INTRODUCTION TO REALIZABILITY OF MODULES OVER TATE COHOMOLOGY

DAVID BENSON, HENNING KRAUSE, AND STEFAN SCHWEDE

Abstract. This paper is a companion to our paper, “Realizability of modules over Tate cohomology,” in which we described an obstruction theory which applies in a number of contexts. This companion paper restricts attention to Tate cohomology, and gives constructions and proofs within that framework. A special case of our main theorem is as follows. Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γ ∈ HH^{3,−1}Ĥ^{∗}(G, k) with the following property. Given a graded Ĥ^{∗}(G, k)-module X, the image of γ in Ext_{Ĥ^{∗}(G,k)}^{3,−1}(X, X) vanishes if and only if X is isomorphic to a direct summand of Ĥ^{∗}(G, M) for some kG-module M. If X is realizable in this way, then the essentially different ways of realizing it form an affine space whose associated vector space is Ext_{Ĥ^{∗}(G,k)}^{2,−1}(X, X).

1. Introduction

This paper is a companion to our paper [2], in which we introduced an obstruction theory which applies in a number of different contexts.

Our investigation began with a study of the following question. Let k be a field and let G be a finite group. Given a graded module X over the Tate cohomology ring Ĥ^{∗}(G, k), how do we decide whether there exists a kG-module M such that Ĥ^{∗}(G, M) ≅ X? We described an element γ ∈ HH^{3,−1}Ĥ^{∗}(G, k) with the property that its image γ̄ ∈ Ext_{Ĥ^{∗}(G,k)}^{3,−1}(X, X) is zero if and only if X

is a direct summand of some module of the form Ĥ^{∗}(G, M). The element γ in some sense encodes all possible information about Massey triple products of elements of Tate cohomology.

The theory was then generalized to various other contexts, including modules over the cohomology of a differential graded algebra, and a general obstruction theory in the context of triangulated categories. Unfortunately, in the process of generalization, it became hard for someone only interested in the case of Tate cohomology to extract from that paper a short, self contained version of the obstruction theory in that context. The purpose of this paper is to give the motivation, a description of the obstruction, and the proof of the theorem purely in the context of Tate cohomology, without any extra baggage. The main theorem of this paper is the following. The case where N = k is the situation described above. We refer the reader to [2] for generalizations and further comments.

The first author is partly supported by a grant from the NSF.

Theorem 1.1. Let k be a field and let G be a finite group. Let N be a finite dimensional kG-module, and write E = Êxt_{kG}^{∗}(N, N) for the Tate Ext algebra of N, regarded as a graded k-algebra (so if N = k then E = Ĥ^{∗}(G, k)). Then there exists a canonical element in Hochschild cohomology of E,

γ ∈ HH^{3,−1}E,

with the following property. Given any graded E-module X, the following are equivalent:

(i) The image γ̄ of γ in Ext_{E}^{3,−1}(X, X) is zero.

(ii) There exists a kG-module M such that X is isomorphic to a direct summand of the graded E-module Êxt_{kG}^{∗}(N, M).

The element γ has an interpretation as the degree three part of the A_{∞}- structure on Ĥ^{∗}(G, k). The general context is that by a theorem of Kadeishvili [5], the cohomology of any differential graded algebra is an A_{∞}-algebra, well defined up to quasi-isomorphism. The Tate cohomology ring Ĥ^{∗}(G, k) is the cohomology of a suitable A_{∞}-algebra. The structure map m_{3} for the cohomology of a differential graded algebra is in general a Hochschild (3, −1)-cocycle, and changing the quasi-isomorphism class changes m_{3} by a coboundary. The element γ is the Hochschild cohomology class represented by m_{3}. All this is explained further in our paper [2], but not here; we refer to Keller [6] for background material on A_{∞}-algebras and modules.

The element γ̄ is the first of a sequence of obstructions in Ext_{E}^{n,2−n}(X, X) (n ≥ 3) which decide when X is realizable, not only as a summand. However, we have no nice interpretation of these higher obstructions and no predecessor in Hochschild cohomology. The obstruction γ̄ works in the context of a triangulated category with arbitrary direct sums. In this context, N is a compact object in the category.

We also include a discussion of the parametrization of the realizations, in the case where γ̄ is the zero element of Ext_{E}^{3,−1}(X, X).

Consider the collection Split(X) (too large to be a set) of ordered triples (M, i, π) consisting of a kG-module M and maps of E-modules

X ⟶^{i} Êxt_{kG}^{∗}(N, M) ⟶^{π} X

whose composite is the identity map Id_{X}. We put an equivalence relation on this collection as follows. Two such triples (M, i, π) and (M′, i′, π′) are said to be equivalent if there is a kG-module homomorphism ρ: M → M′ making the following diagram commute.

\xymatrix{ & Êxt_{kG}^{∗}(N, M) \ar[dd]^{ρ_{∗}} \ar@{-->}[dr]^{π} \\ X \ar[ur]^{i} \ar[dr]_{i′} &  & X \\  & Êxt_{kG}^{∗}(N, M′) \ar@{-->}[ur]_{π′}}

As it stands, this does not define an equivalence relation because it is not symmetric, but we complete it to one by taking the smallest equivalence relation on Split(X) containing all pairs of equivalent triples (M, i, π). It is not obvious that the equivalence classes in Split(X) form a set, but that is part of the content of the following theorem, whose proof can be found in Section 5.

Theorem 1.2. If γ̄ = 0 then the equivalence classes in Split(X) form an affine space whose associated vector space is Ext_{E}^{2,−1}(X, X).

To say that a set forms an affine space 𝔸 with a given associated vector space V means that there is a free and transitive addition map V × 𝔸 → 𝔸. So the difference between two elements of 𝔸 is a well defined element of V, and a choice of which element to call zero in 𝔸 defines a bijection with V.

As in the case of the obstruction γ̄, this theorem works in any triangulated category with arbitrary direct sums. In algebraic topology, there are analogous appearances of Ext^{2} groups in classification problems of realizable (co-)homology modules. Some comments on this can be found at the end of Section 5.

In a related paper, Beligiannis and Krause [1] investigate the question of when all maps from the Tate cohomology of a module are realizable. They develop an extended Milnor sequence which computes the obstructions in this context.

2. A motivating example

In this section, we motivate the discussion with an example. Let p be a prime, G be a cyclic group of order p, and k be a field of characteristic p.

In the case p = 2, the Tate cohomology ring is Ĥ^{∗}(G, k) = k[x, x^{−1}] with deg(x) = 1. In the graded sense, this is a field: every module over it is free. In particular, every module over Ĥ^{∗}(G, k) can be realized as Ĥ^{∗}(G, M) for a suitable kG-module M. Namely, M can be taken as a kG-module with trivial G-action, whose k-dimension is equal to the cardinality of a free generating set for the given Ĥ^{∗}(G, k)-module.

The case p odd is somewhat different. The Tate cohomology Ĥ^{∗}(G, k) is the tensor product of k[x]/(x^{2}) with a Laurent polynomial ring k[y, y^{−1}]. Here, the degrees are given by deg(x) = 1 and deg(y) = 2. The indecomposable kG-modules in this case correspond to Jordan blocks of length at most p. We write M_{n} (1 ≤ n ≤ p) for the indecomposable module of length n. It has a unique composition series, with just one submodule of each dimension from zero to n; we express this by saying that M_{n} is uniserial. The indecomposable projective kG-module is M_{p}, so the short exact sequence

0 → M_{p−n} → M_{p} → M_{n} → 0

shows that Ω(M_{n}) ≅ M_{p−n}. So Ω^{i}(M_{n}) is isomorphic to M_{n} if i is even and M_{p−n} if i is odd. So for all i we have

Ĥ^{i}(G, M_{n}) ≅ \underline{Hom}_{kG}(k, Ω^{−i}(M_{n})) ≅ k.

Multiplication by y ∈ Ĥ^{2}(G, k) is an isomorphism, but multiplication by x is harder to compute. For 2 ≤ n ≤ p − 2, x acts as zero. For n = 1 and n = p − 1, M_{n} is k, respectively Ω(k), so Ĥ^{∗}(G, M_{n}) is either Ĥ^{∗}(G, k) or its shift in degree by one.

Let us consider the Ĥ^{∗}(G, k)-module X = k[y, y^{−1}]. As long as p > 3, we can choose a value of n satisfying 2 ≤ n ≤ p − 2, and then since x acts as zero, Ĥ^{∗}(G, M_{n}) will decompose as a direct sum of X and its shift in degree by one. However, if p = 3 then the only nonprojective indecomposables are k and Ω(k), and there is not enough room to realize X as a direct summand of the Tate cohomology of a module. The problem is that the uniserial module of length three is already projective.

The relationship with Massey products is as follows. In order to build a module M_{n} whose cohomology has k[y, y^{−1}] as a direct summand, we needed to be able to string together at least four copies of the trivial module to make a uniserial module of length four, so that we can form a module M_{n} with length at least two, and so that Ω(M_{n}) also has length at least two.

The obstruction theory for uniserial modules is well understood in terms of cup products and Massey products. If A, B, C and D are simple modules, then a uniserial extension with submodule B and quotient A (which we denote A/B) is represented by a nonzero element of Ext_{kG}^{1}(A, B). The cup product of an element b ∈ Ext_{kG}^{1}(B, C) with an element a ∈ Ext_{kG}^{1}(A, B) is zero in Ext_{kG}^{2}(A, C) precisely when the two extensions A/B and B/C can be fitted together to make a uniserial module of length three of the form A/B/C. If the cup product is zero, there may be a number of different ways of fitting together the twofold extensions to form a threefold extension. They are parametrized by elements of Ext_{kG}^{1}(A, C); but if A happens to be isomorphic to C then there can be unexpected isomorphisms between the resulting length three modules.

Given elements a ∈ Ext_{kG}^{1}(A, B), b ∈ Ext_{kG}^{1}(B, C), c ∈ Ext_{kG}^{1}(C, D), if the products ba and cb vanish so that there are uniserial modules A/B/C and B/C/D, we may ask whether they can be fitted together to form a uniserial module of the form A/B/C/D. The obstruction to doing this is the Massey triple product ⟨c, b, a⟩ ∈ Ext_{kG}^{2}(A, D). There is some indeterminacy involved; we can change the length three modules using elements of Ext_{kG}^{1}(A, C) and Ext_{kG}^{1}(B, D). So the Massey product is really only well defined up to adding an element of c Ext_{kG}^{1}(A, C) + Ext_{kG}^{1}(B, D) a. So in the case of the cyclic group of order three in characteristic three, there is a nonvanishing Massey triple product corresponding to the nonexistence of a uniserial module of length four. It is ⟨x, x, x⟩ = y. For the cyclic group of order p in characteristic p with p ≠ 3, there are no nonvanishing Massey triple products, but the p-fold Massey product ⟨x, …, x⟩ is equal to y.

The preceding discussion should at least make some sense of the following theorem, which slightly generalizes the above setup and describes what happens in Theorem 1.1 if N = k and G has cyclic Sylow p-subgroups. We refer to §7 of [2] for proofs.

Theorem 2.1. Suppose that k has characteristic p and G has cyclic Sylow p-subgroups of order p^{n}. Then the following describes the obstruction γ ∈ HH^{3,−1}Ĥ^{∗}(G, k).

(i) Unless p^{n} = 3, we have γ = 0.

(ii) If p^{n} = 3 and G is p-nilpotent then Ĥ^{∗}(G, k) = k[y, y^{−1}, x]/(x^{2}) with deg(x) = 1, deg(y) = 2. In this case, γ is represented by the (3, −1)-cocycle m which satisfies

m(y^{i}x ⊗ y^{j}x ⊗ y^{ℓ}x) = y^{i+j+ℓ+1} i, j, ℓ ∈ ℤ

and which vanishes on all other tensor products of monomials in x and y.

(iii) If p^{n} = 3 and G is not p-nilpotent then Ĥ^{∗}(G, k) = k[w, w^{−1}, v]/(v^{2}) with deg(v) = 3, deg(w) = 4. In this case, γ is represented by the (3, −1)- cocycle m which satisfies

m(vw^{i} ⊗ vw^{j} ⊗ vw^{ℓ}) = w^{i+j+ℓ+2} i, j, ℓ ∈ ℤ

and which vanishes on all other tensor products of monomials in v and w. □

3. Notations and conventions.

Unless otherwise specified, when we talk about kG-modules, we mean left kG-modules. Let Mod(kG) be the category of all (not necessarily finitely generated) kG-modules. If M and N are left kG-modules, then Êxt_{kG}^{∗}(N, M) is a right Êxt_{kG}^{∗}(N, N)-module by Yoneda composition. So unless otherwise specified, Êxt_{kG}^{∗}(N, N)-modules are right modules.

We write StMod(kG) for the stable category of kG-modules. The objects in this category are the same as in Mod(kG), but the arrows are given by

\underline{Hom}_{kG}(N, M) = Hom_{kG}(N, M)/PHom_{kG}(N, M)

where PHom_{kG}(N, M) is the linear subspace consisting of homomorphisms which factor through some projective module. The category StMod(kG) is a triangulated category, in which the triangles come from short exact sequences in Mod(kG). As usual, we denote by ΩM the kernel of a map from a projective module onto M, and by Ω^{−1}M the cokernel of an embedding of M into an injective module. These operations are well defined and mutually inverse on StMod(kG).

If V is a ℤ-graded vector space, we write V [n] for the graded vector space with V [n]^{i} = V^{n+i}. An element v of V^{n+i}, when regarded as an element of V [n]^{i}, is written Σ^{n}v. If V carries a differential d: V^{i} → V^{i+1} then the differential on V [n] is defined by d(Σ^{n}v) = (−1)^{n}Σ^{n}(dv). If a graded ring Λ acts on V on the left then it also acts on V [n] via λ(Σ^{n}v) = (−1)^{mn}Σ^{n}(λv) where m = deg(λ). If Λ acts on V on the right then the action on V [n] is given by (Σ^{n}v)λ = Σ^{n}(vλ).

Cohomology of a graded module over a graded algebra is bigraded. The first index gives the cohomological degree and the second gives the internal degree. So for example if E = Ext_{kG}^{∗}(N, N), we write Ext_{E}^{s,t}(X, Y) for Ext_{E}^{s}(X, Y [t]). So an element of Ext_{E}^{3,−1}(X, Y) is represented by an exact sequence of graded modules

0 → Y [−1] → X_{2} → X_{1} → X_{0} → X → 0.

Whenever convenient, we shall write a tensor product λ_{1} ⊗ ⋯ ⊗ λ_{n} as an n-tuple (λ_{1}, …, λ_{n}).

4. The obstruction γ̄

We fix a kG-module N, and we write E for Êxt_{kG}^{∗}(N, N). In this section, given a graded E-module X, we define an obstruction

γ̄ ∈ Ext_{E}^{3,−1}(X, X) = Ext_{E}^{3}(X, X[−1]).

We prove that γ̄ = 0 if and only if X is isomorphic to a direct summand of an E-module of the form Êxt_{kG}^{∗}(N, M), for some kG-module M. In Section 7, we define the element

γ ∈ HH^{3,−1}E = HH^{3}(E, E[−1]),

and then in Section 8 we show that γ̄ is the image of γ under the natural map from Hochschild cohomology to Ext.

Let X be a graded E-module, and let

0 → K → F_{1} ⟶^{ρ} F_{0} ⟶^{ε} X → 0  (4.1)

be the beginning of a free resolution of X over E. In other words, each of F_{0} and F_{1} is a direct sum of copies of E, with degree shifts as necessary to hit generators and relations for X, and K is defined as the kernel of ρ.

Since Êxt_{kG}^{∗}(N, Ω^{n}N) ≅ Êxt_{kG}^{∗}(N, N)[−n], we can find kG-modules R_{0} and R_{1}, each a direct sum of modules of the form Ω^{n}N, n ∈ ℤ, such that Êxt_{kG}^{∗}(N, R_{0}) = F_{0} and Êxt_{kG}^{∗}(N, R_{1}) = F_{1}. Furthermore, there is a map α: R_{1} → R_{0} such that

\xymatrix{Êxt_{kG}^{∗}(N, R_{1}) \ar[r]^{α_{∗}} \ar[d]^{≅} & Êxt_{kG}^{∗}(N, R_{0}) \ar[d]^{≅} \\ F_{1} \ar[r]^{ρ} & F_{0}}

commutes. Complete α: R_{1} → R_{0} to a triangle in StMod(kG),

ΩB → R_{1} → R_{0} → B.  (4.2)

The long exact sequence in cohomology then gives

⋯ → Êxt_{kG}^{∗}(N, B)[−1] → F_{1} ⟶^{ρ} F_{0} → Êxt_{kG}^{∗}(N, B) → F_{1}[1] → ⋯

so that we obtain a short exact sequence

0 → X ⟶^{σ} Êxt_{kG}^{∗}(N, B) → K[1] → 0.

Let γ̄ be the element of

Ext_{E}^{1}(K[1], X) ≅ Ext_{E}^{1}(K, X[−1]) ≅ Ext_{E}^{3}(X, X[−1])

defined by this short exact sequence. The second isomorphism here is the dimension shift given by Yoneda splice with the exact sequence (4.1). The element γ̄ is well defined, by the following (more general) theorem.

Theorem 4.3. Let

M_{2} → M_{1} → M_{0} → Ω^{−1}M_{2}

be any triangle in StMod(kG) and let X be the cokernel of

Êxt_{kG}^{∗}(N, M_{1}) → Êxt_{kG}^{∗}(N, M_{0}).

Then the exact sequence of the triangle,

0 → X[−1] → Êxt_{kG}^{∗}(N, M_{2}) → Êxt_{kG}^{∗}(N, M_{1}) → Êxt_{kG}^{∗}(N, M_{0}) → X → 0

represents the element γ̄ ∈ Ext_{E}^{3,−1}(X, X).

Proof. In the diagram of E-modules

[figure: F_{1}; F_{0}; X; 0; Êxt_{kG}^{∗}(N, M_{1}); Êxt_{kG}^{∗}(N, M_{0}); X; 0]

the vertical arrows, which have been constructed using the standard lifting argument, can be realized by a map of triangles

\xymatrix{ΩB \ar[r] \ar[d] & R_{1} \ar[r] \ar[d] & R_{0} \ar[r] \ar[d] & B \ar[d] \\ M_{2} \ar[r] & M_{1} \ar[r] & M_{0} \ar[r] & Ω^{−1}M_{2}.}

This in turn gives rise to a map of extensions

[figure: 0; X[−1]; Êxt_{kG}^{∗}(N, ΩB); F_{1}; F_{0}; X; 0; 0; X[−1]; Êxt_{kG}^{∗}(N, M_{2}); Êxt_{kG}^{∗}(N, M_{1}); Êxt_{kG}^{∗}(N, M_{0}); X; 0]

which proves that these extensions represent the same element of the extension group Ext_{E}^{3,−1}(X, X), namely γ̄; see Mac Lane [7], §III.5. □

Theorem 4.4. The following are equivalent:

(i) γ̄ = 0 in Ext_{E}^{3,−1}(X, X).

(ii) The sequence

0 → X ⟶^{σ} Êxt_{kG}^{∗}(N, B) → K[1] → 0  (4.5)

splits.

(iii) The E-module X is isomorphic to a direct summand of Êxt_{kG}^{∗}(N, M) for some kG-module M.

Proof. For (i) ⇔ (ii), we notice that by construction, under the isomorphism

Ext_{E}^{3,−1}(X, X) ≅ Ext_{E}^{1}(K[1], X),

the class γ̄ corresponds to the extension (4.5). So γ̄ = 0 if and only if this extension splits.

It is obvious that (ii) ⇒ (iii), so we shall prove that (iii) ⇒ (ii). Since the construction of γ̄ is additive in X, we may assume that X ≅ Êxt_{kG}^{∗}(N, M). Then the map F_{0} → X is realized by a map R_{0} → M whose composite with α is zero in StMod(kG), so this map lifts to a map B → M. The induced map Êxt_{kG}^{∗}(N, B) → Êxt_{kG}^{∗}(N, M) = X splits the sequence (4.5). □

5. Parametrizing the realizations

In this section, we prove Theorem 1.2, parametrizing the realizations in the case where γ̄ is the zero element of Ext_{E}^{3,−1}(X, X).

Recall from Section 1 that Split(X) consists of the ordered triples (M, i, π) consisting of a kG-module M and maps of E-modules

X ⟶^{i} Êxt_{kG}^{∗}(N, M) ⟶^{π} X

whose composite is the identity map Id_{X}. There is an equivalence relation on Split(X) described Section 1, and Theorem 1.2 says that the equivalence classes form an affine space whose associated vector space is Ext_{E}^{2,−1}(X, X). In order to prove this theorem, we associate to two triples (M, i, π) and (M′, i′, π′) a difference element in d(M, i, π; M′, i′, π′) ∈ Ext_{E}^{2,−1}(X, X). To do this, we use the constructions described in Section 4. The composite i ∘ ε: F_{0} → Êxt_{kG}^{∗}(N, M) can be realized by a map of kG-modules θ: R_{0} → M whose composite with α: R_{1} → R_{0} is zero in StMod(kG). It therefore lifts to a map j: B → M with the property that the composite j_{∗} ∘ σ: X → Êxt_{kG}^{∗}(N, M) is equal to i. So composing with π gives the identity map on X.

\xymatrix{ &  &  &  &  & 0 \\  &  &  &  & K[1] \ar[dr] \ar@{-->}[ur] \\ F_{1} \ar[r] & F_{0} \ar[rr] \ar[dr]^{ε} \ar@{.>}[ddr]_{θ_{∗}} &  & Êxt_{kG}^{∗}(N, B) \ar@{.>}[ddl]^{j_{∗}} \ar[rr] \ar@{-->}[ur] &  & F_{1}[1] \\  &  & X \ar[d]_{i} \ar[ur]^{σ} \\  & 0 \ar[ur] & Êxt_{kG}^{∗}(N, M) \ar[u]_{π}}

It follows that (π ∘ j_{∗} − π′ ∘ j′_{∗}) ∘ σ = 0, so that π ∘ j_{∗} − π′ ∘ j′_{∗} defines a difference cocycle K[1] → X. The difference element d(M, i, π; M′, i′, π′) is defined to be the image of this element of Hom_{E}(K[1], X) in Ext_{E}^{2,−1}(X, X). It is a routine exercise to check that this difference element does not depend on the choices involved in the construction, and that the following identities hold.

d(M′, i′, π′; M, i, π) = −d(M, i, π; M′, i′, π′)

d(M, i, π; M′, i′, π′) + d(M′, i′, π′; M′′, i′′, π′′) = d(M, i, π; M′′, i′′, π′′)

If two triples (M, i, π) and (M′, i′, π′) are equivalent via a map ρ: M → M′ then the map θ′ may be chosen to be ρ ∘ θ and j′ can be chosen to be ρ ∘ j. Then π ∘ j = π′ ∘ j′ and so the difference element d(M, i, π; M′, i′, π′) is zero.

We can find a set of representatives of the equivalence classes as follows. A triple (M, i, π) is equivalent to the triple (B, σ, π∘j_{∗}) via the map j: B → M. So every triple is equivalent to a splitting of the sequence (4.5). It remains to examine when two such splittings are equivalent.

If λ, λ′: Êxt_{kG}^{∗}(N, B) → X are splittings of (4.5), then λ − λ′ vanishes on X, and defines the map K[1] → X representing the difference element in Ext_{E}^{2,−1}(X, X). Conversely, any map K[1] → X can be used to adjust the splitting. If (B, σ, λ) and (B, σ, λ′) are equivalent then the difference element is zero, so λ − λ′: K[1] → X factors as K[1] → F_{1}[1] → X. We can realize F_{1}[1] → X via a map Ω^{−1}R_{1} → B. The composite ϕ: B → Ω^{−1}R_{1} → B then satisfies λ ∘ ϕ_{∗} = λ′ and ϕ_{∗} ∘ σ = σ. It follows that two splittings λ, λ′ are equivalent if and only if there is a map ϕ: B → B with λ ∘ ϕ_{∗} = λ′ and ϕ_{∗} ∘ σ = σ. If this holds, then λ ∘ (ϕ − 1)_{∗} = λ − λ′ and (ϕ − 1)_{∗} ∘ σ = 0. So ϕ − 1 composes with R_{0} → B to give zero, and it follows that ϕ − 1 factors as B → Ω^{−1}R_{1} → B. Taking cohomology, we get a map F_{1}[1] → Êxt_{kG}^{∗}(N, B) such that the composite Ext_{kG}^{∗}(N, B) → F_{1}[1] → Êxt_{kG}^{∗}(N, B) is (ϕ − 1)_{∗}. This vanishes on X, and so gives a factorization K[1] → F_{1}[1] → X of the difference cocycle. This means that the difference element is zero in Ext_{E}^{2,−1}(X, X). This argument is reversible, so the splittings are equivalent if and only if the difference element is zero in Ext_{E}^{2,−1}(X, X). This completes the proof of Theorem 1.2.

Remarks 5.1. In algebraic topology, there are analogous appearances of Ext^{2} groups in classification problems of realizations of (co-)homology modules. For example, given two p-local spectra X and Y and an isomorphism f: H^{∗}(X; 𝔽_{p}) → H^{∗}(Y; 𝔽_{p}) of modules over the Steenrod algebra 𝒜_{p}, there is a corresponding element in E_{2}^{0,0} of the Adams spectral sequence

E_{2}^{s,t} = Ext_{𝒜_{p}}^{s,t} (H^{∗}(X; 𝔽_{p}), H^{∗}(Y; 𝔽_{p})) ⇒ [Y; X]_{t−s}.

Ignoring convergence issues, the first of an infinite sequence of obstructions to realizability as a map is the d_{2} differential to E_{2}^{2,1} in this spectral sequence.

Here we study a slightly different problem, namely realizations up to summands, and it turns out that then the only invariant is a class in Ext^{2}. This situation also comes up in stable homotopy theory in Bousfield’s classification of K-local spectra at an odd prime [3]. Roughly speaking, Bousfield’s classification is by the K-homology of a spectrum, including the Adams operations; these objects take values in an abelian category with injective dimension two. This implies that every algebraic homology object is realized as the K-homology of a spectrum, but there may be genuinely different realizations (i.e., by K-local spectra which are not stably equivalent).

Because the injective dimension is two, in the associated Adams spectral sequence, the only obstruction to realizing an algebraic morphism between the K-homologies by a geometric morphism is the d_{2} differential, which lives in the respective Ext^{2} group. Bousfield refers to this obstruction as the “K_{∗}-k-invariant.”

6. The map from Hochschild cohomology to Ext

If k is a field of coefficients and Λ is a k-algebra, we write Λ^{e} for Λ ⊗_{k} Λ^{op}, so that Λ^{e}-modules are the same as Λ-Λ-bimodules with scalars from k acting the same way on both sides. If M is a Λ-Λ-bimodule, then Hochschild cohomology of Λ with coefficients in M is defined to be HH^{∗}(Λ, M) = Ext_{Λ^{e}}^{∗}(Λ, M), In case M = Λ, we write HH^{∗}Λ for HH^{∗}(Λ, Λ).

If Λ is graded and M is a graded Λ-Λ-bimodule, then we use degree preserving maps in the above discussion and define

HH^{i,j}(Λ, M) = Ext_{Λ^{e}}^{i}(Λ, M[j]),

and HH^{i}(Λ, M) = HH^{i,0}(Λ, M). So the index i is the Hochschild cohomological index, and j comes from the internal grading. Our sign conventions for working with this definition are given in Section 3.

Hochschild cohomology can be calculated using the free resolution, often called the bar resolution,

⋯ ⟶^{d} Λ^{⊗(n+2)} ⟶^{d} Λ^{⊗(n+1)} → ⋯ ⟶^{d} Λ^{⊗2} → Λ → 0

where the nth term in the resolution is Λ^{⊗(n+2)}. The Λ^{e}-module structure is given by

(μ, μ′)(λ_{0}, …, λ_{n+1}) = (μλ_{0}, λ_{1}, …, λ_{n}, λ_{n+1}μ′).

The differential is defined by

d(λ_{0}, …, λ_{n+1}) = ∑_{i=0}^{n}(−1)^{i}(λ_{0}, …, λ_{i}λ_{i+1}, …, λ_{n+1}).

For more details, see Cartan and Eilenberg [4], §IX.6.

If M is a Λ-Λ-bimodule, then we have

Hom_{Λ^{e}}(Λ^{⊗(n+2)}, M) ≅ Hom_{k}(Λ^{⊗n}, M).

So a Hochschild n-cochain with coefficients in M is given by a function from n-tuples of elements of Λ to M. The differential is then given by

df(λ_{1}, …, λ_{n}) = (−1)^{|λ_{1}||f|}λ_{1}f(λ_{2}, …, λ_{n}) +

∑_{i=1}^{n−1}(−1)^{i}f(λ_{1}, …, λ_{i}λ_{i+1}, …, λ_{n}) + (−1)^{n}f(λ_{1}, …, λ_{n−1})λ_{n}.

In the ungraded situation, all elements λ_{i} are interpreted as having degree zero, so the sign on the first term on the right of this equation disappears.

If X is a right Λ-module, then Hom_{k}(X, X) is a Λ-Λ-bimodule, there is a natural isomorphism

HH^{∗}(Λ, Hom_{k}(X, X)) ≅ Ext_{Λ}^{∗}(X, X).

So the map of bimodules giving the Λ-action Λ → Hom_{k}(X, X) induces a map

HH^{∗}Λ → Ext_{Λ}^{∗}(X, X).

An explicit way to describe the map is as follows. Given a projective resolution of Λ as a Λ^{e}-module and a Hochschild cochain α:

\xymatrix{… \ar[r] & P_{n} \ar[r] \ar[d]^{α} & ⋯ \ar[r] & P_{0} \ar[r] & Λ \ar[r] & 0 \\  & Λ}

we can apply X ⊗_{Λ} − to obtain a cochain α_{X}:

\xymatrix{… \ar[r] & X ⊗_{Λ} P_{n} \ar[r] \ar[d]^{α_{X}} & X ⊗_{Λ} P_{0} \ar[r] & X \ar[r] & 0 \\  & X}

for this projective resolution of X. This construction commutes with the differential, so an element of Hochschild cohomology gives rise to a well defined element of Ext_{Λ}^{∗}(X, X).

7. Description of the element γ

Let P̂_{∗} be a complete resolution of N as a kG-module. In other words, P̂_{∗} is obtained by splicing together a projective resolution and an injective resolution of N (recall that a kG-module is injective if and only if it is projective)

\xymatrix{… \ar[r] & P̂_{2} \ar[r] & P̂_{1} \ar[r] & P̂_{0} \ar[rr] \ar[dr] &  & P̂_{−1} \ar[r] & P̂_{−2} \ar[r] & … \\  &  &  &  & N \ar[dr] \ar@{-->}[ur] \\  &  &  & 0 \ar@{-->}[ur] &  & 0}

We write Ω^{n}N for the kernel of P̂_{n−1} → P̂_{n−2} for n ∈ ℤ.

Denote by A the cochain complex

A = Hom_{kG}^{∗}(P̂_{∗}, P̂_{∗})

so that

A^{n} = ∏_{j} Hom_{kG}(P̂_{n+j}, P̂_{j})

with differential d: A^{n} → A^{n+1} defined by

(df)(x) = ∂(f(x)) − (−1)^{n}f(∂(x))

where ∂ denotes the differential of P̂_{∗}. With this definition, the cocycles in A are the chain homomorphisms (negated if the degree is odd), and cocycles differ by a coboundary if and only if the corresponding chain maps are homologous. So A is a differential graded algebra whose cohomology H^{∗}(A) is Tate cohomology Êxt_{kG}^{∗}(N, N). So we can apply the construction described in §3.3 of Keller [6], which we repeat here for the convenience of the reader.

First, note that Keller writes m_{2} for the multiplication maps,

m_{2} : A ⊗ A → A, m_{2} : H^{∗}(A) ⊗ H^{∗}(A) → H^{∗}(A).

We regard H^{∗}(A) as a differential graded algebra with zero differential, and we choose a morphism of complexes f_{1} : H^{∗}(A) → A which induces the identity in cohomology. This amounts to choosing a representative cocycle for each Tate cohomology class, in a linear fashion.

Now f_{1} usually cannot be chosen to commute with multiplication, but at least it commutes up to coboundaries. So we choose a graded map of degree −1

f_{2} : H^{∗}(A) ⊗ H^{∗}(A) → A

satisfying

df_{2}(x, y) = f_{1}(xy) − f_{1}(x)f_{1}(y).  (7.1)

Given x, y, z ∈ H^{∗}(A), we have

d[(−1)^{|x|}f_{1}(x)f_{2}(y, z) − f_{2}(xy, z) + f_{2}(x, yz) − f_{2}(x, y)f_{1}(z)] = 0,

so the element inside the bracket is a cocycle. This means that we can find maps m_{3} : H^{∗}(A)^{⊗3} → H^{∗}(A) of degree −1 and f_{3} : H^{∗}(A)^{⊗3} → A of degree −2, such that

f_{1}m_{3} − df_{3} = m_{2}(f_{1} ⊗ f_{2}) − f_{2}(m_{2} ⊗ 1) + f_{2}(1 ⊗ m_{2}) − m_{2}(f_{2} ⊗ f_{1}),

or

f_{1}(m_{3}(x, y, z)) − df_{3}(x, y, z) =

(−1)^{|x|}f_{1}(x)f_{2}(y, z) − f_{2}(xy, z) + f_{2}(x, yz) − f_{2}(x, y)f_{1}(z). (7.2)

Lemma 7.3. The map m_{3} : H^{∗}(A)^{⊗3} → H^{∗}(A) of degree −1 described above is a Hochschild cocycle.

Proof. This follows from the identity

m_{2}(1 ⊗ m_{3}) − m_{3}(m_{2} ⊗ 1 ⊗ 1) + m_{3}(1 ⊗ m_{2} ⊗ 1) − m_{3}(1 ⊗ 1 ⊗ m_{2}) + m_{2}(m_{3} ⊗ 1) = 0,

or

(−1)^{|w|}wm_{3}(x, y, z) − m_{3}(wx, y, z) + m_{3}(w, xy, z) − m_{3}(w, x, yz) + m_{3}(w, x, y)z = 0. (7.4)

To check this identity, apply f_{1} to the left-hand side. Using equations 7.1 and 7.2, we see that the result is a coboundary. Since f_{1} is the identity map on cohomology, the identity 7.4 is proved. □

Now the above construction of the Hochschild cocycle m_{3} depends on some choices, but our next task is to prove that the Hochschild cohomology class it determines does not depend on these choices.

Proposition 7.5. The Hochschild cohomology class

γ ∈ HH^{3,−1}H^{∗}(A)

determined by m_{3} is independent of the choices made in defining m_{3}.

Proof. First let f′_{1} be another choice for the map f_{1}. Then there is a map g_{1} : H^{∗}(A) → A of degree −1 such that

dg_{1}(x) = f′_{1}(x) − f_{1}(x).

Setting

f′_{2}(x, y) = f_{2}(x, y) + g_{1}(xy) − (−1)^{|x|}f′_{1}(x)g_{1}(y) − g_{1}(x)f_{1}(y)
f′_{3}(x, y, z) = f_{3}(x, y, z) + g_{1}(m_{3}(x, y, z))
− (−1)^{|x|}g_{1}(x)f_{2}(y, z) − (−1)^{|x|+|y|}f′_{2}(x, y)g_{1}(z)

it is easy to check that equations (7.1) and (7.2) hold for f′_{1}, f′_{2}, f′_{3} and m_{3}.

Next, we keep f_{1} fixed and let f′_{2} be another choice for the map f_{2}. Then there are maps n_{2} : H^{∗}(A) ⊗ H^{∗}(A) → H^{∗}(A) of degree −1 and g_{2} : H^{∗}(A) ⊗ H^{∗}(A) → A of degree −2 such that

f_{1}(n_{2}(x, y)) − dg_{2}(x, y) = f′_{2}(x, y) − f_{2}(x, y).

Setting

f′_{3}(x, y, z) = f_{3}(x, y, z) + (−1)^{|x|}f_{2}(x, n_{2}(y, z)) + f_{1}(x)g_{2}(y, z)
− g_{2}(xy, z) + g_{2}(x, yz) − f_{2}(n_{2}(x, y), z) − g_{2}(x, y)f_{1}(z)
m′_{3}(x, y, z) = m_{3}(x, y, z) + (−1)^{|x|}xn_{2}(y, z)  (7.6)
− n_{2}(xy, z) + n_{2}(x, yz) − n_{2}(x, y)z,

we again check that equations (7.1) and (7.2) hold for f_{1}, f′_{2}, f′_{3} and m′_{3}.

Finally, we keep f_{1} and f_{2} fixed, and let f′_{3} be another choice for the map f_{3}. Then f′_{3} differs from f_{3} by a cocycle, so df′_{3} = df_{3} and equations (7.1) and (7.2) hold for f_{1}, f_{2}, f′_{3} and m_{3}.

The upshot of this analysis is that the extent to which m_{3} is not well defined is expressed by equation (7.6). This equation says that the difference between the cocycles m′_{3} and m_{3} is the Hochschild coboundary of the Hochschild cochain n_{2} : H^{∗}(A) ⊗ H^{∗}(A) → H^{∗}(A). □

8. Comparing γ with γ̄

In order to make the comparison between the definitions of γ and γ̄, our goal is to construct a commutative diagram of E-modules

[figure: X ⊗_{k} E^{⊗4}; d; γ_{X}; X ⊗_{k} E^{⊗3}; d; λ_{∗}; X ⊗_{k} E^{⊗2}; d; X ⊗_{k} E; X; 0; 0; X[−1]; Êxt_{kG}^{∗}(N, ΩB); F_{1}; ρ; F_{0}; X; 0]

(8.1)

where the top row is the Hochschild complex and the bottom row is the sequence defining γ̄. An obvious simplification of this task is to choose F_{1} and F_{0} to be X ⊗_{k} E^{⊗2} and X ⊗_{k} E respectively, and to make the corresponding vertical maps the identity. The hard part of the proof is then to construct the map marked λ_{∗} in this diagram in such a way that the two squares in which it is involved commute. Accomplishing this will complete the proof of Theorem 1.1.

We use the maps f_{1}, f_{2}, f_{3} and m_{3} defined in Section 7 to define explicit maps in Mod(kG) for use in the constructions which were used in Section 4 to define the bottom row of the above diagram. To this end, we define two functors R and Q from ℤ-graded vector spaces to Mod(kG) via

R(V) = ⨁_{n∈ℤ}V^{n} ⊗_{k} Ω^{n}N, Q(V) = ⨁_{n∈ℤ}V^{n} ⊗_{k} P̂_{n}.

The G action is defined via the right tensor factor. There is a short exact sequence of kG-modules

0 → R(V [−1]) ⟶^{i} Q(V) ⟶^{∂} R(V) → 0,

where

∂(x, ζ_{1}, …, ζ_{n}, α) = (−1)^{|xζ_{1}...ζ_{n}|}(x, ζ_{1}, …, ζ_{n}, ∂(α)).

We also write ∂: Q(V) → Q(V [1]) for the map defined by the same formula. The module

R_{n} = R(X ⊗_{k} E^{⊗n})

has the property that

Êxt_{kG}^{∗}(N, R_{n}) ≅ X ⊗_{k} E^{⊗(n+1)}

which is the nth term of the complex described at the end of Section 6. We also write R_{n}[m] for R(X ⊗_{k} E^{⊗n}[m]), Q_{n}[m] for Q(X ⊗_{k} E^{⊗n}[m]) and Q_{n} for Q_{n}[0]. We define d̃: R_{n}[m] → R_{n−1}[m] by

d̃(x, ζ_{1}, …, ζ_{n}, α) = (xζ_{1}, ζ_{2}, …, ζ_{n}, α)

+ ∑_{i=1}^{n−1}(−1)^{i}(x, ζ_{1}, …, ζ_{i}ζ_{i+1}, …, ζ_{n}, α) + (−1)^{n}(x, ζ_{1}, …, f_{1}(ζ_{n})(α)).

After applying Êxt_{kG}^{∗}(N, −), this induces the bar complex differential.

The same formula defines a map which we also denote d̃ from Q_{n}[m] to Q_{n−1}[m]. The following diagram then commutes.

\xymatrix{0 \ar[r] & R_{n} \ar[r]^{i} \ar[d]^{d̃} & Q_{n}[1] \ar[r]^{∂} \ar[d]^{d̃} & R_{n}[1] \ar[r] \ar[d]^{d̃} & 0 \\ 0 \ar[r] & R_{n−1} \ar[r]^{i} & Q_{n−1}[1] \ar[r]^{∂} & R_{n−1}[1] \ar[r] & 0}

An easy calculation using equation (7.1) shows that we have

d̃d̃(x, ζ_{1}, …, ζ_{n}, α) = (x, ζ_{1}, …, ζ_{n−2}, (df_{2}(ζ_{n−1}, ζ_{n}))(α))
= (x, ζ_{1}, …, ζ_{n−2}, ∂(f_{2}(ζ_{n−1}, ζ_{n})(α))
+ (−1)^{|ζ_{n−1}ζ_{n}|}f_{2}(ζ_{n−1}, ζ_{n})(∂α)).

Define

ϕ = 1 ⊗ ⋯ ⊗ 1 ⊗ m_{2}(f_{2} ⊗ 1): Q_{n}[1] → Q_{n−2}

so that

ϕ(x, ζ_{1}, …, ζ_{n}, α) = (−1)^{|xζ_{1}...ζ_{n−2}|}(x, ζ_{1}, …, ζ_{n−2}, f_{2}(ζ_{n−1}, ζ_{n})(α)).

Then we have

∂ϕ(x, ζ_{1}, …, ζ_{n}, α) = (x, ζ_{1}, …, ζ_{n−2}, ∂(f_{2}(ζ_{n−1}, ζ_{n})(α)))
ϕ∂(x, ζ_{1}, …, ζ_{n}, α) = (−1)^{|xζ_{1}...ζ_{n}|}ϕ(x, ζ_{1}, …, ζ_{n}, ∂α)
= (−1)^{|ζ_{n−1}ζ_{n}|}(x, ζ_{1}, …, ζ_{n−2}, f_{2}(ζ_{n−1}, ζ_{n})(∂α)),

and so

d̃d̃= ∂ϕ + ϕ∂.  (8.2)

Next, define

ψ = 1 ⊗ ⋯ ⊗ 1 ⊗ (f_{1}m_{3} − df_{3}): Q_{n}[1] → Q_{n−3}

so that

ψ(x, ζ_{1}, …, ζ_{n}, α) = (−1)^{|xζ_{1}...ζ_{n−3}|}(x, ζ_{1}, …, ζ_{n−3},

[f_{1}(m_{3}(ζ_{n−2}, ζ_{n−1}, ζ_{n})) − df_{3}(ζ_{n−2}, ζ_{n−1}, ζ_{n})](α)).

Then a similar calculation using (7.2) shows that

d̃ϕ − ϕd̃= (−1)^{n}ψ.  (8.3)

It is also easy to check that ψ∂ = ∂ψ, and so ψ induces a well defined map which we also denote ψ: R_{n}[1] → R_{n−3}.

Define B to be the pushout of

\xymatrix{R_{1} \ar[r]^{d̃} \ar[d]^{i} & R_{0} \\ Q_{1}[1]}

in the category of kG-modules, so that there are exact sequences

0 → R_{1} → R_{0} ⊕ Q_{1}[1] → B → 0  (8.4)

and

0 → R_{0} → B → R_{1}[1] → 0  (8.5)

giving rise to a triangle (4.2).

In order to define a map

λ: Q_{2}[1] → B,

we first define a map

λ̃ = ( ϕ ; d̃) : Q_{2}[1] → Q_{0} ⊕ Q_{1}[1]

or more explicitly,

λ̃(x, ζ_{0}, ζ_{1}, α) = ((−1)^{|x|}(x, f_{2}(ζ_{0}, ζ_{1})(α)) ,

(xζ_{0}, ζ_{1}, α) − (x, ζ_{0}ζ_{1}, α) + (x, ζ_{0}, f_{1}(ζ_{1})(α))).

Equation (8.2) shows that the left hand square of the following diagram commutes, so that there is a map λ such that the right hand square also commutes.

\xymatrix{Q_{2} \ar[r]^{∂} \ar[d]^{_{(−ϕ ; d̃)}} & Q_{2}[1] \ar[r]^{∂} \ar[d]_{λ̃ =}^{_{( ϕ ; d̃)}} & R_{2}[1] \ar[r] \ar[d]^{λ} & 0 \\ Q_{0}[−1] ⊕ Q_{1} \ar[r]^{_{(\begin{matrix}∂ & d̃ \\ 0 & ∂ \end{matrix})}} & Q_{0} ⊕ Q_{1}[1] \ar[r] & B \ar[r] & 0}

The map λ induces a map

λ_{∗}: X ⊗_{k} E^{⊗3} → Êxt_{kG}^{∗}(N, B),

[figure: Q_{2}; d̃; _{(−ϕ ; d̃)}; ∂; 0; Q_{0}[−1]; ∂; Q_{0}[−1] ⊕ Q_{1}; _{(\begin{matrix}∂ & d̃ \\ 0 & ∂ \end{matrix})}; Q_{1}; ∂; 0; Q_{3}[1]; d̃; ∂; ψ; Q_{2}[1]; ∂; λ̃; d̃; 0; Q_{0}; ∂; Q_{0} ⊕ Q_{1}[1]; _{(\begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix})}; Q_{1}[1]; ∂; 0; R_{3}[1]; d̃; ψ; R_{2}[1]; λ; d̃; 0; R_{0}; _{(\begin{matrix}1 \\ 0 \end{matrix})}; B; (0 ∂); R_{1}[1]; 0; 0; 0; 0] Figure 1

which is the desired map for diagram (8.1). It remains to check that the two squares in diagram (8.1) in which λ_{∗} is involved commute. For this purpose, we examine the diagram in Figure 1. This diagram does not quite commute. The four front squares and the back square commute, the triangles commute, and the “side walls” commute. But the two “horizontal” squares only commute after applying Êxt_{kG}^{∗}(N, −). To see this, we calculate with maps from R_{3}[1] to B. Using equations (8.2) and (8.3), we have

λd̃∂ = λ∂d̃= ( \begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix} )( ϕ ; d̃)d̃= ( \begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix} )( \begin{matrix}d̃ϕ+ψ \\ ∂ϕ+ϕ∂ \end{matrix} )

= ( \begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix} )( \begin{matrix}∂ & d̃ \\ 0 & ∂ \end{matrix})( \begin{matrix}0 \\ ϕ \end{matrix} ) + ( \begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix} )( \begin{matrix}ψ \\ ϕ∂ \end{matrix} ) = ( \begin{matrix}∂ & 0 \\ 0 & 1 \end{matrix} )( \begin{matrix}ψ \\ ϕ∂ \end{matrix} ) = ( \begin{matrix}ψ∂ \\ ϕ∂ \end{matrix} ).

Since ∂: Q_{3}[1] → R_{3}[1] is surjective, it follows that

λd̃= ( \begin{matrix}ψ \\ ϕ \end{matrix} ): R_{3}[1] → B.

Now the map ( \begin{matrix}0 \\ ϕ \end{matrix} ) factors through the projective module Q_{1}[1]. So the map in cohomology is zero, and it follows that the map

λ_{∗}d: X ⊗_{k} E^{⊗4}[1] = Êxt_{kG}^{∗}(N, R_{3}[1]) → Êxt_{kG}^{∗}(N, B)

is the same as the map induced by ( \begin{matrix}ψ \\ 0 \end{matrix} ). Finally, it is easy to see that the following diagram commutes.

[figure: X ⊗_{k} E^{⊗4}[1]; ψ_{∗}; γ_{X}; _{(\begin{matrix}ψ \\ 0 \end{matrix} )_{∗}}; F_{0}; Êxt_{kG}^{∗}(N, B); X; 0; 0]

References

1. A. Beligiannis and H. Krause, Realizing maps between modules over Tate cohomology, Preprint, 2002.

2. D. J. Benson, H. Krause, and S. Schwede, Realizability of modules over Tate cohomology, Submitted to Trans. Amer. Math. Soc., 2002.

3. A. K. Bousfield, On the homotopy theory of k-local spectra at an odd prime, Amer. J. Math. 107 (1985), 895–932.

4. H. Cartan and S. Eilenberg, Homological algebra, Princeton Mathematical Series, no. 19, Princeton Univ. Press, 1956.

5. T. V. Kadeishvili, The algebraic structure in the homology of an A(∞)-algebra, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), 249–252.

6. B. Keller, Introduction to A-infinity algebras and modules, Homology, Homotopy & Appl. 3 (2001), 1–35, — Addendum, ibid. 4 (2002), 25–28.

7. S. Mac Lane, Homology, Grundlehren der mathematischen Wissenschaften, vol. 114, Springer-Verlag, Berlin/New York, 1963.

Department of Mathematics, University of Georgia, Athens GA 30602, USA

   E-mail address: djb@byrd.math.uga.edu

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: schwede@mathematik.uni-bielefeld.de