abstract. In this paper, we use the notion of strictly Mittag-Leffler modules, in order to study the flat length of injective modules over a ring R. We show that the supremum of these flat lengths is closely related to the invariants silp R and spli R, that were defined by Gedrich and Gruenberg, as well as to the finitistic dimension of R and the injective length of the regular module. Finally, we examine the special case where R = ℤG is the integral group ring of a group G.

Contents

0. Introduction 1 1. Strictly Mittag-Leffler modules 4 2. The invariant sfli R and related invariants 10 3. On the relation between id_{R}R and silp R 15 4. The special case of group rings 19 References 24

0. Introduction

Gedrich and Gruenberg introduced in [17] the invariants spli R and silp R of a ring R, in connection with the existence of complete cohomological functors in the category of left Rmodules. Here, spli R is the supremum of the projective lengths of injective left R-modules, whereas silp R is the supremum of the injective lengths of projective left R-modules. In the same way, we may consider right R-modules and define the right invariants spli R^{op} and silp R^{op} respectively. Gedrich and Gruenberg note that the relation between spli R and 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 spli R = silp R; indeed, it is easily seen that if both invariants are finite, then they are equal. As shown by Faith and Walker in [15] and [16], the class of quasi-Frobenius rings is characterized by the vanishing of any one of the four invariants spli R, silp R, spli R^{op} and silp R^{op}. On the other hand, Jensen has proved in [25, 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 spli R = 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].

In the special case where R = ℤG is the integral group ring of a group G, there is no distinction between left and right modules, in view of the Hopficity of ℤG, whereas the invariants spli ℤG and silp ℤG are always equal to each other (cf. [13]). The finiteness of these invariants has important geometric consequences. For example, if the group G has periodic cohomology after some steps, then the finiteness of these invariants is equivalent to the existence of a free action of G on a finite dimensional CW-complex, which is homotopy equivalent to a sphere (cf. [1] and [36]). In general, it is conjectured in [37] that the finiteness of these invariants is equivalent to the existence of a finite dimensional model for the classifying space \underline{E}G for proper actions of G. The latter conjecture has been proved for a big class of groups in [38].

In this paper, we study the above invariants and, in particular, their finiteness, by means of sfli R, which is defined for any ring R as the supremum of the flat lengths of injective left R-modules. Even though we are mainly interested in the relation between spli R and silp R, the invariant sfli R seems to be a very useful intermediary.

The comparison between spli R and sfli R is essentially a problem about the relation between the projective and flat dimension of modules and eventually reduces to questions about the projectivity of flat modules. In that direction, it is worth mentioning a result due to Raynaud and Gruson. Even though they were mainly interested in modules over commutative rings (and, more generally, quasi-coherent sheaves on schemes), they established in [32] a criterion for a countably presented flat left module M over a not necessarily commutative ring R to be projective. As shown by Lazard [29], such a module M can be expressed as the direct limit of a countable direct system (M_{i})_{i} of finitely generated free left R-modules. Raynaud and Gruson proved that M is projective if and only if the inverse system of abelian groups (Hom_{R}(M_{i}, R))_{i} satisfies the (strict) Mittag-Leffler condition. The corresponding notion of (strictly) Mittag-Leffler modules has been successfully used during the last few years, in order to solve problems in homological algebra and representation theory (cf. [6], [7], [13] and [19]). For a thorough and systematic study of this notion, the reader is referred to [20].

On the other hand, one may relate the invariants sfli R and silp R, by means of a certain natural transformation Φ, which involves both Tor and Ext groups. This natural transformation has been introduced by Cartan and Eilenberg in [10, Chapter VI, §5], who used it in order to obtain duality results. It is very interesting to note that the strictly Mittag-Leffler modules of Raynaud and Gruson can be characterized by means of the injectivity of the natural transformation Φ. This characterization of strictly Mittag-Leffler modules, which is due to Raynaud and Gruson [32], Zimmermann [39] and Hügel and Herbera [20], is the main technical tool used in this paper.

We shall now describe some of the main results that are proved in the paper. It is worth pointing out that even though these results can be stated without mentioning the invariant sfli or the notion of strictly Mittag-Leffler modules, the proofs that we provide make essential use precisely of the invariant sfli and the notion of strictly Mittag-Leffler modules.

It is well-known that the left and right global dimensions of a ring R may be different; examples of such rings have been found by Small in [33] and [34]. Nevertheless, as shown by Jensen in [23], if the ring R is left and right ℵ_{0}-Noetherian, then these invariants are either equal or else they differ by one. Jensen proved this result by relating these global dimensions to the weak dimension of the ring, which is itself left-right symmetric. Using the invariant sfli along the same lines, we shall prove the following result (cf. Theorem 2.9):

Theorem A. Let R be a ring, which is left and right ℵ_{0}-Noetherian. If the invariants spli R and spli R^{op} are finite, then they are either equal to each other or else they differ by one.

The finitistic dimension 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. It is easily seen that we always have fin.dim R ≤ silp R. On the other hand, we also have id_{R}R ≤ silp R and the latter inequality may be strict. Using the fact that projective modules are strictly Mittag-Leffler, we shall prove the following result (cf. Corollaries 2.20 and 2.26):

Theorem B. If the ring R is either left perfect or countable, then fin.dim R ≤ id_{R}R.

As we noted above, the inequality id_{R}R ≤ silp R may very well be strict. In order to study the relation between id_{R}R and silp R, one has to examine the relation between the functors Ext_{R}^{n}( , R) and Ext_{R}^{n}( , R^{(Λ)}), where Λ is any set. Using a characterization of the modules whose syzygies satisfy the strict Mittag-Leffler condition, we shall prove the following result (cf. Corollary 3.2):

Theorem C. Let R be a left ℵ_{0}-Noetherian ring. Then, silp R = id_{R}R^{(ℕ)}.

As we have pointed out above, Jensen has proved in [25, 5.9] that spli R = silp R if R is a commutative Noetherian ring. As a partial generalization of that result, it is proved in [13] that spli R = silp R for any ℵ_{0}-Noetherian commutative ring R if, in addition, the invariant silp R is finite. Using the natural transformation Φ, we shall prove the following result (cf. Corollary 3.8):

Theorem D. Let R be a ring, which is isomorphic with its opposite R^{op} and is left (and hence right) ℵ_{0}-Noetherian. If id_{R}R < ∞, then spli R = silp R.

Let G be a group of type F P, i.e. assume that the trivial ℤG-module ℤ admits a resolution of finite length by finitely generated projective ℤG-modules. Then, it is well-known that the cohomological dimension of G is the largest integer n such that H^{n}(G, ℤG) ≠ 0. Using the natural transformation Φ, we shall prove the following result (cf. Theorem 4.8(ii)):

Theorem E. Let G be a countable group and assume that cd G = n. Then, H^{n}(G, ℤG) ≠ 0.

Using the equality between the invariants spli ℤG and silp ℤG, together with Theorem B, we can show that for any countable group G the invariants id_{ℤG}ℤG and silp ℤG can differ only if the former is finite and the latter infinite. In other words, we shall prove the following result (cf. Theorem 4.8(iii)), which should be compared with Theorem C above:

Theorem F. Let G be a countable group and assume that silp ℤG is finite. Then, silp ℤG = id_{ℤG}ℤG.

Ikenaga has defined [21] the generalized cohomological dimension \underline{cd} G of a group G as the supremum of those integers n, for which there exist a ℤ-free ℤG-module M and a projective ℤG-module P, such that Ext_{ℤG}^{n}(M, P) ≠ 0. Ikenaga has also defined a generalized homological dimension for G. Modifying slightly Ikenaga’s definition, we define the generalized homological dimension \underline{hd} G as the supremum of those integers n, for which there exist a ℤ-free ℤGmodule M and an injective ℤG-module I, such that Tor_{n}^{ℤG}(I, M) ≠ 0. Using the natural transformation Φ, we shall prove the following result (cf. Theorem 4.11):

Theorem G. For any group G we have \underline{hd} G ≤ \underline{cd} G.

The contents of the paper can be described as follows: In Section 1, we first present a few equivalent descriptions of the strict Mittag-Leffler condition on inverse systems of abelian groups. We then examine the strictly Mittag-Leffler modules, which are defined in terms of a certain property that the representation of them as a direct limit of finitely presented modules may enjoy, and then relate that property to the injectivity of a natural transformation. By deriving the latter natural transformation, we obtain a useful means of relating certain Tor and Ext groups. In Section 2, we consider the invariant sfli and relate its finiteness to that of other invariants of the ring, in particular in the ℵ_{0}-Noetherian case. We examine the relation between the invariants sfli and spli and then use our results, in order to bound the finitistic dimension of a ring R by the injective dimension of the left regular module, in the case where R is either left perfect or countable. In Section 3, we study the relation between the invariant silp and the injective dimension of the left regular module, in the case where R is left ℵ_{0}-Noetherian, and then examine the relation between the finiteness of the invariants silp and spli as well as the relation between the left and right invariants. Finally, in Section 4, we consider the special case where R is the integral group ring of a group G and relate the generalized homological and cohomological dimensions of G.

Notations and terminology. For any two abelian groups M, N we shall denote by Hom(M, N) the group Hom_{ℤ}(M, N) of all additive maps from M to N. In the same way, we shall denote the tensor product M ⊗_{ℤ} N simply by M ⊗ N. If R is a ring, then we shall denote by R^{op} its opposite ring. We shall not distinguish between right R-modules and left R^{op}-modules. In particular, if λ(R) is an invariant, which is defined for any ring R in terms of a certain class of left R-modules, then we shall denote by λ(R^{op}) the corresponding invariant, which is defined for R in terms of the appropriate class of right R-modules.

Acknowledgements. It is a pleasure to thank A. Beligiannis for his helpful comments and suggestions.

1. Strictly Mittag-Leffler modules

In this Section, we examine a few properties that a module may enjoy, as far as the representation of it as a direct limit of finitely presented modules is concerned. These properties have been successfully used during the last few years, in order to solve problems in homological algebra and representation theory (cf. [6], [7], [13] and [19]). For a thorough and systematic study of these properties, the reader is referred to [20].

I. The strict Mittag-Leffler condition. We shall begin by recording some properties of inverse systems of abelian groups that are related to the Mittag-Leffler condition. We shall consider a directed set I and assume that all direct and inverse systems are indexed by I.

Let (A_{i})_{i} be such an inverse system of abelian groups with structural maps σ_{ij} : A_{j} ⟶ A_{i}, i ≤ j, and consider the inverse limit A = lim_{⟵i}A_{i}, which is endowed with canonical maps s_{i} : A ⟶ A_{i}, i ∈ I. The inverse system (A_{i})_{i} is said to satisfy the Mittag-Leffler condition if for all i ∈ I there exists an index j = j(i) ∈ I with j ≥ i, such that

im(A_{j} ⟶^{σ_{ij}} A_{i})= im(A_{k} ⟶^{σ_{ik}} A_{i})

for all k ∈ I with k ≥ j. Assuming that the inverse system (A_{i})_{i} satisfies the Mittag-Leffler condition, we shall refer to the subgroup A′_{i} = im(A_{j} ⟶^{σ_{ij}} A_{i}), where j = j(i) as above, as the stable image. We say that the inverse system (A_{i})_{i} satisfies the strict Mittag-Leffler condition if for all i ∈ I there exists an index j = j(i) ∈ I with j ≥ i, such that

im(A_{j} ⟶^{σ_{ij}} A_{i})= im(A ⟶^{s_{i}} A_{i}) .

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

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

(ii) The inverse system (A_{i})_{i} satisfies the Mittag-Leffler condition and the stable image A′_{i} coincides with the image of the canonical map s_{i} : A ⟶ A_{i} for all i ∈ I.

(iii) For all i ∈ I there exists an index j = j(i) ∈ I with j ≥ i, such that for any abelian group B the kernel of the additive map

(1)  Hom(A_{i}, B) ⟶ Hom(A, B),

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

(2)  Hom(A_{i}, B) ⟶ Hom(A_{j}, B),

which is induced by σ_{ij} : A_{j} ⟶ A_{i}.

Proof. The implication (i)→(ii) follows since for any i, j, k ∈ I with i ≤ j ≤ k we have

im(A_{j} ⟶^{σ_{ij}} A_{i})⊇ im(A_{k} ⟶^{σ_{ik}} A_{i})⊇ im(A ⟶^{s_{i}} A_{i}),

whereas the implication (ii)→(i) follows from the definitions.

In order to show that (i)→(iii), let us fix an abelian group B. Then, the kernel of (1) consists of those additive maps f : A_{i} ⟶ B, which vanish when restricted to the image of the canonical map s_{i} : A ⟶ A_{i}. Similarly, the kernel of (2) consists of those additive maps f : A_{i} ⟶ B, which vanish when restricted to the image of σ_{ij} : A_{j} ⟶ A_{j}. Therefore, (iii) follows readily from (i).

In order to show that (iii)→(i), we fix i ∈ I and choose an index j ∈ I with j ≥ i as in (iii). We consider the cokernel of the canonical map s_{i} : A ⟶ A_{i} and note that the projection π : A_{i} ⟶ coker s_{i} is contained in the kernel of the additive map

Hom(A_{i}, coker s_{i}) ⟶ Hom(A, coker s_{i}).

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

Hom(A_{i}, coker s_{i}) ⟶ Hom(A_{j}, coker s_{i}).

In other words, π vanishes in the image of σ_{ij} : A_{j} ⟶ A_{i}, i.e. we have

im(A_{j} ⟶^{σ_{ij}} A_{i})⊆ ker π = im(A ⟶^{s_{i}} A_{i}).

The reverse inclusion holds as well (as we noted in the beginning of the proof) and hence we conclude that

im(A_{j} ⟶^{σ_{ij}} A_{i})= im(A ⟶^{s_{i}} A_{i}),

as needed. □

Remark 1.2. We may obtain a statement equivalent to those of Lemma 1.1, by imposing the condition in assertion (iii) therein only to the case where the abelian group B is divisible. Indeed, in the proof of the implication (iii)→(i), we could have taken B to be a divisible group containing coker s_{i} and work with the composition

A_{i} ⟶^{π} coker s_{i} ↪ B,

where π is the projection map.

The Mittag-Leffler condition was introduced by Grothendieck in [18, §13.1.2]. In the special case where the index set I is countable, Grothendieck showed that the Mittag-Leffler condition implies the vanishing of lim_{⟵}^{1}, the first right derived functor of the inverse limit functor lim_{⟵}. Moreover, in the case where I is countable, an inverse system (A_{i})_{i} satisfies the Mittag-Leffler condition if and only if it satisfies the strict Mittag-Leffler condition. This is implicit in [18, §13.2.2]; see also [20, Lemma 3.3] and [13, Lemma 1.1].

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

(3)  Φ : Hom(N, D) ⊗_{R} M ⟶ Hom(Hom_{R}(M, N), D),

by letting Φ(f ⊗ m) be the operator which is given by g ↦ f(g(m)), g ∈ Hom_{R}(M, N), for all f ∈ Hom(N, D) and m ∈ M. It is easily seen that Φ is a well-defined additive map, which is natural in N, D and M. The natural transformation Φ appears in [10, Chapter VI, §5] and [8, Chapter II, §4, Exercise 6] and has been also used by Raynaud and Gruson in [32]. As we are mainly interested in the dependence of Φ upon M, we shall denote the map (3) above by Φ_{M}. It is easily seen that Φ_{M} is an isomorphism if the abelian group D is divisible and the R-module M is finitely presented.

The following result provides us with a characterization of those left R-modules M for which Φ_{M} is injective for any divisible abelian group D. Its proof, which is essentially obtained by putting together results of Raynaud and Gruson [32], Zimmermann [39] and Hügel and Herbera [20], is given below for the convenience of the reader.

Theorem 1.3. The following conditions are equivalent for a pair of left R-modules M, N: (i) The natural map Φ_{M} defined above is injective for any divisible abelian group D. (ii) For any representation of M as the direct limit of a direct system (M_{i})_{i} of finitely

generated left R-modules, the inverse system of abelian groups (Hom_{R}(M_{i}, N))_{i}, whose structural maps are induced by the structural maps of the direct system (M_{i})_{i}, satisfies the strict Mittag-Leffler condition.

(iii) There exists a representation of M as the direct limit of a direct system (M_{i})_{i} of finitely presented left R-modules, such that the inverse system of abelian groups (Hom_{R}(M_{i}, N))_{i}, whose structural maps are induced by the structural maps of the direct system (M_{i})_{i}, satisfies the strict Mittag-Leffler condition.

Proof. (i)→(ii): Assume that M is the direct limit of a direct system (M_{i})_{i} of finitely generated left R-modules. We shall denote by τ_{ij} : M_{i} ⟶ M_{j} the structural maps of the direct system for all i ≤ j and let t_{i} : M_{i} ⟶ M be the canonical map. We fix an index i ∈ I and a set of generators m_{1}^{(i)}, m_{2}^{(i)}, …, m_{n}^{(i)} of M_{i}. For any index j ≥ i we also consider the elements m_{1}^{(j)}, m_{2}^{(j)}, …, m_{n}^{(j)} ∈ M_{j}, where m_{s}^{(j)} = τ_{ij}(m_{s}^{(i)}) for all s = 1, 2, …, n. Finally, we

shall denote by m_{1}, m_{2}, …, m_{n} ∈ M the elements, which are defined by letting m_{s} = t_{i}(m_{s}^{(i)}) for all s = 1, 2, …, n.

Let H ⊆ N^{n} be the additive subgroup of N^{n}, which is defined by

H = {(g(m_{1}), g(m_{2}), …, g(m_{n})) : g ∈ Hom_{R}(M, N)}.

We choose a divisible abelian group D containing the quotient group N^{n}/H and define for all s = 1, 2, …, n the additive map f_{s} : N ⟶ D as the composition

N ⟶^{ν_{s}} N^{n} ⟶^{π} N^{n}/H ↪ D.

Here, ν_{s} is the natural embedding of N as the s-th summand of N^{n}, π is the quotient map and the unlabeled arrow is the inclusion of N^{n}/H into D.

We now consider the element ξ = ∑_{s=1}^{n} f_{s} ⊗ m_{s} ∈ Hom(N, D) ⊗_{R} M. We note that for all g ∈ Hom_{R}(M, N) we have

[Φ_{M}(ξ)](g) = ∑_{s=1}^{n} [Φ_{M}(f_{s} ⊗ m_{s})](g) = ∑_{s=1}^{n}f_{s}(g(m_{s})) ∈ D.

The latter element vanishes, being precisely the image of (g(m_{1}), g(m_{2}), …, g(m_{n})) ∈ N^{n} under the composition N^{n} ⟶^{π} N^{n}/H ↪ D. Since [Φ_{M}(ξ)](g) = 0 ∈ D for all g ∈ Hom_{R}(M, N), we conclude that Φ_{M}(ξ) = 0 ∈ Hom(Hom_{R}(M, N), D). Therefore, in view of the injectivity of Φ, we conclude that ξ = 0 ∈ Hom(N, D) ⊗_{R} M.

For any j ≥ i we consider the element ξ^{(j)} = ∑_{s=1}^{n} f_{s} ⊗ m_{s}^{(j)} ∈ Hom(N, D) ⊗_{R} M_{j}. Since the group Hom(N, D) ⊗_{R} M is the direct limit of the direct system (Hom(N, D) ⊗_{R} M_{i})_{i} and ξ^{(j)} = (1 ⊗ τ_{ij})(ξ^{(i)}) for all j ≥ i, the vanishing of ξ = (1 ⊗ t_{i})(ξ^{(i)}) implies that there is an index j ≥ i, such that ξ^{(j)} = 0 ∈ Hom(N, D) ⊗_{R} M_{j}. We fix such an index j. Then, we have Φ_{M_{j}}(ξ^{(j)})= 0 ∈ Hom(Hom_{R}(M_{j}, N), D) and hence

∑_{s=1}^{n}f_{s}(g′(m_{s}^{(j)})) = ∑_{s=1}^{n} [Φ_{M_{j}}(f_{s} ⊗ m_{s}^{(j)})](g′) = Φ_{M_{j}}(ξ^{(j)})(g′) = 0 ∈ D

for all g′ ∈ Hom_{R}(M_{j}, N). Therefore, the image of (g′(m_{1}^{(j)}), g′(m_{2}^{(j)}), …, g′(m_{n}^{(j)})) ∈ N^{n}

vanishes under the composition N^{n} ⟶^{π} N^{n}/H ↪ D for all g′ ∈ Hom_{R}(M_{j}, N). In other words, for any g′ ∈ Hom_{R}(M_{j}, N) there exists g ∈ Hom_{R}(M, N), such that

(g′(m_{1}^{(j)}), g′(m_{2}^{(j)}), …, g′(m_{n}^{(j)}))= (g(m_{1}), g(m_{2}), …, g(m_{n})) ∈ N^{n}.

The latter equality implies that

(g′ ∘ τ_{ij})(m_{s}^{(i)}) = g′(m_{s}^{(j)}) = g(m_{s}) = (g ∘ t_{i})(m_{s}^{(i)}) ∈ N

for all s = 1, 2, …, n. Since the R-linear maps g′ ∘τ_{ij}, g ∘t_{i} : M_{i} ⟶ N agree on the generators m_{1}^{(i)}, m_{2}^{(i)}, …, m_{n}^{(i)} of M_{i}, we conclude that g′ ∘ τ_{ij} = g ∘ t_{i} ∈ Hom_{R}(M_{i}, N). Hence, we have proved that the inclusion

im(Hom_{R}(M, N) ⟶^{t_{i}^{∗}} Hom_{R}(M_{i}, N))⊆ im(Hom_{R}(M_{j}, N) ⟶^{τ_{ij}^{∗}} Hom_{R}(M_{i}, N))

is actually an equality, showing thereby that the inverse system (Hom_{R}(M_{i}, N))_{i} satisfies the strict Mittag-Leffler condition.

(ii)→(iii): This is clear.

(iii)→(i): Assume that M can be represented as the direct limit of a direct system (M_{i})_{i} of finitely presented left R-modules, in such a way that the inverse system of abelian groups (Hom_{R}(M_{i}, N))_{i}, whose structural maps are induced by the structural maps of the direct system (M_{i})_{i}, satisfies the strict Mittag-Leffler condition. We denote by t_{i} : M_{i} ⟶ M the canonical map for any index i ∈ I. We now consider a divisible abelian group D and the commutative diagram

Hom(N, D) ⊗_{R} M_{i} ⟶^{Φ_{M_{i}}} Hom(Hom_{R}(M_{i}, N), D)
1⊗t_{i} ↓  ↓ t_{i∗}
Hom(N, D) ⊗_{R} M ⟶^{Φ_{M}} Hom(Hom_{R}(M, N), D)

We note that Hom(N, D) ⊗_{R} M is the direct limit of the direct system (Hom(N, D) ⊗_{R} M_{i})_{i}. Therefore, in order to prove the injectivity of Φ_{M}, it suffices to show that

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

for all i ∈ I. To that end, we fix an index i. We note that the abelian group Hom_{R}(M, N) can be naturally identified with the inverse limit of the system (Hom_{R}(M_{i}, N))_{i}, in such a way that the canonical map from the inverse limit to Hom_{R}(M_{i}, N) is identified with the map

Hom_{R}(M, N) ⟶ Hom_{R}(M_{i}, N),

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

t_{i∗} : Hom(Hom_{R}(M_{i}, N), D) ⟶ Hom(Hom_{R}(M, N), D)

coincides with the kernel of the additive map

τ_{i,j∗} : Hom(Hom_{R}(M_{i}, N), D) ⟶ Hom(Hom_{R}(M_{j}, N), D) ,

which is induced by the structural map τ_{ij} : M_{i} ⟶ M_{j}. We now consider the commutative diagram

Hom(N, D) ⊗_{R} M_{i} ⟶^{Φ_{M_{i}}} Hom(Hom_{R}(M_{i}, N), D)
1⊗τ_{ij} ↓  ↓ τ_{i,j∗}
Hom(N, D) ⊗_{R} M_{j} ⟶^{Φ_{M_{j}}} Hom(Hom_{R}(M_{j}, N), D)

The R-module M_{j} being finitely presented, the additive map Φ_{M_{j}} is bijective and hence

(4)  ker (τ_{i,j∗} ∘ Φ_{M_{i}}) = ker (1 ⊗ τ_{ij}).

Since t_{j} ∘ τ_{ij} = t_{i}, it follows that (1 ⊗ t_{j}) ∘ (1 ⊗ τ_{ij}) = 1 ⊗ t_{i} and hence we conclude that

(5)  ker (1 ⊗ τ_{ij}) ⊆ ker (1 ⊗ t_{i}).

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

(6)  ker (t_{i∗} ∘ Φ_{M_{i}}) = Φ_{M_{i}}^{−1}(ker t_{i∗})= Φ_{M_{i}}^{−1}(ker τ_{i,j∗})= ker (τ_{i,j∗} ∘ Φ_{M_{i}}).

Combining (4), (5) and (6), it follows that ker (t_{i∗} ∘ Φ_{M_{i}}) ⊆ ker (1 ⊗ t_{i}), as needed. □

If M, N is a pair of left R-modules, such that the equivalent conditions of Theorem 1.3 are satisfied, then we shall say that M is a strictly Mittag-Leffler module over N.

Remarks 1.4. (i) Let (M_{λ})_{λ} be a family of left R-modules and consider the direct sum M = ⨁_{λ} M_{λ}. Then, M is strictly Mittag-Leffler over a left R-module N if and only if M_{λ} is strictly Mittag-Leffler over N for any λ. (This is an immediate consequence of Theorem 1.3, since it is easily seen that Φ_{M} is injective if and only if Φ_{M_{λ}} is injective for all λ.)

(ii) Let N be a left R-module. Since the left R-module R is obviously strictly Mittag-Leffler over N, it follows from (i) above that any projective left R-module is strictly Mittag-Leffler over N as well.

III. Deriving the natural transformation Φ. Having fixed a left R-module N and a divisible abelian group D, we consider for any left R-module M a projective resolution P_{∗} ⟶ M ⟶ 0. Then, the natural transformation Φ induces a chain map

Φ_{P_{∗}} : Hom(N, D) ⊗_{R} P_{∗} ⟶ Hom(Hom_{R}(P_{∗}, N), D).

By applying homology, we obtain additive maps

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom(N, D), M)⟶ Hom(Ext_{R}^{n}(M, N), 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(N, D), )⟶ Hom(Ext_{R}^{∗}( , N), D)

(cf. [10, Chapter VI, §5]). Given a projective resolution of M as above, we shall denote by K_{n} the corresponding n-th syzygy module. Of course, K_{n} depends upon the choice of the particular resolution. Nevertheless, Schanuel’s lemma implies that for any two n-th syzygy modules K_{n} and K′_{n} of M there are projective modules P and P′, such that K_{n}⊕P ≃ K′_{n}⊕P′. Hence, invoking Remark 1.4, we conclude that K_{n} is strictly Mittag-Leffler over N if and only if K′_{n} is. In that case, we shall say that the n-th syzygy module of M is strictly Mittag-Leffler over N.

Proposition 1.5. Let M, N be two left R-modules and fix a non-negative integer n. Then, the following conditions are equivalent:

(i) The natural map Φ_{M}^{(n)} : Tor_{n}^{R}(Hom(N, D), M) ⟶ Hom(Ext_{R}^{n}(M, N), D) is injective for any divisible adelian group D.

(ii) The n-th syzygy module of M is strictly Mittag-Leffler over N.

Proof. Let P_{∗} ⟶ M ⟶ 0 be a projective resolution of M and consider the corresponding n-th syzygy module K_{n}. Then, K_{n} fits to an exact sequence

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

By applying dimension shifting, we obtain for any divisible abelian group D exact sequences

0 ⟶ Tor_{n}^{R}(Hom(N, D), M) ⟶^{a_{n}} Hom(N, D) ⊗_{R} K_{n} ⟶ Hom(N, D) ⊗_{R} P_{n−1}

and

0 ⟶ Hom(Ext_{R}^{n}(M, N), D) ⟶^{b_{n}} Hom(Hom_{R}(K_{n}, N), D) ⟶ Hom(Hom_{R}(P_{n−1}, N), D).

Since the additive maps a_{n}, b_{n} are defined by composing connecting homomorphisms and Φ^{(∗)} commutes with connecting homomorphisms, we obtain a commutative diagram of abelian groups with exact rows

0 → Tor_{n}^{R}(Hom(N, D), M) →^{a_{n}} Hom(N, D) ⊗_{R} K_{n} → Hom(N, D) ⊗_{R} P_{n−1}
Φ_{M}^{(n)} ↓  ↓ Φ_{Kn}^{(0)}  ↓ Φ_{P_{n−1}}^{(0)}
0 → Hom(Ext_{R}^{n}(M, N), D) →^{b_{n}} Hom(Hom_{R}(K_{n}, N), D) → Hom(Hom_{R}(P_{n−1}, N), D)

In view of Remark 1.4(ii), the projective left R-module P_{n−1} is strictly Mittag-Leffler over N and hence the map Φ_{P_{n−1}}^{(0)} = Φ_{P_{n−1}} is injective. Then, a diagram chase shows that Φ_{M}^{(n)} is injective if and only if this is the case for Φ_{K_{n}}^{(0)} = Φ_{K_{n}}, proving the equivalence between (i) and (ii) in the statement. □

2. The invariant sfli R and related invariants

I. Generalities on sfli. 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 [17] the invariant silp R as the supremum of the injective lengths of projective left R-modules and the invariant spli R as the supremum of the projective lengths of injective left R-modules. We note that, instead of considering projective resolutions, we may consider flat resolutions and define the invariant sfli R as the supremum of the flat lengths of injective left R-modules. It is easily seen that sfli R is finite if and only if any injective left R-module has finite flat dimension. Colby has studied in [11] the special case of those rings R for which sfli R = 0 (i.e. for which all injective left R-modules are flat); he proved that these are precisely the rings R over which all finitely presented left R-modules can be embedded in a free module. Since projective modules are flat, it is clear that we always have sfli R ≤ spli R. In the following Section, we shall examine the extent to which that inequality is strict. In an analogous way, we may define the invariant silf R as the supremum of the injective lengths of flat left R-modules.

We recall that the finitistic dimension fin.dim R of R is defined as the supremum of the projective dimensions of those left R-modules that have finite projective dimension. The finiteness of fin.dim R is then equivalent to the assertion that there is a uniform upper bound on the projective dimensions of those left R-modules that have finite projective dimension. It is easily seen that we always have fin.dim R ≤ silp R. Following Raynaud and Gruson [32, §II.3.3], we shall also consider the invariant splf R, which is defined as the supremum of the projective lengths of flat left R-modules. (The invariant splf R is denoted by d(R) in [loc.cit.]; our notation seems to be more in line with the other invariants of R that are defined above.)

Proposition 2.1. Let R be a ring. Then, silf R = silp R.

Proof. Since projective modules are flat, it is clear that silp R ≤ silf R. Therefore, it only remains to prove that silf R ≤ silp R. The latter inequality is obvious if silp R = ∞ and hence we may assume that silp R = n < ∞. Since we always have fin.dim R ≤ silp R, we conclude that fin.dim R is also finite. Then, as shown by Jensen in [24, Proposition 6], any flat left R-module has finite projective dimension. We now consider a flat left R-module M and fix a projective resolution P_{∗} ⟶ M ⟶ 0 of finite length. Since all of the projective left R-modules that are involved in the resolution have injective dimension ≤ silp R = n, it is easily seen that id_{R}M ≤ n as well. It follows that silf R ≤ n, as needed. □

Proposition 2.2. If R is a ring, then spli R ≤ sfli R + splf R ≤ sfli R + fin.dim R ≤ sfli R + silp R.

Proof. In order to prove the first of the three inequalities, we may assume that sfli R = n < ∞ and splf R = m < ∞. Let I be an injective left R-module and consider a projective resolution of it. Since fd_{R}I ≤ sfli R = n, the n-th syzygy module K_{n} of the resolution is flat. Since splf R = m, the flat left R-module K_{n} admits a projective resolution of length m. It follows that I has a projective resolution of length n+m and hence pd_{R}I ≤ n+m. Therefore, spli R ≤ n + m, as needed.

The second inequality follows from Jensen’s result [24, Proposition 6], which may be restated as the assertion that splf R ≤ fin.dim R, whereas the third inequality follows since fin.dim R ≤ silp R. □

Corollary 2.3. Let R be a ring. Then, the following conditions are equivalent: (i) spli R = silp R < ∞. (ii) sfli R < ∞ and silp R < ∞. Proof. The implication (i)→(ii) is obvious and the implication (ii)→(i) follows from Propo-

sition 2.2, in view of the following well-known observation (cf. [17, §1.6]): If both invariants spli R and silp R are finite, then they are equal. □

II. The relation between sfli and the finitistic flat dimension. The invariant sfli R is related to the finitistic flat dimension fin.f.dim R of the ring R, which is defined as the supremum of the flat dimensions of those left R-modules that have finite flat dimension.

Proposition 2.4. Let R be a ring. Then:

(i) fin.f.dim R ≤ sfli R^{op}.

(ii) If sfli R < ∞, then sfli R ≤ fin.f.dim R.

Proof. (i) Let M be a left R-module of finite flat dimension and assume that fd_{R}M = n. Then, the functor Tor_{n+1}^{R}( , M) vanishes identically, whereas Tor_{n}^{R}(A, M) ≠ 0, for some right R-module A. If I is an injective right R-module containing A, then the exact sequence

0 = Tor_{n+1}^{R}(I/A, M) ⟶ Tor_{n}^{R}(A, M) ⟶ Tor_{n}^{R}(I, M)

shows that Tor_{n}^{R}(I, M) ≠ 0. In particular, we have sfli R^{op} ≥ fd_{R^{op}}I ≥ n and hence sfli R^{op} ≥ fin.f.dim R, as needed.

(ii) If sfli R = n < ∞, then there exists an injective left R-module I with fd_{R}I = n and hence fin.f.dim R ≥ n. □

Corollary 2.5. Let R be a ring and assume that the invariants sfli R and sfli R^{op} are finite. Then, sfli R = fin.f.dim R = sfli R^{op} = fin.f.dim R^{op}. □

Remark 2.6 If R is a ring, then the invariants sfli R and sfli R^{op} are not necessarily equal to each other. Indeed, Colby has exhibited in [11, Example 2] a ring R for which sfli R > 0 and sfli R^{op} = 0. If R is Colby’s example, then Proposition 2.4(i) implies that fin.f.dim R = 0, i.e. that any left R-module of finite flat dimension is necessarily flat. It follows that not all injective left R-modules are flat and any injective left R-module which is not flat has infinite flat dimension. In particular, we have sfli R = ∞ (in accordance with Corollary 2.5).

Corollary 2.7. Let R be a ring, which is isomorphic with its opposite R^{op}. If sfli R < ∞, then sfli R = fin.f.dim R. □

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 2.8. Let R be a left ℵ_{0}-Noetherian ring. Then, fin.dim R ≤ fin.f.dim R + 1.

Proof. As shown in [32, §II.3.2], the finitistic dimension fin.dim R of R is the supremum of the projective dimensions of those countably generated left R-modules that have finite projective dimension. Let M be a countably generated left R-module with pd_{R}M = n < ∞; then, fd_{R}M = m ≤ n. We consider a resolution of M consisting of countably generated free left R-modules and let K_{m} be the m-th syzygy module. Then, K_{m} is flat and countably presented and hence pd_{R}K_{m} ≤ 1 (cf. [29, Théorème I.3.2]). It follows that M has a projective resolution of length m + 1 and hence n = pd_{R}M ≤ m + 1 ≤ fin.f.dim R + 1. Therefore, fin.dim R ≤ fin.f.dim R + 1, as needed. □

Theorem 2.9. Let R be a ring, which is both left and right ℵ_{0}-Noetherian. Then, the following conditions are equivalent:

(i) The invariants spli R and spli R^{op} are finite.

(ii) The invariants sfli R and sfli R^{op} are finite. If these conditions are satisfied, then sfli R = sfli R^{op}, whereas the invariants spli R and spli R^{op} are either equal or else they differ by one; more precisely, if sfli R = sfli R^{op} = n, then we have n ≤ spli R ≤ fin.dim R ≤ n + 1 and n ≤ spli R^{op} ≤ fin.dim R^{op} ≤ n + 1.

Proof. The implication (i)→(ii) is clear. Conversely, if both invariants sfli R and sfli R^{op} are finite, then Corollary 2.5 implies that they are equal; let sfli R = sfli R^{op} = n. Then, using Propositions 2.4(i) and 2.8, it follows that

fin.dim R ≤ fin.f.dim R + 1 ≤ sfli R^{op} + 1 = n + 1.

Since both of sfli R and fin.dim R are finite, it follows from Proposition 2.2 that spli R is also finite. Then, the invariant spli R being finite, it is clear that spli R ≤ fin.dim R. A symmetric argument shows that spli R^{op} ≤ fin.dim R^{op} ≤ n + 1. □

Corollary 2.10. Let R be a ring, which is isomorphic with its opposite R^{op} and is left (and right) ℵ_{0}-Noetherian. Then, the following conditions are equivalent:

(i) spli R < ∞.

(ii) sfli R < ∞. Moreover, if these conditions are satisfied, then sfli R ≤ spli R ≤ fin.dim R ≤ sfli R + 1. □

III. On the relation between sfli and spli. Let R be a ring and M a left R-module. Since projective modules are flat, the projective dimension pd_{R}M of M is always greater than or equal the flat dimension fd_{R}M. We shall examine the extent to which the inequality fd_{R}M ≤ pd_{R}M is strict.

Theorem 2.11. Let R be a ring and consider a left R-module M and a positive integer n. If pd_{R}M ≤ n and Ext_{R}^{n}(M, R) = 0, then fd_{R}M ≤ n − 1.

Proof. In order to show that fd_{R}M ≤ n − 1, we have to show that the functor Tor_{n}^{R}( , M) vanishes identically. Since fd_{R}M ≤ pd_{R}M, our assumption about pd_{R}M implies that fd_{R}M ≤ n. It follows that Tor_{n+1}^{R}( , M) vanishes identically and hence the functor Tor_{n}^{R}( , M) is left exact. Since any right R-module is a submodule of a cofree right R-module, the vanishing of Tor_{n}^{R}( , M) will follow if we show that Tor_{n}^{R}(Hom(R, D), M) = 0 for any divisible abelian group D.

Let D be a divisible abelian group. Since pd_{R}M ≤ n, there is a projective resolution of M of length n. Then, the corresponding n-th syzygy module is projective and hence strictly Mittag-Leffler over R (cf. Remark 1.4(ii)). Invoking Proposition 1.5 (with P = R), we may conclude that the additive map

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom(R, D), M)⟶ Hom(Ext_{R}^{n}(M, R), D)

is injective. Then, our assumption about the vanishing of the group Ext_{R}^{n}(M, R) implies that Tor_{n}^{R}(Hom(R, D), M) = 0, as needed. □

Corollary 2.12. Let R be a ring and consider a left R-module M and a positive integer n. If pd_{R}M ≤ n and id_{R}R ≤ n − 1, then fd_{R}M ≤ n − 1. □

Corollary 2.13. Let R be a ring and consider a left R-module M. If fd_{R}M = pd_{R}M < ∞, then fd_{R}M = pd_{R}M ≤ id_{R}R.

Proof. Let pd_{R}M = n < ∞ and assume that id_{R}R < n. Then, Corollary 2.12 implies that fd_{R}M < n, a contradiction. □

Corollary 2.14. Let R be a left self-injective ring (i.e. a ring for which id_{R}R = 0) and M a left R-module with pd_{R}M ≤ 1. Then, M is flat.

Proof. This is an immediate consequence of Corollary 2.12. □

As an application of the results above, we shall now examine the extent to which the inequality sfli R ≤ spli R is strict.

Corollary 2.15. Let R be a ring and consider a positive integer n, such that spli R ≤ n and id_{R}R ≤ n − 1. Then, sfli R ≤ n − 1.

Proof. Let I be an injective left R-module. Then, pd_{R}I ≤ spli R ≤ n and hence Corollary 2.12 implies that fd_{R}I ≤ n − 1, as needed. □

Corollary 2.16. Let R be a ring.

(i) If sfli R = spli R < ∞, then sfli R = spli R ≤ id_{R}R.

(ii) If id_{R}R < spli R < ∞, then sfli R < spli R.

Proof. (i) Let spli R = n < ∞ and assume that id_{R}R < n. Then, Corollary 2.15 implies that sfli R < n, a contradiction.

(ii) This is merely a restatement of (i). □

Corollary 2.17. Let R be a left self-injective ring, which is not quasi-Frobenius. If spli R < ∞, then sfli R < spli R.

Proof. Since R is not quasi-Frobenius, a result by Faith and Walker (cf. [15], [16]) implies that spli R > 0. Then, the result is an immediate consequence of Corollary 2.16(ii). □

Remark 2.18. Besides the class of left self-injective rings, which are not quasi-Frobenius and have finite spli, other examples of rings R that satisfy the inequality sfli R < spli R can be obtained by considering the integral group ring of certain groups. For example, we have sfli R < spli R if R is the integral group ring of the additive group ℚ of rational numbers (cf. Example 4.10).

IV. The finitistic dimension of certain rings. As an application of the results above, we shall now obtain some information about the finitistic dimension of certain types of rings.

We have already noted before that fin.dim R ≤ silp R for any ring R. On the other hand, the left R-module R being projective, we also have id_{R}R ≤ silp R. Even though id_{R}R = silp R if the ring R is left Noetherian, there are examples of rings R for which id_{R}R < silp R; see the discussion at the beginning of Section 3. We shall prove that fin.dim R ≤ id_{R}R if the ring R is either left perfect or countable.

Bass has defined in [4] the class of left perfect rings as those rings that satisfy the descending chain condition on principal right ideals. He showed that a ring R is left perfect if and only if any flat left R-module is projective; for a proof of this result, see also [27, Theorem 24.25].

Theorem 2.19. The following conditions are equivalent for a ring R:

(i) Any flat left R-module of finite projective dimension is projective.

(ii) If M is a left R-module of finite projective dimension, then fd_{R}M = pd_{R}M. Moreover, if these conditions hold, then fin.dim R ≤ id_{R}R.

Proof. In order to show that (i)→(ii), let M be a left R-module with pd_{R}M = n < ∞ and assume that fd_{R}M = m < n. We consider a projective resolution of M of length n and note that the corresponding m-th syzygy module K_{m} is flat and has projective dimension ≤ n−m. Therefore, in view of (i), the left R-module K_{m} is projective and hence M has a projective resolution of length ≤ m. This is a contradiction, since pd_{R}M = n > m.

It is clear that (ii)→(i), whereas the final assertion follows from Corollary 2.13. □

Corollary 2.20. If R is a left perfect ring, then fin.dim R ≤ id_{R}R. □

Corollary 2.21. If R is a right Artinian ring, then fin.dim R ≤ id_{R}R. In particular, if id_{R}R < ∞, then fin.dim R < ∞.

Proof. Any right Artinian ring R satisfies the descending chain condition on right ideals; hence, R is left perfect. The result therefore follows from Corollary 2.20. □

Remark 2.22. Let R be a commutative Noetherian ring. Then, as shown by Bass [5] and Raynaud and Gruson [32, Théorème II.3.2.6], the finitistic dimension fin.dim R is equal to the Krull dimension of R. In particular, in the special case where R is Gorenstein, we have fin.dim R = id_{R}R < ∞; this equality was already obtained by Bass in [5]. Therefore, we may regard Corollary 2.21 as an analogue of Bass’ result to the class of right Artinian rings that satisfy the left Gorenstein condition (namely, the condition id_{R}R < ∞).

Let R be a ring and M a left R-module. Then, Theorem 2.11 implies that M is flat if pd_{R}M ≤ 1 and Ext_{R}^{1}(M, R) = 0. This result can be strengthened in the case where R and M are countable, as follows:

Theorem 2.23. Let R be a countable ring and M a countable left R-module. Then, the following conditions are equivalent:

(i) M is projective.

(ii) pd_{R}M ≤ 1 and Ext_{R}^{1}(M, R) = 0.

Proof. It is clear that (i)→(ii). Assuming that (ii) holds, we may invoke Theorem 2.11 and conclude that M is a flat module. In view of the countability assumptions made on R and M, the projectivity of M follows from [26, Theorem 1] (see also [14, Theorem 1.3]). □

Corollary 2.24. Let R be a countable ring and M a countable left R-module. Then, the following conditions are equivalent for a non-negative integer n:

(i) pd_{R}M ≤ n.

(ii) pd_{R}M ≤ n + 1 and Ext_{R}^{n+1}(M, R) = 0.

Proof. Let P_{∗} ⟶ M ⟶ 0 be a resolution of M consisting of countably generated free left R-modules and consider the corresponding n-th syzygy module K_{n}. Then, condition (i) (resp. (ii)) is equivalent to the assertion that K_{n} is projective (resp. to the assertion that pd_{R}K_{n} ≤ 1 and Ext_{R}^{1}(K_{n}, R) = 0). Hence, the result follows from Theorem 2.23. □

Corollary 2.25. Let R be a countable ring and M a non-zero countable left R-module with pd_{R}M = n < ∞. Then, Ext_{R}^{n}(M, R) ≠ 0 and hence id_{R}R ≥ n.

Proof. Assume on the contrary that Ext_{R}^{n}(M, R) = 0. Since M ≠ 0 and pd_{R}M = n, the integer n must be strictly positive. We may then invoke Corollary 2.24, in order to conclude that pd_{R}M ≤ n − 1, a contradiction. □

Corollary 2.26. If R is a countable ring, then fin.dim R ≤ id_{R}R.

Proof. Since a countable ring is left ℵ_{0}-Noetherian, it follows from [32, §II.3.2] that fin.dim R is the supremum of the projective dimensions of those countably generated left R-modules that have finite projective dimension. Then, the result follows from Corollary 2.25. □

3. On the relation between id_{R}R and silp R

Let R be a ring. Then, the left R-module R being projective, we have id_{R}R ≤ silp R. The latter inequality may very well be strict. For example, let R be a left self-injective ring, which is not left or right Noetherian.^{1} Then, we have id_{R}R = 0, whereas silp R > 0. Indeed, if silp R = 0, then R would be quasi-Frobenius (cf. [15], [16]) and, in particular, both left and right Noetherian, contradicting our hypothesis. On the other hand, as we shall later show, if R is the integral group ring of a countable group and both of the invariants id_{R}R and silp R are finite, then they are equal (cf. Theorem 4.8(iii)).

In this Section, we shall examine conditions under which the inequality id_{R}R ≤ silp R is actually an equality. Since the invariant silp R is easily seen to be equal to the supremum of the injective lengths of free left R-modules, the problem consists in studying the relation between the functors Ext_{R}^{n}( , R) and Ext_{R}^{n}( , R^{(Λ)}), where Λ is any set.

As an example, we note that if the ring R is left Noetherian, then id_{R}R = silp R. Indeed, assuming that id_{R}R = n < ∞, we may construct an injective resolution of R of length n. Since a direct sum of injective left R-modules is injective, in the case of a left Noetherian ring R (see [5, Theorem 1.1] and also [28, Theorem 3.46]), we may easily construct an injective resolution of R^{(Λ)} of length n for any set Λ.

We shall examine the inequality id_{R}R ≤ silp R, in the case where R is left ℵ_{0}-Noetherian.

Theorem 3.1. Let R be a left ℵ_{0}-Noetherian ring. Then, the following conditions are equivalent for a non-negative integer n:

(i) silp R ≤ n.

(ii) id_{R}R^{(ℕ)} ≤ n.

(iii) id_{R}R ≤ n and the n-th syzygy module of any left R-module is strictly Mittag-Leffler over R.

(iv) id_{R}R ≤ n and the n-th syzygy module of any countably generated left R-module is strictly Mittag-Leffler over R.

(v) id_{R}R ≤ n and the n-th syzygy module of any cyclic left R-module is strictly Mittag-Leffler over R.

1As shown by Osofsky in [30, Example 1], there are commutative local rings that satisfy these conditions.

Proof. We shall prove that (i)→(ii)→(iv)→(iii)→(v)→(i). Of course, the implications (i)→(ii) and (iii)→(v) are obvious.

(ii)→(iv): Since the left R-module R is a direct summand of R^{(ℕ)}, it is clear that id_{R}R ≤ n. Let M be a countably generated left R-module and consider a resolution of it by countably generated free left R-modules. If K = K_{n} is the corresponding n-th syzygy module, then

Ext_{R}^{1}(K, R^{(ℕ)})= Ext_{R}^{n+1}(M, R^{(ℕ)})= 0.

Since the left R-module K is countably presented, we may represent it as the direct limit of a countable direct system (K_{i})_{i} of finitely presented left R-modules. Then, the vanishing of the group Ext_{R}^{1}(K, R^{(ℕ)}) implies that the inverse system of abelian groups (Hom_{R}(K_{i}, R))_{i} satisfies the Mittag-Leffler condition (cf. [19, Example 2.4(iv)] and [13, Proposition 1.2]) and hence, the index set of the inverse system being countable, the strict Mittag-Leffler condition.

(iv)→(iii): Let M be a left R-module. In order to show that the n-th syzygy module of M is strictly Mittag-Leffler over R, we have to show (cf. Proposition 1.5) the injectivity of the natural map

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom(R, D), M)⟶ Hom(Ext_{R}^{n}(M, R), D),

for any divisible abelian group D. To that end, we fix a divisible abelian group D and consider an element ξ ∈ ker Φ_{M}^{(n)}. We may express M as the directed union of its countably generated submodules and use the continuity of the functor Tor_{n}^{R}(Hom(R, D), ), in order to express the abelian group Tor_{n}^{R}(Hom(R, D), M) as the direct limit of the groups Tor_{n}^{R}(Hom(R, D), M′), where M′ ranges through the countably generated submodules of M. In particular, there is a countably generated submodule M′ of M and an element ξ′ ∈ Tor_{n}^{R}(Hom(R, D), M′) mapping onto ξ under the additive map

ı : Tor_{n}^{R}(Hom(R, D), M′)⟶ Tor_{n}^{R}(Hom(R, D), M),

which is induced by the inclusion M′ ↪ M. On the other hand, since id_{R}R ≤ n, the group Ext_{R}^{n+1}(M/M′, R) is trivial. Therefore, the inclusion M′ ↪ M induces a surjective map

Ext_{R}^{n}(M, R) ⟶ Ext_{R}^{n}(M′, R),

which in turn induces an injective map

ȷ : Hom(Ext_{R}^{n}(M′, R), D)⟶ Hom(Ext_{R}^{n}(M, R), D).

We now invoke the naturality of Φ^{(n)} and consider the commutative diagram

Tor_{n}^{R}(Hom(R, D), M′) ⟶^{ı} Tor_{n}^{R}(Hom(R, D), M)
Φ_{M′}^{(n)} ↓  ↓ Φ_{M}^{(n)}
Hom(Ext_{R}^{n}(M′, R), D) ⟶^{ȷ} Hom(Ext_{R}^{n}(M, R), D)

Since (ȷ ∘ Φ_{M′}^{(n)})(ξ′) = (Φ_{M}^{(n)} ∘ ı)(ξ′) = Φ_{M}^{(n)}(ξ) = 0, the injectivity of both ȷ and Φ_{M′}^{(n)} implies

that ξ′ = 0. It follows that ξ = ı(ξ′) = 0 and hence the group ker Φ_{M}^{(n)} is trivial, as needed.

(v)→(i): We have to show that id_{R}R^{(Λ)} ≤ n for any set Λ. To that end, we fix a set Λ and consider an injective resolution of R^{(Λ)} and the corresponding n-th cosyzygy module Q = Q^{n}. We have to show that the left R-module Q is injective. Since the functors Ext_{R}^{n+1}( , R^{(Λ)}) and Ext_{R}^{1}( , Q) are naturally isomorphic, Baer’s criterion for the injectivity of Q shows that we only have to show that Ext_{R}^{n+1}(M, R^{(Λ)}) = 0 for all cyclic left R-modules M. We now fix a cyclic left R-module M and consider a resolution of it by countably generated free left Rmodules and the corresponding n-th syzygy module K = K_{n}. Then, K is countably presented and hence we may express it as the direct limit of a countable direct system (K_{i})_{i} of finitely presented left R-modules.

It is well-known that one may describe the Ext-groups of K in terms of the Ext-groups of the K_{i}’s, by means of short exact sequences

0 ⟶ lim_{⟵i}^{1} Ext_{R}^{∗−1}(K_{i}, ) ⟶ Ext_{R}^{∗}(K, ) ⟶ lim_{⟵i} Ext_{R}^{∗}(K_{i}, ) ⟶ 0.

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

0 ⟶ lim_{⟵i}^{1} Hom_{R}(K_{i}, R^{(Λ)})⟶ Ext_{R}^{1}(K, R^{(Λ)})⟶ lim_{⟵i} Ext_{R}^{1}(K_{i}, R^{(Λ)})⟶ 0.

Therefore, in order to show that the group Ext_{R}^{n+1}(M, R^{(Λ)}) = Ext_{R}^{1}(K, R^{(Λ)}) is trivial, we have to show that this is the case for both groups lim_{⟵i}^{1}Hom_{R}(K_{i}, R^{(Λ)}) and lim_{⟵i}Ext_{R}^{1}(K_{i}, R^{(Λ)}).

Since K_{i} is a finitely generated left R-module for all i, the inverse system (Hom_{R}(K_{i}, R^{(Λ)}))_{i} is naturally identified with the direct sum (Hom_{R}(K_{i}, R)^{(Λ)})_{i} of copies of the inverse system (Hom_{R}(K_{i}, R))_{i}. Hence, in view of our hypothesis about the inverse system (Hom_{R}(K_{i}, R))_{i}, the inverse system (Hom_{R}(K_{i}, R^{(Λ)}))_{i} satisfies the (strict) Mittag-Leffler condition; cf. [19, Proposition 1.2]. In particular, the group lim_{⟵i}^{1} Hom_{R}(K_{i}, R^{(Λ)}) is trivial.

Since K_{i} is a finitely presented left R-module for all i, the inverse system (Ext_{R}^{1}(K_{i}, R^{(Λ)}))_{i} is naturally identified with the direct sum (Ext_{R}^{1}(K_{i}, R)^{(Λ)})_{i} of copies of the inverse system (Ext_{R}^{1}(K_{i}, R))_{i}. Hence, in view of the left exactness of the inverse limit functor and the fact that it commutes with direct products, we conclude that

lim_{⟵i} Ext_{R}^{1}(K_{i}, R^{(Λ)})= lim_{⟵i} (Ext_{R}^{1}(K_{i}, R))^{(Λ)} ⊆ lim_{⟵i} (Ext_{R}^{1}(K_{i}, R))^{Λ} =(lim_{⟵i} Ext_{R}^{1}(K_{i}, R))^{Λ}.

On the other hand, the short exact sequence of abelian groups

0 ⟶ lim_{⟵i}^{1} Hom_{R}(K_{i}, R) ⟶ Ext_{R}^{1}(K, R) ⟶ lim_{⟵i} Ext_{R}^{1}(K_{i}, R) ⟶ 0

and the vanishing of the group Ext_{R}^{1}(K, R) = Ext_{R}^{n+1}(M, R) shows that lim_{⟵i} Ext_{R}^{1}(K_{i}, R) = 0.

Hence, it follows that lim_{⟵i} Ext_{R}^{1}(K_{i}, R^{(Λ)})= 0, as needed. □

Corollary 3.2. Let R be a left ℵ_{0}-Noetherian ring. Then, silp R = id_{R}R^{(ℕ)}.

Proof. Since we always have id_{R}R^{(ℕ)} ≤ silp R, it only remains to show that silp R ≤ id_{R}R^{(ℕ)}. To that end, we may assume that id_{R}R^{(ℕ)} = n < ∞. Then, the inequality silp R ≤ n follows from Theorem 3.1. □

Corollary 3.3. Let R be a left ℵ_{0}-Noetherian ring and assume that id_{R}R = n < ∞. Then, the following conditions are equivalent:

(i) silp R = n.

(ii) The n-th syzygy module of any left R-module is strictly Mittag-Leffler over R. □

Corollary 3.4. Let R be a left ℵ_{0}-Noetherian ring. Then, the following conditions are equivalent:

(i) R is quasi-Frobenius,

(ii) R is left self-injective and any left R-module is strictly Mittag-Leffler over R.

Proof. In view of the characterization by Faith and Walker of the class of quasi-Frobenius rings in terms of the vanishing of silp (cf. [15], [16]), the result follows from Theorem 3.1, by letting n = 0 therein. □

Proposition 3.5. Let R be a left ℵ_{0}-Noetherian ring.

(i) If n is a positive integer, such that id_{R}R ≤ n and sfli R^{op} ≤ n − 1, then silp R ≤ n. (ii) If id_{R}R < silp R, then silp R = sfli R^{op} + 1. Proof. (i) Since sfli R^{op} ≤ n−1, the functor Tor_{n}^{R}(Hom(R, D), ) vanishes identically for any

divisible abelian group D and hence the n-th syzygy module of any left R-module is strictly Mittag-Leffler over R (cf. Proposition 1.5). Hence, the result follows from Theorem 3.1.

(ii) If silp R is infinite, then sfli R^{op} must be infinite as well, since the finiteness of both id_{R}R and sfli R^{op} implies the finiteness of silp R (as shown in (i) above).

We now assume that silp R = n < ∞; then, id_{R}R ≤ n − 1. If sfli R^{op} ≤ n − 2, then it would follow from (i) above that silp R ≤ n − 1, a contradiction. This shows that we must have sfli R^{op} ≥ n − 1. Therefore, it only remains to show that sfli R^{op} ≤ n − 1. Since any injective right R-module is a direct summand of a cofree right R-module, it suffices to show that the functor Tor_{n}^{R}(Hom(R, D), ) vanishes identically for any divisible abelian group D. To that end, we consider a divisible abelian group D and a left R-module M. Since silp R = n, Theorem 3.1 implies that the natural map

Φ_{M}^{(n)} : Tor_{n}^{R}(Hom(R, D), M)⟶ Hom(Ext_{R}^{n}(M, R), D)

is injective. On the other hand, we also have id_{R}R ≤ n − 1 and hence the group Ext_{R}^{n}(M, R) is trivial. It follows that Tor_{n}^{R}(Hom(R, D), M) = 0, as needed. □

Corollary 3.6. If R is a left ℵ_{0}-Noetherian ring, then the following conditions are equivalent: (i) silp R < ∞. (ii) id_{R}R < ∞ and sfli R^{op} < ∞. Proof. The result is an immediate consequence of the inequalities

max{sfli R^{op}, id_{R}R} ≤ silp R ≤ max{sfli R^{op} + 1, id_{R}R},

which follow themselves from [13, Proposition 3.2] and Proposition 3.5(i). □

The following result complements Theorem 2.9.

Theorem 3.7. Let R be a ring, which is both left and right ℵ_{0}-Noetherian. Then, the following conditions are equivalent:

(i) The invariants silp R and silp R^{op} are finite.

(ii) The invariants spli R, spli R^{op}, id_{R}R and id_{R^{op}}R^{op} are finite.

(iii) The invariants sfli R, sfli R^{op}, id_{R}R and id_{R^{op}}R^{op} are finite. If these conditions are satisfied, then sfli R = sfli R^{op}, spli R = silp R and spli R^{op} = silp R^{op}. Moreover, if sfli R = sfli R^{op} = n, then each of spli R(= silp R) and spli R^{op}(= silp R^{op}) is either equal to n or else to n + 1.

Proof. The equivalence between (i) and (iii) follows from Corollary 3.6, whereas the equivalence between (ii) and (iii) is an immediate consequence of Theorem 2.9. If conditions (i), (ii) and (iii) hold, then the equalities spli R = silp R and spli R^{op} = silp R^{op} follow from [17, §1.6]. Finally, the remaining assertions in the statement follow from Theorem 2.9. □

Corollary 3.8. Let R be a ring, which is isomorphic with its opposite R^{op} and is left (and hence right) ℵ_{0}-Noetherian. Then, the following conditions are equivalent:

(i) silp R < ∞.

(ii) spli R < ∞ and id_{R}R < ∞.

(iii) sfli R < ∞ and id_{R}R < ∞. In particular, if id_{R}R < ∞, then spli R = silp R. □

4. The special case of group rings

Let G be a group and consider the associated integral group ring ℤG. A special feature of ℤG is that it is actually a Hopf algebra. We list two well-known consequences of the Hopficity of ℤG below:

(i) The anti-isomorphism of ℤG, which is induced by the map g ↦ g^{−1}, g ∈ G, enables us to view any right ℤG-module M as a left ℤG-module M′ and vice versa. This correspondence defines an exact functor which preserves projective, injective and flat modules. In particular, M and M′ have the same homological properties.

(ii) The algebra homomorphism ℤG ⟶ ℤG ⊗ ℤG, which is induced by the diagonal map g ↦ (g, g), g ∈ G, enables us to define the structure of a ℤG-module to the tensor product M ⊗ N (respectively, to the group of all additive maps Hom(M, N)) for any two ℤG-modules M and N. We now consider a subgroup H ⊆ G. Then, G/H is a (left) G-set and we may consider the permutation ℤG-module ℤ(G/H). If M is any ℤG-module, then there is a natural isomorphism of ℤG-modules

ℤ(G/H) ⊗ M ⟶^{∼} ℤG ⊗_{ℤH} res_{H}^{G}M,

where the action of G on the left (resp. right) hand side is the diagonal action (resp. the action induced by left multiplication on ℤG). The isomorphism is given by letting gH ⊗x ↦ g⊗g^{−1}x, gH ∈ G/H, x ∈ M, whereas its inverse is given by g ⊗ x ↦ gH ⊗ gx, g ∈ G, x ∈ M. In fact, the existence of the above isomorphism is a special case of a more general result, for which the reader is referred to [9, Chapter III, §5, Exercise 2].

Lemma 4.1. Let G be a group.

(i) Let H ⊆ G be a normal subgroup and consider the quotient Q = G/H. We also consider a flat ℤQ-module M, which may be viewed as a ℤG-module via the quotient map G ⟶ Q, and a ℤG-module N, which is flat as a ℤH-module. Then, the diagonal ℤG-module M ⊗ N is flat.

(ii) Let M, N be two ℤG-modules, such that M is flat and N is ℤ-torsion-free. Then, the diagonal ℤG-module M ⊗ N is flat.

Proof. (i) Since the flat ℤQ-module M is a direct limit of free ℤQ-modules (cf. [29, Théorème I.1.2]), whereas a direct limit of flat modules is flat, we may reduce to the case where M is actually ℤQ-free. In fact, since a direct sum of flat modules is also flat, we may further assume that M = ℤQ. But, as we noted above, the diagonal ℤG-module ℤQ ⊗ N is isomorphic with the induced ℤG-module ℤG ⊗_{ℤH} res_{H}^{G}N and hence the result follows since N is ℤH-flat.

(ii) This is a special case of (i), by letting H = 1 therein. □

We note that the short exact sequence of ℤG-modules

0 ⟶ I_{G} ⟶ ℤG ⟶ ℤ ⟶ 0,

where I_{G} is the augmentation ideal of G, splits as an exact sequence of abelian groups. Therefore, letting J = Hom(ℤG, ℤ) and C = Hom(I_{G}, ℤ), we obtain a short exact sequence of ℤG-modules

(7)  0 ⟶ ℤ ⟶ J ⟶ C ⟶ 0,

which splits as an exact sequence of abelian groups as well.

Gedrich and Gruenberg have essentially proved in [17] the following result:

Proposition 4.2. Let G be a group and J = Hom(ℤG, ℤ). Then, the following conditions are equivalent:

(i) spli ℤG < ∞.

(ii) pd_{ℤG}J < ∞.

(iii) For any ℤ-free ℤG-module M the ℤG-module J ⊗ M has finite projective dimension. (iv) For any ℤG-module M the ℤG-module J ⊗ M has finite projective dimension. □

Using Lemma 4.1 and following step-by-step the arguments by Gedrich and Gruenberg, we obtain the next result, whose proof is provided for the convenience of the reader.

Proposition 4.3. Let G be a group and J = Hom(ℤG, ℤ). Then, the following conditions are equivalent:

(i) sfli ℤG < ∞.

(ii) fd_{ℤG}J < ∞.

(iii) For any ℤ-torsion-free ℤG-module M the ℤG-module J ⊗ M has finite flat dimension. (iv) For any ℤG-module M the ℤG-module J ⊗ M has finite flat dimension. Proof. (i)→(ii): The short exact sequence of abelian groups

0 ⟶ ℤ ⟶ ℚ ⟶ ℚ/ℤ ⟶ 0

induces the short exact sequence of ℤG-modules

0 ⟶ Hom(ℤG, ℤ) ⟶ Hom(ℤG, ℚ) ⟶ Hom(ℤG, ℚ/ℤ) ⟶ 0.

Since both ℤG-modules Hom(ℤG, ℚ) and Hom(ℤG, ℚ/ℤ) are injective, they have finite flat dimension. It follows that the ℤG-module J = Hom(ℤG, ℤ) has finite flat dimension as well.

(ii)→(iii): Let P_{∗} ⟶ J ⟶ 0 be a flat resolution of J of length n. If M is a ℤGmodule, which is ℤ-torsion-free, then there is an associated exact sequence of ℤG-modules (with diagonal action) P_{∗} ⊗ M ⟶ J ⊗ M ⟶ 0. In view of Lemma 4.1(ii), the latter exact sequence is a flat resolution of J ⊗ M of length n and hence fd_{ℤG}(J ⊗ M) ≤ n.

(iii)→(iv): Let M be a ℤG-module and consider a short exact sequence of ℤG-modules

0 ⟶ K ⟶ P ⟶ M ⟶ 0,

where K and P are ℤ-torsion-free (for example, we may consider a ℤG-free presentation of M). Since the ℤG-module J is ℤ-torsion-free, there is an associated short exact sequence of ℤG-modules

0 ⟶ J ⊗ K ⟶ J ⊗ P ⟶ J ⊗ M ⟶ 0.

Since the ℤG-modules J ⊗K and J ⊗P have finite flat dimension, the same is true for J ⊗M.

(iv)→(i): Let I be a ℤG-module. Tensoring with I the short exact sequence (7), which splits as an exact sequence of abelian groups, we obtain a short exact sequence of ℤG-modules

0 ⟶ I ⟶ J ⊗ I ⟶ C ⊗ I ⟶ 0.

Assuming that I is an injective ℤG-module, the short exact sequence above splits over ℤG. Therefore, I is a direct summand of J ⊗ I and hence fd_{ℤG}I ≤ fd_{ℤG}(J ⊗ I) < ∞. Since any injective ℤG-module has finite flat dimension, it is easily seen that sfli ℤG < ∞. □

Remarks 4.4. (i) It follows from the proof of the implication (i)→(ii) in Proposition 4.3 that fd_{ℤG}J ≤ sfli ℤG.

(ii) It follows from the proof of Proposition 4.3 that assertion (ii) therein may be replaced with the following (apparently weaker, but, in fact, equivalent) one: There exists a ℤ-torsion-free ℤG-module J′ of finite flat dimension, such that there is a ℤ-split monomorphism of ℤG-modules ℤ ⟶ J′.

Corollary 4.5. Let G be an infinite group of cardinality ℵ and assume that 2^{ℵ} = ℵ_{k} for some k ∈ ℕ. Then, sfli ℤG < ∞ if and only if spli ℤG < ∞.

Proof. Let J = Hom(ℤG, ℤ). In view of Propositions 4.2 and 4.3, we have to show that fd_{ℤG}J < ∞ if and only if pd_{ℤG}J < ∞. It is clear that the finiteness of pd_{ℤG}J implies that fd_{ℤG}J is also finite. Conversely, assume that fd_{ℤG}J = n < ∞ and consider a resolution of J consisting of ℵ_{k}-generated free ℤG-modules. (This is possible, since ℤG is a ring of cardinality ℵ and J has cardinality 2^{ℵ} = ℵ_{k}.) Then, the associated n-th syzygy module K = K_{n} is flat and ℵ_{k}-presented. As shown by Osofsky [31], we then have pd_{ℤG}K ≤ k + 1. It follows that J has a projective resolution of length ≤ n + k + 1 and hence pd_{ℤG}J < ∞. □

Gedrich and Gruenberg have proved in [17, Theorem 5.5] that the invariant spli is subadditive under extensions. More precisely, they showed that for any group G and any normal subgroup H ⊆ G we have spli ℤG ≤ spli ℤH + spli ℤQ, where Q = G/H is the corresponding quotient group. Once more, we may adapt their arguments for the invariant sfli and prove the following result.

Proposition 4.6. Let G be a group, H ⊆ G a normal subgroup and Q = G/H the corresponding quotient. Then, sfli ℤG ≤ sfli ℤH + sfli ℤQ.

Proof. The inequality to be proved is obvious if at least one of the invariants sfli ℤH and sfli ℤQ is infinite and hence we may assume that sfli ℤH = n < ∞ and sfli ℤQ = m < ∞. Then, we have to prove that sfli ℤG ≤ n + m.

To that end, let us consider an injective ℤG-module I. Then, the ℤ-split exact sequence of ℤQ-modules

0 ⟶ I_{Q} ⟶ ℤQ ⟶ ℤ ⟶ 0,

where I_{Q} is the augmentation ideal of Q, induces the ℤ-split exact sequence of ℤQ-modules

0 ⟶ ℤ ⟶ Hom(ℤQ, ℤ) ⟶ Hom(I_{Q}, ℤ) ⟶ 0.

We may regard the latter exact sequence as an exact sequence of ℤG-modules and obtain thereby an exact sequence of ℤG-modules

0 ⟶ I ⟶ Hom(ℤQ, ℤ) ⊗ I ⟶ Hom(I_{Q}, ℤ) ⊗ I ⟶ 0.

Since I is injective, that exact sequence splits over ℤG and hence I is a direct summand of Hom(ℤQ, ℤ) ⊗ I. It is therefore enough to show that Hom(ℤQ, ℤ) ⊗ I has flat dimension ≤ n + m.

Since sfli ℤQ = m, the ℤQ-module Hom(ℤQ, ℤ) has flat dimension ≤ m (cf. Remark 4.4(i)) and hence there is a resolution P_{∗} ⟶ Hom(ℤQ, ℤ) ⟶ 0 of Hom(ℤQ, ℤ) by flat ℤQ-modules of length m. Of course, the latter resolution may be also regarded as an exact sequence of ℤG-modules. On the other hand, the ℤH-module res_{H}^{G}I being injective, our assumption about the value of sfli ℤH implies that there is an exact sequence F_{∗} ⟶ I ⟶ 0 of ℤG-modules, which, when restricted to ℤH, is a resolution of res_{H}^{G}I by flat ℤH-modules of length n. Since Hom(ℤQ, ℤ) is ℤ-torsion-free, it follows from Künneth’s formula that the total complex of the double complex P_{∗} ⊗ F_{∗} provides us with a resolution of Hom(ℤQ, ℤ) ⊗ I by ℤG-modules of length n + m. In view of Lemma 4.1(i), this resolution consists of flat ℤG-modules and hence we conclude that the ℤG-module Hom(ℤQ, ℤ) ⊗ I has flat dimension ≤ n + m, as needed. □

We shall now present some immediate consequences of the results obtained in Section 2. We note that, as shown in [13], the invariants silp ℤG and spli ℤG are always equal. In particular, if spli ℤG is finite, then we have silp ℤG = spli ℤG = fin.dim ℤG.

Proposition 4.7. Let G be a group. Then, the following conditions are equivalent: (i) spli ℤG < ∞. (ii) sfli ℤG < ∞ and fin.dim ℤG < ∞. Proof. The implication (i)→(ii) follows since the finiteness of spli ℤG implies that spli ℤG =

fin.dim ℤG, as we noted above. On the other hand, the implication (ii)→(i) follows from Proposition 2.2. □

If the group G is countable, then the result of Proposition 4.7 above may be strengthened as in assertion (i) of the following theorem (see also Corollary 4.5). We note that assertion (ii) below is well-known in the special case of groups of type F P (cf. [9, Chapter VIII, Proposition 6.7]). Finally, assertion (iii) should be compared to Corollary 3.2. We recall that the cohomological dimension cd G of G is defined as the projective dimension of the trivial ℤG-module ℤ.

Theorem 4.8. Let G be a countable group.

(i) We always have sfli ℤG ≤ spli ℤG ≤ sfli ℤG + 1. In particular, spli ℤG < ∞ if and only if sfli ℤG < ∞.

(ii) If cd G = n < ∞, then H^{n}(G, ℤG) ≠ 0.

(iii) If silp ℤG < ∞, then id_{ℤG}ℤG = silp ℤG.

Proof. Assertion (i) (resp. (ii)) is a special case of Corollary 2.10 (resp. of Corollary 2.25). In order to prove (iii), assume that silp ℤG is finite. Then, as noted above, we have silp ℤG = fin.dim ℤG and hence the result follows from the inequalities

id_{ℤG}ℤG ≤ silp ℤG = fin.dim ℤG ≤ id_{ℤG}ℤG

(cf. Corollary 2.26). □

Ikenaga has introduced in [21, §II] the generalized cohomological dimension \underline{cd} G of a group G, by defining \underline{cd} G to be the supremum of those integers n, for which there exist a ℤ-free ℤG-module M and a projective ℤG-module P, such that Ext_{ℤG}^{n}(M, P) ≠ 0. It is known that:

(i) If G has finite cohomological dimension, then \underline{cd} G = cd G (cf. [21, Proposition 3]). (ii) We have inequalities \underline{cd} G ≤ silp ℤG ≤ \underline{cd} G + 1 (cf. [22, Lemma 1.8]). (iii) We always have \underline{cd} G = Gcd G, where Gcd G is the Gorenstein projective dimension of

the trivial ℤG-module ℤ (cf. [3]). Ikenaga has also defined in [21, §III] a generalized homological dimension of G. We shall slightly modify Ikenaga’s definition (even though we shall keep his notation) and define the generalized homological dimension \underline{hd} G of G to be the supremum of those integers n, for which there exist a ℤ-free ℤG-module M and an injective ℤG-module I, such that Tor_{n}^{ℤG}(I, M) ≠ 0.^{2} It is clear that \underline{hd} G ≤ n if and only if Tor_{n+1}^{ℤG}(I, M) = 0 for any two ℤG-modules I and M, with M ℤ-free and I injective.

2In Ikenaga’s definition, M is any ℤ-torsion-free ℤG-module. It is easily seen that the two definitions can differ by at most one; it would be of some interest to know if they actually agree.

We now list some properties of the generalized homological dimension, which are essentially due to Ikenaga. (These properties are also satisfied by Ikenaga’s generalized homological dimension and the proofs that we present below follow closely his arguments.) Recall that the homological dimension hd G of G is defined as the flat dimension of the trivial ℤG-module ℤ.

Proposition 4.9. Let G be a group.

(i) We have \underline{hd} G ≤ hd G.

(ii) If G has finite homological dimension, then \underline{hd} G = hd G.

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

(iv) If G is the union of a directed family of subgroups (G_{i})_{i}, then \underline{hd} G ≤ sup \underline{hd} G_{i}.

Proof. (i) Let M be a ℤ-free ℤG-module. Then, for any ℤG-module I, there is a natural isomorphism between Tor_{n}^{ℤG}(I, M) and H_{n}(G, I ⊗ M). It follows that \underline{hd} G ≤ hd G.

(ii) Assume that hd G = n < ∞. Then, the functor H_{n+1}(G, ) is identically zero and H_{n}(G, A) ≠ 0 for some ℤG-module A. If I is an injective ℤG-module containing A, then the exact sequence

0 = H_{n+1}(G, I/A) ⟶ H_{n}(G, A) ⟶ H_{n}(G, I)

shows that H_{n}(G, I) ≠ 0. Since H_{n}(G, I) = Tor_{n}^{ℤG}(I, ℤ), it follows that \underline{hd} G ≥ n. We also have \underline{hd} G ≤ n, in view of (i) above, and hence we conclude that \underline{hd} G = n, as needed.

(iii) If the group Tor_{n}^{ℤG}(I, M) is non-trivial for two ℤG-modules M and I, where I is injective, then n ≤ fd_{ℤG}I ≤ sfli ℤG. It follows readily that \underline{hd} G ≤ sfli ℤG. In order to prove the inequality sfli ℤG ≤ \underline{hd} G + 1, we may assume that \underline{hd} G = n < ∞. Then, for any two ℤG-modules N and I, where I is injective, we have Tor_{n+2}^{ℤG}(I, N) = Tor_{n+1}^{ℤG}(I, ΩN), where ΩN is the kernel of a free presentation of N. Since ΩN is ℤ-free, we conclude that the latter group vanishes and hence fd_{ℤG}I ≤ n + 1. It follows that sfli ℤG ≤ n + 1, as needed.

(iv) We assume that sup \underline{hd} G_{i} = n < ∞ and consider two ℤG-modules M and I, with M ℤ-free and I injective. For any index i we consider the restricted ℤG_{i}-modules res_{G_{i}}^{G}M and res_{G_{i}}^{G}I; the former is (obviously) ℤ-free and the latter is injective. Since \underline{hd} G_{i} ≤ n, it follows that the group Tor_{n+1}^{ℤG_{i}}(I, M) is trivial for all i. Therefore, we conclude that Tor_{n+1}^{ℤG}(I, M) = lim_{⟶i} Tor_{n+1}^{ℤG_{i}}(I, M) = 0 and hence \underline{hd} G ≤ n, as needed. □

We shall now present a class of examples of groups G, for which the inequality sfli ℤG ≤ spli ℤG is strict.

Example 4.10. Let G be a countable group, which is locally free, but not free. (For example, G may be the additive group ℚ of rational numbers.) Then, hd G = 1 and cd G = 2. In particular, \underline{hd} G = 1 and hence 1 ≤ sfli ℤG ≤ 2 (cf. Proposition 4.9(ii),(iii)). Let p be a prime number and consider the finite field 𝔽_{p} with p elements. Since the group G is torsion-free but not free, its cohomological dimension over 𝔽_{p} is ≥ 2 (cf. [35, Theorem A]). Hence, the group G being countable and locally free, its cohomological dimension over 𝔽_{p} is equal to 2. Therefore, we may use [12, Lemma 2.3] and conclude that spli ℤG = cd G + 1 = 3. In particular, we have sfli ℤG < spli ℤG.

The following relation between the generalized homological and cohomological dimensions of a group justifies (to our opinion) the definition of the generalized homological dimension that we proposed above.

Theorem 4.11. If G is a group, then \underline{hd} G ≤ \underline{cd} G.

Proof. The inequality is clear if \underline{cd} G = ∞ and hence we may assume that \underline{cd} G = n < ∞. We shall first consider the special case where the group G is countable. Since silp ℤG ≤

\underline{cd} G + 1 = n + 1, we may invoke Theorem 3.1, in order to conclude that the (n + 1)-th syzygy module of any ℤG-module is strictly Mittag-Leffler over ℤG. Therefore, in view of Proposition 1.5, we know that for any ℤG-module M and any divisible abelian group D, the natural map

Φ_{M}^{(n+1)} : Tor_{n+1}^{ℤG}(Hom(ℤG, D), M)⟶ Hom(Ext_{ℤG}^{n+1}(M, ℤG), D)

is injective. We now assume that the ℤG-module M is ℤ-free. Then, in view of our assumption about the value of \underline{cd} G, we conclude that the group Ext_{ℤG}^{n+1}(M, ℤG) is trivial. Hence, it follows that Tor_{n+1}^{ℤG}(Hom(ℤG, D), M) = 0 as well. Since any injective ℤG-module is a direct summand of Hom(ℤG, D) for a suitable divisible abelian group D, we may conclude that Tor_{n+1}^{ℤG}(I, M) = 0 for any injective ℤG-module I. Hence, it follows that \underline{hd} G ≤ n.

In the general case, we may express G as the directed union of its countable subgroups (G_{i})_{i}. We note that for any i we have \underline{cd} G_{i} ≤ \underline{cd} G = n, in view of [21, Proposition 4], and hence \underline{hd} G_{i} ≤ \underline{cd} G_{i} ≤ n, in view of the special case that we have already considered above. Then, the inequality \underline{hd} G ≤ n follows from Proposition 4.9(iv). □

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

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

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

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