The Burnside dimension of projective Mackey functors

Serge Bouc

Abstract : In this note, I will apply methods exposed by J. P. May ([7]) to the special case of Mackey functors for a finite group G over a commutative ring R. In particular, any finitely generated projective Mackey functor has a Burnside dimension, which is an element of the Burnside algebra RB(G) of G over R.

1. Mackey functors

There are several equivalent possible definitions of Mackey functors. In this note, I will use two of them. In both of them R is a commutative ring (with identity element), and G is a finite group :

1.1. Definition in terms of G-sets. The first definition of Mackey functors is due to A. Dress ([5]) :

A Mackey functor M for G over R is a bivariant functor M = (M_{∗}, M^{∗}), from the category G-set of finite G-sets to the category R-Mod of R-modules, satisfying the following two conditions :

1. The functor M maps disjoint unions to direct sums : if X and Y are finite G-sets, if i_{X} and i_{Y} are the canonical inclusions from X and Y to the disjoint union

X ⊔ Y, then the maps (M_{∗}(i_{X}), M_{∗}(i_{Y})) and ( \begin{matrix}M^{∗}(i_{X}) \\ M^{∗}(i_{Y}) \end{matrix} ) are mutual inverse

isomorphisms of R-modules between M(X) ⊕ M(Y) and M(X ⊔ Y).

2. If

X \underrightarrow{a} Y
b ↓  ↓ c
Z \overrightarrow{d} T

is a cartesian (pullback) square of finite G-sets, then M_{∗}(b)M^{∗}(a) = M^{∗}(d)M_{∗}(c).

A morphism of Mackey functors is a natural transformation of bivariant functors. The Mackey functors for G over R form a category, denoted by Mack_{R}(G).

1.2. The Mackey algebra. The second definition of Mackey functors is due to J. Thévenaz and P. Webb ([9]), who defined the Mackey algebra. The present exposition follows Chapter 4 of [2].

Let

Ω_{G} = ⨆_{H⊆G} G/H = {xH | x ∈ G, H ⊆ G}

denote the disjoint union of all transitive left G-sets G/H, where H runs through the set of subgroups of G.

If X is a finite (left) G-set, denote by B(X) the Burnside group of X, i.e. the Grothendieck group of the category G-set↓_{X} of G-sets over X. Similarly denote by RB(X) the tensor product R ⊗_{ℤ} B(X).

Date : March 31, 2005

The Mackey algebra μ_{R}(G) of the group G over R is defined by

μ_{R}(G) = RB(Ω_{G}^{2}) ,

where Ω_{G}^{2} denotes the G-set Ω_{G} × Ω_{G} (for diagonal G-action). The multiplication on μ_{R}(G) is defined by R-linear extension from the pullback product

\xymatrix{ & X \ar[ddl]_{a} \ar[ddr]^{b} &  &  &  & Y \ar[ddl]_{c} \ar[ddr]^{d} &  &  &  & X ×_{Ω_{G}} Y \ar[ddl]_{e} \ar[ddr]^{f} \\  &  &  & × &  &  &  & ↦ &  &  &  & , \\ Ω_{G} &  & Ω_{G} &  & Ω_{G} &  & Ω_{G} &  & Ω_{G} &  & Ω_{G}}

where X×_{Ω_{G}} Y = {(x, y) ∈ X×Y | b(x) = c(y)}, and e(x, y) = a(x), and f(x, y) = d(y). The identity element for this multiplication is the G-set

\xymatrix{ & Ω_{G} \ar[dl] \ar[dr] \\ Ω_{G} &  & Ω_{G}}

where both maps are the identity map of Ω_{G}.

Now a Mackey functor for G over R is a left μ_{R}(G)-module.

1.3. Equivalence. Saying that the above two definitions are equivalent means that the categories Mack_{R}(G) and μ_{R}(G)-Mod are equivalent, and this equivalence can be seen as follows : suppose that V is a μ_{R}(G)-module, and that X is a finite G-set. Then the R-module RB_{X} = RB(Ω_{G} × X) has a natural structure of left μ_{R}(G)-module, obtained by R-linear extension from the obvious pullback product. Thus setting

F_{V} (X) = Hom_{μ_{R}(G)}(RB_{X}, V)

defines a Mackey functor F_{V}, in the sense of Dress.

Conversely, if M is a Mackey functor in this sense, then M(Ω_{G}) has a natural structure of μ_{R}(G)-module (see Section 4.3 of [2] for details).

1.4. Tensor product of Mackey functors. If M, N, and P are Mackey functors for G over R, a bilinear morphism φ : M, N → P is a collection of R-bilinear φ_{X,Y} : M(X)×N(Y) → P (X×Y), for any finite G-sets X and Y, which are moreover bivariant with respect to X and Y. The tensor product M ˆ⊗N can be defined as the solution to the universal problem of bilinear morphisms : this means that the set of bilinear morphisms from M, N to P is in one to one correspondence with the set of morphisms of Mackey functors from M ˆ⊗N to P (Proposition 1.8.2 of [2]).

If X is a finite G-set, then M ˆ⊗N(X) can be computed as follows :

M ˆ⊗N(X) = ( ⊕_{Y →^{f}X} M(Y) ⊗ N(Y)) /J ,

where J is the R-submodule generated by expressions [M_{∗}(a)(u)⊗v]_{Z,g}−[u⊗N^{∗}(a)(v)]_{Y,f} and [M^{∗}(a)(u′) ⊗ v′]_{Y,f} − [u′ ⊗ N_{∗}(a)(v′)]_{Z,g}, for every commutative triangle of finite G-sets

\xymatrix{Y \ar[rr]^{a} \ar[dr]_{f} &  & Z \ar[dl]^{g} \\  & X}

for every u ∈ M(Y), v ∈ N(Z), u′ ∈ M(Z), v′ ∈ N(Y), where e.g. [M_{∗}(a)(u) ⊗ v]_{Z,g} denotes the element M_{∗}(a)(u)⊗v of the component M(Z)⊗N(Z) indexed by g : Z → X in the direct sum.

The tensor product of Mackey functors is commutative (or symmetric), and associative. The Burnside functor RB is an identity for this tensor product, which means that the functors RB ˆ⊗− and − ˆ⊗RB are both isomorphic to the identity functor of Mack_{R}(G) (see Section 2.4 of [2] for details).

1.5. The Dress construction. Let M be a Mackey functor for G over R, and let X be a finite G-set. The bivariant functor M_{X} obtained by composition of M with the endofunctor Y ↦ Y × X of G-set is a Mackey functor for G over R. This construction Id_{X} : M ↦ M_{X} is an endofunctor of the category Mack_{R}(G), called the Dress construction associated to X. This functor Id_{X} is self adjoint (Lemma 3.1.1 of [2]).

1.6. Internal Hom. If M and N are Mackey functors for G over R, the functor H(M, N) was defined in Section 1.3 of [2]. It is another Mackey functor for G over R, whose value at the G-set X is

H(M, N)(X) = Hom_{Mack_{R}(G)}(M, N_{X}) .

The construction (M, N) ↦ H(M, N) is an internal Hom in the category Mack_{R}(G). It is right adjoint to the tensor product of Mackey functors, in the following sense : if M, N, and P are Mackey functors for G over R, then there are isomorphisms of Mackey functors

H(M ˆ⊗N, P) ≅ H(N, H(M, P))

which are natural in M, N, and P.

In the same situation, there is also a composition morphism

γ : H(M, N) ˆ⊗H(N, P) → H(M, P)

defined as follows : let X be a finite G-set. Then H(M, P)(X) = Hom_{Mack_{R}(G)}(M, P_{X}), whereas H(M, N) ˆ⊗H(N, P)(X) is a quotient of the direct sum

Σ = ⊕_{Y →^{f}X} Hom_{Mack_{R}(G)}(M, N_{Y}) ⊗ Hom_{Mack_{R}(G)}(N, P_{Y}) .

Fix some G-set (Y, f) over X (where f : Y → X), and let a : M → N_{Y} and b : N → P_{Y} be some morphisms of Mackey functors. Then the image by γ_{X} of the element a ⊗ b of the component of Σ indexed by (Y, f) is the morphism M → P_{X} whose evaluation at a G-set Z is the map M(Z) → P (Z × X) obtained by the composition

M(Z) ⟶^{a_{Z}} N(ZY) ⟶^{b_{ZY}} P (ZY^{2}) ⟶^{P^{∗}(\begin{matrix}zy \\ ↓ \\ zyy \end{matrix})} P (ZY) ⟶^{P_{∗}(\begin{matrix}zy \\ ↓ \\ zf(y) \end{matrix})} P (ZX) ,

where for short (\begin{matrix}zy \\ ↓ \\ zyy \end{matrix}) denote the map (z, y) ∈ Z × Y ↦ (z, y, y) ∈ Z × Y × Y.

Finally, the Burnside functor is a left unit for H : for any Mackey functor, there is an isomorphism H(RB, M) ≅ M.

2. Burnside trace and dimension

The previous section recalls various constructions in Mack_{R}(G), and shows that this category is a closed symmetric monoidal category. In this general framework, J.P May has developed a theory of Euler characteristics and Burnside rings (see [7]), and one can try to see how this theory applies in this particular example.

2.1. Dualizable objects. The dual DM of a Mackey functor M for G over R is defined by

DM = H(M, RB) .

(this notion is different from the notion of dual over R defined in Section 6.2.2 of [2]) The isomorphism H(RB, M) ≅ M gives a composition morphism

j_{M} : DM ˆ⊗M → H(M, M) ,

and M is called dualizable if j_{M} is an isomorphism.

Conversely, there is a morphism

e_{M} : M ˆ⊗DM → H(RB, RB) ≅ RB .

2.2. Lemma : The Mackey functor M is dualizable if and only if M is finitely generated and projective. Proof: (Sketch) Suppose that M is dualizable. Evaluating j_{M} at the trivial G-set • gives an isomorphism

(DM ˆ⊗M)(•) → End_{Mack_{R}(G)}(M) .

Choosing an element in the left hand side mapping to the identity of M shows that there exist a positive integer n, finite G-sets Y_{i}, and morphisms a^{(i)} : M → RB_{Y_{i}} and b^{(i)} : RB → M_{Y_{i}}, for i ∈ {1, …, n} such that for any G-set Z

(2.3)  Id_{M(Z)} = ∑_{i=1}^{n} M_{∗} (\begin{matrix}zy_{i} \\ ↓ \\ z \end{matrix}) M^{∗} (\begin{matrix}zy_{i} \\ ↓ \\ zy_{i}y_{i} \end{matrix}) ∘ b_{ZY_{i}}^{(i)} ∘ a_{Z}^{(i)} .

Using the adjunction Hom_{Mack_{R}(G)}(RB, M_{Y_{i}}) ≅ Hom_{Mack_{R}(G)}(RB_{Y_{i}}, M), the morphisms b^{(i)} give morphisms b̃^{(i)} : RB_{Y_{i}} → M, and one can check that equality 2.3 is equivalent to

Id_{M} = ∑_{i=1}^{n} b̃^{(i)} ∘ a^{(i)} .

Setting X = ⊔_{i=1}^{n} Y_{i}, this shows that M is a direct summand of RB_{X}. Since RB_{X} is finitely generated and projective, so is M.

Conversely, if M is finitely generated and projective, then M is a direct summand of some functor RB_{X}, for a finite G-set X. Since any direct summand of a dualizable object is a dualizable object, it suffices to show that RB_{X} is dualizable. And this is easy, because

H(RB_{X}, RB) ˆ⊗RB_{X} ≅ H(RB_{X}, RB)_{X} ≅ H(RB_{X}, RB_{X}) .

Here the first isomorphism is a consequence of Lemma 7.2.3 of [2], which implies that for any Mackey functors M and N, and any finite G-set X, one has that (M ˆ⊗N)_{X} ≅ M_{X} ˆ⊗N ≅ M ˆ⊗N_{X}, and from the fact that RB is a unit for the tensor product ˆ⊗. The second isomorphism follows easily from the definitions of H.

2.4. Burnside trace and dimension. Let M be a dualizable (i.e. finitely generated and projective) Mackey functor, and let f ∈ End_{Mack_{R}(G)}(M). There is a sequence of morphisms

RB →^{i_{M}} H(M, M) →^{j_{M}^{−1}} DM ˆ⊗M →^{σ_{M}} M ˆ⊗DM →^{f⊗Id} M ˆ⊗DM →^{e_{M}} RB .

Here the morphism i_{M} is the unique morphism of Green functors from RB to H(M, M) : there is a unique such morphism, because H(M, M) is a Green functor (Proposition 2.1.1 of [2]), and RB is an initial object in the category of Green functors for G over R (Proposition 2.4.4 of [2]). The morphism σ_{M} comes from the commutativity of ˆ⊗.

The composition of these morphisms of Mackey functors is an endomorphism of RB. Since End_{Mack_{R}(G)}(RB) is isomorphic to the evaluation RB(•) at the trivial G-set, i.e. to the Burnside algebra RB(G) of G over R, this gives an element denoted by Btr(f) of RB(G).

2.5. Definition and Notation : This element Btr(f) of RB(G) will be called the Burnside trace of the endomorphism f. When f = Id, it will be called the Burnside dimension of M, and denoted by Bdim(M).

2.6. Proposition : Let M be a dualizable Mackey functor for G over R, let X be a finite G set, and let p : RB_{X} → M and s : M → RB_{X} be morphisms of Mackey functors such that p ∘ s = Id_{M}. Let ϵ_{X} the element of RB(X^{2}) corresponding to the diagonal inclusion x ↦ (x, x) of X into X^{2}. If f ∈ End_{Mack_{R}(G)}, then :

Btr(f) = RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) s_{X}f_{X}p_{X}(ϵ_{X}) .

Proof: First of all, the isomorphism End_{Mack_{R}(G)}(RB) ≅ RB(•) is the map sending the endomorphism f of RB to f_{•}(•) ∈ RB(•). Now the image of • ∈ RB(•) in H(M, M)(•) = End_{Mack_{R}(G)}(M) is the identity map of M. The hypotheses imply that the image by (j_{M}^{−1})_{•} of the identity map of M is the element s ⊗ p̃ of the component X → • in the direct sum

⊕_{X→•} Hom_{Mack_{R}(G)}(M, RB_{X}) ⊗ Hom_{Mack_{R}(G)}(RB, M_{X})

defining (DM ˆ⊗M)(•), where p̃ is the morphism RB → M_{X} obtained by adjunction from the morphism p : RB_{X} → M. The image of this element s ⊗ p̃ by (σ_{M})_{•} is the element p̃ ⊗ s of the component X → • in the direct sum

⊕_{X→•} Hom_{Mack_{R}(G)}(RB, M_{X}) ⊗ Hom_{Mack_{R}(G)}(M, RB_{X})

defining (M ˆ⊗DM)(•). By the map (f ⊗ Id)_{•}, this element p̃ ⊗ s is sent to f_{X} p̃ ⊗ s in the same component of the direct sum. And finally, by the map (e_{M})_{•}, this is sent to the endomorphism s̃f_{X} p̃ of RB, where s̃ : M_{X} → RB is the morphism deduced by adjunction from s : M → RB_{X}.

It follows that

Btr(f) = (s̃f_{X} p̃)_{•}(•) .

So it is the image of • ∈ RB(•) by the map

RB(•) ⟶^{p̃_{•}} M(X) ⟶^{f_{X}} M(X) ⟶^{s̃_{•}} RB(•) .

Now the map p̃_{•} is the map

RB(•) ⟶^{RB^{∗}(\begin{matrix}x \\ ↓ \\ • \end{matrix})} RB(X) ⟶^{RB_{∗}(\begin{matrix}x \\ ↓ \\ xx \end{matrix})} RB(X^{2}) ⟶^{p_{X}} M(X) ,

and RB_{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) (•) = ϵ_{X}. Moreover the map s̃_{•} is the map

M(X) ⟶^{s_{X}} RB(X^{2}) ⟶^{RB^{∗}(\begin{matrix}x \\ ↓ \\ xx \end{matrix})} RB(X) ⟶^{RB_{∗}(\begin{matrix}x \\ ↓ \\ • \end{matrix})} RB(•) .

It follows that Btr(f) = RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) s_{X}f_{X}p_{X}(ϵ_{X}), as was to be shown.

The following is the special case M = RB_{X} : then End_{Mack_{R}(G)}(M) ≅ RB(X^{2}) :

2.7. Corollary : Let X and Z be finite G-sets, and let a and b be maps of G-sets from Z to X. Let

f =\xymatrix{ & Z \ar[dl]_{a} \ar[dr]^{b} \\ X &  & X}

be the corresponding element of RB(X^{2}), viewed as an endomorphism of RB_{X}. Then

Btr(f) = {z ∈ Z | a(z) = b(z)} .

In particular Bdim(RB_{X}) = X. Proof: In this case, one can suppose that the maps p and s are the identity map. The result for Btr(f) follows from a straightforward computation, and the result for Bdim(RB_{X}) is the special case Z = X and a = b = Id_{X}.

2.8. Example : Suppose that X = G/1. Then End_{Mack_{R}(G)}(RB_{X}) is isomorphic to the group algebra RG : this isomorphism RG → RB((G/1)^{2}), denoted by x ↦ x̂, maps the element g ∈ G of the canonical basis of RG to the element

ĝ = \xymatrix{ & G/1 \ar[dl]_{Id} \ar[dr]^{g} \\ G/1 &  & G/1}

of RB((G/1)^{2}), where the right hand side arrow is right multiplication by g on the G-set G/1. In this case

Btr(ĝ) = δ_{g,1} · G/1 ,

where δ_{g,1} is a Kronecker symbol, so in general Btr(x̂) = tr_{RG}(x) · G/1, where tr_{RG}(x) is the usual trace map on the group algebra RG.

2.9. Remark : Through the equivalence 1.3, the category of Mackey functors is equivalent to the category of μ_{R}(G)-modules. For μ_{R}(G)-modules, there is the Hattori-Stallings trace map Tr_{μ_{R}(G)} (see [6] and [8]), which associates to any endomorphism f of a finitely generated projective μ_{R}(G)-module, an element Tr_{μ_{R}(G)}(f) in the zero-th Hochschild homology group of μ_{R}(G), i.e.

Tr_{μ_{R}(G)}(f) ∈ HH_{0}(μ_{R}(G)) = μ_{R}(G)/[μ_{R}(G), μ_{R}(G)] .

One can show easily that with this equivalence, the Burnside trace Btr(f) is the image of Tr_{μ_{R}(G)}(f) by the map

HH_{0}(μ_{R}(G)) → RB(G)

induced by the “equalizer map” from μ_{R}(G) = RB(Ω_{G}^{2}) to RB(G), sending the element

\xymatrix{ & Z \ar[dl]_{a} \ar[dr]^{b} \\ Ω_{G} &  & Ω_{G}}

to the equalizer {z ∈ Z | a(z) = b(z)}, viewed as an element of RB(G).

3. Functorial properties

3.1. Composition with a biset. Let G and H be finite groups, and let U be a finite (H, G)-biset. If X is a finite G-set, define

U ∘ X = {(u, x) ∈ U × X | ∀g ∈ G, u · g = u ⇒ g · x = x} ,

and denote by U ∘_{G} X the quotient of U ∘ X by the right action of G given by

(u, x) · g = (u · g, g^{−1} · x), ∀(u, x, g) ∈ U × X × G .

This construction extends to a map X ↦ U ∘_{G} X from B(G) to B(H).

The construction X ↦ U ∘_{G} X is a functor γ_{U} from G-set to H-set, which preserves disjoint unions and pullback squares. Conversely, any functor G-set → H-set with these two properties is isomorphic to a functor γ_{U}, for some finite (H, G)-biset U (see [1] for details).

By composition, the functor γ_{U} induces a functor

Γ_{U} : Mack_{R}(H) → Mack_{R}(G) ,

defined by Γ_{U}(N) = N ∘ γ_{U}, for any Mackey functor N for H over R.

3.2. Adjoint functors. These functors Γ_{U} have left and right adjoints, respectively denoted by L_{U} and R_{U} : an explicit, but rather complicated description of the functors L_{U} was given in Chapter 9 of [2]. A simpler description ([4]) can be obtained as follows, using the equivalence 1.3 : consider

RB_{U} = RB(Ω_{H} × (U ∘_{G} Ω_{G})) .

This is a (μ_{R}(H), μ_{R}(G))-bimodule, for the actions extending linearly the following

products : suppose that (X, (a, b)) is an H-set over Ω_{H} × Ω_{H}, that (Y, (c, d)) is a

G-set over Ω_{G} × Ω_{G}, and that (Z, (e, f)) is an H-set over Ω_{H} × (U ∘_{G} Ω_{G}). Build the following diagram

\xymatrix{ &  &  & E \ar[dl]_{k} \ar[dr]^{l} \\  &  & C \ar[dl]_{g} \ar[dr]^{h} &  & D \ar[dl]_{i} \ar[dr]^{j} \\  & X \ar[dl]_{a} \ar[dr]^{b} &  & Z \ar[dl]_{e} \ar[dr]_{f} &  & U ∘_{G} Y \ar[dl]_{U∘_{H}c} \ar[dr]^{U∘_{H}d} \\ Ω_{H} &  & Ω_{H} &  & U ∘_{G} Ω_{G} &  & U ∘_{G} Ω_{G}}

where all the squares are pull-back squares. Then the left and right actions on RB_{U} are defined by

(X, (a, b)).(Z, (e, f)).(Y, (c, d)) = (E, (agk, (U ∘_{H} d)jl)) .

It is easy to this from this definition that there is an isomorphism of left μ_{R}(H)-modules

RB_{U} ≅ RB_{U∘_{G}Ω_{G}}(Ω_{H}) .

In particular RB_{U} is projective and finitely generated as μ_{R}(H)-module. Moreover, one can show that if N is a Mackey functor for H over R, then the natural isomorphism of R-modules

(N ∘ U)(Ω_{G}) = N(U ∘_{G} Ω_{G}) ≅ Hom_{μ_{R}(H)}(RB_{U}, N(Ω_{H}))

is an isomorphism of μ_{R}(G)-modules. Thus if M is a Mackey functor for G over R, this gives by standard arguments

L_{U}(M)(Ω_{H}) ≅ RB_{U} ⊗_{μ_{R}(G)} M(Ω_{G}) .

More generally, if Z is a finite H-set, the same argument shows that

(3.3)  L_{U}(M)(Z) ≅ RB(Z × U ∘_{G} Ω_{G}) ⊗_{μ_{R}(G)} M(Ω_{G}) ,

where the right μ_{R}(G)-module structure on RB(Z × U ∘_{G} Ω_{G}) is given by pullback as above.

3.4. Remark : A similar argument (see [4]), considering the (μ_{R}(G), μ_{R}(H))-bimodule

RB_{U}^{♯} = Hom_{μ_{R}(H)}(RB_{U}, μ_{R}(H))

gives the description of the right adjoint R_{U}, by

R_{U}(M)(Ω_{H}) ≅ Hom_{μ_{R}(G)}(RB_{U}^{♯}, M(Ω_{G})) .

3.5. Proposition : Let G be a finite group. If M and N are finitely generated projective Mackey functors for G over R, then DM is finitely generated and projective, and

Bdim(M ⊕ N) = BdimM + BdimN
Bdim(M ˆ⊗N) = BdimM · BdimN
Bdim(DM) = BdimM .

Proof: This is Proposition 2.7 and Proposition 4.3 of [7].

3.6. Proposition : Let G and H be finite groups, and let U be a finite (H, G)- biset. Let M be a finitely generated projective Mackey functor for G over R, and let f ∈ End_{Mack_{R}(G)}(M). Then

Btr(L_{U}(f)) = U ∘_{G} Btr(f) .

Proof: Since M is finitely generated and projective, there exist a finite G-set X and morphisms p : RB_{X} → M and s : M → RB_{X} such that p ∘ s = Id_{M}. In this case, with the notation of Proposition 2.6

Btr(f) = RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) s_{X}f_{X}p_{X}(ϵ_{X}) .

Applying the functor L_{U} gives morphisms

L_{U}(p) : L_{U}(RB_{X}) → L_{U}(M) L_{U}(s) : L_{U}(M) → L_{U}(RB_{X})

such that L_{U}(p) ∘ L_{U}(s) = Id_{L_{U} (M)}. Now there is an isomorphism of Mackey functors (see Lemma 5.4 of [3])

L_{U}(RB_{X}) ≅ RB_{U∘_{G}X} ,

which can be seen as follows : if Z is a finite H-set, then there is a map

π_{Z,X} : L_{U}(RB_{X})(Z) ≅ RB(Z × U ∘_{G} Ω_{G}) ⊗_{μ_{R}(G)} RB(Ω_{G} × X) → RB(Z × U ∘_{G} X)

sending S ⊗ T, where S is some H-set over Z × U ∘_{G} Ω_{G} and T is some G-set over Ω_{G} × X, to the pullback product of S and γ_{U}(T) over U ∘_{G} Ω_{G}. To check that this map π_{Z,X} is an isomorphism, first consider the case X = Ω_{G}, where it is trivial, and then the case where X is a disjoint union of copies of Ω_{G}. The general case follows, because any G-set is a subset of such a disjoint union, and the corresponding map π_{Z,X} is then the retract of an isomorphism.

Denote by T the H-set U ∘_{G} X. By definition now

Btr(L_{U}(f)) = RB_{∗} (\begin{matrix}t \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}t \\ ↓ \\ tt \end{matrix}) L_{U}(s)_{T} L_{U}(f)_{T} L_{U}(p)_{T} (ϵ_{T}) .

To compute this, the first thing to do is to find a preimage of ϵ_{T} by the above isomorphism π_{T,X} : suppose that ω : X → ⊔_{j=1}^{n} Ω_{G} is an inclusion. For j ∈ {1, …, n},

denote by X_{j} the inverse image by ω of the j-th component of ⊔_{j=1}^{n} Ω_{G}. Denote by ω_{j}

the restriction of ω to X_{j}, and by i_{j} the inclusion of X_{j} into X. Then

[figure: π_{T,X}^{−1}(ϵ_{T}) = ∑_{j=1}^{n}; U ∘_{G} X_{j}; U∘_{G}i_{j}; U∘_{G}ω_{j}; U ∘_{G} X; U ∘_{G} Ω_{G}; ⊗; X_{j}; ω_{j}; i_{j}; Ω_{G}; X; .]

The image of this element by L_{U}(s)_{T} L_{U}(f)_{T} L_{U}(p)_{T} is equal to

[figure: ∑_{j=1}^{n}; U ∘_{G} X_{j}; U∘_{G}i_{j}; U∘_{G}ω_{j}; U ∘_{G} X; U ∘_{G} Ω_{G}; ⊗ s_{Ω_{G}}f_{Ω_{G}}p_{Ω_{G}}(; X_{j}; ω_{j}; i_{j}; Ω_{G}; X; ) ,]

and the image of this by the isomorphism π_{T,X} is equal to the element

\xymatrix{ &  & U ∘_{G} X_{j} \ar[ddl]_{U∘_{G}i_{j}} \ar[ddr]^{U∘_{G}ω_{j}} &  &  &  & X_{j} \ar[ddl]_{ω_{j}} \ar[ddr]^{i_{j}} \\ S = ∑_{j=1}^{n} &  &  &  & ×_{U∘_{G}Ω_{G}} U ∘_{G} [s_{Ω_{G}}f_{Ω_{G}}p_{Ω_{G}}( &  &  &  & )] , \\  & U ∘_{G} X &  & U ∘_{G} Ω_{G} &  & Ω_{G} &  & X}

of RB(T^{2}), where ×_{U∘_{G}Ω_{G}} denotes the pullback product over U ∘_{G} Ω_{G}. Then

Btr(L_{U}(f)) = RB_{∗} (\begin{matrix}t \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}t \\ ↓ \\ tt \end{matrix}) (S) .

Since the functor γ_{U} preserves pullbacks, this is equal to

[figure: U ∘_{G} RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}); _{[∑_{j=1}^{n}}; X_{j}; i_{j}; ω_{j}; X; Ω_{G}; ×_{Ω_{G}} s_{Ω_{G}}f_{Ω_{G}}p_{Ω_{G}}(; X_{j}; ω_{j}; i_{j}; Ω_{G}; X; )] ,]

where ×_{Ω_{G}} denotes the pullback product over Ω_{G}. Now

\xymatrix{ & X_{j} \ar[dl]_{ω_{j}} \ar[dr]^{i_{j}} \\ Ω_{G} & & X}= RB_{X∗}(ω_{j})RB_{X}^{∗}(i_{j})(ϵ_{X}) ,

and since s, f, and p are morphisms of Mackey functors, it follows that

s_{Ω_{G}}f_{Ω_{G}}p_{Ω_{G}}(\xymatrix{ & X_{j} \ar[dl]_{ω_{j}} \ar[dr]^{i_{j}} \\ Ω_{G} &  & X}) = RB_{X∗}(ω_{j})RB_{X}^{∗}(i_{j})s_{X}f_{X}p_{X}(ϵ_{X}) ,

and this is also equal to

\xymatrix{ & X_{j} \ar[dl]_{ω_{j}} \ar[dr]^{i_{j}} \\ Ω_{G} & & X}×_{X} s_{X}f_{X}p_{X}(ϵ_{X}) ,

where ×_{X} is the pullback over X. Finally Btr(L_{U}(f)) is equal to

\xymatrix{ &  & X_{j} \ar[ddl]_{i_{j}} \ar[ddr]^{ω_{j}} &  &  &  & X_{j} \ar[ddl]_{ω_{j}} \ar[ddr]^{i_{j}} \\ U ∘_{G} RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) ∑_{j=1}^{n} &  &  &  & ×_{Ω_{G}}( &  &  &  & ×_{X} s_{X}f_{X}p_{X}(ϵ_{X})) , \\  & X &  & Ω_{G} &  & Ω_{G} &  & X}

and by associativity of pullback products, this is equal to

[figure: U ∘_{G} RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) ∑_{j=1}^{n}(; X_{j}; i_{j}; ω_{j}; X; Ω_{G}; ×_{Ω_{G}}; X_{j}; ω_{j}; i_{j}; Ω_{G}; X; ) ×_{X} s_{X}f_{X}p_{X}(ϵ_{X}) .]

Now

[figure: _{∑_{j=1}^{n} (}; X_{j}; i_{j}; ω_{j}; X; Ω_{G}; ×_{Ω_{G}}; X_{j}; ω_{j}; i_{j}; Ω_{G}; X; ) = ϵ_{X} ,]

and ϵ_{X} is the identity element for ×_{X}. Hence

Btr(L_{U}(f)) = U ∘_{G} RB_{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) RB^{∗} (\begin{matrix}x \\ ↓ \\ xx \end{matrix}) s_{X}f_{X}p_{X}(ϵ_{X}) = U ∘_{G} Btr(f) ,

as was to be shown.

3.7. p-permutation modules. Suppose that R = k is a field of characteristic p. It was shown in Section 12 of [9], that evaluation at the trivial subgroup induces a one to one correspondence between the isomorphism classes of indecomposable projective Mackey functors for G over k, which are moreover projective relative to p-subgroups of G, and isomorphism classes of indecomposable trivial source kG-modules.

Thus if V is such an indecomposable trivial source module, denote by P_{V} the projective Mackey functor for G over k such that P_{V} (1) = V. It is natural to look at the Burnside dimension of P_{V}.

Proposition 3.6 involves the special case of restriction to a subgroup : if H is a subgroup of G, and if U = G, viewed as an (H, G)-biset by left and right multiplication, then the corresponding functor L_{U} is the restriction functor Res_{H}^{G} : Mack_{R}(G) → Mack_{R}(H) (see Section 9.9.1 of [2]). The functor γ_{U} : G-set → H-set is also the restriction functor.

Suppose that Q is a p-subgroup of G. Then the module Res_{Q}^{G}V is a permutation kQ-module. So there is a finite Q-set X_{Q}, such that

Res_{Q}^{G}V ≅ kX_{Q} ,

and up to isomorphism, the Q-set X_{Q} does not depend on the choice of such a Q-stable basis. In particular, this gives a well defined element X_{Q} ∈ kB(Q). Then obviously, if Q′ ⊆ Q

Res_{Q′}^{Q}X_{Q} = X_{Q′} ,

and if x ∈ P, then ^{x}X_{Q} = X_{^{x}Q}.

Hence the sequence β_{V} = (X_{Q})_{Q∈\underline{s}_{p}(G)}, indexed by the set \underline{s}_{p}(G) of all p-subgroups

of G, is an element of ( lim ; ⟵_{Q∈\underline{s}_{p}(G)} kB(Q))^{G}. The map

λ_{G} : (u_{Q}) ∈ ( lim ; ⟵_{Q∈\underline{s}_{p}(G)} kB(Q))^{G} ↦ − ∑_{Q∈\underline{s}_{p}(G)/G} \frac{χ̃]Q, .[_{\underline{s}_{p}(G)}}{|N_{G}(Q) : Q|}Ind_{Q}^{G}u_{Q} ∈ kB(G) ,

where χ̃]Q, .[_{\underline{s}_{p}(G)} is the reduced Euler-Poincaré characteristics of the poset of p-subgroups of G containing Q as a proper subgroup, is injective, and right inverse to the map

ρ_{G} : X ∈ kB(G) ↦ (Res_{Q}^{G}X)_{Q∈\underline{s}_{p}(G)} .

Now back to the projective Mackey functor P_{V} : since Res_{Q}^{G} and evaluation at the trivial subgroup commute, the evaluation at the trivial subgroup of the Mackey functor Res_{Q}^{G}P_{V} is isomorphic to kX_{Q}. Thus

Res_{Q}^{G}P_{V} ≅ kB_{X_{Q}} .

It follows that Res_{Q}^{G}Bdim(P_{V}) = X_{Q}, and this gives finally

ρ_{G}(Bdim(P_{V})) = β_{V} .

3.8. Remark : It is natural to ask if the stronger result

Bdim(P_{V}) = λ_{G}(β_{V})

holds. One can show it is the case if k is the residue field of a discrete valuation ring of characteristic 0 (e.g. if k is perfect).

3.9. Tensor induction. In this section, the ground ring R is the ring ℤ of integers. If G and H are finite groups, if U is a finite (H, G)-biset, one can define another functor T_{U} : Mack_{ℤ}(G) → Mack_{ℤ}(H), called tensor induction, associated to U (see [3] for details). This functor is not additive, but rather multiplicative (i.e. it commutes with the tensor product of Mackey functors).

It is defined by extending the functor B_{X} ↦ B_{Hom_{G}(U^{op},X)}, defined on the subcategory of permutation Mackey functors, to a right exact (non additive) functor defined on Mack_{ℤ}(G). It follows in particular from this construction that

T_{U}(B_{X}) = B_{Hom_{G}(U^{op},X)} ,

and that T_{U} maps finitely generated projective Mackey functors to finitely generated projective Mackey functors. So it is natural to look at the connection between this tensor induction and Burnside traces and dimensions. One can show the following :

3.10. Proposition : Let G and H be finite groups, and let U be a finite (H, G)-biset. Let moreover M be a finitely generated projective Mackey functor for G over ℤ. If f ∈ End_{Mack_{ℤ}(G)}(M), then

Btr(T_{U}(f)) = Hom_{G}(U^{op}, Btr(f)) .

3.11. Remark : Here in the right hand side, the map Hom_{G}(U^{op}, −) : B(G) → B(H) is the extension to the Burnside ring of the natural map defined on G-sets. This extension can be achieved by considering polynomial maps, or as in Section 3 of [3], by considering finite G-posets and associated Lefschetz invariants.

References

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[2] S. Bouc. Green-functors and G-sets, volume 1671 of Lecture Notes in Mathematics. Springer, October 1997.

[3] S. Bouc. Non-additive exact functors and tensor induction for Mackey functors, volume 144 of Memoirs. A.M.S., 2000. n683.

[4] S. Bouc. Bisets ans associated functors. Lectures at C.I.R.M (Luminy)

   http://www.math.jussieu.fr/∼bouc/biset.pdf, 2002.

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[6] A. Hattori. Rank element of a projective module. Nagoya J. Maths, 25:113–120, 1965.

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[8] J. R. Stallings. Centerless groups - an algebraic formulation of Gottlieb’s theorem. Topology, 4:129–134, 1965.

[9] J. Thévenaz and P. Webb. The structure of Mackey functors. Trans. Amer. Math. Soc., 347(6):1865–1961, June 1995.