BALANCE IN COMPLETE COHOMOLOGY

IOANNIS EMMANOUIL

Abstract. The balance of complete cohomology for modules that admit complete resolutions has been established by Christensen and Jorgensen in [2], as well as by Enochs, Estrada and Iacob in [4], by using two types of constructions on a given bicomplex. In this paper, we show that these constructions are closely related to each other. We also present an alternative proof of balance, which is based on the corresponding assertion for ordinary cohomology.

0. Introduction

It is well-known that the Ext groups Ext_{R}^{n}(M, N) of two modules M, N over a ring R may be computed using either a projective resolution P_{∗} ⟶^{ε} M ⟶ 0 of M, as the cohomology groups of the complex Hom_{R}(P_{∗}, N), or else using an injective resolution 0 ⟶ N ⟶^{η} I^{∗} of N, as the cohomology groups of the complex Hom_{R}(M, I^{∗}). This result, which is usually referred to by saying that the Ext functor is balanced, can be easily proved by considering the first quadrant bicomplex Hom_{R}(P_{∗}, I^{∗}) and showing that η and ε induce quasi-isomorphisms

Hom_{R}(P_{∗}, N) ⟶^{η_{∗}} Tot Hom_{R}(P_{∗}, I^{∗}) ⟵^{ε^{∗}} Hom_{R}(M, I^{∗}).

Here, we denote by Tot X the total complex associated with a bicomplex X.

Mislin has defined in [6] for any module M complete cohomology functors Êxt_{R}^{∗}(M, ) and a natural transformation Ext_{R}^{∗}(M, ) ⟶ Êxt_{R}^{∗}(M, ), as the projective completion of the Ext functors Ext_{R}^{∗}(M, ). The complete cohomology groups Êxt_{R}^{∗}(M, N) vanish if the module N has finite projective dimension. Equivalent definitions of the complete cohomology functors have been independently formulated by Vogel in [5] and by Benson and Carlson in [1].

Under an additional hypothesis on the module M, the groups Êxt_{R}^{∗}(M, N) can be described in more familiar terms, as the cohomology groups of a suitable complex: We say that M admits a complete projective resolution if there exists a doubly infinite acyclic chain complex of projective modules T_{∗}, which remains acyclic after applying the functor Hom_{R}( , Q) for any projective module Q and coincides with an ordinary projective resolution P_{∗} of M in sufficiently large degrees. If M admits such a complete projective resolution and τ_{∗} : T_{∗} ⟶ P_{∗} is a chain map which is the identity in sufficiently large degrees, then the complete cohomology groups Êxt_{R}^{∗}(M, N) may be computed as the cohomology groups of the complex Hom_{R}(T_{∗}, N), while the canonical map Ext_{R}^{∗}(M, N) ⟶ Êxt_{R}^{∗}(M, N) is that induced by the chain map τ_{∗}; this is shown in [3]. In particular, if T_{∗} coincides with P_{∗} in degrees ≥ n (in that case, we say that the coincidence index of the complete projective resolution is ≤ n), the canonical map Ext_{R}^{∗}(M, N) ⟶ Êxt_{R}^{∗}(M, N) is bijective in degrees > n. Moreover, the groups Êxt_{R}^{∗}(M, N) vanish if N has finite injective dimension, since the complex Hom_{R}(T_{∗}, N) is then acyclic.

Nucinkis has defined in [7] for any module N complete cohomology functors Ẽxt_{R}^{∗}( , N) and a natural transformation Ext_{R}^{∗}( , N) ⟶ Ẽxt_{R}^{∗}( , N), as the injective completion of the Ext functors Ext_{R}^{∗}( , N). The complete cohomology groups Ẽxt_{R}^{∗}(M, N) vanish if the module M has finite injective dimension. Equivalent descriptions of these functors may be obtained by using the dual approaches of those by Vogel and Benson and Carlson.

Under an additional hypothesis on the module N, the groups Ẽxt_{R}^{∗}(M, N) can be described in more familiar terms, as the cohomology groups of a suitable complex: We say that N admits a complete injective resolution if there exists a doubly infinite acyclic cochain complex of injective modules E^{∗}, which remains acyclic after applying the functor Hom_{R}(J, ) for any injective module J and coincides with an ordinary injective resolution I^{∗} of N in sufficiently large degrees. If N admits such a complete injective resolution and ϱ^{∗} : I^{∗} ⟶ E^{∗} is a cochain map which is the identity in sufficiently large degrees, then the complete cohomology groups Ẽxt_{R}^{∗}(M, N) may be computed as the cohomology groups of the complex Hom_{R}(M, E^{∗}), while the canonical map Ext_{R}^{∗}(M, N) ⟶ Ẽxt_{R}^{∗}(M, N) is that induced by the cochain map ϱ^{∗}; this is shown in [7]. In particular, if E^{∗} coincides with I^{∗} in degrees ≥ n (in that case, we say that the coincidence index of the complete injective resolution is ≤ n), the canonical map Ext_{R}^{∗}(M, N) ⟶ Ẽxt_{R}^{∗}(M, N) is bijective in degrees > n. Moreover, the groups Ẽxt_{R}^{∗}(M, N) vanish if M has finite projective dimension, since the complex Hom_{R}(M, E^{∗}) is then acyclic.

In analogy with the case of the ordinary Ext functors, one may ask whether the groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are always (naturally) isomorphic. In that direction, Nucinkis has proved in [7] that the following conditions on the ring R are equivalent:

(i) any R-module M admits a complete projective resolution,

(ii) any R-module N admits a complete injective resolution and

(iii) the groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are naturally isomorphic for all R-modules M, N. As there are examples of rings R over which conditions (i) and (ii) fail, one may not expect to prove balance in complete cohomology without any additional assumptions on the modules under consideration.

The result above was strengthened by Christensen and Jorgensen, who proved in [2] that the groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are naturally isomorphic for any pair of R-modules M, N, provided that M admits a complete projective resolution and N admits a complete injective resolution. To that end, they considered a complete projective resolution T_{∗} of M and a complete injective resolution E^{∗} of N, as above, and constructed the pinched Homcomplex Hom_{R}^{▷◁}(T_{∗}, E^{∗}), as the total complex of a variation of the bicomplex Hom_{R}(T_{∗}, E^{∗}). They proved the balance of complete cohomology in this case, by showing that both groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) may be computed as the cohomology groups of Hom_{R}^{▷◁}(T_{∗}, E^{∗}). An alternative proof of the same result has been obtained by Enochs, Estrada and Iacob in [4]. Instead of modifying the bicomplex Hom_{R}(T_{∗}, E^{∗}) and then considering the cohomology of the associated total complex, they considered a different kind of cohomology groups that may be attached to Hom_{R}(T_{∗}, E^{∗}), not those arrising from the total complex construction. Then, they proved balance as a consequence of a periodicity phenomenon that this new kind of cohomology groups enjoy.

Our first goal in this paper is to establish a connection between the techniques of [2] and [4], by showing that these are closely related to each other. For any cochain bicomplex X we shall define the pinched complex X^{▷◁} and the bigraded cohomology module h(X) = (h^{i,j}(X)), as in [2] and [4] respectively. In the special case where X has exact rows and columns, we show that h^{i,j}(X) = H^{i+j}(X^{▷◁}) for all integers i, j. This formula shows that the cohomology of the pinched complex X^{▷◁} may be expressed in terms of the h^{i,j}(X)’s and explains the (1, −1)- periodicity of the latter modules. The second goal of this paper is to present yet another proof of the balance of complete cohomology for modules that admit complete resolutions, which is elementary, in the sense that it only uses dimension shifting.

The contents of the paper are as follows: In section 1, we describe the cohomology modules that may be associated with a bicomplex, following [4], and elaborate on the (1, −1)-periodicity that these modules satisfy, in the case where the bicomplex has exact rows and columns. Then, in section 2, we use these modules in order to compute the cohomology of the pinched complex, under the presence of certain exactness hypotheses on the bicomplex. Finally, in section 3, we give an alternative proof of the balance of complete cohomology for modules that admit complete resolutions, which is based on the balance of the ordinary Ext functor.

Notations and terminology. Unless otherwise specified, all modules considered in this paper will be left modules over a fixed ring R. If P_{∗} ⟶ M ⟶ 0 is a projective resolution of a module M, we denote by Ω^{n}M the image of the map P_{n} ⟶ P_{n−1} for all n ≥ 1; it is the n-th syzygy module of M. Even though Ω^{n}M is not uniquely determined by M, Schanuel’s lemma implies uniqueness up to addition of a projective module. Dually, if 0 ⟶ N ⟶ I^{∗} is an injective resolution of a module N, we denote by Σ^{n}N the kernel of the map I^{n} ⟶ I^{n+1} for all n ≥ 1; it is the n-th cosyzygy module of N.

1. Cohomology modules associated with a bicomplex

We consider a cochain bicomplex X = (X^{i,j}) of left R-modules, where the indices i, j range over the set of integers, and denote by ∂′ (resp. ∂′′) its horizontal (resp. vertical) differential. We assume that these differentials anti-commute, i.e. we have ∂′∂′′ + ∂′′∂′ = 0. Then, we may state the following elementary result.

Lemma 1.1. (i) The differential ∂′ induces an isomorphism of bigraded modules

\frac{ker ∂′∂′′}{ker ∂′ + im ∂′′} ⟶^{∼} \frac{im ∂′ ∩ ker ∂′′}{im ∂′∂′′}

of bidegree (1, 0).

(ii) The differential ∂′′ induces an isomorphism of bigraded modules

\frac{ker ∂′∂′′}{im ∂′ + ker ∂′′} ⟶^{∼} \frac{ker ∂′ ∩ im ∂′′}{im ∂′∂′′}

of bidegree (0, 1).

Proof. (i) If x ∈ X^{i,j} is contained in the kernel of ∂′∂′′ = −∂′′∂′, then ∂′x ∈ im ∂′ ∩ ker ∂′′. Moreover, if x ∈ ker ∂′ + im ∂′′, then ∂′x ∈ im ∂′∂′′. Therefore, ∂′ induces a well-defined linear map

(1)  \frac{ker ∂′∂′′}{ker ∂′ + im ∂′′} ⟶ \frac{im ∂′ ∩ ker ∂′′}{im ∂′∂′′}

of bidegree (1, 0). If x ∈ ker ∂′∂′′ is such that ∂′x = ∂′∂′′y for some y, then ∂′(x − ∂′′y) = 0. It follows that x − ∂′′y ∈ ker ∂′ and hence x ∈ ker ∂′ + im ∂′′. This proves the injectivity of (1). As far as its surjectivity is concerned, we note that for any element z ∈ im ∂′ ∩ ker ∂′′ we may write z = ∂′x for some x ∈ ker ∂′′∂′ = ker ∂′∂′′.

(ii) This follows from (i) by symmetry. □

We may attach two bigraded (cohomology) modules 𝔥(X) = (𝔥^{i,j}(X)) and h(X) = (h^{i,j}(X)) to the bicomplex X, by letting

𝔥(X) = \frac{ker ∂′∂′′}{im ∂′ + im ∂′′} and h(X) = \frac{ker ∂′ ∩ ker ∂′′}{im ∂′∂′′} .

In this way, the components 𝔥^{i,j}(X) and h^{i,j}(X) may be computed by studying the diagram

X^{i,j+1} ⟶^{∂′} X^{i+1,j+1}
∂′′ ↑  ∂′′ ↑
X^{i−1,j} ⟶^{∂′} X^{i,j} ⟶^{∂′} X^{i+1,j}
∂′′ ↑  ∂′′ ↑
X^{i−1,j−1} ⟶^{∂′} X^{i,j−1}

Proposition 1.2. (i) If the bicomplex X has exact rows, then the differential ∂′ induces isomorphisms 𝔥^{i,j}(X) ⟶ h^{i+1,j}(X) for all i, j.

(ii) If the bicomplex X has exact columns, then the differential ∂′′ induces isomorphisms 𝔥^{i,j}(X) ⟶ h^{i,j+1}(X) for all i, j.

(iii) (cf. [4, Theorem 2.1]) If the bicomplex X has exact rows and columns, then there is a natural automorphism of the bigraded module h(X) of bidegree (1, −1), i.e. there is a natural isomorphism h^{i,j+1}(X) ⟶ h^{i+1,j}(X) for all i, j.

Proof. Assertions (i) and (ii) are immediate consequences of Lemma 1.1, while (iii) follows from (i) and (ii). □

Remarks 1.3. (i) Let

\begin{matrix}B & ⟶^{γ} & D \\ f ↑ & & ↑ φ \\ A & ⟶^{g} & C \end{matrix}

be a commutative diagram of left R-modules and linear maps. Then, it is easily seen that g induces an isomorphism

\frac{ker φg}{ker g + ker f} ⟶^{∼} \frac{im g ∩ ker φ}{g(ker f)}

and that f induces an isomorphism

\frac{ker φg}{ker g + ker f} ⟶^{∼} \frac{im f ∩ ker γ}{f(ker g)} .

In particular, there is an isomorphism

\frac{im f ∩ ker γ}{f(ker g)} ⟶^{∼} \frac{im g ∩ ker φ}{g(ker f)},

which is natural with respect to the obvious notion of morphisms of commutative squares.

(ii) Let X = (X^{i,j}) be a cochain bicomplex. Then, applying the result of (i) above to the commutative square

X^{i,j+1} ⟶^{∂′} X^{i+1,j+1}
−∂′′ ↑  ∂′′ ↑
X^{i,j}  ⟶^{∂′} X^{i+1,j}

we conclude that there exists a natural isomorphism

\frac{im ∂′′ ∩ ker ∂′}{∂′′(ker ∂′)} ⟶^{∼} \frac{im ∂′ ∩ ker ∂′′}{∂′(ker ∂′′)} .

If X has exact rows and columns, then this isomorphism reduces (up to a sign) to that of Proposition 1.2(iii).

(iii) The existence of the periodicity isomorphism of Proposition 1.2(iii) is the key ingredient in the proof of balance for complete cohomology, which is presented in [4]. Indeed, as shown in the discussion following [4, Corollary 3.4], it implies that Êxt_{R}^{∗}(M, N) ≃ Ẽxt_{R}^{∗}(M, N) if M admits a complete projective resolution and N admits a complete injective resolution, which both have coincidence index 0. The general case may be easily reduced to that one, by dimension shifting. Indeed, assume that M admits a complete projective resolution of coincidence index ≤ m and N admits a complete injective resolution of coincidence index ≤ n. Then, there are natural isomorphisms

Êxt_{R}^{i}(M, N) ≃ Êxt_{R}^{i−m}(Ω^{m}M, N) ≃ Êxt_{R}^{i−m−n}(Ω^{m}M, Σ^{n}N)

for all i. The first isomorphism above is obvious, whereas the second one follows by dimension shifting, since the functors Êxt_{R}^{∗}(Ω^{m}M, ) vanish on injective modules. Similarly, there are natural isomorphisms

Ẽxt_{R}^{i}(M, N) ≃ Ẽxt_{R}^{i−n}(M, Σ^{n}N) ≃ Ẽxt_{R}^{i−n−m}(Ω^{m}M, Σ^{n}N)

for all i. The first isomorphism above is obvious, whereas the second one follows by dimension shifting, since the functors Ẽxt_{R}^{∗}( , Σ^{n}N) vanish on projective modules. By the coincidence index 0 case, the groups Êxt_{R}^{i−m−n}(Ω^{m}M, Σ^{n}N) and Ẽxt_{R}^{i−n−m}(Ω^{m}M, Σ^{n}N) are naturally isomorphic and hence we may conclude that Êxt_{R}^{i}(M, N) ≃ Ẽxt_{R}^{i}(M, N) for all i, as needed.

(iv) One may define in a similar way for any chain bicomplex X = (X_{i,j}) bigraded (homology) modules 𝔥(X) = (𝔥_{i,j}(X)) and h(X) = (h_{i,j}(X)), and obtain a result analogous to Proposition 1.2. In particular, one deduces as in (iii) above a proof of the balance of the Tate Tor functor T̂or_{∗}^{R}(M, N) for modules M, N that admit complete projective resolutions (cf. the discussion at the end of [4]). The details are left to the reader.

2. Pinched complexes

As in the previous section, we consider a cochain bicomplex X = (X^{i,j}) of left R-modules with anti-commuting horizontal and vertical differentials ∂′ and ∂′′ respectively. The pinched complex X^{▷◁} is then defined, following [2, Construction 4.4], to be the total complex of the cochain bicomplex

(2)  \begin{aligned}& & & & & & ⋮ & & ⋮ \\ & & & & & & ∂′′ ↑ & & ∂′′ ↑ \\ & & & & & & X^{0,1} & ⟶^{∂′} & X^{1,1} & ⟶^{∂′} & ⋯ \\ & & & & & & ∂′′ ↑ & & ∂′′ ↑ \\ ⋯ & ⟶^{∂′} & X^{−2,−1} & ⟶^{∂′} & X^{−1,−1} & ⟶^{∂′∂′′} & X^{0,0} & ⟶^{∂′} & X^{1,0} & ⟶^{∂′} & ⋯ \\ & & ∂′′ ↑ & & ∂′′ ↑ \\ ⋯ & ⟶^{∂′} & X^{−2,−2} & ⟶^{∂′} & X^{−1,−2} \\ & & ∂′′ ↑ & & ∂′′ ↑ \\ & & ⋮ & & ⋮ \end{aligned}

Here, the module X^{i,j} is located in bidegree (i, j) (resp. in bidegree (i, j + 1)) if i, j ≥ 0 (resp. if i, j < 0). It is clear that the cohomology modules of the pinched complex X^{▷◁} in degrees 0 and −1 are given by H^{0}(X^{▷◁}) = h^{0,0}(X) and H^{−1}(X^{▷◁}) = 𝔥^{−1,−1}(X), where h(X) and 𝔥(X) are the bigraded (cohomology) modules that are associated with X as in section 1. Under the presence of certain additional exactness hypotheses on X, we may compute the cohomology modules of X^{▷◁}, as follows:

Theorem 2.1. (i) If the bicomplex X has exact rows, then there is a natural isomorphism H^{n}(X^{▷◁}) ≃ h^{0,n}(X) for all n.

(ii) If the bicomplex X has exact columns, then there is a natural isomorphism H^{n}(X^{▷◁}) ≃ h^{n,0}(X) for all n.

(iii) If the bicomplex X has exact rows and columns, then there is a natural isomorphism H^{n}(X^{▷◁}) ≃ h^{i,n−i}(X) for all n, i.

Proof. (i) As we have noted above, we always have an equality H^{0}(X^{▷◁}) = h^{0,0}(X). Since H^{−1}(X^{▷◁}) = 𝔥^{−1,−1}(X), we may invoke Proposition 1.2(i) and conclude that H^{−1}(X^{▷◁}) ≃ h^{0,−1}(X). Let us denote by Y^{n} the kernel of the horizontal differential ∂′ : X^{0,n} ⟶ X^{1,n} for all n. Then, a standard spectral sequence argument, applied to the first quadrant part of the bicomplex (2), implies that the cohomology modules H^{n}(X^{▷◁}) may be identified with the cohomology modules H^{n}(Y) of the cochain complex

⋯ ⟶^{∂′′|} Y^{i−1} ⟶^{∂′′|} Y^{i} ⟶^{∂′′|} Y^{i+1} ⟶^{∂′′|} ⋯

for all n > 0. Since H^{n}(Y) is precisely the bidegree (0, n) component of the bigraded module

\frac{ker ∂′ ∩ ker ∂′′}{∂′′(ker ∂′)} = \frac{ker ∂′ ∩ ker ∂′′}{im ∂′′∂′} = \frac{ker ∂′ ∩ ker ∂′′}{im ∂′∂′′} = h(X),

it follows that H^{n}(X^{▷◁}) ≃ h^{0,n}(X) for all n > 0. We use a similar argument, applied to the third quadrant part of the bicomplex (2), to deal with degrees n < −1: The point is that the cokernel of the horizontal differential ∂′ : X^{−2,n} ⟶ X^{−1,n} may be identified with the kernel Y^{n} of ∂′ : X^{0,n} ⟶ X^{1,n}, whereas the vertical differential ∂′′ : X^{−1,n} ⟶ X^{−1,n+1} induces by passage to the quotient a map, which may be identified with the restriction ∂′′| : Y^{n} ⟶ Y^{n+1} of ∂′′ : X^{0,n} ⟶ X^{0,n+1} to Y^{n}.

(ii) This follows from (i) by symmetry.

(iii) The result follows from (i) (or (ii)), in view of the (1, −1)-periodicity of the bigraded module h(X) (cf. Proposition 1.2(iii)). □

Remarks 2.2. (i) Let T_{∗} be an acyclic chain complex of projective modules and E^{∗} an acyclic cochain complex of injective modules. Then, the bicomplex of abelian groups Hom_{R}(T_{∗}, E^{∗}) has exact rows and columns. Therefore, the computation of the bigraded (cohomology) module h(Hom_{R}(T_{∗}, E^{∗})), which is given in [4, Theorem 3.3], together with Theorem 2.1 above, recover [2, Theorem 4.7 and Proposition 5.3].

(ii) One may define in a similar way for any chain bicomplex X = (X_{i,j}) the pinched complex X^{▷◁}, as in [2, Construction 3.2], and prove a result analogous to Theorem 2.1, relating the homology modules of X^{▷◁} to the bigraded (homology) module h(X) (cf. Remark 1.3(iv)). In particular, if T_{∗} and T′_{∗} are acyclic chain complexes of flat left and right modules respectively, then we may apply that result to the bicomplex of abelian groups T_{∗} ⊗_{R} T′_{∗} (which has exact rows and columns), and use the computation of the bigraded (homology) module h(T_{∗} ⊗_{R} T′_{∗}), which is given in [4, Theorem 3.6], in order to recover [2, Theorem 3.5]. The details are left to the reader.

3. A proof by reduction to the balance of Ext

We fix two modules M, N and assume that M (resp. N) admits a complete projective (resp. a complete injective) resolution. In this final section, we shall present a proof of the assertion that the complete cohomology groups Êxt_{R}^{∗}(M, N) and Ẽxt_{R}^{∗}(M, N) are naturally isomorphic, which is conceptually simpler than those presented in [2] and [4], as it only uses dimension shifting and the balance of the ordinary Ext functor.

We begin by considering a complete injective resolution of N of coincidence index ≤ n

⋯ ⟶^{∂^{i−2}} E^{i−1} ⟶^{∂^{i−1}} E^{i} ⟶^{∂^{i}} E^{i+1} ⟶^{∂^{i+1}} ⋯

and denote by Z^{i} the kernel of ∂^{i} : E^{i} ⟶ E^{i+1} for all i. Then, E^{∗} coincides with an injective resolution of N in degrees ≥ n; in other words, N admits an injective resolution of the form

0 ⟶ N ⟶ I^{0} ⟶ I^{1} ⟶ ⋯ ⟶ I^{n−1} ⟶ E^{n} ⟶^{∂^{n}} E^{n+1} ⟶^{∂^{n+1}} ⋯

The proof will follow from the three claims below: Claim 1: There is a natural isomorphism Êxt_{R}^{i}(M, N) ≃ Êxt_{R}^{i−s}(M, Z^{s}) for all integers i, s. Claim 2: There is a natural isomorphism Ẽxt_{R}^{i}(M, N) ≃ Ẽxt_{R}^{i−s}(M, Z^{s}) for all integers i, s. Claim 3: If M admits a complete projective resolution of coincidence index ≤ m and j, s are integers with j > m, then there is a natural isomorphism Êxt_{R}^{j}(M, Z^{s}) ≃ Ẽxt_{R}^{j}(M, Z^{s}). Assuming that these claims have been established, we fix an integer i and assume that M admits a complete projective resolution of coincidence index ≤ m. We also choose an integer s, such that s < i − m. We now have a chain of isomorphisms

Êxt_{R}^{i}(M, N) ≃ Êxt_{R}^{i−s}(M, Z^{s}) ≃ Ẽxt_{R}^{i−s}(M, Z^{s}) ≃ Ẽxt_{R}^{i}(M, N),

which finishes the proof. The first isomorphism above follows from claim 1, the second one follows from claim 3 (since i − s > m) and the third one follows from claim 2. Hence, it only remains to provide proofs for claims 1, 2 and 3.

Proof of claim 1: Since the functors Êxt_{R}^{∗}(M, ) may be computed using a complete projective resolution of M, we conclude that these vanish identically on the class of injective modules. Using dimension shifting and the exact sequence

0 ⟶ N ⟶ I^{0} ⟶ I^{1} ⟶ ⋯ ⟶ I^{n−1} ⟶ Z^{n} ⟶ 0,

it follows that Êxt_{R}^{i}(M, N) ≃ Êxt_{R}^{i−n}(M, Z^{n}) for all i. On the other hand, for any two integers a, b with a < b we may use the exact sequence

0 ⟶ Z^{a} ⟶ E^{a} ⟶^{∂^{a}} E^{a+1} ⟶^{∂^{a+1}} ⋯ ⟶^{∂^{b−2}} E^{b−1} ⟶ Z^{b} ⟶ 0

and dimension shifting as above, in order to conclude that Êxt_{R}^{i−a}(M, Z^{a}) ≃ Êxt_{R}^{i−b}(M, Z^{b}) for all i. This completes the proof of claim 1.

Proof of claim 2: We fix s and let E^{∗}[s] be the cochain complex obtained from E^{∗} by shifting degrees by s, i.e. we let E^{j}[s] = E^{j+s} for all j. It is clear that E^{∗}[s] is a complete injective resolution of Z^{s} of coincidence index 0 and hence the groups Ẽxt_{R}^{∗}(M, Z^{s}) may be computed as the cohomology groups of the complex Hom_{R}(M, E^{∗}[s]). In particular, we have

Ẽxt_{R}^{i−s}(M, Z^{s}) = H^{i−s}(Hom_{R}(M, E^{∗}[s])) = H^{i}(Hom_{R}(M, E^{∗})) = Ẽxt_{R}^{i}(M, N)

for all i, as needed.

Proof of claim 3: Since j > m, we have an isomorphism Êxt_{R}^{j}(M, Z^{s}) ≃ Ext_{R}^{j}(M, Z^{s}), where the latter group is computed using a projective resolution of M. On the other hand, as we noted in the proof of claim 2 above, the module Z^{s} admits a complete injective resolution of coincidence index 0. Since j > m ≥ 0, we also have Ẽxt_{R}^{j}(M, Z^{s}) ≃ Ext_{R}^{j}(M, Z^{s}), where the latter group is computed using an injective resolution of Z^{s}. Therefore, invoking the balance of the ordinary Ext functor, we may conclude that the groups Êxt_{R}^{j}(M, Z^{s}) and Ẽxt_{R}^{j}(M, Z^{s}) are naturally isomorphic.

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

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