ON THE STABLE HOMOLOGY OF MODULES

IOANNIS EMMANOUIL AND PANAGIOTA MANOUSAKI

Abstract. In this paper, we study certain properties of the stable homology groups of modules over an associative ring, which were defined by Vogel [12]. We compute the kernel of the natural surjection from stable homology to complete homology, which was itself defined by Triulzi [21]. This computation may be used in order to formulate conditions under which the two theories are isomorphic. Duality considerations reveal a connection between stable homology and the complete cohomology theory defined by Nucinkis [19]. Using this connection, we show that the vanishing of the stable homology functors detects modules of finite flat or injective dimension over Noetherian rings. As another application, we characterize the coherent rings over which stable homology is balanced, in terms of the finiteness of the flat dimension of injective modules.

Contents

0. Introduction 1 1. The injective completion of the contravariant Hom functor 4 2. The relation between stable and complete homology 7 3. Isomorphism criteria 10 4. Vanishing criteria 13 5. Vanishing criteria over Noetherian rings 15 6. Balance of stable homology 18 Appendix A. Hausdorff filtrations on complexes and homology 20 References 22

0. Introduction

The classical Tate cohomology theory of finite groups (cf. [3, Chapter XII]) has been generalized to modules over any ring R by Mislin [18]. Using an approach that involves the notion of satellites, he defined for any left R-module M the complete cohomology functor Êxt_{R}^{∗}(M, ) and a natural transformation Ext_{R}^{∗}(M, ) ⟶ Êxt_{R}^{∗}(M, ) as the projective completion of the ordinary Ext functor Ext_{R}^{∗}(M, ); Mislin’s approach is heavily influenced by Gedrich and Gruenberg’s theory of terminal completions [11]. As an immediate consequence of the definition, we note that the complete cohomology groups Êxt_{R}^{∗}(M, N) vanish if pd_{R}N < ∞. Equivalent descriptions of the complete cohomology functors have been independently formulated by Vogel in [12] and by Benson and Carlson in [2]. Using the approach by Benson and Carlson, it follows that the elements in the kernel of the canonical map Hom_{R}(M, N) ⟶ Êxt_{R}^{0}(M, N) are those R-linear maps f : M ⟶ N, which are such that the map Ω^{n}f : Ω^{n}M ⟶ Ω^{n}N induced by f between the n-th syzygy modules of M and N factors through a projective module for n ≫ 0. In particular, M has finite projective dimension if and only if the complete cohomology functor Êxt_{R}^{∗}(M, ) is identically zero; in fact, it only suffices to assume that the group Êxt_{R}^{0}(M, M) be trivial (cf. [16, §4.2]).

Research supported by a GSRT/Greece excellence grant, cofunded by the ESF/EU and National Resources.

Using the dual approach, Nucinkis has defined in [19] for any left R-module N the complete cohomology functor Ẽxt_{R}^{∗}( , N) and a natural transformation Ext_{R}^{∗}( , N) ⟶ Ẽxt_{R}^{∗}( , N), as the injective completion of the Ext functor Ext_{R}^{∗}( , N). The complete cohomology groups Ẽxt_{R}^{∗}(M, N) vanish if M has finite injective dimension. Equivalent descriptions of these functors may be obtained by using the dual approaches to those by Vogel and Benson and Carlson. It follows that the elements in the kernel of the canonical map Hom_{R}(M, N) ⟶ Ẽxt_{R}^{0}(M, N) are those R-linear maps f : M ⟶ N, which are such that the map Σ^{n}f : Σ^{n}M ⟶ Σ^{n}N induced by f between the n-th cosyzygy modules of M and N factors through an injective module for n ≫ 0. In particular, N has finite injective dimension if and only if the complete cohomology functor Ẽxt_{R}^{∗}( , N) is identically zero; in fact, it only suffices to assume that the group Ẽxt_{R}^{0}(N, N) be trivial (cf. [19, Theorem 3.7]).

If M, N are two left R-modules, then a natural question to ask is whether the complete cohomology groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are isomorphic to each other. The inherent asymmetry in the definition of these groups suggests that a positive answer to this question shouldn’t come for free; instead, it should reflect the presence of a homological finiteness condition of some kind. This is indeed the case: Nucinkis has proved in [19, Theorem 5.2] that the following conditions are equivalent on R:

(i) the groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are (naturally) isomorphic for all left R-modules M, N and

(ii) any projective (resp. injective) left R-module has finite injective (resp. projective) dimension. If condition (ii) holds, then (as shown by Gedrich and Gruenberg in [11]) the invariants silp R, the supremum of the injective lengths of projective left R-modules, and spli R, the supremum of the projective lengths of injective left R-modules, are both finite and equal to each other. The common value of these invariants is closely related to the finiteness of the so-called Gorenstein projective and injective dimensions of left R-modules; the reader may consult [8, §4] for more details on this issue.

Things appear to be more complicated for homology. Given a right R-module M and a left R-module N, Vogel has defined in [12] the stable homology groups T̃or_{∗}^{R}(M, N), by using a projective (or flat) resolution of M and an injective resolution of N. It is an immediate consequence of their definition, that the groups T̃or_{∗}^{R}(M, N) vanish if M has finite flat dimension or else if N has finite injective dimension. Motivated by the corresponding results concerning complete cohomology, one may ask whether these homological finiteness conditions are equivalent to the vanishing of the functors T̃or_{∗}^{R}(M, ) and T̃or_{∗}^{R}( , N). Some special cases of this problem have been examined in [4]. We shall prove the following result, which gives an affirmative answer to both questions, in the case of Noetherian rings.

Theorem A. Let R be a left Noetherian ring.

(i) A right R-module M has finite flat dimension if and only if the functors T̃or_{∗}^{R}(M, ) are identically zero.

(ii) A left R-module N has finite injective dimension if and only if the functors T̃or_{∗}^{R}( , N) are identically zero.

Given a right R-module M and a left R-module N, we may define the stable homology groups T̃or_{∗}^{R^{o}}(N, M), using an injective resolution of M and a projective (or flat) resolution of N. A natural question to ask is whether the stable homology groups T̃or_{∗}^{R}(M, N) and T̃or_{∗}^{R^{o}}(N, M) are isomorphic to each other, i.e. whether stable homology is balanced. As in the case of complete cohomology, the inherent asymmetry in the definition of these groups suggests that a homological finiteness condition of some kind should be involved in the answer. In this direction, we shall prove the following result.

Theorem B. Let R be a ring, which is left and right coherent. Then, the following conditions are equivalent:

(i) The stable homology groups T̃or_{∗}^{R}(M, N) and T̃or_{∗}^{R^{o}}(N, M) are (naturally) isomorphic for any right R-module M and any left R-module N.

(ii) Any injective left or right R-module has finite flat dimension.

The implication (ii)→(i) is valid over any ring; this is essentially shown in [4, Theorem 4.2]. We note that if condition (ii) holds, then (as shown in [9, Corollary 2.5] and under no assumption on R) the invariants sfli R and sfli R^{o}, the suprema of the flat lengths of injective left (resp. right) R-modules are both finite and equal to each other. The common value of these invariants is closely related to the finiteness of the so-called Gorenstein flat dimensions of left and right R-modules; the reader may consult [8, §5] for more details on this issue.

Our main technical tool for proving Theorems A and B is a description of the stable homology groups, in terms of a certain inverse system of abelian groups. In order to be more precise, we note that Triulzi has defined in [21] for any right R-module M the complete homology functor T̂or_{∗}^{R}(M, ) and a natural transformation T̂or_{∗}^{R}(M, ) ⟶ Tor_{∗}^{R}(M, ), as the injective completion of the Tor functor Tor_{∗}^{R}(M, ). The complete homology groups T̂or_{∗}^{R}(M, N) vanish if the left R-module N has finite injective dimension. It turns out that the group T̂or_{n}^{R}(M, N) may be computed as the limit of the inverse system (Tor_{n+i}^{R}(M, Σ^{i}N))_{i}, where Σ^{i}N denotes the i-th cozyzygy module of N. The universal property of complete homology provides us with a natural map T̃or_{∗}^{R}(M, N) ⟶ T̂or_{∗}^{R}(M, N), which is always surjective; this has been proved by Triulzi. We shall identify its kernel as a certain lim_{⟵}^{1} -group; it will turn out that

the kernel in degree n is the group lim_{⟵i}^{1}Tor_{n+i+1}^{R}(M, Σ^{i}N). First of all, this computation gives a systematic way of examining conditions under which stable homology is isomorphic with complete homology. In this way, we are able to reinterpret some of the results obtained in [5, §2]. On the other hand, this description of the stable homology groups may be also used in order to analyze their vanishing. This circle of ideas, coupled with the duality between stable homology and Nucinkis’ complete cohomology, will lead to the proofs of Theorems A and B.

The contents of the paper are as follows: In Section 1, we record a description of Nucinkis’ complete cohomology, in terms of a decreasing filtration of the total Hom complex, which is associated with injective resolutions of both variables. This description will be used in order to establish the duality with stable homology. In the next section, we examine the kernel of the natural map from stable to complete homology, by using a description of the former, in terms of a decreasing filtration of the total tensor product complex, which is associated with a flat (resp. injective) resolution of the first (resp. second) variable. We use some generalities involving the homology of complexes endowed with such decreasing filtrations, that have been collected in the Appendix at the end of the paper. In Section 3, we examine conditions that imply the vanishing of the lim_{⟵}^{1} -term appearing as the kernel of the natural map from stable to complete homology. In Sections 4 and 5, we examine the vanishing of the stable homology groups and prove Theorem A, by using the duality between stable homology and Nucinkis’ complete cohomology. Finally, in Section 6, we study the balance of stable homology and use the vanishing criteria obtained in the previous sections, in order to prove the characterization presented in Theorem B.

Notations and terminology.

(i) All direct and inverse systems examined in this paper are indexed by the ordered set ℕ of natural numbers.

(ii) If R is a ring, then we denote by R^{o} its opposite ring. We do not distinguish between left (resp. right) R-modules and right (resp. left) R^{o}-modules.

(iii) For any left R-module M we shall denote by ΩM the kernel of an epimorphism from a projective module onto M. Even though ΩM is not uniquely determined by M, Schanuel’s lemma implies uniqueness up to projective direct summands. The iterates Ω^{i}M, i ≥ 0, are defined inductively by letting Ω^{0}M = M and Ω^{i}M = ΩΩ^{i−1}M for all i > 0. If P_{∗} ⟶ M ⟶ 0 is a projective resolution of M, then Ω^{i}M is identified with the image of the differential P_{i} ⟶ P_{i−1} for all i > 0.

(iv) Dually, for any left R-module M we shall denote by ΣM the cokernel of a monomorphism from M into an injective module; Schanuel’s lemma implies uniqueness up to injective direct summands. The iterates Σ^{i}M, i ≥ 0, are defined inductively by letting Σ^{0}M = M and Σ^{i}M = ΣΣ^{i−1}M for all i > 0. If 0 ⟶ M ⟶ I^{∗} is an injective resolution of M, then Σ^{i}M is identified with the image of the differential I^{i−1} ⟶ I^{i} for all i > 0.

(v) We denote by D the Pontryagin duality functors from the category of left (resp. right) R-modules to the category of right (resp. left) R-modules, which are both defined by letting M ↦ Hom(M, ℚ/ℤ). We shall often use in the sequel Lambek’s equality fd_{R^{o}}M = id_{R}DM, which is valid for any right R-module M; cf. [17, Theorem 4.9]. We shall also use the fact that the functor D maps injective right R-modules onto flat left R-modules, in the special case where the ring R is right coherent; for a proof of that assertion, the reader may consult [22, Lemma 3.1.4].

(vi) For any complex X of left R-modules and any integer i, we shall denote by X[−i] the i-fold suspension of X. In particular, if M is a left R-module, then we denote by M[i] the complex consisting of M in degree i and zeroes elsewhere.

1. The injective completion of the contravariant Hom functor

Let R be a ring and consider two left R-modules N, L. Nucinkis has defined in [19] the complete cohomology groups Ẽxt_{R}^{∗}(N, L), by evaluating the injective completion Ẽxt_{R}^{∗}( , L) of the cohomological functor Ext_{R}^{∗}( , L) to N. These groups vanish if N is injective and are universal (in a certain sense) with that property.

Nucinkis’ construction of these complete cohomology groups involves the left satellites of the contravariant Ext functor Ext_{R}^{∗}( , L), a notion due to Cartan and Eilenberg [3, Chapter III]. There is a concrete description of complete cohomology, which may be expressed in terms of an injective resolution 0 ⟶ N ⟶ I^{∗} of N. If Σ^{i}N is the corresponding i-th cosyzygy module of N for all i ≥ 0, then the short exact sequence

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0

induces connecting homomorphisms

(1)  d_{n}^{i} : Ext_{R}^{n+i}(Σ^{i}N, L) ⟶ Ext_{R}^{n+i+1}(Σ^{i+1}N, L)

for all n, i with i ≥ 0. We may consider the direct system (Ext_{R}^{n+i}(Σ^{i}N, L))_{i} with structure maps given by the connecting homomorphisms d_{n}^{i} as above; it turns out that

(2)  Ẽxt_{R}^{n}(N, L) = lim_{⟶i} Ext_{R}^{n+i}(Σ^{i}N, L)

for all n (see the proof of [19, Theorem 3.6]). The complete cohomology groups defined above admit two other equivalent descriptions: one of them follows an idea by Benson and Carlson [2] and the other one is due to Vogel [12]. Using the approach by Benson and Carlson, one may prove that Ẽxt_{R}^{0}(N, N) = 0 if and only if id_{R}N < ∞ (cf. [19, Theorem 3.7]).

In the sequel, we shall use yet another description of Nucinkis’ complete cohomology groups Ẽxt_{R}^{∗}(N, L), which may be formulated in terms of an injective resolution 0 ⟶ N ⟶ I^{∗} of N and an injective resolution 0 ⟶ L ⟶ J^{∗} of L. For all i ≥ 0 we consider the subcomplex I^{≥i} ⊆ I^{∗}, which coincides with I in (cohomological) degrees j ≥ i and vanishes in degrees j < i. We note that the exactness of the sequence

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ I^{i+1} ⟶ …

may be reformulated by saying that the inclusion Σ^{i}N ↪ I^{i} induces a quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} for all i ≥ 0. The short exact sequence of cochain complexes of left R-modules

0 ⟶ I^{≥i+1} ⟶ I^{≥i} ⟶ I^{i}[i] ⟶ 0

induces a short exact sequence of complexes of abelian groups

0 ⟶ Hom_{R}(I^{i}[i], J^{∗})⟶ Hom_{R}(I^{≥i}, J^{∗})⟶^{c_{i}} Hom_{R}(I^{≥i+1}, J^{∗})⟶ 0.

We note that the cochain map denoted by c_{i} above is just the restriction along the inclusion I^{≥i+1} ⊆ I^{≥i}. We are interested in the additive maps

c_{i}^{n} : H^{n}(Hom_{R}(I^{≥i}, J^{∗}))⟶ H^{n}(Hom_{R}(I^{≥i+1}, J^{∗})) ,

which are induced in cohomology by the cochain map c_{i} for all n, i with i ≥ 0. These induce a direct system of abelian groups (H^{n}(Hom_{R}(I^{≥i}, J^{∗})))_{i}, whose colimit we describe below.

Proposition 1.1. Let the notation be as above.

(i) The quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} induces an isomorphism of abelian groups

H^{n}(Hom_{R}(I^{≥i}, J^{∗}))⟶^{∼} Ext_{R}^{n+i}(Σ^{i}N, L)

for all n, i with i ≥ 0.

(ii) There is a commutative diagram of abelian groups

H^{n}(Hom_{R}(I^{≥i}, J^{∗})) ⟶^{∼} Ext_{R}^{n+i}(Σ^{i}N, L)
c_{i}^{n} ↓  ↓ d_{i}^{n}
H^{n}(Hom_{R}(I^{≥i+1}, J^{∗})) ⟶^{∼} Ext_{R}^{n+i+1}(Σ^{i+1}N, L)

for all n, i with i ≥ 0. Here, the horizontal isomorphisms are those defined in (i), whereas d_{i}^{n} is the connecting homomorphism (1).

(iii) There is a natural isomorphism Ẽxt_{R}^{n}(N, L) ≃ lim_{⟶i} H^{n}(Hom_{R}(I^{≥i}, J^{∗})) for all n.

For the proof, we shall need the following auxiliary result.

Lemma 1.2. Let J^{∗} be a left bounded cochain complex of injective left R-modules.

(i) For any acyclic complex of left R-modules C^{∗} the complex of abelian groups Hom_{R}(C^{∗}, J^{∗}) is acyclic as well.

(ii) Any quasi-isomorphism f : X^{∗} ⟶ Y^{∗} of complexes of left R-modules induces a quasiisomorphism f^{t} : Hom_{R}(Y^{∗}, J^{∗}) ⟶ Hom_{R}(X^{∗}, J^{∗}) of complexes of abelian groups.

Proof. (i) A cocycle of degree n of the complex Hom_{R}(C^{∗}, J^{∗}) is precisely a cochain map form C^{∗} to the n-fold suspension of J^{∗}. Moreover, the cocycles which are coboundaries are precisely those cochain maps that are null-homotopic. It follows that the acyclicity of Hom_{R}(C^{∗}, J^{∗}) is merely a reformulation of the fact that any cochain map from an acyclic complex to a left bounded complex of injective modules is null-homotopic.

(ii) This is an immediate consequence of (i), since the mapping cone C^{∗} = cone(f) of the quasi-isomorphism f is acyclic and the complex Hom_{R}(C^{∗}, J^{∗}) is identified with the mapping cone of the cochain map f^{t} : Hom_{R}(Y^{∗}, J^{∗}) ⟶ Hom_{R}(X^{∗}, J^{∗}). □

Proof of Proposition 1.1. (i) Since J^{∗} is a left bounded cochain complex of injective modules, Lemma 1.2(ii) implies that the quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} induces a quasi-isomorphism

Hom_{R}(I^{≥i}, J^{∗})⟶Hom_{R}(Σ^{i}N[i], J^{∗})

for all i ≥ 0. This finishes the proof, since

H^{n}(Hom_{R}(Σ^{i}N[i], J^{∗}))= H^{n+i}(Hom_{R}(Σ^{i}N, J^{∗}))= Ext_{R}^{n+i}(Σ^{i}N, L)

for all n, i with i ≥ 0.

(ii) The isomorphism H^{n}(Hom_{R}(I^{≥i}, J^{∗}))≃ Ext_{R}^{n+i}(Σ^{i}N, L) established in (i) identifies an element ξ ∈ Ext_{R}^{n+i}(Σ^{i}N, L), which is represented by a linear map f : Σ^{i}N ⟶ Σ^{n+i}L, with the class of the (unique up to homotopy) cochain map (f^{j})_{j≥i} : I^{≥i} ⟶ J^{∗}[−n] lifting f

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ I^{i+1} ⟶ I^{i+2} ⟶ …
f ↓ f^{i} ↓ f^{i+1} ↓ f^{i+2} ↓
0 ⟶ Σ^{n+i}L ⟶ J^{n+i} ⟶ J^{n+i+1} ⟶ J^{n+i+2} ⟶ …

The restriction c_{i}[(f^{j})_{j≥i}] is the cochain map (f^{j})_{j≥i+1} : I^{≥i+1} ⟶ J^{∗}[−n] which lifts the linear map g : Σ^{i+1}N ⟶ Σ^{i+n+1}L induced from f^{i} by passage to the quotients

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0
f ↓ f^{i} ↓ g ↓
0 ⟶ Σ^{n+i}L ⟶ J^{n+i} ⟶ Σ^{n+i+1}L ⟶ 0

The proof is complete, since the linear map g represents the image in Ext_{R}^{n+i+1}(Σ^{i+1}N, L) of ξ under the connecting homomorphism

d_{n}^{i} : Ext_{R}^{n+i}(Σ^{i}N, L) ⟶ Ext_{R}^{n+i+1}(Σ^{i+1}N, L).

(iii) In view of equation (2), this is an immediate consequence of (ii) above, which identifies the direct system (H^{n}(Hom_{R}(I^{≥i}, J^{∗})))_{i}, with structure maps given by the c_{i}^{n}’s, with the direct system (Ext_{R}^{n+i}(Σ^{i}N, L))_{i}, with structure maps given by the d_{i}^{n}’s. □

Remark 1.3. There is a similar description of Mislin’s complete cohomology Êxt, in terms of a decreasing filtration of the total Hom complex, which is associated with projective resolutions of both variables; cf. [5, Lemma A.3].

2. The relation between stable and complete homology

Let R be a ring and consider a right R-module M and a left R-module N. Vogel has defined in [12] the stable homology groups T̃or_{∗}^{R}(M, N). If P_{∗} ⟶ M ⟶ 0 is a projective resolution of M and 0 ⟶ N ⟶ I^{∗} an injective resolution of N, then

T̃or_{n}^{R}(M, N) = H_{n+1}(P_{∗} ˜⊗_{R} I^{∗})

for all n. Here, P_{∗} ˜⊗_{R} I^{∗} is the complex of abelian groups whose group of chains in degree n is the quotient of the direct product ∏_{i−j=n} P_{i} ⊗_{R} I^{j} modulo the direct sum ⨁_{i−j=n} P_{i} ⊗_{R} I^{j}. The reader may consult [12] and [4] for several basic properties of the stable homology groups. We shall now recall some of these properties: The stable homology groups T̃or_{∗}^{R}(M, N) may be computed by replacing the projective resolution P_{∗} ⟶ M ⟶ 0 of M in the definition above by any flat resolution F_{∗} ⟶ M ⟶ 0 of M; we have

T̃or_{n}^{R}(M, N) = H_{n+1}(F_{∗} ˜⊗_{R} I^{∗})

for all n. Consequently, the groups T̃or_{∗}^{R}(M, N) vanish if either M has finite flat dimension or else if N has finite injective dimension.

The complete homology groups T̂or_{∗}^{R}(M, N) were defined by Triulzi in [21] as the evaluation of the injective completion T̂or_{∗}^{R}(M, ) of the ordinary Tor functor Tor_{∗}^{R}(M, ) to the left Rmodule N. These groups vanish if N is injective and are universal (in a certain sense) with that property. Triulzi’s construction involves the notion of right satellites of functors, which is due to Cartan and Eilenberg [3, Chapter III]. There is a concrete description of complete homology, which may be expressed in terms of an injective resolution 0 ⟶ N ⟶ I^{∗} of N. Indeed, given such a resolution, we may consider the corresponding cosyzygy modules Σ^{i}N, i ≥ 0, and note that the short exact sequence

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0

induces connecting homomorphisms

(3)  δ_{n}^{i} : Tor_{n+i+1}^{R}(M, Σ^{i+1}N) ⟶ Tor_{n+i}^{R}(M, Σ^{i}N)

for all n, i with i ≥ 0. We may consider the inverse system (Tor_{n+i}^{R}(M, Σ^{i}N))_{i} with structure maps given by the connecting homomorphisms δ_{n}^{i} as above; it turns out that

(4)  T̂or_{n}^{R}(M, N) = lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N),

whereas the canonical map T̂or_{n}^{R}(M, N) ⟶ Tor_{n}^{R}(M, N) is identified with the natural projection of the limit

lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N) ⟶ Tor_{n+0}^{R}(M, Σ^{0}N)

for all n (cf. [5, Lemma 1.7]).

In order to relate the stable homology groups T̃or_{∗}^{R}(M, N) to the complete homology groups T̂or_{∗}^{R}(M, N), we shall use yet another description of the latter, which may be formulated in terms of a flat resolution F_{∗} ⟶ M ⟶ 0 of M and an injective resolution 0 ⟶ N ⟶ I^{∗} of N (cf. [4, §2.4]). As in the previous section, we consider for all i ≥ 0 the subcomplex I^{≥i} ⊆ I^{∗}, which coincides with I^{∗} in (cohomological) degrees j ≥ i and vanishes in degrees j < i. We recall that the inclusion Σ^{i}N ↪ I^{i} induces a quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} for all i ≥ 0.

Since I^{≥i+1} is a subcomplex of I^{≥i}, it follows that F_{∗} ⊗_{R} I^{≥i+1} is a subcomplex of F_{∗} ⊗_{R} I^{≥i}; we shall denote by γ^{i} the inclusion F_{∗} ⊗_{R} I^{≥i+1} ↪ F_{∗} ⊗_{R} I^{≥i} for all i ≥ 0. We also consider the additive maps γ_{n}^{i} : H_{n}(F_{∗} ⊗_{R} I^{≥i+1})⟶ H_{n}(F_{∗} ⊗_{R} I^{≥i}), which are induced by the γ^{i}’s in homology for all n, i with i ≥ 0.

Proposition 2.1. Let the notation be as above.

(i) The quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} induces an isomorphism of abelian groups

Tor_{n+i}^{R}(M, Σ^{i}N) ⟶^{∼} H_{n}(F_{∗} ⊗_{R} I^{≥i})

for all n, i with i ≥ 0.

(ii) There is a commutative diagram of abelian groups

Tor_{n+i+1}^{R}(M, Σ^{i+1}N) ⟶^{∼} H_{n}(F_{∗} ⊗_{R} I^{≥i+1})
δ_{n}^{i} ↓  ↓ γ_{n}^{i}
Tor_{n+i}^{R}(M, Σ^{i}N) ⟶^{∼} H_{n}(F_{∗} ⊗_{R} I^{≥i})

for all n, i with i ≥ 0. Here, the horizontal isomorphisms are those defined in (i), whereas δ_{n}^{i} is the connecting homomorphism (3).

(iii) There is a natural isomorphism T̂or_{n}^{R}(M, N) ≃ lim_{⟵i} H_{n}(F_{∗} ⊗_{R} I^{≥i}) for all n.

Proof. The flat resolution F_{∗} ⟶ M ⟶ 0 of the right R-module M induces, by applying the Pontryagin duality functor D, an injective resolution 0 ⟶ DM ⟶ DF_{∗} of the left R-module DM.

(i) Proposition 1.1(i) implies that the quasi-isomorphism Σ^{i}N[i] ⟶ I^{≥i} induces an isomorphism of abelian groups

H^{n}(Hom_{R}(I^{≥i}, DF_{∗}))⟶^{∼} Ext_{R}^{n+i}(Σ^{i}N, DM)

for all n, i with i ≥ 0. In view of the standard Hom-tensor duality, the latter isomorphism is precisely that obtained by applying the Pontryagin duality functor D to the additive map

Tor_{n+i}^{R}(M, Σ^{i}N) ⟶ H_{n}(F_{∗} ⊗_{R} I^{≥i})

in the statement. Since ℚ/ℤ is faithfully injective (as an abelian group), it follows that the latter map is bijective, as needed.

(ii) We argue as in (i) above: Since the abelian group ℚ/ℤ is faithfully injective, it suffices to verify the commutativity of the diagram, which is obtained from that in the statement of (ii), by applying the Pontryagin duality functor D. Invoking the standard Hom-tensor duality, the latter diagram is identified with

Ext_{R}^{n+i+1}(Σ^{i+1}N, DM) ⟵^{∼} H^{n}(Hom_{R}(I^{≥i+1}, DF_{∗}))
d_{i}^{n} ↑  ↑ c_{i}^{n}
Ext_{R}^{n+i}(Σ^{i}N, DM) ⟵^{∼} H^{n}(Hom_{R}(I^{≥i}, DF_{∗}))

Here, the functor D maps the connecting homomorphism δ_{n}^{i} (resp. γ_{n}^{i}) onto the connecting homomorphism d_{i}^{n} (resp. onto c_{i}^{n}). The proof is completed, since the latter diagram is commutative, as shown in Proposition 1.1(ii).

(iii) In view of equation (4), this is an immediate consequence of (ii) above, which identifies the inverse system (H_{n}(F_{∗} ⊗_{R} I^{≥i}))_{i}, with structure maps given by the γ_{n}^{i}’s, with the inverse system (Tor_{n+i}^{R}(M, Σ^{i}N))_{i}, with structure maps given by the δ_{n}^{i}’s. □

We are now ready to state and prove the following result, establishing the relation between stable and complete homology.

Theorem 2.2. Let R be a ring and consider a right R-module M and a left R-module N. Then, for all n there is a short exact sequence

0 ⟶ lim_{⟵i}^{1}Tor_{n+i+1}^{R}(M, Σ^{i}N) ⟶ T̃or_{n}^{R}(M, N) ⟶^{∂} T̂or_{n}^{R}(M, N) ⟶ 0,

which is natural in both M and N. Here, Σ^{i}N, i ≥ 0, are the cosyzygy modules of N that are associated with any injective resolution of it, whereas the structure maps of the inverse systems (Tor_{n+i}^{R}(M, Σ^{i}N))_{i} are the connecting homomorphisms δ_{n}^{i} in (3) for all n.

Proof. Let F_{∗} ⟶ M ⟶ 0 be a flat resolution of M and 0 ⟶ N ⟶ I^{∗} an injective resolution of N. We consider the subcomplexes I^{≥i} ⊆ I^{∗}, i ≥ 0, which were defined above and note that these provide us with a decreasing filtration of I^{∗}

I^{∗} = I^{≥0} ⊇ I^{≥1} ⊇ I^{≥2} ⊇ … ⊇ I^{≥i} ⊇ ….

We also consider the induced filtration of the complex F_{∗} ⊗_{R} I^{∗}

F_{∗} ⊗_{R} I^{∗} = F_{∗} ⊗_{R} I^{≥0} ⊇ F_{∗} ⊗_{R} I^{≥1} ⊇ F_{∗} ⊗_{R} I^{≥2} ⊇ … ⊇ F_{∗} ⊗_{R} I^{≥i} ⊇ ….

Having fixed the degree n, the associated decreasing filtration of the abelian group of n-chains (F_{∗} ⊗_{R} I^{∗})_{n} = ⨁_{j} F_{j+n} ⊗_{R} I^{j} = ⨁_{j≥0} F_{j+n} ⊗_{R} I^{j} of the complex F_{∗} ⊗_{R} I^{∗} is presented below

⨁_{j≥0}F_{j+n} ⊗_{R} I^{j} ⊇ ⨁_{j≥1}F_{j+n} ⊗_{R} I^{j} ⊇ ⨁_{j≥2}F_{j+n} ⊗_{R} I^{j} ⊇ … ⊇ ⨁_{j≥i}F_{j+n} ⊗_{R} I^{j} ⊇ ….

It follows readily that this filtration of the complex F_{∗} ⊗_{R} I^{∗} is Hausdorff (cf. Definition A.1 of the Appendix). We wish to identify the complex lim_{⟵i}^{1} (F_{∗} ⊗ I^{≥i}). As noted in the discussion preceding Lemma A.2, the abelian group of n-chains of the latter complex is the cokernel of the natural map

(5)  (F_{∗} ⊗_{R} I^{∗})_{n} ⟶ lim_{⟵i} [(F_{∗} ⊗_{R} I^{∗})_{n}/(F_{∗} ⊗_{R} I^{≥i})_{n}].

Since (F_{∗} ⊗_{R} I^{∗})_{n} = ⨁_{j≥0} F_{j+n} ⊗_{R} I^{j} and (F_{∗} ⊗_{R} I^{≥i})_{n} = ⨁_{j≥i} F_{j+n} ⊗_{R} I^{j}, we conclude that

(F_{∗} ⊗_{R} I^{∗})_{n}/(F_{∗} ⊗_{R} I^{≥i})_{n} = ⨁_{j=0}^{i−1}F_{j+n} ⊗_{R} I^{j} = ∏_{j=0}^{i−1}F_{j+n} ⊗_{R} I^{j}

for all n, i with i ≥ 0. It follows easily from this that the group lim_{⟵i}[(F_{∗} ⊗_{R} I^{∗})_{n}/(F_{∗} ⊗_{R} I^{≥i})_{n}]

is isomorphic with the direct product ∏_{j≥0} F_{j+n} ⊗_{R} I^{j}, in such a way that the map (5) above is identified with the natural inclusion

⨁j≥0F_{j+n} ⊗_{R} I^{j} ↪ _{∏j≥0}F_{j+n} ⊗_{R} I^{j}.

We therefore conclude that the complex lim_{⟵i}^{1}(F_{∗} ⊗ I^{≥i}) coincides with the complex F_{∗} ˜⊗_{R} I^{∗},

that computes the stable homology groups T̃or_{∗}^{R}(M, N). Hence, applying Corollary A.3 of the Appendix to the special case of the Hausdorff filtration of F_{∗} ⊗_{R} I^{∗} given by the F_{∗} ⊗ I^{≥i}’s, we obtain for all n a short exact sequence of abelian groups

0 ⟶ lim_{⟵i}^{1} H_{n+1}(F_{∗} ⊗_{R} I^{≥i})⟶ H_{n+1}(F_{∗} ˜⊗_{R} I^{∗})⟶^{∂} lim_{⟵i} H_{n}(F_{∗} ⊗_{R} I^{≥i})⟶ 0.

Proposition 2.1(ii) implies that the inverse system (H_{n}(F_{∗} ⊗_{R} I^{≥i}))_{i}, with structure maps given by the γ_{n}^{i}’s, is isomorphic with the inverse system (Tor_{n+i}^{R}(M, Σ^{i}N))_{i}, with structure maps given by the δ_{n}^{i}’s. Hence, the exact sequence above reduces to that in the statement of the Theorem and the proof is complete. □

Remark 2.3. Keeping the same notation as above, we note that for all n the map

∂ : T̃or_{n}^{R}(M, N) ⟶ T̂or_{n}^{R}(M, N),

i.e. the map

∂ : H_{n+1}(F_{∗} ˜⊗_{R} I^{∗})⟶ lim_{⟵i} H_{n}(F_{∗} ⊗_{R} I^{≥i})

resulting from the short exact sequence in Corollary A.3 of the Appendix, coincides with that defined by Triulzi; cf. [21, Lemma 6.2.9]. An explicit description of this map can be also found in [5, §2.5].

3. Isomorphism criteria

Having proved Theorem 2.2, one may look for conditions implying that the natural surjection from stable homology to complete homology is bijective. An answer to this problem may have important consequences for the two homology theories. For example, if the functors T̃or_{∗}^{R}(M, ) and T̂or_{∗}^{R}(M, ) are naturally isomorphic, then:

(i) stable homology has the universal property inherited by the fact that it is naturally isomorphic with the injective completion of the functor Tor_{∗}^{R}(M, ) and

(ii) complete homology is a homological δ-functor, since it is naturally isomorphic with a functor that is computed as the homology of a complex (cf. [4, §2.4 and §2.5]).

In the present section, we shall obtain two types of conditions under which the natural map from stable homology to complete homology is bijective. We should point out that the results presented in Propositions 3.1 and 3.3 below are not new; they have been already proved in [5]. Our goal is to provide alternative proofs for these results, by using the identification of the kernel of the canonical map ∂ : T̃or ⟶ T̂or. In other words, we shall approach the problem by looking for conditions that imply the vanishing of the lim_{⟵}^{1} -term in the short exact sequence of Theorem 2.2.

Grothendieck introduced in [13] a simple condition for an inverse system of abelian groups (A_{i})_{i} that implies the vanishing of lim_{⟵i}^{1} A_{i}, the so-called Mittag-Leffler condition. We say that the inverse system (A_{i})_{i} satisfies the Mittag-Leffler condition if the decreasing filtration induced on A_{i} by the images of the structure maps from A_{j}, j ≥ i, is eventually constant for all i. In other words, we demand that for all i there exists j(i) ≥ i, such that the image of the structure map A_{j(i)} ⟶ A_{i} is equal to the image of the structure map A_{k} ⟶ A_{i} for all k ≥ j(i); in that case, the image A′_{i} of the map A_{j(i)} ⟶ A_{i} is referred to as the stable image. It is clear that the Mittag-Leffler condition is satisfied if the structure maps A_{i} ⟶ A_{i−1} are surjective for all i ≫ 0.

Proposition 3.1. Let R be a ring and consider a right R-module M. We assume that one of the following two conditions is satisfied:

(i) There exists an integer m, such that the functors Tor_{i}^{R}(M, ) vanish on all injective left R-modules for all i > m.^{1}

(ii) All injective left R-modules have finite flat dimension.^{2} Then, the natural map ∂ : T̃or_{∗}^{R}(M, N) ⟶ T̂or_{∗}^{R}(M, N) is bijective for all left R-modules N.

1In the terminology of [10], this condition says that the module M has finite copure flat dimension.

2In that case, there is an upper bound on the flat lengths of injective left R-modules; in the terminology of [9] (see also [6] and [15]), this condition says that the invariant sfli R is finite.

Proof. Let 0 ⟶ N ⟶ I^{∗} be an injective resolution of the left R-module N. Then, both assumptions (i) and (ii) imply that there exists an integer m, such that the groups Tor_{i}^{R}(M, I^{j}) vanish for all i > m and all j. Denoting by Σ^{i}N, i ≥ 0, the corresponding cosyzygy modules of N, the exact sequence of abelian groups

Tor_{n+i+1}^{R}(M, Σ^{i+1}N) ⟶^{δ_{n}^{i}} Tor_{n+i}^{R}(M, Σ^{i}N) ⟶ Tor_{n+i}^{R}(M, I^{i}),

which is induced by the short exact sequence of left R-modules

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0,

implies that the connecting homomorphism

δ_{n}^{i} : Tor_{n+i+1}^{R}(M, Σ^{i+1}N) ⟶ Tor_{n+i}^{R}(M, Σ^{i}N)

is surjective for all i > m − n. It follows from the preceding discussion that the inverse system (Tor_{n+i}^{R}(M, Σ^{i}N))_{i} satisfies the Mittag-Leffler condition and hence lim_{⟵i}^{1}Tor_{n+i}^{R}(M, Σ^{i}N) = 0 for all n. Invoking Theorem 2.2, this completes the proof. □

Another type of condition on an inverse system that implies the vanishing of the corresponding lim_{⟵}^{1} is obtained by dualizing a direct system, by means of a contravariant exact functor which maps direct sums onto direct products. We shall illustrate this by considering the Pontryagin duality functor D. We note that any direct system (B_{i})_{i} of abelian groups with structure maps λ_{i} : B_{i} ⟶ B_{i+1} induces an inverse system of abelian groups (DB_{i})_{i}, with structure maps Dλ_{i} : DB_{i+1} ⟶ DB_{i} for all i.

Lemma 3.2. Let (B_{i})_{i} be a direct system of abelian groups and consider the associated inverse system (DB_{i})_{i}, as above. Then, lim_{⟵i} DB_{i} ≃ D(lim_{⟶i}B_{i}) and lim_{⟵i}^{1}DB_{i} = 0.

Proof. The colimit B = lim_{⟶i} B_{i} fits into a short exact sequence

0 ⟶ ⨁_{i}B_{i} ⟶^{1−λ} ⨁_{i}B_{i} ⟶ B ⟶ 0,

where 1 − λ is the map (b_{i})_{i} ↦ (b_{i} − λ_{i−1}b_{i−1})_{i}, (b_{i})_{i} ∈ ⨁_{i} B_{i}. Applying the duality functor D to the above sequence, we obtain the short exact sequence

0 ⟶ DB ⟶ ∏_{i}DB_{i} ⟶^{1−Dλ} ∏_{i}DB_{i} ⟶ 0.

It follows readily that lim_{⟵i} DB_{i} = DB and lim_{⟵i}^{1}DB_{i} is the trivial group. □

If M, K are two right R-modules, then we may define the additive map

Φ : M ⊗_{R} DK ⟶ DHom_{R}(M, K),

by letting Φ(m ⊗ f) be the operator which is given by g ↦ f(g(m)), g ∈ Hom_{R}(M, K), for all m ∈ M and f ∈ DK. The map Φ is natural in M, K and has been introduced by Cartan and Eilenberg in [3, Chapter VI, §5]. It is bijective if M is finitely presented.

Having fixed a right R-module K, we consider for any right R-module M a projective resolution P_{∗} ⟶ M ⟶ 0. Then, the natural transformation Φ induces a chain map

Φ : P_{∗} ⊗_{R} DK ⟶ DHom_{R}(P_{∗}, K).

By applying homology, we obtain additive maps

Φ_{i} : Tor_{i}^{R}(M, DK) ⟶ DExt_{R}^{i}(M, K),

i ≥ 0, which do not depend upon the particular choice of the projective resolution of M. The Φ_{i}’s are natural in both M and K and commute with the connecting homomorphisms which are associated with any short exact sequence of right R-modules

0 ⟶ K′ ⟶ K ⟶ K′′ ⟶ 0.

If M is of type FP_{∞}, i.e. if M admits a projective resolution P_{∗} ⟶ M ⟶ 0, which consists of finitely generated (projective) modules in each degree, then the Φ_{i}’s are bijective for all i.

Proposition 3.3. Let R be a ring and consider a right R-module M of type FP_{∞}. Then, the natural map ∂ : T̃or_{∗}^{R}(M, DK) ⟶ T̂or_{∗}^{R}(M, DK) is bijective for any right R-module K.

Proof. Let F_{∗} ⟶ K ⟶ 0 be a flat resolution of the right R-module K and denote by K_{i} the image of the differential F_{i} ⟶ F_{i−1} for all i ≥ 1. Then, 0 ⟶ DK ⟶ DF_{∗} is an injective resolution of the left R-module DK and DK_{i} is identified with the corresponding i-th cosyzygy module Σ^{i}DK for all i ≥ 1. It follows from the discussion above that the inverse system (Tor_{n+i}^{R}(M, DK_{i}))_{i}, whose structure maps are connecting homomorphisms induced by the short exact sequences

0 ⟶ DK_{i} ⟶ DF_{i} ⟶ DK_{i+1} ⟶ 0,

is isomorphic with the inverse system (DExt_{R}^{n+i}(M, K_{i}))_{i}, which is induced by applying the duality functor D to the direct system (Ext_{R}^{n+i}(M, K_{i}))_{i}, whose structure maps are connecting homomorphisms induced by the short exact sequences

0 ⟶ K_{i+1} ⟶ F_{i} ⟶ K_{i} ⟶ 0.

It follows from Lemma 3.2 that

lim_{⟵i}^{1}Tor_{n+i}^{R}(M, DK_{i}) = lim_{⟵i}^{1}DExt_{R}^{n+i}(M, K_{i}) = 0

for all n and hence the proof is finished by invoking Theorem 2.2. □

Remark 3.4. Keeping the same notation as in the proof of Proposition 3.3, we note that the groups T̃or_{n}^{R}(M, DK) ≃ T̂or_{n}^{R}(M, DK) are isomorphic with the limit

lim_{⟵i} Tor_{n+i}^{R}(M, DK_{i}) = lim_{⟵i} DExt_{R}^{n+i}(M, K_{i}).

As noted in Lemma 3.2, the latter limit is identified with the Pontryagin dual of the colimit lim_{⟶i} Ext_{R}^{n+i}(M, K_{i}) of the direct system (Ext_{R}^{n+i}(M, K_{i}))_{i}. This identification is valid for any flat resolution F_{∗} ⟶ K ⟶ 0 of the right R-module K. Assuming that the flat modules F_{i} are in fact projective (so that F_{∗} ⟶ K ⟶ 0 is a projective resolution of K), it follows that K_{i} = Ω^{i}K is the i-th syzygy module of K for all i ≥ 1. Then, as shown in [18, §4], the colimit

lim_{⟶i} Ext_{R}^{n+i}(M, K_{i}) = lim_{⟶i} Ext_{R}^{n+i}(M, Ω^{i}K)

is isomorphic with Mislin’s complete cohomology group Êxt_{R}^{n}(M, K). The identification of the D-dual of that group with T̃or_{n}^{R}(M, DK) ≃ T̂or_{n}^{R}(M, DK) has been noted during the proof of [5, Theorem 2.14].

4. Vanishing criteria

As another application of Theorem 2.2, we may study the vanishing of stable homology. It will turn out that duality considerations reveal an interesting connection between the vanishing of the stable homology functors T̃or and that of Nucinkis’ complete cohomology functors Ẽxt.

Let (A_{i})_{i} be an inverse system of abelian groups with structure maps τ_{i} : A_{i+1} ⟶ A_{i}, i ≥ 0. The duality functor D may be used in order to construct a direct system of abelian groups (DA_{i})_{i}, with structure maps Dτ_{i} : DA_{i} ⟶ DA_{i+1}, i ≥ 0. We say that the inverse system (A_{i})_{i} is essentially zero if for all i there exists j(i) ≥ i, such that the structure map A_{j(i)} ⟶ A_{i} is the zero map. The limit of an essentially zero inverse system is clearly trivial; moreover, since such a system satisfies the Mittag-Leffler condition, its lim_{⟵}^{1} is also trivial.

Lemma 4.1. Let (A_{i})_{i} be an inverse system of abelian groups. Then, the following conditions are equivalent:

(i) lim_{⟵i} A_{i}^{(X)} = lim_{⟵i}^{1}A_{i}^{(X)} = 0 for any set X.

(ii) lim_{⟵i} A_{i} = lim_{⟵i}^{1}A_{i}^{(ℕ)} = 0.

(iii) The inverse system (A_{i})_{i} is essentially zero.

(iv) lim_{⟶i} D(A_{i}^{(X)})= 0 for any set X.

Proof. It is clear that (i)→(ii). We shall complete the proof, by showing that (ii)→(iii)→(i) and (iii)↔(iv).

(ii)→(iii): As shown in [7], the triviality of the group lim_{⟵i}^{1}A_{i}^{(ℕ)} implies that (A_{i})_{i} satisfies the Mittag-Leffler condition. Let (A′_{i})_{i} be the inverse system of stable images of (A_{i})_{i}; we note that the structure maps A′_{i+1} ⟶ A′_{i} are surjective for all i. Since (A′_{i})_{i} is a subsystem of (A_{i})_{i}, we have lim_{⟵i} A′_{i} ⊆ lim_{⟵i} A_{i} and hence lim_{⟵i} A′_{i} = 0. It follows readily that A′_{i} = 0 for all i and hence the inverse system (A_{i})_{i} is essentially zero, as needed.

(iii)→(i): Since (A_{i})_{i} is assumed to be essentially zero, the system (A_{i}^{(X)})_{i} is essentially

zero for any set X. As we have noted above, this implies that lim_{⟵i} A_{i}^{(X)} = lim_{⟵i}^{1}A_{i}^{(X)} = 0.

(iii)→(iv): In view of our assumption, for any i there exists j(i) ≥ i, such that the structure map A_{j(i)} ⟶ A_{i} is trivial. It follows that the map A_{j(i)}^{(X)} ⟶ A_{i}^{(X)} is trivial for any set X. By

applying the duality functor D, we conclude that the map D(A_{i}^{(X)}) ⟶ D(A_{j(i)}^{(X)}) is trivial

as well. It follows readily that lim_{⟶i} DA_{i}^{(X)} = 0.

(iv)→(iii): Since the abelian group ℚ/ℤ is faithfully injective, we may choose X, in such a way that A_{i} admits a monomorphism f_{i} into the direct product Δ =(ℚ/ℤ)^{X} for all i.^{3} We note that the direct system (D(A_{i}^{(X)}))_{i} may be identified with the direct system (Hom(A_{i}, Δ))_{i}, which is obtained from (A_{i})_{i} by applying the contravariant functor Hom( , Δ). Having fixed i, the vanishing of the colimit lim_{⟶i} Hom(A_{i}, Δ) = lim_{⟶i} D(A_{i}^{(X)}) implies that the image of the element f_{i} ∈ Hom(A_{i}, Δ) must vanish in the group Hom(A_{j}, Δ) for a suitable j ≥ i; in other words, the composition A_{j} ⟶ A_{i} ⟶^{f_{i}} Δ must be the zero map. Since f_{i} is injective, this can be true only if the structure map A_{j} ⟶ A_{i} is already zero. Therefore, we have proved that the inverse system (A_{i})_{i} is essentially zero, as needed. □

3Any abelian group A may be embedded into the direct product of Hom(A, ℚ/ℤ) copies of ℚ/ℤ. Hence, X may be chosen to be any set of cardinality exceeding ∑_{i} card Hom(A_{i}, ℚ/ℤ).

Proposition 4.2. Let R be a ring and consider a right R-module M and a left R-module N. If T̃or_{n}^{R}(M, N) = T̃or_{n−1}^{R}(M^{(ℕ)}, N)= 0 for some n, then Ẽxt_{R}^{n}(N, DM) = 0.

Proof. We fix an injective resolution 0 ⟶ N ⟶ I^{∗} of the left R-module N and consider the corresponding cosyzygy modules Σ^{i}N, i ≥ 0. We also consider the inverse system of abelian groups (A_{i})_{i}, with A_{i} = Tor_{n+i}^{R}(M, Σ^{i}N) for all i ≥ 0 and structure maps given by connecting homomorphisms which are induced by the short exact sequences

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0.

In view of Theorem 2.2, the vanishing of the groups T̃or_{n}^{R}(M, N) and T̃or_{n−1}^{R}(M^{(ℕ)}, N) implies that lim_{⟵i} A_{i} = lim_{⟵i}^{1}A_{i}^{(ℕ)} = 0. On the other hand, the group Ẽxt_{R}^{n}(N, DM) is isomorphic with

the colimit of the direct system (Ext_{R}^{n+i}(Σ^{i}N, DM))_{i}, whose structure maps are connecting homomorphisms which are induced by the short exact sequences above. Since the latter system may be naturally identified with the direct system (DA_{i})_{i}, we may invoke Lemma 4.1 in order to conclude that Ẽxt_{R}^{n}(N, DM) = lim_{⟶i} Ext_{R}^{n+i}(Σ^{i}N, DM)= lim_{⟶i} DA_{i} = 0. □

Corollary 4.3. Let R be a ring and consider a right R-module M. Then, the following conditions are equivalent:

(i) T̃or_{∗}^{R}( , DM) = 0,

(ii) T̃or_{0}^{R}(M, DM) = T̃or_{−1}^{R}(M^{(ℕ)}, DM)= 0 and

(iii) fd_{R^{o}}M < ∞.

Proof. (i)→(ii): This is obvious.

(ii)→(iii): The vanishing of the stable homology groups implies, in view of Proposition 4.2, that Ẽxt_{R}^{0}(DM, DM) = 0 and hence id_{R}DM < ∞. The result follows since fd_{R^{o}}M = id_{R}DM.

(iii)→(i): Since id_{R}DM = fd_{R^{o}}M < ∞, the stable homology functors T̃or_{∗}^{R}( , DM) vanish identically. □

Proposition 4.4. Let R be a ring and consider a right R-module M and a left R-module N. Then, the following conditions are equivalent:

(i) T̃or_{∗}^{R}(M^{(X)}, N)= 0 for any set X,

(ii) T̃or_{∗}^{R}(M^{(ℕ)}, N)= 0 and

(iii) Ẽxt_{R}^{∗}(N, D (M^{(X)}))= 0 for any set X.

Proof. The proof proceeds along the same lines as that of Proposition 4.2. We fix an injective resolution 0 ⟶ N ⟶ I^{∗} of the left R-module N and consider the corresponding cosyzygy modules Σ^{i}N, i ≥ 0, and the inverse system of abelian groups (Tor_{n+i}^{R}(M, Σ^{i}N))_{i} for all n. Then, Theorem 2.2 implies that conditions (i) and (ii) in the statement are equivalent to the following two conditions respectively:

(a) lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)} = lim_{⟵i}^{1} Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)} = 0 for all n and any set X,

(b) lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N)^{(ℕ)} = lim_{⟵i}^{1} Tor_{n+i}^{R}(M, Σ^{i}N)^{(ℕ)} = 0 for all n.

On the other hand, for any set X the direct system (Ext_{R}^{n+i}(Σ^{i}N, D (M^{(X)})))_{i}, whose colimit is the complete cohomology group Ẽxt_{R}^{n}(N, D (M^{(X)})), may be naturally identified with the induced direct system (D(Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)}))_{i}. Hence, condition (iii) in the statement is equivalent to

(c) lim_{⟶i} D(Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)})= 0 for all n and any set X. It follows that the equivalence of the three conditions in the statement is an immediate consequence of Lemma 4.1. □

Corollary 4.5. Let R be a ring and consider a left R-module N. Then, the following conditions are equivalent:

(i) T̃or_{∗}^{R}( , N) = 0 and

(ii) Ẽxt_{R}^{∗}(N, D ) = 0. □

5. Vanishing criteria over Noetherian rings

In this section, we shall restrict our attention to the case where the ring R is left Noetherian, in order to obtain clean characterizations of modules for which the partial stable homology functors vanish. As we have already noted before, it is an immediate consequence of the definition of stable homology that the functors T̃or_{∗}^{R}(M, ) vanish identically if M is a right R-module of finite flat dimension, whereas the functors T̃or_{∗}^{R}( , N) vanish identically if N is a left R-module of finite injective dimension. We shall prove that both of these implications may be reversed, in the case where R is left Noetherian.

We begin with the functors T̃or_{∗}^{R}(M, ), where M is a right R-module. In order to examine their vanishing, we shall need the following variants of Propositions 4.2 and 4.4.

Proposition 5.1. Let R be a left Noetherian ring and consider a right R-module M and a left R-module N. If T̃or_{n}^{R}(M, N) = T̃or_{n−1}^{R}(M, N^{(ℕ)})= 0 for some n, then Ẽxt_{R}^{n}(N, DM) = 0.

Proof. We fix an injective resolution 0 ⟶ N ⟶ I^{∗} of the left R-module N and consider the corresponding cosyzygy modules Σ^{i}N, i ≥ 0. In view of our hypothesis on R, the direct sum of injective left R-modules is also injective (cf. [1]). Therefore, 0 ⟶ N^{(ℕ)} ⟶ I^{∗(ℕ)} is an injective resolution of the left R-module N^{(ℕ)}; the corresponding cosyzygy modules Σ^{i}(N^{(ℕ)}) are thereby identified with (Σ^{i}N)^{(ℕ)} for all i ≥ 0.

From this point on, the proof proceeds as that of Proposition 4.2. We consider the inverse system of abelian groups (A_{i})_{i} with A_{i} = Tor_{n+i}^{R}(M, Σ^{i}N) for all i ≥ 0 and structure maps given by connecting homomorphisms which are induced by the short exact sequences

0 ⟶ Σ^{i}N ⟶ I^{i} ⟶ Σ^{i+1}N ⟶ 0.

In view of Theorem 2.2, the vanishing of the groups T̃or_{n}^{R}(M, N) and T̃or_{n−1}^{R}(M, N^{(ℕ)}) implies that lim_{⟵i} A_{i} = lim_{⟵i}^{1}A_{i}^{(ℕ)} = 0. On the other hand, the group Ẽxt_{R}^{n}(N, DM) is isomorphic with

the colimit of the direct system (Ext_{R}^{n+i}(Σ^{i}N, DM))_{i}, whose structure maps are connecting homomorphisms which are induced by the short exact sequences above. Since the latter system may be naturally identified with the induced direct system (DA_{i})_{i}, we may invoke Lemma 4.1 in order to conclude that Ẽxt_{R}^{n}(N, DM) = lim_{⟶i} Ext_{R}^{n+i}(Σ^{i}N, DM)= lim_{⟶i} DA_{i} = 0. □

Proposition 5.2. Let R be a left Noetherian ring and consider a right R-module M and a left R-module N. Then, the following conditions are equivalent:

(i) T̃or_{∗}^{R}(M, N^{(X)})= 0 for any set X, (ii) T̃or_{∗}^{R}(M, N^{(ℕ)})= 0 and (iii) Ẽxt_{R}^{∗}(N^{(X)}, DM)= 0 for any set X.

Proof. We fix an injective resolution 0 ⟶ N ⟶ I^{∗} of the left R-module N and consider the corresponding cosyzygy modules Σ^{i}N, i ≥ 0. As in the proof of Proposition 5.1, our hypothesis on R implies that 0 ⟶ N^{(X)} ⟶ I^{∗(X)} is an injective resolution of the left Rmodule N^{(X)} for any set X and hence the corresponding cosyzygy modules Σ^{i}(N^{(X)}) are identified with (Σ^{i}N)^{(X)} for all i ≥ 0.

From this point on, the proof proceeds as that of Proposition 4.4. Invoking Theorem 2.2, we conclude that conditions (i) and (ii) in the statement are equivalent to the following two conditions respectively:

(a) lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)} = lim_{⟵i}^{1} Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)} = 0 for all n and any set X.

(b) lim_{⟵i} Tor_{n+i}^{R}(M, Σ^{i}N)^{(ℕ)} = lim_{⟵i}^{1} Tor_{n+i}^{R}(M, Σ^{i}N)^{(ℕ)} = 0 for all n.

On the other hand, for any set X the direct system (Ext_{R}^{n+i}((Σ^{i}N)^{(X)}, DM))_{i}, whose colimit is the complete cohomology group Ẽxt_{R}^{n}(N^{(X)}, DM), may be naturally identified with the induced direct system (D(Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)}))_{i}. Hence, condition (iii) in the statement is equivalent to

(c) lim_{⟶i} D(Tor_{n+i}^{R}(M, Σ^{i}N)^{(X)})= 0 for all n and any set X. It follows that the equivalence of the three conditions in the statement is an immediate consequence of Lemma 4.1. □

We can now state and prove the following result, which complements Corollary 4.3 (under the presence of the additional assumption that R is left Noetherian).

Theorem 5.3. Let R be a left Noetherian ring. Then, the following conditions are equivalent for a right R-module M:

(i) T̃or_{∗}^{R}(M, ) = 0,

(i’) T̃or_{0}^{R}(M, DM) = T̃or_{−1}^{R}(M, (DM)^{(ℕ)})= 0,

(ii) T̃or_{∗}^{R}( , DM) = 0,

(ii’) T̃or_{0}^{R}(M, DM) = T̃or_{−1}^{R}(M^{(ℕ)}, DM)= 0,

(iii) Ẽxt_{R}^{∗}( , DM) = 0 and

(iv) fd_{R^{o}}M < ∞.

Proof. The equivalence between conditions (ii), (ii’) and (iv) was established in Corollary 4.3 (without any Noetherian assumption on R), whereas the equivalence between (i) and (iii) follows from Proposition 5.2. Of course, it is obvious that (i)→(i’) and he have already noted that the implication (iv)→(i) is an immediate consequence of the definition of stable homology (without any Noetherian assumption on R). Therefore, it only remains to prove that (i’)→(iv). In order to prove the latter implication, assume that (i’) holds. Then, Proposition 5.1 implies that Ẽxt_{R}^{0}(DM, DM) = 0 and hence fd_{R^{o}}M = id_{R}DM < ∞, as needed. □

We shall now turn our attention to the case of the stable homology functors T̃or_{∗}^{R}( , N), where N is a left R-module. In order to examine whether their vanishing implies the finiteness of id_{R}N, we shall employ an injectivity criterion, which involves the notion of purity. We recall that a short exact sequence of left R-modules

0 ⟶ L′ ⟶^{i} L ⟶ L′′ ⟶ 0

is called pure if the induced sequence of abelian groups

0 ⟶ Hom_{R}(C, L′) ⟶ Hom_{R}(C, L) ⟶ Hom_{R}(C, L′′) ⟶ 0

is exact for any finitely presented left R-module C. In that case, we also say that i is a pure monomorphism. The condition defining purity is easily seen to be equivalent to the following assertion: If f : A ⟶ B is a linear map whose cokernel coker f is finitely presented and

\begin{matrix}A & ⟶^{f} & B \\ a ↓ & & ↓ b \\ L′ & ⟶^{i} & L \end{matrix}

is a commutative diagram of left R-modules, then there exists a linear map g : B ⟶ L′ with a = gf. We note that for any left R-module L the natural map ν_{L} : L ⟶ D^{2}L is a pure monomorphism; for a proof of that assertion, the reader may consult [20, §II.1.1.5].

Lemma 5.4. Let R be a left Noetherian ring. Then, the following conditions are equivalent for a left R-module L:

(i) L is injective,

(ii) the natural map ν_{L} : L ⟶ D^{2}L factors through an injective module and

(iii) there is a pure monomorphism i : L ⟶ N, which factors through an injective module. Proof. Since the implications (i)→(ii)→(iii) are obvious, it only remains to prove that

(iii)→(i). To that end, we fix a pure monomorphism i : L ⟶ N that factors as the composition L ⟶^{a} I ⟶^{b} N, where I is an injective module. We shall prove that L is injective, using Baer’s criterion: We consider a left ideal 𝔄 ⊆ R and a linear map f : 𝔄 ⟶ L. Since I is injective, the composition af : 𝔄 ⟶ I extends to a linear map g : R ⟶ I. We now consider the following diagram

𝔄 ⟶^{j} R
f ↓  ↓ bg
L ⟶^{i} N

where j : 𝔄 ⟶ R is the inclusion map. We note that this diagram is commutative, since if = (ba)f = b(af) = b(gj) = (bg)j. The ring R being left Noetherian, the left ideal 𝔄 is finitely generated and hence the cokernel coker j = R/𝔄 is finitely presented. Therefore, the purity of i implies the existence of a linear map h : R ⟶ L, such that f = hj, as needed. □

We can now state and prove the following result, which complements Corollary 4.5 (under the presence of the additional assumption that R is left Noetherian).

Theorem 5.5. Let R be a left Noetherian ring. Then, the following conditions are equivalent for a left R-module N:

(i) T̃or_{∗}^{R}( , N) = 0,

(i’) T̃or_{0}^{R}(DN, N) = T̃or_{−1}^{R}((DN)^{(ℕ)}, N)= 0,

(ii) Ẽxt_{R}^{∗}(N, D ) = 0,

(ii’) Ẽxt_{R}^{0}(N, D^{2}N) = 0 and

(iii) id_{R}N < ∞.

Proof. The equivalence between conditions (i) and (ii) was established in Corollary 4.5. The implication (i)→(i’) is obvious, whereas Proposition 4.2 implies that (i’)→(ii’). We have already pointed out that the implication (iii)→(i) is an immediate consequence of the definition of stable homology. All of the above are valid without any Noetherian assumption on the ring R. We shall complete the proof by showing that (ii’)→(iii); it is for this implication that we shall use our assumption that R is left Noetherian.

Let us therefore assume that Ẽxt_{R}^{0}(N, D^{2}N) = 0 and proceed to show that id_{R}N is finite. We consider an injective resolution 0 ⟶ N ⟶ I^{∗} of N and the corresponding cosyzygy modules Σ^{i}N, i ≥ 0. Being left Noetherian, the ring R is left coherent; therefore, it follows that DI^{∗} ⟶ DN ⟶ 0 is a flat resolution of the right R-module DN. Applying the functor D once more, it follows that 0 ⟶ D^{2}N ⟶ D^{2}I^{∗} is an injective resolution of the left Rmodule D^{2}N. The corresponding cosyzygy modules Σ^{i}(D^{2}N) are identified with the modules D^{2}(Σ^{i}N) for all i ≥ 0. Let ν_{N} : N ⟶ D^{2}N be the natural map; invoking the naturality of ν, we may lift ν_{N} to the cochain map ν_{I^{∗}} : I^{∗} ⟶ D^{2}I^{∗}. In this way, the linear map which is induced by ν_{N} between the corresponding cosyzygy modules is identified with the natural map ν_{Σ^{i}N} : Σ^{i}N ⟶ D^{2}(Σ^{i}N) for all i ≥ 0. Since the complete cohomology group Ẽxt_{R}^{0}(N, D^{2}N) is trivial, the image of ν_{N} under the canonical map Hom_{R}(N, D^{2}N) ⟶ Ẽxt_{R}^{0}(N, D^{2}N) must necessarily vanish. Hence, using the approach to complete cohomology by Benson and Carlson, we may conclude that the linear map which is induced by ν_{N} between the corresponding cosyzygy modules factors through an injective module in sufficiently large degrees. In other words, the linear map ν_{Σ^{i}N} : Σ^{i}N ⟶ D^{2}(Σ^{i}N) factors through an injective module for all i ≫ 0. Since R is left Noetherian, Lemma 5.4 implies that Σ^{i}N is an injective module for all i ≫ 0; it follows that N has finite injective dimension, as needed. □

Remark 5.6. If R is an Artin algebra and N a finitely generated left R-module, the equivalence between conditions (i) and (iii) in Theorem 5.5 was proved in [4, Proposition 5.1].

6. Balance of stable homology

The definition of the stable homology groups T̃or_{∗}^{R}(M, N), where M is a right R-module and N is a left R-module, has a built-in asymmetry, as it involves a flat resolution of M and an injective resolution of N. We could have used an injective resolution of M and a flat resolution of N, in order to define the stable homology groups T̃or_{∗}^{R^{o}}(N, M). There is no a priori reason for the latter groups to coincide with the former ones; we say that stable homology is balanced for the pair (M, N) if the groups T̃or_{n}^{R}(M, N) and T̃or_{n}^{R^{o}}(N, M) are isomorphic for all n.

An analogous lack of symmetry is also present in the definition of the complete cohomology groups. The theory introduced by Mislin, which is based on projective modules (projective completions of functors, projective resolutions of modules), is not necessarily isomorphic with the dual theory, which was introduced by Nucinkis and is itself based on injective modules (injective completions of functors, injective resolutions of modules). In fact, Nucinkis has proved in [19, Theorem 5.2] that the two complete cohomology theories agree for all pairs of left R-modules if and only if all injective (resp. projective) left R-modules have finite projective (resp. injective) dimension. If this is the case, Gedrich and Gruenberg have shown in [11] that the invariants silpR, the supremum of the injective lengths of projective left R-modules, and spliR, the supremum of the projective lengths of injective left R-modules, are both finite and equal to each other.

Our goal in this section is to examine the homological counterpart of the above result, describing conditions under which stable homology is balanced for any pair of (left and right) coefficient modules. If this is the case, we say that stable homology is balanced over R. It appears that this problem is closely related to the finiteness of the flat dimension of injective modules. If R is a ring, such that all injective left and all injective right R-modules have finite flat dimension, then the invariants sfliR and sfliR^{o}, the suprema of the flat lengths of injective left and right R-modules respectively, are both finite and equal to each other; for a proof, see [9, Corollary 2.5].

A Tate flat resolution of the left R-module N is an acyclic complex of flat left R-modules, which remains acyclic upon tensoring with any injective right R-module and coincides with a flat resolution of N in sufficiently large degrees. Tate flat resolutions of right R-modules are defined analogously.

Proposition 6.1. Let R be a ring and assume that the invariants sfliR and sfliR^{o} are both finite. Then, stable homology is balanced over R.

Proof. Assume that sfli R = n. Then, as shown in [8, Lemma 5.2], for any left R-module N there exists an acyclic complex of flat left R-modules T_{∗}, which coincides with a projective resolution of N in degrees ≥ n. The finiteness of sfli R^{o} implies that T_{∗} remains acyclic upon tensoring with any injective right R-module; therefore, T_{∗} is a Tate flat resolution of N. A symmetric argument shows that any right R-module M admits a Tate flat resolution as well. Then, the balance of stable homology for the pair (M, N) follows from [4, Theorem 4.2]. □

We wish to prove that the sufficient condition for balance in Proposition 6.1 above is also necessary, i.e. that stable homology is balanced over R only if all injective left and all injective right R-modules have finite flat dimension. We shall prove below that this is indeed the case if the ring R is coherent on both sides.

Proposition 6.2. Assume that stable homology is balanced over a ring R.

(i) If R is either left Noetherian or right coherent, then sfli R^{o} < ∞.

(ii) If R is either right Noetherian or left coherent, then sfli R < ∞.

Proof. Since the balance of stable homology over a ring is a left-right symmetric condition, it only suffices to prove assertion (i). To that end, we consider an injective right R-module M and aim at proving that it has finite flat dimension.

The injectivity of M implies that the functors T̃or_{∗}^{R}(M, ) ≃ T̃or_{∗}^{R^{o}}( , M) vanish identically. If R is left Noetherian, then Theorem 5.3 implies that fd_{R^{o}}M < ∞.

Assuming that R is right coherent, the left R-module DM is flat. Then, the stable homology functors T̃or_{∗}^{R}( , DM) ≃ T̃or_{∗}^{R^{o}}(DM, ) vanish identically and hence the finiteness of the flat dimension of M follows invoking Corollary 4.3. □

Theorem 6.3. Let R be a ring, which is both left and right coherent. Then, the following conditions are equivalent:

(i) stable homology is balanced over R and

(ii) sfli R = sfli R^{o} < ∞.

Proof. The implication (i)→(ii) follows from Proposition 6.2, whereas the implication (ii)→(i), which is valid over any ring, is proved in Proposition 6.1. □

Let R be a left Noetherian ring. Then, as shown by Iwanaga in [14], the left self-injective dimension of R (i.e. the injective dimension of the left regular module R) is equal to sfli R^{o}.

We say that a ring R is Iwanaga-Gorenstein if R is left and right Noetherian and has finite left and right self-injective dimension.

Corollary 6.4. Let R be a ring, which is both left and right Noetherian. Then, the following conditions are equivalent:

(i) stable homology is balanced over R and

(ii) R is Iwanaga-Gorenstein. □

Remarks 6.5. (i) For an Artin algebra R, Corollary 6.4 was proved in [4, Corollary 4.5].

(ii) Let R be a ring over which stable homology is balanced and consider a right R-module M, for which either one of the following two conditions is satisfied:

(a) the left R-module DM has finite flat dimension or

(b) the right R-module M^{(ℕ)} has finite injective dimension. Then, fd_{R^{o}}M < ∞. Indeed, assumption (a) implies that the functors T̃or_{∗}^{R}( , DM) ≃ T̃or_{∗}^{R^{o}}(DM, ) vanish identically and hence Corollary 4.3 implies that M has finite flat dimension. On the other hand, assumption (b) implies that the functors T̃or_{∗}^{R}(M^{(ℕ)}, ) ≃ T̃or_{∗}^{R^{o}}( , M^{(ℕ)}) vanish identically. Hence, Proposition 4.4 implies that the functors Ẽxt_{R}^{∗}( , DM) vanish identically. In particular, Ẽxt_{R}^{0}(DM, DM) = 0 and hence fd_{R^{o}}M = id_{R}DM < ∞.

Appendix A. Hausdorff filtrations on complexes and homology

We consider an inverse system of complexes of abelian groups (X^{i})_{i} with structure chain maps denoted by τ^{i} : X^{i+1} ⟶ X^{i} for all i ≥ 0. Let

1 − τ : ∏_{i}X^{i} ⟶ ∏_{i}X^{i}

be the chain map, whose component in degree n is the additive map ∏_{i} X_{n}^{i} ⟶ ∏_{i} X_{n}^{i}, which is given by (x^{i})_{i} ↦ (x^{i} − τ^{i}x^{i+1})_{i}, (x_{i})_{i} ∈ ∏_{i} X_{n}^{i}. Then, the complexes lim_{⟵i} X^{i} and lim_{⟵i}^{1}X^{i} are defined by means of the exact sequence

0 ⟶ lim_{⟵i} X^{i} ⟶ ∏_{i}X^{i} ⟶^{1−τ} ∏_{i}X^{i} ⟶ lim_{⟵i}^{1}X^{i} ⟶ 0.

We are interested in the special case where lim_{⟵i} X^{i} is the zero complex; in other words, we

assume that lim_{⟵i} X_{n}^{i} = 0 for all n. Then, we obtain a short exact sequence of complexes

0 ⟶ ∏_{i}X^{i} ⟶^{1−τ} ∏_{i}X^{i} ⟶ lim_{⟵i}^{1}X^{i} ⟶ 0,

which induces a long exact sequence in homology

… ⟶ H_{n+1}(lim_{⟵i}^{1}X^{i})⟶^{∂} ∏_{i}H_{n}(X^{i}) ⟶^{1−τ} ∏_{i}H_{n}(X^{i}) ⟶ H_{n}(lim_{⟵i}^{1}X^{i})⟶ …

Identifying the kernel and the cokernel of the additive map 1 − τ : ∏_{i} H_{n}(X^{i}) ⟶ ∏_{i} H_{n}(X^{i}) as the lim_{⟵} and lim_{⟵i}^{1} of the inverse system of abelian groups (H_{n}(X^{i}))_{i} respectively, we obtain for all n a short exact sequence of abelian groups

0 ⟶ lim_{⟵i}^{1}H_{n+1}(X^{i}) ⟶ H_{n+1}(lim_{⟵i}^{1}X^{i})⟶^{∂} lim_{⟵i} H_{n}(X^{i}) ⟶ 0.

One source of examples of inverse systems of complexes of abelian groups with a vanishing limit, is obtained by considering complexes endowed with a Hausdorff filtration, in the sense of the following definition.

Definition A.1. A Hausdorff filtration of a complex of abelian groups X is a decreasing sequence of subcomplexes X = X^{0} ⊇ X^{1} ⊇ X^{2} ⊇ … ⊇ X^{i} ⊇ …, such that ⋂_{i} X^{i} = 0 (i.e., such that ⋂_{i} X_{n}^{i} = 0 for all n).

Let X be a complex of abelian groups endowed with a Hausdorff filtration (X^{i})_{i} as above. Then, we may consider (X^{i})_{i} as an inverse system of complexes, with structure chain maps the inclusions X^{i+1} ↪ X^{i} for all i ≥ 0; in view of our assumption on the filtration, we have lim_{⟵i} X^{i} = ⋂_{i} X^{i} = 0. In order to identify lim_{⟵i}^{1}X^{i}, we consider the inverse system of complexes

(X/X^{i})_{i} with structure chain maps X/X^{i+1} ⟶ X/X^{i} induced by the identity of X for all i ≥ 0. The projections π^{i} : X ⟶ X/X^{i}, i ≥ 0, induce a chain map

π : X ⟶ lim_{⟵i} X/X^{i}

and we claim that the complex lim_{⟵i}^{1}X^{i} is isomorphic with the cokernel of π. In other words, we claim that there is an exact sequence of complexes

(6)  X ⟶^{π} lim_{⟵i} X/X^{i} ⟶ lim_{⟵i}^{1}X^{i} ⟶ 0.

Since kernels, cokernels and products of complexes are computed degreewise, it suffices to show that for any degree n there is an exact sequence of abelian groups

X_{n} ⟶^{π} lim_{⟵i} X_{n}/X_{n}^{i} ⟶ lim_{⟵i}^{1}X_{n}^{i} ⟶ 0.

Since X_{n} = X_{n}^{0} ⊇ X_{n}^{1} ⊇ X_{n}^{2} ⊇ … ⊇ X_{n}^{i} ⊇ … is a filtration of the abelian group X_{n}, the identification of lim_{⟵i}^{1}X_{n}^{i} claimed above is a consequence of the following folklore result.

Lemma A.2. Let A be an abelian group, which is endowed with a decreasing filtration by subgroups A = A_{0} ⊇ A_{1} ⊇ A_{2} ⊇ … ⊇ A_{i} ⊇ . . .. Then, lim_{⟵i}^{1}A_{i} is the cokernel of the canonical

map A ⟶ lim_{⟵i} A/A_{i}.

Proof. This follows from the 6-term lim_{⟵} -lim_{⟵}^{1} exact sequence, which is induced from the short exact sequence of inverse systems

0 ⟶ (A_{i})_{i} ⟶ (A)_{i} ⟶ (A/A_{i})_{i} ⟶ 0,

since the constant system (A)_{i} has lim_{⟵i} A = A and lim_{⟵i}^{1}A = 0. □

The following result summarizes the discussion above.

Corollary A.3. Let X be a complex of abelian groups, endowed with a Hausdorff filtration X = X^{0} ⊇ X^{1} ⊇ X^{2} ⊇ … ⊇ X^{i} ⊇ . . .. Then, the complex lim_{⟵i}^{1}X^{i} is computed by the exact sequence (6) and there is a short exact sequence of abelian groups

0 ⟶ lim_{⟵i}^{1}H_{n+1}(X^{i}) ⟶ H_{n+1}(lim_{⟵i}^{1}X^{i})⟶^{∂} lim_{⟵i} H_{n}(X^{i}) ⟶ 0

for all n. □

References

[1] Bass, H.: lnjective dimension in Noetherian rings. Trans. Amer. Math. Soc. 102, 18-29 (1962).

[2] Benson, D.J., Carlson, J.F.: Products in negative cohomology. J. Pure Appl. Algebra 82, 107-129 (1992)

[3] Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press 1956

[4] Celikbas, O., Christensen, L.W., Liang, L., Piepmeyer, G.: Stable homology. arXiv:1409.3605v1

[5] Celikbas, O., Christensen, L.W., Liang, L., Piepmeyer, G.: Complete homology. arXiv:1501.00297v1 [6] Ding, N., Chen, J.: The flat dimensions of injective modules. Manuscripta Math. 78, 165- 177 (1993)

[7] Emmanouil, I.: Mittag-Leffler condition and the vanishing of lim_{⟵}^{1}. Topology 35, 267-271 (1996)

[8] Emmanouil, I.: On the finiteness of Gorenstein homological dimensions. J. Algebra 372, 376-396 (2012)

[9] Emmanouil, I., Talelli, O.: On the flat length of injective modules. J. London Math. Soc. 84, 408-432 (2011)

[10] Enochs, E.E., Jenda, O.M.G.: Copure injective resolutions, flat resolvents and dimensions. Comment. Math. Univ. Carolin. 34, 203-211 (1993)

[11] Gedrich, T.V., Gruenberg, K.W.: Complete cohomological functors on groups. Topology Appl. 25, 203-223 (1987)

[12] Goichot, F.: Homologie de Tate-Vogel équivariante. J. Pure Appl. Algebra 82, 39-64 (1992)

[13] Grothendieck, A.: EGA III. Publ. Math. IHES 11 (1961)

[14] Iwanaga, Y.: On rings with finite self-injective dimension II. Tsukuba J. Math. 4, 107-113 (1980)

[15] Jo, J.H.: Complete homology and related dimensions of groups. J. Group Theory 12, 431-448 (2009)

[16] Kropholler, P.H.: On groups of type (FP)_{∞}. J. Pure Appl. Algebra 90, 55-67 (1993)

[17] Lam, T.Y.: Lectures on Modules and Rings. Grad. Texts Math. 189, Springer, Berlin 1999

[18] Mislin, G.: Tate cohomology for arbitrary groups via satellites. Topology Appl. 56, 293-300 (1994)

[19] Nucinkis, B.E.A.: Complete cohomology of arbitrary rings using injectives. J. Pure Appl. Algebra 131, 297-318 (1998)

[20] Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Invent. Math. 13, 1-89 (1971)

[21] Triulzi, M.: Completion constructions in homological algebra and finiteness conditions. Ph.D. Thesis, Swiss Federal Institute of Technology, 1998

[22] Xu, J.: Flat Covers of Modules. Lect. Notes Math. 1634, Springer, Berlin 1996

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

  E-mail addresses: emmanoui@math.uoa.gr and i.manousaki@math.uoa.gr