Generalized Burnside rings and group cohomology

Robert Hartmann ^{∗} University of Leicester, Department of Mathematics University Road, Leicester LE1 7RH, UK E-mail: rh91@mcs.le.ac.uk

Ergün Yalçın ^{†} Bilkent University, Department of Mathematics Ankara, 06800, Turkey E-mail: yalcine@fen.bilkent.edu.tr

August 5, 2005

Abstract

We define the cohomological Burnside ring B^{n}(G, M) of a finite group G with coefficients in a ℤG-module M as the Grothendieck ring of the isomorphism classes of pairs [X, u] where X is a G-set and u is a cohomology class in a cohomology group H_{X}^{n}(G, M). The cohomology groups H_{X}^{∗}(G, M) are defined in such a way that H_{X}^{∗}(G, M) ≅ ⊕_{i}H^{∗}(H_{i}, M) when X is the disjoint union of transitive G-sets G/H_{i}. If A is an abelian group with trivial action, then B^{1}(G, A) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B^{0}(G, M) is equal to the crossed Burnside ring B^{c}(G, M). We discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological Burnside rings. We also give an interpretation of B^{2}(G, M) in terms of twisted group rings when M = k^{×} is the unit group of a commutative ring. 2000 Mathematics Subject Classification. Primary: 19A22, 20J06. Key words and phrases. Cohomology of groups, monomial G-sets, generalized Burnside rings.

1 Introduction

Let G be a finite group and X be a finite G-set. Given a ℤG-module M, we define H_{X}^{∗}(G, M), the cohomology of G associated to X with coefficients in M, as the cohomology of a cochain complex, where the n-cochains are the maps f : G^{n} ×X → M and derivations are given by

∗Research supported by the European Community, through Marie Curie fellowship MCFI 2002-01325.

†Partially supported by the Turkish Academy of Sciences, in the framework of the Young Scientist Award Program (TÜBA-GEBİP/2005-16).

(δf)(g_{0}, …, g_{n}; x) =g_{0} · f(g_{1}, …, g_{n}; x)
− f(g_{0}g_{1}, …, g_{n}; x)
…
+ (−1)^{n}f(g_{0}, …, g_{n−1}g_{n}; x)
+ (−1)^{n+1}f(g_{0}, …, g_{n−1}; g_{n}x).

The cohomology group H_{X}^{n}(G, M) can be described in terms of the usual group cohomology of subgroups of G. In particular, when X is the disjoint union of transitive G-sets G/H_{i} for i = 1, …, k, then we have

H_{X}^{n}(G, M) ≅ ⨁_{i=1}^{k} H^{n}(H_{i}, M).

Given a G-set map f : X → Y, we define f^{∗} : H_{Y}^{n}(G, M) → H_{X}^{n}(G, M) and f_{∗} : H_{X}^{n}(G, M) → H_{Y}^{n}(G, M) on the chain level in such a way that the assignment X → H_{X}^{n}(G, M) together with ( )^{∗} and ( )_{∗} defines a Mackey functor in the sense described in [11]. In section 3, we show that this Mackey functor is naturally equivalent to the cohomology of groups Mackey functor H^{n}(?, M).

The motivation for this definition comes from a classification problem for monomial G-sets, where the cocycles defined above appear in a natural way. This example also motivates the definition of cohomological Burnside rings. Recall that the Burnside ring B(G) of a finite group G is defined as the Grothendieck ring of the isomorphism classes of G-sets, with addition given by disjoint unions and multiplication by cartesian products. We generalize this definition as follows:

A positioned G-set is a pair of the form (X, u), where X is a G-set and u is a class in H_{X}^{n}(G, M). A map f : (X, u) → (Y, v) is called a positioned G-set map if f : X → Y is a G-set map such that f^{∗}(v) = u. We say that two positioned G-sets (X, u) and (Y, v) are isomorphic if there is a positioned G-set map f : (X, u) → (Y, v) such that f : X → Y is an isomorphism as a G-set map. We denote the isomorphism class of a positioned G-set (X, u) simply by [X, u]. The set of isomorphism classes of positioned G-sets is a semi-ring with addition and multiplication defined by

[X, u] + [Y, v] = [X ⨿ Y, u ⊕ v]
[X, u] · [Y, v] = [X × Y, u ⊗ v].

where u ⊕ v ∈ H_{X⨿Y}^{n} (G, M) and u ⊗ v ∈ H_{X×Y}^{n} (G, M) are defined in the following way: For u ∈ H_{X}^{n}(G, M) and v ∈ H_{Y}^{n}(G, M),

u ⊕ v = (i_{X})_{∗}(u) + (i_{Y})_{∗}(v)

where i_{X} : X → X ⨿ Y and i_{Y} : Y → X ⨿ Y are the usual inclusion maps of X and Y, and

u ⊗ v = (π_{X})^{∗}(u) + (π_{Y})^{∗}(v)

where π_{X} : X × Y → X and π_{Y} : X × Y → Y are the projection maps.

The cohomological Burnside ring B^{n}(G, M) of degree n of the group G with coefficients in M is defined as the Grothendieck ring of this semi-ring. The motivation for this definition comes from earlier constructions of generalized Burnside rings such as the crossed Burnside ring and the monomial Burnside ring.

Our construction of cohomological Burnside rings includes these two special cases. For example, if we take n = 0 and M as an arbitrary G-monoid, then B^{0}(G, M) is equal to the crossed Burnside ring B^{c}(G, M) described by Oda and Yoshida in [8], [9] (see also Bouc [4]). In the case n = 1, if M is taken as an abelian group A with trivial G-action, then the cohomological Burnside ring B^{1}(G, A) coincides with the monomial Burnside ring over A defined by Dress [6] (see also Boltje [3] and Barker [1]).

This is all proved in Section 5. We also give a ℤ-basis for B^{n}(G, M) and prove that

B^{n}(G, M) ≅ ⨁_{[H]∈Cl(G)}(ℤH^{n}(H, M))_{W_{G}(H)},

where Cl(G) denotes the set of conjugacy classes of subgroups of G and W_{G}(H) = N_{G}(H)/H is the Weyl group of H in G. Motivated by this description, we define the ghost ring of the cohomological Burnside ring as

β^{n}(G, M) ≅ ⨁_{[H]∈Cl(G)}(ℤH^{n}(H, M))^{W_{G}(H)}

and describe the mark homomorphism explicitly.

Section 7 is devoted to the proof of the fundamental theorem for cohomological Burnside rings. Here we use a different approach than the earlier results for monomial Burnside rings and crossed Burnside rings. Instead of choosing a basis of the ghost ring as an abelian group, we use the direct sum decomposition coming from the conjugacy classes of subgroups and express the mark homomorphism as a matrix of homomorphisms instead of a matrix of scalars. In other words, we replace the classical table of marks for Burnside rings with a table of marks where each mark is a homomorphism. We calculate the cokernel of the mark homomorphism as the direct sum over conjugacy classes of subgroups where each summand is the 0-th Tate cohomology of the Weyl group W_{G}(H) = N_{G}(H)/H with coefficients in the ℤW_{G}(H)-module ℤH^{n}(H, M). We state the fundamental theorem for cohomological Burnside rings in the following form.

Theorem 1.1 (Fundamental Theorem). Let G be a finite group, let M be a ℤG-module, and let n be a non-negative integer. Then the following sequence of abelian groups is exact

0 → B^{n}(G, M) ⟶^{φ} β^{n}(G, M) ⟶^{ψ} Obs^{n}(G, M) → 0

with

Obs^{n}(G, M) = ⨁_{[H]∈Cl(G)} Ĥ^{0}(W_{G}(H), ℤH^{n}(H, M)),

where φ is the mark homomorphism, and the [K]-th component of ψ is defined by

ψ_{[K]}(f) = ∑_{K≤L} μ(K, L)res_{K}^{L}f(L)

modulo the image of the trace map T r_{1}^{W_{G}(K)}.

It is easy to calculate the Tate group Ĥ^{0}(W_{G}(H), ℤH^{n}(H, M)) for each [H] ∈ Cl(G) using an appropriate basis for ℤH^{n}(H, M). We obtain

Proposition 1.2. We have

Obs^{n}(G, M)) ≅ ⨁_{[G/H,u]} ℤ/|N_{G}(H, u) : H|ℤ

where the sum is taken over the isomorphism classes of pointed G-sets with transitive G-sets.

Thus, our obstruction groups for n = 0 and n = 1 are the same as the ones given earlier for crossed Burnside rings and monomial Burnside rings respectively. This is all done in Section 7.

Finally, in Section 8 we give an interpretation of B^{2}(G, M) in terms of twisted group rings when M = k^{×} is the unit group of a commutative ring.

2 The definition of H_{X}^{∗}(G, M)

Let G be a finite group, and let X be a finite G-set. For a given ℤG-module M, we define the cochain complex (C_{X}^{∗}(G, M), δ) as follows: The n-cochains are the functions

f : G^{n} × X → M

and the coboundary δ : C_{X}^{n}(G, M) → C_{X}^{n+1}(G, M) is defined by

(δf)(g_{0}, …, g_{n}; x) =g_{0} · f(g_{1}, …, g_{n}; x)
− f(g_{0}g_{1}, …, g_{n}; x)
…
+ (−1)^{n}f(g_{0}, …, g_{n−1}g_{n}; x)
+ (−1)^{n+1}f(g_{0}, …, g_{n−1}; g_{n}x).

It is easy to check that δ^{2} = 0, so we define

H_{X}^{∗}(G, M) := H^{∗}(C_{X}^{∗}(G, M), δ),

and call it the cohomology of G associated to X with coefficients in M. Note that the cohomology of G associated to the trivial G-set G/G is just the usual cohomology of the group G.

Given a 0-cochain f : X → M, we see that (δf)(g; x) = gf(x) − f(gx). So, f is a 0-cocycle if and only if f is a G-map. Thus, H_{X}^{0}(G, M) = Map_{G}(X, M). Note that a 1-cochain f : G × X → M is a cocycle (or a derivation) if and only if

f(g_{0}g_{1}; x) = g_{0}f(g_{1}; x) + f(g_{0}; g_{1}x)  (1)

for every g_{0}, g_{1} ∈ G and x ∈ X. We say that f : G × X → M is a trivial derivation (or a inner derivation) if f = δt for some function t : X → M.

The motivation for this definition comes from the classification problem for monomial G-sets. A monomial G-set with coefficients in A is an A-free A × G-set Γ with Γ/A isomorphic to a given G-set X. As a set, Γ is isomorphic to A × X, where the action can be described as

(a, g)(a′, x) = (a + a′ + α(g, x), gx)  (2)

with α : G × X → M being a derivation in the above sense. In fact, Equation 1 holds for α if and only if Equation 2 defines an action. We study this problem in detail in Section 4.

In the rest of the section we will prove the following:

Theorem 2.1. Suppose that G is a finite group, X is a G-set, and M is a ℤG-module. Then, we have

H_{X}^{∗}(G, M) ≅ H^{∗}(G, Map(X, M)),

where Map(X, M) is the abelian group of functions f : X → M considered as a (left) ℤG-module with the action given by (gf)(x) = gf(g^{−1}x).

We will prove Theorem 2.1 using the Hochschild cohomology of ℤG. Given a (ℤG, ℤG)- bimodule B, the Hochschild cohomology HH^{∗}(ℤG, B) is defined as the cohomology of the cochain complex

C^{n}(ℤG, B) = Hom_{ℤ}((ℤG)^{⊗n}, B)

with coboundary

(δf)(a_{0}, …, a_{n}) =a_{0} · f(a_{1}, …, a_{n})
− f(a_{0}a_{1}, …, a_{n})
…
+ (−1)^{n}f(a_{0}, …, a_{n−1}a_{n})
+ (−1)^{n+1}f(a_{0}, …, a_{n−1}) · a_{n}.

Proposition 2.2. Suppose that G is a finite group, X is a G-set, and M is a ℤG-module. Then, we have

H_{X}^{∗}(G, M) ≅ HH^{∗}(ℤG, Map(X, M))

where Map(X, M) is the abelian group of functions f : X → M considered as a (ℤG, ℤG)- bimodule with the ℤG-action given by

(g_{1} · f · g_{2})(x) = g_{1}f(g_{2}x).

Proof. Consider the map

Φ^{n} : C_{X}^{n}(G, M) → C^{n}(ℤG, Map(X, M))

defined for all n ≥ 0 by the formula Φ^{n} : f → \overline{f} where

\overline{f}(g_{0}, …, g_{n−1})(x) = f(g_{0}, …, g_{n−1}; x).

Note that

\begin{aligned}[\overline{f}(g_{0}, …, g_{n−1}) · g_{n}](x) & = \overline{f}(g_{0}, …, g_{n−1})(g_{n}x) \\ & = f(g_{0}, …, g_{n−1}; g_{n}x) \end{aligned}

So, we have \overline{δf} = δ\overline{f}, i.e., Φ is a cochain map. Note that the obvious inverse is also a cochain map, hence Φ induces an isomorphism on cohomology.

We recall the following fact about Hochschild cohomology:

Lemma 2.3. Let B be a (ℤG, ℤG)-bimodule. Then the Hochschild cohomology HH^{∗}(ℤG, B) is isomorphic to the usual group cohomology H^{∗}(G, L(B)) where L(B) = B is the (left) ℤG-module with the action given by g · b = gbg^{−1}.

Proof. See Theorem 5.5 on page 292 in Mac Lane [7].

Theorem 2.1 follows now from Proposition 2.2 and Lemma 2.3.

Remark 2.4. Note that the isomorphism in Lemma 2.3 is induced by the chain isomorphism Ψ^{n} : C^{n}(ℤG, B) → C^{n}(G, L(B)) defined by f → f̃ where

f̃(g_{1}, …, g_{n}) = f(g_{1}, …, g_{n})g_{n}^{−1} ⋯ g_{1}^{−1}.

So, the isomorphism given in Theorem 2.1 is induced by the chain map

ΨΦ : C_{X}^{n}(G, M) → C^{n}(G, Map(X, M))

defined by f → \overline{f} where

(\overline{f})(g_{1}, …, g_{n})(x) = f(g_{1}, …, g_{n}; g_{n}^{−1} ⋯ g_{1}^{−1}x).

One can also prove Theorem 2.1 using this chain map.

Note that as a ℤG-module

Map(X, M) ≅ Hom_{ℤ}(ℤX, M)

where ℤX denotes the permutation module with basis given by X. When X is equal to G/H = {gH | g ∈ G}, then we have

Map(G/H, M) ≅ Hom_{ℤ}(ℤ[G/H], M)

as ℤG-modules.

Lemma 2.5. Let G be a finite group, H ≤ G a subgroup, and M a ℤG-module. Then there is an isomorphism of ℤG-modules

σ : Hom_{ℤ}(ℤ[G/H], M) → Hom_{ℤH}(ℤG, M)

defined by σ : f → \overline{f} where \overline{f}(g) = gf(g^{−1}H).

Proof. It is easy to check that f → \overline{f} is a ℤG-module homomorphism and its inverse can be defined by f → f̃ where f̃(gH) = gf(g^{−1}). Note that the (left) G-action on Hom_{ℤH}(ℤG, M) is given by [g · f](g′) = f(g′g).

We conclude this section with the following corollary.

Corollary 2.6. Suppose that G is a finite group, H ≤ G a subgroup, and M a ℤG-module. Then, we have

H_{G/H}^{∗}(G, M) ≅ H^{∗}(H, M).

Proof. By Theorem 2.1 and Lemma 2.5, we have

H_{G/H}^{n}(G, M) ≅ H^{n}(G, Map(G/H, M)) ≅ H^{n}(G, Hom_{ℤH}(ℤG, M)).

Using the Eckmann-Shapiro isomorphism

H^{n}(G, Hom_{ℤH}(ℤG, M)) ≅ H^{n}(H, M)

we conclude that H_{G/H}^{n}(G, M) ≅ H^{n}(H, M).

3 Functorial properties of H_{X}^{∗}(G, M)

In this section we will be listing the basic functorial properties of the cohomology of groups associated to a G-set. We will see later that the assignment X ↦ H_{X}^{n}(G, M) is just the description of the cohomology of groups Mackey functor for G as a bifunctor from G-sets to abelian groups. But note that all our constructions are done explicitly on the chain level.

Throughout this section we assume that G is a fixed finite group and M is a fixed ℤG-module. We also fix a non-negative integer n, and consider the cohomology groups H_{X}^{n}(G, M) of degree n for various G-sets.

Let X and Y be two G-sets and let f : X → Y be a G-set map. There are two ways to obtain maps between H_{X}^{n}(G, M) and H_{Y}^{n}(G, M) associated to f. Given an n-cocycle γ : G^{n} × Y → M, we define f^{∗}(γ) : G^{n} × X → M by

f^{∗}(γ)(g_{1}, …, g_{n}; x) = γ(g_{1}, …, g_{n}; f(x))

for g_{1}, …, g_{n} ∈ G and x ∈ X. Since f is a G-map, this defines a chain map f^{∗} : C_{Y} (G, M) → C_{X}(G, M), hence a group homomorphism

f^{∗} : H_{Y}^{n}(G, M) → H_{X}^{n}(G, M).

For the other direction, let α : G^{n}×X → M be an n-cocycle, we define f_{∗}(α) : G^{n}×Y → M by

f_{∗}(α)(g_{1}, …, g_{n}; y) = ∑_{x∈f^{−1}(y)} α(g_{1}, …, g_{n}; x)

for g_{1}, …, g_{n} ∈ G and y ∈ Y. It is easy to check that this defines a chain map f_{∗} : C_{X}(G, M) → C_{Y} (G, M), hence a group homomorphism

f_{∗} : H_{X}^{n}(G, M) → H_{Y}^{n}(G, M).

Lemma 3.1. For every pullback diagram of G-sets

X_{1} ⟶^{f_{1}} X_{2}
↓^{f_{2}}  ↓^{f_{3}}
X_{3} ⟶^{f_{4}} X_{4}

we have (f_{4})^{∗}(f_{3})_{∗} = (f_{2})_{∗}(f_{1})^{∗}.

Proof. Let α : G^{n} × X_{2} → M be a cocycle. For g_{1}, …, g_{n} ∈ G and x ∈ X_{3}, we have

(f_{4})^{∗}(f_{3})_{∗}(α)(g_{1}, …, g_{n}; x) = ∑_{z∈X_{2}, f_{3}(z)=f_{4}(x)} α(g_{1}, …, g_{n}; z)

whereas

(f_{2})_{∗}(f_{1})^{∗}(α)(g_{1}, …, g_{n}; x) = ∑_{y∈X_{1}, f_{2}(y)=x} α(g_{1}, …, g_{n}; f_{1}(y)).

The equality of these sums follows from the pullback condition.

We also note that H_{X}^{n}(G, M) decomposes as an abelian group if X decomposes as a G-set. We state this fact as follows:

Lemma 3.2. For every pair of finite G-sets X and Y and inclusion maps X ⟶^{i_{X}} X ⨿ Y and Y ⟶^{i_{Y}} X ⨿ Y, the homomorphism

Φ : H_{X}^{n}(G, M) ⊕ H_{Y}^{n}(G, M) → H_{X⨿Y}^{n} (G, M)

defined by Φ(u, v) = (i_{X})_{∗}(u) + (i_{Y})_{∗}(v) is an isomorphism.

Proof. It is easy to check that the homomorphism

H_{X⨿Y}^{n} (G, M) → H_{X}^{n}(G, M) ⊕ H_{Y}^{n}(G, M)

defined by w → ((i_{X})^{∗}(w), (i_{Y})^{∗}(w)) is the inverse of Φ.

The above two lemmas show that the assignment X → H_{X}^{n}(G, M) together with ( )^{∗} and ( )_{∗} defines a Mackey functor in the sense described in page 5 of “A Guide to Mackey Functors” by Peter Webb [11]. Let us denote this Mackey functor by H_{?}^{n}(G, M). We will show later that the Mackey functor H_{?}^{n}(G, M) is equivalent to the cohomology of groups Mackey functor H^{n}(?, M) via the isomorphism

H_{G/H}^{n}(G, M) ≅ H^{n}(H, M)

given in Corollary 2.6.

For a Mackey functor defined as a functor from G-sets to abelian groups, there is a standard way to obtain restriction, induction, and conjugation maps. If we apply these definitions, we obtain the following maps:

Let K ≤ H ≤ G and g ∈ G, consider the G-maps f_{H,K} : G/K → G/H defined by xK → xH and f_{H,g} : G/H → G/^{g}H defined by xH → xg^{−1}^{g}H. Then, the induced maps

r_{H,K} : H_{G/H}^{n}(G, M) ⟶^{(f_{H,K})_{∗}} H_{G/K}^{n}(G, M),

i_{H,K} : H_{G/K}^{n}(G, M) ⟶^{(f_{H,K})^{∗}} H_{G/H}^{n}(G, M),

c_{H,g} : H_{G/H}^{n}(G, M) ⟶^{(f_{H,g})_{∗}} H_{G/^{g}H}^{n}(G, M)

are the restriction, induction, and conjugation maps for H_{?}^{n}(G, M). By the equivalence of the different definitions for Mackey functors, we can also consider H_{?}^{n}(G, M) as a Mackey functor H → H_{G/H}^{n}(G, M) with the above restriction, induction, and conjugation maps.

Theorem 3.3. The Mackey functor H_{?}^{n}(G, M) is equivalent to the cohomology of groups Mackey functor H^{n}(?, M).

Proof. For every H ≤ G, we have an isomorphism

H_{G/H}^{n}(G, M) ≅ H^{n}(H, M)

by Corollary 2.6. We just need to show that this isomorphism commutes with restriction, induction, and conjugation maps. This follows from the following lemma.

Lemma 3.4. Let K ≤ H ≤ G, and g ∈ G. Then, the induced maps

res_{K}^{H} : H^{n}(H, M) ≅ H_{G/H}^{n}(G, M) ⟶^{r_{H,K}} H_{G/K}^{n}(G, M) ≅ H^{n}(K, M),

tr_{K}^{H} : H^{n}(K, M) ≅ H_{G/K}^{n}(G, M) ⟶^{i_{H,K}} H_{G/H}^{n}(G, M) ≅ H^{n}(H, M),

c_{g}^{H} : H^{n}(H, M) ≅ H_{G/H}^{n}(G, M) ⟶^{c_{H,g}} H_{G/^{g}H}^{n}(G, M) ≅ H^{n}(^{g}H, M)

are the usual restriction, transfer, and conjugation maps in group cohomology.

Proof. First let us consider the restriction map. We have

H_{G/H}^{n}(G, M)  ⟶^{f^{∗}}  H_{G/K}^{n}(G, M)
≅↓_{Thm 2.1}  ≅↓_{Thm 2.1}
H^{n}(G, Map(G/H, M)) ⟶^{f^{∗}} H^{n}(G, Map(G/K, M))
≅↓^{σ_{∗}}  ≅↓^{σ_{∗}}
H^{n}(G, Hom_{ℤH}(ℤG, M)) ⟶^{(res_{K}^{H})^{∗}} H^{n}(G, Hom_{ℤK}(ℤG, M))
≅↓^{Eck−Shap}  ≅↓^{Eck−Shap}
H^{n}(H, M)  ⟶^{res_{K}^{H}}  H^{n}(K, M)

where the vertical composition is the isomorphism given in Corollary 2.6, and (res_{K}^{H})^{∗} is the homomorphism induced from

res_{K}^{H} : Hom_{ℤH}(ℤG, M) → Hom_{ℤK}(ℤG, M)

which is defined by mapping a ℤH-homomorphism f : ℤG → M to itself considered as a ℤK-homomorphism. The first diagram commutes for obvious reasons, and the third diagram commutes by standard results in homological algebra. One can show that the second diagram commutes by showing that the corresponding diagram for modules

Map(G/H, M) ⟶^{f^{∗}} Map(G/K, M)
≅↓^{σ}  ≅↓^{σ}
Hom_{ℤH}(ℤG, M) ⟶^{(res_{K}^{H})^{∗}} Hom_{ℤK}(ℤG, M)

commutes. This can be done by a direct calculation as follows: Let φ : G/H → M be a function. Then,

[σf^{∗}](φ)(x) = xf^{∗}(φ)(x^{−1}K) = xφ(x^{−1}H) = [(res_{K}^{H})^{∗}σ](φ)(x).

This completes the proof of Lemma 3.4 for the restriction map.

For transfer and conjugation, the arguments are similar. For each of these we will replace the second commuting diagram with an appropriate one and show that they commute on the module level. For the transfer map, we need to check the commutativity of the following diagram

Map(G/K, M) ⟶^{f_{∗}} Map(G/H, M)
≅↓^{σ}  ≅↓^{σ}
Hom_{ℤK}(ℤG, M) ⟶^{tr_{K}^{H}} Hom_{ℤH}(ℤG, M)

where tr_{K}^{H}(ψ)(g) = ∑_{hK∈H/K} hψ(h^{−1}g) for every ℤH-module ψ : ℤG → M. For every φ : G/K → M, we have

[tr_{K}^{H}σ(φ)](g) = ∑_{hK∈H/K} hσ(φ)(h^{−1}g) = ∑_{hK∈H/K} gφ(g^{−1}hK) = [σf_{∗}(φ)](g),

so the diagram commutes.

For conjugation, we have the following diagram

Map(G/H, M) ⟶^{f_{∗}} Map(G/^{g}H, M)
≅↓^{σ}  ≅↓^{σ}
Hom_{ℤH}(ℤG, M) ⟶^{c_{g}^{H}} Hom_{ℤ^{g}H}(ℤG, M)

where c_{g}^{H}(ψ)(x) = gψ(g^{−1}x) for every ℤH-module ψ : ℤG → M. Take φ in Map(G/H, M). We have

[c_{g}^{H}σ](φ)(x) = gσ(φ)(g^{−1}) = gg^{−1}xφ(x^{−1}gH) = xφ(x^{−1}gH)

and

[σ(f_{H,g})_{∗}](φ)(x) = x(f_{H,g})_{∗}(φ)(x^{−1}^{g}H) = xφ(x^{−1}gH).

So, the diagram commutes. This completes the proof of Lemma 3.4.

The cohomology of groups functor H^{n}(?, M) for a group G is known to be a cohomological functor, i.e., it satisfies tr_{K}^{H}res_{K}^{H}u = |H : K|u for every K ≤ H ≤ G and u ∈ H^{n}(H, M). By the above equivalence, the functor H_{?}^{n}(G, M) is also cohomological. One can also see this directly from the definitions of i_{H,K} and r_{H,K}.

In the next section, we will consider the classification problem for A-free A × Gsets where A is an abelian group. The groups H_{X}^{1}(G, A) will appear naturally in this classification.

4 A-free A × G-sets

Let G be a finite group and A be an abelian group. Throughout this section we assume that G acts trivially on A, although most of the results in this section still hold for a non-trivial ℤG-module M when the group A × G is replaced by the semidirect product M ×_{φ} G. We do not state our results in this generality since the case of trivial G-action is sufficient for all the applications we know.

Given an A × G-set Γ with a free A-action, let X = Γ/A denote the quotient of Γ by the A-action. This gives a fibration π : Γ → X with fibers isomorphic to A and base space X which is a G-set.

Observe that there is always a bijection Γ ≅ A × X, but in general it is not an isomorphism of A × G-sets, where A × X is considered as an A × G-set by the product action

(a, g)(a′, x) = (a + a′, gx).

In other words, there is always a set theoretical splitting s : X → Γ, but s is not a map of A × G-sets in general when X is considered as an A × G-set through the projection A × G → G.

Our first result in this section classifies the fibrations π : Γ → X over a fixed G-set X up to isomorphism. We say the fibrations π_{1} : Γ_{1} → X and π_{2} : Γ_{2} → X are isomorphic if there is an A × G-map F : Γ_{1} → Γ_{2} such that π_{1} = π_{2}F.

Proposition 4.1. Suppose G is a finite group, X is a G-set, and A is an abelian group with trivial G-action. There is a one-to-one correspondence between isomorphism classes of fibrations Γ → X with fibre group A and the cohomology classes in H_{X}^{1}(G, A).

Proof. We will first show that given a fibration π : Γ → X, there is a unique cohomology class in H_{X}^{1}(G, A) associated to it. Let s : X → Γ be a set theoretical section. We define α : G × X → A by

(0, g)s(x) = α(g, x)s(gx).

The identity

((0, g_{1})(0, g_{2})) · γ = (0, g_{1}) · ((0, g_{2}) · γ)

gives the derivation condition

α(g_{1}g_{2}; x) = g_{1}α(g_{2}; x) + α(g_{1}; g_{2}x).

So, α : G × X → A is a 1-cocycle of the chain complex (C_{X}^{∗}(G, A), δ) described earlier.

Let s_{1} and s_{2} be two different splittings. Since πs_{1} = πs_{2}, there exists a function t : X → A, such that s_{2}(x) = t(x) · s_{1}(x) for all x ∈ X. Let α_{1} and α_{2} be derivations associated to the sections s_{1} and s_{2} respectively. Then, an easy calculation shows that

α_{2}(g; x) − α_{1}(g; x) = t(gx) − gt(x) = (δt)(g; x).

So, the cohomology class [α] ∈ H_{X}^{∗}(G, A) does not depend on the choice of the section.

Conversely, given a cohomology class [α] ∈ H_{X}^{1}(G, A) represented by a derivation α : G × X → A, we define the A × G-set Γ as the set A × X with the action given by

(a, g) · (a′, x) = (a + a′ + α(g, x), gx)

for a, a′ ∈ A, g ∈ G and x ∈ X. It is easy to verify that this defines an action by using the fact that α is a derivation. Note that if we choose another representative, α′ = δt + α, then we obtain Γ′ = A × X with the action given by

(a, g) · (a′, x) = (a + a′ + α′(g; x), gx) = (a + a′ + α(g; x) + gt(x) − t(gx), gx).

Then, the map F : Γ → Γ′ defined by (a, g) → (a+t(x), x) is an isomorphism of fibrations.

Finally, we need to show that two isomorphic fibrations give the same cohomology class. For this, observe that we can fix splittings for both Γ_{1} → X and Γ_{2} → X and assume that Γ_{i} = A × X with action given by

(a, g) · (a′, x) = (a + a′ + α_{i}(g; x), gx)

for i = 1, 2. Let F : Γ_{1} → Γ_{2} be a isomorphism of fibrations, then there exists a function t : X → A such that F (a, x) = (a + t(x), x) for all x ∈ X. Note that F is a A × G-set map if and only if

α_{1}(g; x) − α_{2}(g; x) = t(x) − t(gx) = (δt)(g; x).

This completes the proof of the proposition.

Next, we will classify all A-free A × G-sets up to isomorphism. Let Γ be an A-free A × G-set, then taking the orbit space of the A-action as before, we obtain a G-set X and a fibration π : Γ → X with fibers isomorphic to A. By fixing a set theoretical splitting, we can assume Γ = A × G and the action is given by

(a, g) · (a′, x) = (a + a′ + α(g; x), gx)

for some derivation α : G × X → A. So, associated to an A-free A × G-set Γ, there is a G-set X and a cohomology class u = [α] ∈ H_{X}^{1}(G, A).

Proposition 4.2. Let Γ_{1} and Γ_{2} be two A-free A × G-sets with corresponding G-sets X_{1} and X_{2} and cohomology classes u_{1} ∈ H_{X_{1}}^{1}(G, A) and u_{2} ∈ H_{X_{2}}^{1}(G, A). Then, Γ_{1} and Γ_{2} are isomorphic as A × G-sets if and only if there is a G-set isomorphism f : X_{1} → X_{2} such that f^{∗}(u_{2}) = u_{1}.

Proof. Let F : Γ_{1} → Γ_{2} be an A × G-set isomorphism. Passing to the orbit spaces, we obtain a G-set isomorphism f : X_{1} → X_{2} such that the diagram

Γ_{1} ⟶^{F} Γ_{2}
↓^{π_{1}}  ↓^{π_{2}}
X_{1} ⟶^{f} X_{2}

commutes. Choosing set theoretical splittings for both Γ_{1} and Γ_{2}, we can assume that Γ_{i} = A × G with the actions given by

(a, g) · (a′, x) = (a + a′ + α_{i}(g; x), gx)

for some derivations α_{i} : G × X_{i} → A for i = 1, 2. Since F : Γ_{1} → Γ_{2} is an A-map, we can write F (a, x) = (a + t(x), f(x)). Now, using the fact that F is an A × G-map, we obtain

α_{1}(g; x) − α_{2}(g; f(x)) = t(x) − t(gx) = (δt)(g; x)

for g ∈ G and x ∈ X. Thus f^{∗}(u_{2}) = u_{1} as desired.

Conversely, given a G-set isomorphism f : X_{1} → X_{2} such that f^{∗}(u_{2}) = u_{1}, we can pick representative derivations α_{1} and α_{2} such that f^{∗}(α_{2}) = α_{1}, and assume that Γ_{i} is an A×G-set with action determined by α_{i} for i = 1, 2. Then, the map F (a, x) = (a, f(x)) defines a A × G-set isomorphism.

Let X be a G-set, and denote by Aut_{G}(X) the group of all G-set isomorphisms f : X → X. For each f ∈ Aut_{G}(X), we have an isomorphism f^{∗} : H_{X}^{1}(G, A) → H_{X}^{1}(G, A). So, Aut_{G}(X) acts on H_{X}^{1}(G, A) as group automorphisms. We have the following:

Corollary 4.3. Suppose that G is a finite group and A is an abelian group with trivial G-action. Then, the isomorphism classes of A-free A × G-sets are in one-to-one correspondence with the pairs ([X], [α]) where [X] runs through the isomorphism classes of G-sets and [α] is a representative of a class in H_{X}^{1}(G, A) under the Aut_{G}(X)-action.

Proof. By Proposition 4.2, the isomorphism classes of A-free A × G-sets are in one-to-one correspondence with the equivalence classes of pairs (X, u), where we say that (X, u) is equivalent to (Y, v) if there is a G-set isomorphism f : X → Y such that f^{∗}(v) = u. Note that the set of equivalence classes of pairs (X, u) is the same as the set given in the Corollary.

The category of A-free A × G-sets admits two operations, called direct sum and direct product over A, which induce well defined addition and multiplication operations on the isomorphism classes of A-free A × G-sets. The Grothendieck ring of isomorphism classes of A-free A × G-sets with these operations is called the monomial Burnside ring over A. Below we briefly describe how direct sum and direct product over A are defined. For more details on these operations and on the monomial Burnside ring, we refer the reader to Dress [6], Boltje [3], and Barker [1].

We assume that all the A-free A × G-sets we consider have a fixed splitting, and hence they are sets of the form A × X with A × G-action given by

(a, g) · (a′, x) = (a + a′ + α(g; x), gx)

for some derivation α : G × X → A. We denote such an A-free A × G-set briefly by A_{α}X. Addition and multiplication of two A-free A × G-sets A_{α}X and A_{β}Y are defined as follows:

A_{α}X ⊕_{A} A_{α}Y = A_{α}X ⨿ A_{β}Y

and

A_{α}X ⊗_{A} A_{β}Y = A_{α}X × A_{β}Y/{ aζ ⊗ η − ζ ⊗ aη | a ∈ A, ζ ∈ A_{α}X, η ∈ A_{β}Y }.

The A × G-action in the first case is defined in the obvious way, and in the second case by diagonal action. One can easily show that these give well defined addition and multiplication on the isomorphism classes of A-free A × G-sets.

To see the effect of these operations on the derivations, observe that

A_{α}X ⊕_{A} A_{α}Y = A_{θ}(X ⨿ Y)

and

A_{α}X ⊗_{A} A_{β}Y = A_{γ}(X × Y)

for some θ : G × (X ⨿ Y) → A and γ : G × (X × Y) → A. One can describe θ and γ in terms of α and β as follows:

θ(g; z) = \begin{cases}α(g; z) if z ∈ X \\ β(g; z) if z ∈ Y \end{cases}

and

γ(g; (x, y)) = α(g; x) + β(g; y).

In fact, one can verify that these define direct sum and tensor product on the onedimensional cohomology classes. We do not give details of this here, since this will be done in the next section in greater generality.

Motivated by this example, in the next section we define cohomological Burnside rings B^{n}(G, M) for each n ≥ 0. In the case n = 1 and M is an abelian group A with trivial G-action, we obtain that B^{1}(G, A) is equal to the monomial Burnside ring over A.

5 Cohomological Burnside ring

Throughout this section G is a finite group, M is a ℤG-module, and n is a non-negative integer.

Definition 5.1. A pair of the form (X, u), where X is a G-set and u is a class in H_{X}^{n}(G, M), is called a positioned G-set of degree n with coefficients in M, or shortly a positioned G-set, when degree and coefficients are well understood. A map f : (X, u) → (Y, v) is called a positioned G-set map if f : X → Y is a G-set map such that f^{∗}(v) = u.

We say that two positioned G-sets (X, u) and (Y, v) are isomorphic if there is a positioned G-set map f : (X, u) → (Y, v) such that f : X → Y is an isomorphism of G-sets. We denote the isomorphism class of a positioned G-set (X, u) simply by [X, u].

The set of isomorphism classes of positioned G-sets is a semi-ring with addition and multiplication defined as follows: Given two positioned G-sets (X, u) and (Y, v), we define the cohomology class

u ⊕ v ∈ H_{X⨿Y}^{n} (G, M) by u ⊕ v = (i_{X})_{∗}(u) + (i_{Y})_{∗}(v)

where i_{X} : X → X ⨿ Y and i_{Y} : Y → X ⨿ Y are the usual inclusion maps of X and Y. Note that if u = [α] and v = [γ], then u ⊕ v = [θ] where

θ(g_{1}, …, g_{n}; z) = \begin{cases}α(g_{1}, …, g_{n}; z) if z ∈ X \\ γ(g_{1}, …, g_{n}; z) if z ∈ Y. \end{cases}

If f_{X} : (X, u) → (X′, u′) and f_{Y} : (Y, v) → (Y′, v′) are two positioned G-set isomorphisms, then

f_{X} ⨿ f_{Y} : (X ⨿ Y, u ⊕ v) → (X′ ⨿ Y′, u′ ⊕ v′)

is a positioned G-set isomorphism. To see this consider the following diagram

X ⟵^{i_{X}} X ⨿ Y ⟶^{i_{Y}} Y
f_{X} ↓  ^{f_{X}⨿f_{Y}} ↓  ^{f_{Y}} ↓
X′ ⟵^{i_{X′}} X′ ⨿ Y′ ⟶^{i_{Y′}} Y′

where both of the diagrams are pullback diagrams. By Lemma 3.1, we have

(f_{X} ⨿ f_{Y})^{∗}(u′ ⊕ v′) = (f_{X} ⨿ f_{Y})^{∗}((i_{X′})_{∗}(u′) + (i_{Y′})_{∗}(v′))
= (i_{X})_{∗}(f_{X})^{∗}(u′) + (i_{Y})_{∗}(f_{Y})^{∗}(v′)
= (i_{X})_{∗}(u) + (i_{Y})_{∗}(v)
= u ⊕ v.

This shows that

[X, u] + [Y, v] = [X ⨿ Y, u ⊕ v]

gives a well defined addition on isomorphism classes.

To define the product of two positioned G-sets (X, u) and (Y, v), we first define the cohomology class

u ⊗ v ∈ H_{X×Y}^{n} (G, M) by u ⊗ v = (π_{X})^{∗}(u) + (π_{Y})^{∗}(v)

where π_{X} : X × Y → X and π_{Y} : X × Y → Y are the projection maps. We can describe u ⊗ v on the chain level as follows: Let u = [α] and v = [γ], then u ⊗ v = [θ] where

θ(g_{1}, …, g_{n}; (x, y)) = α(g_{1}, …, g_{n}; x) + γ(g_{1}, …, g_{n}; y).

If f_{X} : (X, u) → (X′, u′) and f_{Y} : (Y, v) → (Y′, v′) are two positioned G-set isomorphisms, then

f_{X} × f_{Y} : (X × Y, u ⊗ v) → (X′ × Y′, u′ ⊗ v′)

is a positioned G-set isomorphism. One can verify this by forming appropriate commuting diagrams as in the case of addition. It follows that the formula

[X, u] · [Y, v] = [X × Y, u ⊗ v]

defines a multiplication on isomorphism classes of positioned G-sets.

Definition 5.2. The cohomological Burnside ring B^{n}(G, M) of degree n of the group G with coefficients in M is defined as the Grothendieck ring of the semi-ring of isomorphism classes of positioned G-sets (of degree n with coefficients in M) where addition and multiplication are defined by

[X, u] + [Y, v] = [X ⨿ Y, u ⊕ v]
[X, u] · [Y, v] = [X × Y, u ⊗ v].

We have the following proposition, which is immediate from the discussion in Section 4:

Proposition 5.3. If A is an abelian group with trivial G-action, then B^{1}(G, A) is the same as the monomial Burnside ring over A.

Note that for n = 0, the definition given in 5.2 extends to non-abelian coefficients, even to a monoid, so in this case we can take M as a G-monoid (that is, M is a monoid with G acting on it as monoid automorphisms). We observe the following:

Proposition 5.4. If M is a G-monoid, then B^{0}(G, M) is isomorphic to the crossed Burnside ring over M.

Proof. Recall from [8] the definition of the crossed Burnside ring B^{c}(G, M). It is the Grothendieck ring of the category of crossed G-sets over the G-monoid M. A crossed G-set over M is a pair (X, α), with X a G-set and α : X → M a G-map. A morphism of crossed G-sets f : (X, α) → (Y, β) is a G-set map f : X → Y such that α = β ∘ f. Direct sums and tensor products are defined in the same way as we defined them for positioned G-sets. So it it clear that B^{c}(G, M) is the same as B^{0}(G, M).

More details about crossed Burnside rings can be found at Bouc [4] and Yoshida-Oda [8]. For the construction of the Mackey functor B^{c}(−, M) through the Dress construction see Bouc [5] and Yoshida-Oda [9].

Later in Section 8, we will give an interpretation of B^{2}(G, M) in terms of twisted group rings when M = k^{×} is the unit group of a commutative ring k.

In the rest of this section, we calculate the dimension of B^{n}(G, M) as a free abelian group. We have the following.

Lemma 5.5. Let Cl(G) denote the set of conjugacy classes of subgroups of G, and let Aut_{G}(X) denote the group of G-set isomorphisms of X. For each H ≤ G, let U_{H}(G, M) denote a set of orbit representatives for the elements in H_{G/H}^{n}(G, M) under the Aut_{G}(G/H)-action. Then,

B = {[G/H, u] | H ∈ Cl(G), u ∈ U_{H}(G, M)}

is a basis for B^{n}(G, M).

Proof. Given a positioned G-set (X, u) such that X = X_{1} ⨿ X_{2}, then there exist u_{i} ∈ H_{X_{i}}^{n}(G, M) for i = 1, 2 such that (X, u) = (X_{1}, u_{1}) + (X_{2}, u_{2}). In fact, for each i = 1, 2, we can take u_{i} as (i_{X_{i}})^{∗}(u). So, each element in B^{n}(G, M) can be written as a linear combination of [X, u]’s with X a transitive G-set. From standard G-set theory, every transitive G-set is isomorphic to a left coset G-set G/H for some H ≤ G. Moreover, two such G-sets G/H and G/K are isomorphic if and only if H and K are conjugate in G. So, every element x ∈ B^{n}(G, M) can be written as

x = ∑_{[H]∈Cl(G)} ∑_{u∈U_{H}(G,M)} a_{H,u} · [G/H, u]

where U_{H} is the set of representatives of u ∈ H_{G/H}^{n}(G, M) under the equivalence relation u_{1} ∼ u_{2} if and only if [G/H, u_{1}] = [G/H, u_{2}]. Note that u_{1} ∼ u_{2} if and only if there is an G-set isomorphism f : G/H → G/H such that u_{1} = f^{∗}(u_{2}). Hence U is a set of representatives for the elements in H_{G/H}^{n}(G, M) under the Aut_{G}(G/H)-action.

The proof of Lemma 5.5 suggests that

B^{n}(G, M) ≅ ⨁_{[H]∈Cl(G)}(ℤH_{G/H}^{n}(G, M))_{Aut_{G}(G/H)}

where (ℤH_{G/H}^{n}(G, M))_{Aut_{G}(G/H)} denotes the coinvariants of ℤH_{G/H}^{n}(G, M) as a ℤAut_{G}(G/H)- module. A specific homomorphism Ψ inducing the above isomorphism can be given as follows: Let [G/H, u] be a basis element in B^{n}(G, M). Then, we define Ψ([G/H, u]) as a Cl(G)-tuple whose [H]-th coordinate is the image of u ∈ ℤH_{G/H}^{n}(G, M) under the coinvariants epimorphism

ℤH_{G/H}^{n}(G, M) → (ℤH_{G/H}^{n}(G, M))_{Aut_{G}(G/H)}.

To distinguish linear combinations from the addition of cohomology classes, we write H_{G/H}^{n}(G, M) multiplicatively in the group ring ℤH_{G/H}^{n}(G, M). Note that with this convention, then we have

Ψ([G/H ⨿ G/H, u_{1} ⊕ u_{2}])_{[H]} = u_{1} + u_{2}

whereas

Ψ([G/H, u_{1} + u_{2}])_{[H]} = u_{1}u_{2}.

Recall that, as a group, Aut_{G}(G/H) is isomorphic to N_{G}(H)/H where the isomorphism is given by the map N_{G}(H)/H → Aut_{G}(G/H) defined by n → f_{n}, where f_{n}(xH) = xn^{−1}H. We denote N_{G}(H)/H by W_{G}(H) and call it the Weyl group of H ≤ G. We can consider H_{G/H}^{n}(G, M) as a W_{G}(H)-set via this isomorphism.

Note that H^{n}(H, M) is a W_{G}(H)-set where the action of W_{G}(H) is induced by the conjugation homomorphism c_{g}^{H} : H^{n}(H, M) → H^{n}(H, M). We have the following:

Lemma 5.6. The isomorphism H_{G/H}^{n}(G, M) ≅ H^{n}(H, M) given in Corollary 2.6 is an isomorphism of W_{G}(H)-sets.

Proof. Recall that c_{g}^{H} : H_{G/H}^{n}(G, M) → H_{G/^{g}H}^{n}(G, M) is defined as (f_{H,g})_{∗} where f_{H,g} : G/H → G/^{g}H is given by f_{H,g}(xH) = xg^{−1}(^{g}H). So, the action of W_{G}(H) on H_{G/H}^{n}(G, M) induced by the isomorphism N_{G}(H)/H → Aut_{G}(G/H) is the same as the W_{G}(H)-action on H_{G/H}^{n}(G, M) via the conjugation homomorphism c_{H,g} : H_{G/H}^{n}(G, M) → H_{G/H}^{n}(G, M). Since the isomorphism in Corollary 2.6 commutes with the conjugation homomorphisms, it also commutes with the W_{G}(H)-action.

We conclude the following:

B^{n}(G, M) ≅ ⨁_{[H]∈Cl(G)}(ℤH^{n}(H, M))_{W_{G}(H)}

where the W_{G}(H)-action on H^{n}(H, M) is the usual one, induced by conjugation.

6 The mark homomorphism

In this section, we will define the ghost ring and describe the mark homomorphism for cohomological Burnside rings.

Definition 6.1. The ghost ring of the cohomological Burnside ring is defined as

β^{n}(G, M) ≅ ⨁_{[H]∈Cl(G)}(ℤH^{n}(H, M))^{W_{G}(H)}

where W_{G}(H) acts on H^{n}(H, M) by conjugation.

Note that an alternative description for β^{n}(G, M) can be given by using super class functions. Recall that a super class function for G is a function from the set of subgroups of G to the integers ℤ which is constant on the conjugacy classes of subgroups. We generalize this definition as follows.

Definition 6.2. We call a function

f : {H | H ≤ G} → ⨁_{H≤G} ℤH^{n}(H, M)

a cohomological super class function if it satisfies the following two conditions:

(i) The K-th coordinate of f(H) is zero if K ≠ H, and (ii) f(^{g}H) = c_{g}^{H}(f(H)) for all H ≤ G, and g ∈ G.

To describe the mark homomorphism for cohomological Burnside rings, we first define a family of ring homomorphisms

s_{H} : B^{n}(G, M) → ℤH^{n}(H, M)

for every H ≤ G.

Let (X, u) be a positioned G-set, where u ∈ H_{X}^{n}(G, M). Let α : G^{n} × X → M be a cocycle representing u. For each x ∈ X^{H}, the map α_{x} : H^{n} → M defined by

α_{x}(h_{1}, …, h_{n}) = α(h_{1}, …, h_{n}; x)

is a cocycle, giving a cohomology class [α_{x}] ∈ H^{n}(H, M).

Lemma 6.3. [α_{x}] does not depend on the choice of α as a representing cocycle for u.

Proof. Suppose [α] = [β] = u ∈ H_{X}^{n}(G, M). Then, β = α + δλ for some map λ : G^{n−1} × X → M. For any x ∈ X^{H} we have β_{x} = α_{x} + (δλ)_{x} = α_{x} + δ(λ_{x}), where λ_{x} : H^{n−1} → M is defined by λ_{x}(h_{1}, …, h_{n−1}) = λ(h_{1}, …, h_{n−1}; x). Hence [α_{x}] = [β_{x}].

The above lemma shows that we can denote [α_{x}] simply by u_{x} since it only depends on u. We define s_{H}([X, u]) ∈ ℤH^{n}(H, M) as the linear combination

s_{H}([X, u]) = ∑_{x∈X^{H}} u_{x}.

Lemma 6.4. The map s_{H} is a ring homomorphism.

Proof. Let [X, u], [Y, v] ∈ B^{n}(G, M). By definition,

s_{H}([X, u] · [Y, v]) = s_{H}([X × Y, u ⊗ v]) = ∑_{(x,y)∈(X×Y)^{H}}(u ⊗ v)_{(x,y)},

and

s_{H}([X, u]) · s_{H}([Y, v]) = ( ∑_{x∈X^{H}} u_{x}) ( ∑_{y∈Y^{H}} v_{y}) = ∑_{(x,y)∈(X×Y)^{H}} u_{x} ⊗ v_{y}.

Let u = [α] and v = [γ], then

(α ⊗ γ)_{(x,y)}(h_{1}, …, h_{n}) = α(h_{1}, …, h_{n}; x) + γ(h_{1}, …, h_{n}; y) = (α_{x} ⊗ γ_{y})(h_{1}, …, h_{n}).

Hence u_{x} ⊗ v_{y} = (u ⊗ v)_{(x,y)}, concluding the proof.

We also have the following:

Lemma 6.5. For every H ≤ G and g ∈ G, and for [X, u] ∈ B^{n}(G, M), we have

s_{^{g}H}([X, u]) = c_{g}^{H}(s_{H}([X, u])).

Proof. Recall that there is a bijection between X^{H} and X^{^{g}H} given by x → gx. So,

s_{^{g}H}([X, u]) = ∑_{y∈X^{^{g}H}} u_{y} = ∑_{x∈X^{H}} u_{gx}.

It is easy to see that u_{gx} ∈ H^{n}(^{g}H, M) is the same as c_{g}^{H}(u_{x}).

Now, we are ready to define the mark homomorphism.

Definition 6.6. The mark homomorphism

φ : B^{n}(G, M) → β^{n}(G, M)

is defined as the group homomorphism which takes [X, u] to a cohomological class function f_{[X,u]} where

f_{[X,u]}(H) = s_{H}([X, u]) ∈ ℤH^{n}(H, M).

Note that the mark homomorphism is a ring homomorphism, since s_{H} is a ring homomorphism for all H ≤ G.

7 The fundamental theorem for B^{n}(G, M)

The main purpose of this section is to prove the fundamental theorem for the cohomological Burnside rings. By the fundamental theorem we mean a theorem which explains the kernel and cokernel of the mark homomorphism of the cohomological Burnside ring. We first show that the mark homomorphism is injective.

Lemma 7.1. The mark homomorphism

φ : B^{n}(G, M) → β^{n}(G, M)

is an injective ring homomorphism.

Proof. Note that

s_{K}([G/H, u]) = ∑_{gH∈(G/H)^{K}} u_{gH} = ∑_{gH∈G/H, K≤^{g}H} res_{K}^{^{g}H}c_{g}^{H}u

is zero if K is not conjugate to a subgroup of H. We can write the mark homomorphism φ as a family of homomorphisms {φ_{K,H} | [K], [H] ∈ Cl(G) } where

φ_{K,H} : (ℤH^{n}(H, M))_{W_{G}(H)} → (ℤH^{n}(K, M))^{W_{G}(K)}

is the restriction to corresponding summands. Ordering the conjugacy classes of subgroups in a way that the order respects inclusions, i.e., if K is conjugate to a subgroup of H, then [K] ≤ [H], it is easy to see from the above calculation that φ_{K,H} = 0 if [H] < [K]. Hence φ is (upper) triangular if expressed as a matrix of homomorphisms. To show that it is injective it is enough to prove that it is injective on the diagonal, i.e., we need to show

φ_{H,H} : (ℤH^{n}(H, M))_{W_{G}(H)} → (ℤH^{n}(H, M))^{W_{G}(H)}

is injective for all [H] ∈ Cl(G). For some u ∈ H^{n}(H, M), we have

φ_{H,H}(u) = s_{H}([G/H, u]) = ∑_{gH∈N_{G}(H)/H} c_{g}^{H}(u) = T r_{1}^{W_{G}(H)}u

where W_{G}(H) = N_{G}(H)/H as before, and T r_{1}^{W_{G}(H)} is the usual trace map in representation theory, defined by T r_{1}^{W} (x) = ∑_{w∈W} wx. By standard homological algebra arguments, the kernel of the trace map is equal to the (−1)-th Tate cohomology of ℤH^{n}(H, M) as a W_{G}(H)-module. Hence the kernel of φ_{H,H} is equal to Ĥ^{−1}(W_{G}(H), ℤH^{n}(H, M)) for all [H] ∈ Cl(G). Note that as a ℤW_{G}(H)-module, ℤH^{n}(H, M) is a permutation module isomorphic to

⨁_{u∈H^{n}(H,M)/W_{G}(H)} ℤ ↑_{W_{G}(H,u)}^{W_{G}(H)}

where u runs through a set of orbit representatives of the W_{G}(H)-action on the set H^{n}(G, H) and W_{G}(H, u) denotes the stabilizer of u in W_{G}(H). So, we have

\begin{aligned}Ĥ^{−1}(W_{G}(H), ℤH^{n}(H, M)) & ≅ ⨁_{u} Ĥ^{−1}(W_{G}(H), ℤ ↑_{W_{G}(H,u)}^{W_{G}(H)}) \\ & ≅ ⨁_{u} Ĥ^{−1}(W_{G}(H, u), ℤ) = 0. \end{aligned}

The last equality follows from the fact that the trace map on the trivial module ℤ is equal to the map defined by multiplication by the order of the group, and over ℤ this is an injective map. This completes the proof of injectivity of the mark homomorphism.

In the case of Burnside rings, the mark homomorphism has the same cokernel as the diagonal homomorphism ⨁_{[H]∈Cl(G)} φ_{H,H}. In that case the cokernel is a direct sum of the form

⨁_{[H]∈Cl(G)} ℤ/|W_{G}(H)|ℤ

which is equal to the 0-th Tate cohomology group Ĥ^{0}(W_{G}(H), ℤ). This group is usually denoted by Obs(G) since it is the group of obstructions for an element in the ghost ring to come from an element in the Burnside ring. Analogous to the usual Burnside ring case, we define the following :

Definition 7.2. The Obstruction group for the cohomological Burnside ring of degree n is defined as the group

Obs^{n}(G, M) = ⨁_{[H]∈Cl(G)} Ĥ^{0}(W_{G}(H), ℤH^{n}(H, M)).

Now, we will define a map

ψ : β^{n}(G, M) → Obs^{n}(G, M)

which will appear in the fundamental theorem. First we define a function η : β^{n}(G, M) → β^{n}(G, M) such that for all K ≤ G

η(f)(K) = ∑_{K≤L} μ(K, L)res_{K}^{L}f(L)

where μ(K, L) denotes the Möbius function on the poset of subgroups of G. Note that η(f) defines a cohomological super class function. To see this, observe that condition (i) in Definition 6.2 holds trivially, and condition (ii) follows from the following calculation.

η(f)(^{g}K) = ∑_{^{g}K≤L} μ(^{g}K, L)res_{^{g}K}^{L}f(L) = ∑_{K≤L^{g}} μ(K, L^{g})res_{^{g}K}^{L}f(L)

= ∑_{K≤J} μ(K, J)res_{^{g}K}^{^{g}J}f(^{g}J) = ∑_{K≤J} μ(K, J)res_{^{g}K}^{^{g}J}c_{g}^{J}f(J)

= c_{g}^{K}( ∑_{K≤J} μ(K, J)res_{K}^{J}f(J)) = c_{g}^{K}(η(f)(K)).

Note that

β^{n}(G, M) = ∑_{[H]∈Cl(G)}(ℤH^{n}(H, M))^{W_{G}(H)}.

Hence we can define a surjective map π : β^{n}(G, M) → Obs^{n}(G, M) as the projection onto the cokernel of the trace map on the each [H]-component. We define

ψ : β^{n}(G, M) → Obs^{n}(G, M)

as the composition π ∘ η. We have the following:

Theorem 7.3 (Fundamental Theorem). Let G be a finite group, let M be a ℤG-module, and let n be a non-negative integer. Then the following sequence of abelian groups is exact

0 → B^{n}(G, M) ⟶^{φ} β^{n}(G, M) ⟶^{ψ} Obs^{n}(G, M) → 0,

where φ is the mark homomorphism, and the [K]-th component of ψ is defined by

ψ_{K}(f) = ∑_{K≤L} μ(K, L)res_{K}^{L}f(L)

modulo the image of the trace map T r_{1}^{W_{G}(K)}.

Proof. We have already shown that φ is injective. Since η can be represented as an upper triangular matrix of homomorphisms with diagonal entries equal to the identity homomorphism, clearly ψ is surjective. So, the theorem follows from the following statement:

The composition η ∘ φ : B^{n}(G, M) → β^{n}(G, M) is a diagonal matrix with the diagonal entry at [H] equal to the trace map T r_{1}^{W_{G}(H)}. To prove this statement we calculate:

(η ∘ φ)([G/H, u])(K) = ∑_{K≤L} μ(K, L)res_{K}^{L}s_{L}([G/H, u])

= ∑_{K≤L} μ(K, L) ∑_{gH∈G/H, L≤^{g}H} res_{K}^{^{g}H}c_{g}^{H}(u)

= ∑_{gH∈G/H} res_{K}^{^{g}H}c_{g}^{H}(u) ∑_{K≤L≤^{g}H} μ(K, L)

= ∑_{gH∈G/H} res_{K}^{^{g}H}c_{g}^{H}(u)δ_{^{g}H,K}.

So, the sum is zero unless H and K are conjugate to each other. In the case that H and K are conjugate to each other, we can assume that H = K in the above formula by replacing K with a conjugate if necessary. Putting K = H in the last formula, we get

∑_{gH∈G/H} res_{H}^{^{g}H}c_{g}^{H}(u)δ_{^{g}H,H} = ∑_{gH∈N_{G}(H)/H} c_{g}^{H}(u) = T r_{1}^{W_{G}(H)}(u)

This completes the proof of the fundamental Theorem.

Now, we calculate the obstruction group. Since ℤH^{n}(H, M) is a permutation module isomorphic to

⨁_{u∈H^{n}(H,M)/W_{G}(H)} ℤ ↑_{W_{G}(H,u)}^{W_{G}(H)},

we have

\begin{aligned}Obs^{n}(G, M) & = Ĥ^{0}(W_{G}(H), ℤH^{n}(H, M)) ≅ ⨁_{u} Ĥ^{0}(W_{G}(H), ℤ ↑_{W_{G}(H,u)}^{W_{G}(H)}) \\ & ≅ ⨁_{u} Ĥ^{0}(W_{G}(H, u), ℤ) ≅ ⨁_{u} ℤ/|W_{G}(H, u)|ℤ \end{aligned}

where the second isomorphism is the Eckmann-Shapiro isomorphism. Note that W_{G}(H, u) = N_{G}(H, u)/H, so we conclude the following result.

Proposition 7.4. We have

Obs^{n}(G, M)) ≅ ⨁_{[G/H,u]} ℤ/|N_{G}(H, u) : H|ℤ

where the sum is taken over the isomorphism classes of pointed G-sets with transitive G-sets.

Using this, one can easily see that the obstruction groups Obs^{n}(G, M) for n = 0 and n = 1 give the obstruction groups for the crossed Burnside ring B^{c}(G, M) (see Corollary 5.3 in [8]), and the monomial Burnside ring (see Corollary 2.8 in [3]).

In [3], Boltje proves several fundamental theorems with different ψ for the mark homomorphism φ : A_{+}(H) → A^{+}(H) from lower plus constructions to upper plus constructions (see [3] for the definition of lower and upper plus constructions). The obstruction group calculation we gave above shows that there are similarities between our fundamental theorem and the fundamental theorem given in [3]. These similarities are best explained by the following observation.

Proposition 7.5. Let A(H) denote the restriction functor H → ℤH^{n}(H, M). Then, B^{n}(G, M) ≅ A_{+}(G) and β^{n}(G, M) ≅ A^{+}(G) as abelian groups.

Proof. The lower plus construction A_{+}(G) is defined as

( ⨁_{H≤G} ℤH^{n}(H, M))_{G}

where the G-action is by conjugation. So, it is easy to see that as abelian groups A_{+}(G) and

B^{n}(G, M) = ⨁_{[H]∈Cl(G)} ℤH^{n}(H, M)_{W_{G}(H)}

are isomorphic. Similarly, the upper plus construction A^{+}(G) is defined as

( ⨁_{H≤G} ℤH^{n}(H, M))^{G}

where the G action is by conjugation. So, it is easy to see that as an abelian group A^{+}(G) is isomorphic to β^{n}(G, M).

One can also show that there is even an isomorphism of Mackey functors once the Mackey functor structure for B^{n}(G, M) and β^{n}(G, M) are defined in an appropriate way.

8 Twisted group algebras

Let G be a finite group, let X be a finite G-set, and let k be any commutative ring. In this final section, we give an interpretation of B^{2}(G, k^{×}).

Let R be a commutative ring with a G-action, then it is well known that each cohomology class [α] ∈ H^{2}(G, R^{×}) corresponds to an equivalence class of twisted group rings R_{α}G, where R_{α}G has as R-basis the elements \overline{g}, for g ∈ G, and multiplication is given by

r_{1}\overline{g}_{1} · r_{2}\overline{g}_{2} := α(g_{1}, g_{2})r_{1}(^{g_{1}}r_{2})\overline{g_{1}g_{2}}

for r_{1}, r_{2} ∈ R and g_{1}, g_{2} ∈ G. Here two twisted group algebras are called equivalent, if they are isomorphic as G-graded R-algebras.

Now let [α] ∈ H_{X}^{2}(G, k^{×}) for a finite G-set X. Via the isomorphism H_{X}^{2}(G, k^{×}) ≅ H^{2}(G, Map(X, k^{×})) we can view α as a map G × G → Map(X, k^{×}), where Map(G, k^{×}) is viewed as a ℤG-module with g ∈ G acting on a ∈ Map(G, k^{×}) by (^{g}a)(x) = a(g^{−1}x), for x ∈ X. We can define multiplication in Map(G, k) pointwise, that is (ab)(x) = a(x)b(x) for a, b ∈ Map(G, k) and x ∈ X. Then, for R = Map(G, k), the twisted group algebra R_{α}G is defined. To simplify notation, we will write k_{α}G for this algebra, the dependence on X being implicit in α. Then

k_{α}G = ⨁_{g∈G} Map(X, k)\overline{g}.

We will write just g for the element \overline{g} ∈ k_{α}G. The multiplication in k_{α}G is given by

a_{1}g_{1} · a_{2}g_{2} = α(g_{1}, g_{2})a_{1}(^{g_{1}}a_{2})(g_{1}g_{2})

for a_{1}, a_{2} ∈ Map(G, k) and g_{1}, g_{2} ∈ G. Then k_{α}G is a k-algebra with identity 1_{k} · 1_{G}, where 1_{k} is the constant map with value 1 ∈ k. It is clear that k_{α}G is associative, by the aforementioned general theory of twisted group rings.

An explicit k-basis for k_{α}G is given by the set { (x, g) | x ∈ X, g ∈ G }, where (x, g) represents a_{x}g with a_{x} ∈ Map(G, k) defined by a_{x}(y) = δ_{x,y}, for y ∈ X. Multiplication of two basis elements is then given by (x, g) · (y, h) = 0, unless x = gy, and in this case (gy, g) · (y, h) = α(g, h; gy)(gy, gh), now viewing α again as a map G × G × X → k^{×}. In terms of the basis, the identity element in k_{α}G is given by ∑_{x∈X}(x, 1_{G}).

Now assume that X = G/H for a subgroup H of G. By Corollary 2.6 we have an isomorphism

H_{G/H}^{2}(G, k^{×}) → H^{2}(H, k^{×}), α ↦ α̂,

so one can expect a relationship between k_{α}G and the twisted group algebra kα̂H.

Proposition 8.1. The k-algebras k_{α}G and kα̂H are Morita equivalent.

Proof. Let A be a k-algebra, and let e ∈ A be an idempotent. By Theorem 9.9 in [10], the algebras A and eAe are Morita equivalent, if and only if the two-sided ideal AeA equals A. In our situation, observe that kα̂H is isomorphic to the subalgebra B of A = k_{α}G with basis those pairs (x, g) with g ∈ H and x = H being the trivial coset. The identity in this subalgebra is the idempotent e = (H, 1), so that B = eAe. By the above, we only need to show that A = AeA. So let (xH, g) denote an arbitrary basis element of A, with g, x ∈ G. One calculates easily that

(xH, g) = α(x, 1; xH)^{−1}α(x, x^{−1}g; xH)^{−1}(xH, x)(H, 1)(H, x^{−1}g),

so that (xH, g) ∈ AeA.

We can now view B^{2}(G, k^{×}) as the Grothendieck group of a category T w(G), defined as follows. The objects are k_{α}G for cocyles α : G×G×X → k^{×} and G-sets X. Morphisms f : k_{α}G → k_{β}G are induced by G-set maps f_{0} : Y → X satisfying (f_{0})^{∗}[α] = [β]. Any such G-set map gives rise to a ring homomorphism

f_{1} : Map(X, k) → Map(Y, k), a ↦ a ∘ f_{0},

which in turn induces a G-graded k-algebra homomorphism

f : k_{α}G → k_{β}G, ag ↦ f_{1}(a)g.

Then k_{α}G and k_{β}G are isomorphic in T w(G) if and only if [X, [α]] = [Y, [β]] in B^{2}(G, k^{×}). Direct sums in T w(G) are defined as

k_{α}G ⊕_{T w} k_{β}G = k_{α⊕β}G.

Note that k_{α}G ⊕_{T w} k_{β}G is isomorphic to the direct sum of k_{α}G and k_{β}G as algebras (with componentwise addition and multiplication) and also that the Morita equivalence in Proposition 8.1 respects direct sums.

The multiplication in B^{2}(G, k^{×}) corresponds to a product in T w(G), defined by

k_{α}G ⊗_{T w} k_{β}G = k_{α⊗β}G,

but this algebra is not isomorphic to the k-tensor product of the two algebras. In fact, k_{α}G ⊗_{T w} k_{β}G is isomorphic to a diagonal subalgebra of k_{α}G ⊗_{k} k_{β}G, that is

k_{α}G ⊗_{T w} k_{β}G ≅ {∑_{g∈G} a_{g}g ⊗ b_{g}g ∈ k_{α}G ⊗ k_{β}G | a_{g} ∈ Map(X, k), b_{g} ∈ Map(Y, k)}

as G-graded algebras, the latter being equipped with componentwise addition and multiplication.

Acknowledgments: Most of this article had been written during three research visits of the first author to Bilkent University. He wants to express his gratitude for the hospitality he received during these visits.

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