数理解析研究所講究録 1440 巻 2005 年 107-120

107

The Burnside dimension of projective Mackey functors
Serge Boucto the special case of
Abstract : In this note, I will aPply methods exposed by , P. May {Mackey functors for a finite grouP $G$ over a commutative ring $R$ . In particular, any finitely generated
projective Mackey functor has a Burnsede dimension, which is an elem ent of the Burnside algebra$\mathrm{J}$$\zeta 7]\rangle$$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 is a commutative ring (with identity element),
and is a finite group :
$R$

$G$

1.1. Definition in terms of -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’), fromthe category G-set of finite -sets to the category R-Mod of -modtdes, satisfying thefollowing two conditions :1. The functor M maps disjoint unions to direct sums: if X and Y are finite G-sets,ifand
are the canonical inclusions from X and Y to the disjoint union
are mutual inverse
andXuY, then the maps
$G$

$R$

$G$

$\mathrm{i}_{X}$$\dot{\mathrm{s}}_{Y}$$(M_{*}(\mathrm{i}x), M_{*}(\mathrm{i}_{Y}))$$(M^{*}(\mathrm{i}_{X})\mathrm{A}f^{*}(\mathrm{i}_{Y}))$

isomorphisms of $R$-modules between $M(X)\oplus M(\mathrm{Y})$ and

2. If

Y[

$\underline{a}$$bX1^{1}$Z$M\langle X$

u Y).

$c$

$T$

$\overline{d}$is a cartesian (pullback) square of finite $G$-sets, them $M_{*}(b)M^{*}(a)=M’(d)M*(c)$ .
A morphism of Mackey functors is a natural transformation of bivariant functors. The.
Mackey functors for G over R form a category, denoted by$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$1.2. The Mackey algebra. The second definition of Mackey functors is due to
J. Th\’evenaz and P. Webb ([9]), who defined the Mackey algebra. The present exposition follows Chapter 4 of [2].
Let$\Omega_{G}=\mathrm{u}_{G}G/H=\{xH|x\in Gg\subseteq’$H$\subseteq G\}$

denote the disjoint union of all tansitive left -secs $G/H$ , where H runs through theset of subgroups of G.
If X is a finite (left) G-set, denote by $B(X)$ the Burnside group of X, i.e. theGrothendieck group of the category G-setJ.x of -sets over X. Similarly denote by.
$RB(X)$ the tensor product
$G$

$G$

$R\otimes \mathrm{z}^{B(X)}$

Date : March 11, 2005

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The Mackey algebra

where$\mu_{R}(G)$$\mu_{R},(G)$

over

$R$

$\mu_{R}(G)=RB(\Omega_{G}^{2})$,of the group$G$

is defined by(for diagonal $G$-action). The multiplication on
denotes the G-set
extension kom the pullback productis defined by$\Omega_{G}\mathrm{x}$$\Omega_{G}^{2}$$\Omega_{G}$$R\sim \mathrm{h}.\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$$\Omega_{G}\Omega_{G}\mathit{1}^{X}a\backslash b\backslash$$\Omega_{G}\Omega_{G}c\mathit{1}^{\mathrm{Y}}\backslash [searrow]^{d}$$\mathrm{x}$$\mapsto$$\Omega_{G}\Omega_{G}X\mathrm{x}_{\Omega c}\mathrm{Y}el\mathrm{h}^{f}$, and $e(x,y)$
The identity element for this multiplication is the G-set

where$\mathrm{X}\mathrm{x}$$\mathrm{Y}=\{(x,y)\in X\mathrm{x} Y|6(\mathrm{x})=c(y)\}$$=6(\mathrm{X})$, and $f(x,y)=d(y)$ .

$\Omega_{G}$.
where both maps are the identity map of
Now a Mackey functor for over is a left$\Omega_{G}$

$R$

$G$

$\mu_{R}(G)$-module.

1.3. Equivalence. Saying that the above two definitions are equivalent means thatthe categories
and &r(G}-Mod are equivalent, and this equivalence can be-module, and that X is a finite G-set. Then
seen as follows: suppose that V is a$=RB(\Omega_{G}\mathrm{x}X)$$RB_{X}$-module,has a natural structure of leftthe -module
product.
settingThuspullback
obtained by -linear extension from the obvious$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$$\mu_{R}(G)$$R$$\mu_{R}(G)$$R$$6(x)=\mathrm{H}\mathrm{o}\mathrm{m}_{l^{l}R(G)}(RB_{X},$V), in the sense of Dress.
defines a Mackey functor
Conversely, if M is a Mackey functor in this sense, then-module (see Section 4.3 of [2] for details).
structure of$F_{V}$$M(\Omega_{G})$has a natural$\mu_{R}(G)$1.4. Tensor product of Mackey functors. If M, N, and P are Mackey functors:
G over R, a bilinear morphisrn : M, N $arrow P$ is a collection of R bilinear$M(X)$ x, for any finite $G$-sets X and Y, which are moreover bivariant
can be defined as the solution
with respect to X and Y. The tensor productto the universal problem of bilinear morphisms :this means that the set of bilinear
morphisms from M,N to P is in one to one correspondence with the set of morphismsof Mackey functors fiiom
to P (Proposition 1,8.2 of [2]).
If X is a finite G-set, thencan be computed as follows:$\mathrm{f}\mathrm{i}\}\mathrm{r}$$\varphi X,Y$$\varphi$$N(\mathrm{Y})arrow P(X\mathrm{x}\mathrm{Y})$$M\otimes N\wedge$$M\otimes N\wedge$$M\otimes N\wedge(X)$$M \otimes N(X)=\wedge(\bigoplus_{Yarrow X},M(\mathrm{Y}\}\otimes N(Y))/J$

,

I 09

where $J$ is the -submodule generated by expressions
and $[M^{*}(a)(u’)\otimes v’]_{Y,f}-[u’\otimes N_{*}(a)(v’)]_{Z,g}$ , for every commutative triangle of finite
G-sets$[M_{*}(a)(u)\otimes v]z_{\mathit{9}},-[u\otimes N^{*}(a)(v)]_{Y,f}$$R$$\mathrm{Y}\frac{a}{\mathrm{h}_{X}fl_{g}}Z$

for every $u\in M(Y)$ , $v\in N(Z)$ , $u’\in M(Z)$ , $v’\in N(Y)$ , where e.g. $[M_{*}(a)(u)\otimes v]_{Z,g}$denotes the element $M_{*}(a)$ (u)&v of the component $M(Z)\otimes N(Z)$ indexed by : $Zarrow X$
in the direct sum.
The tensor product of Mackey functors is commutative (or symm etric), and also
ciative. The Burnside functor $RB$ is an identity for this tensor product, which means
are both isomorphic to the identity functor ofandthat the functors
Mackfl(G) (see Section 2.4 of [2] for details),$g$$-\otimes RB\wedge$$RB\otimes^{\wedge}-$

1.5- The Dress construction. Let M be a Mackey functor for G over R, and let
obtained by composition of M with the
X be a finite G-set. The bivariant functorx X of G-set is a Mackey functor for G over R. This construcendofunctor Y}, called the Dressis an endofunctor of the categorytion Idx : hf
(Lemma3-1.1 of [2]).
is self adjoint
construction associated to X. This functor$M_{X}$$arrow \mathrm{Y}$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$$\mapsto M_{X}$$\mathrm{I}\mathrm{d}_{X}$

1.6. Internal Hon. If M and N are Mackey functors for G over R, the functor
It is another Mackey functor for G over R,
whose value at the G-set X is

$7t(M,$N) was defined in Section 1.3 of [2].

$\mathcal{H}(M, N)(X)=\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)}(M, N_{X})$. ItN) is an internal Hom in the category
The construction (M, N)M,
iffollowing
sense:
functors,intheMackeyis right adjoint to the tensor product ofMackeyofisomorphisms
are
R,
there
then
N, and P are Mackey functors &r G overfunctors$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$$\mathrm{k}arrow?t(M,$$\mathcal{H}(M\otimes N, P)\cong \mathcal{H}(\wedge N, H(M, P))$

which are natural in M, N, and P.
In the same situation, there is also a composition morphismP)$\gamma:\mathcal{H}(M, N)\otimes \mathcal{H}(N, P)\wedgearrow H(M,$,defined as follows:let X be a finite G-set. Then
is a quotient of the direct s um
whereas$?\{(M,P)(X)=\mathrm{H}_{\mathrm{o}\mathrm{m}_{\mathrm{M}\mathrm{a}\ _{R}(G)}}(M, P_{\mathrm{Y}}.)$$H(M, N)\otimes \mathcal{H}(N, P)\wedge(X)$I$= \bigoplus_{Yarrow {}^{t}x}\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)}(M, N_{Y})\otimes \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\rangle(N,fl\prime 1}$

and b : N
Fix some G-set (Y, f) over X (where f : Y $arrow X$ ), and let a : Mofelement
the
of
by
image
the
be some morphisms of Mackey functors. Then
$M\wedge$aatevaluation
whosemorphism
f)
(Y,
is
the
Px
by
the component of indexed
composition
the$arrow
P(Z$byxX)$M(Z)$
obtainedG -set Z is the map$arrow P_{Y}$$arrow N_{Y}$$a\otimes b$$\gamma_{X}$$\Sigma$$\underline{bz\mathrm{v}}P(ZY^{2})P(Z\mathrm{Y})P(ZX)\underline{P^{\cdot}(_{zyy}^{z_{1}y})}\underline{P_{*}(_{zf}^{z}\iota_{\mathrm{t}v\}}^{y})}$$M(Z)arrow N(a_{Z}Z\mathrm{Y})$

,

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Y.
denote the map $(z, y)\in Z\mathrm{x}$where for short: for any Mackey functor, there is
Finally, the Burnside functor is a left unit for$\mathrm{Y}\mapsto(z, y, y)\in Z\mathrm{x}\mathrm{Y}\mathrm{x}$$\mathrm{t}_{zyy}^{zy}\downarrow)$$?\mathrm{f}$an isomorphism $H(RB, M)\cong M$ .

2. Burnside trace and dimension, and shows that thisThe previous section recalls various constructions in
category is a closed symmetric monoidal category. In this general ffamework, 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.
$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$2.1. Dualizable objects. The dual DM of a Mackey functor NI for G over R is
defined byRB)DM$=\mathcal{H}(M,$(this notion is different ffom the notion of dual over R defined in Section 6.2.2 of [2])The isomorphism $\mathcal{H}(RB, M)\cong M$ gives a composition morphism

and M is called dualizable if
Conversely, there is a$j_{M}$: $DM\otimes M\wedgearrow 7\{(M,$M) ,$j_{M}$

is an isomorphism.

$\mathrm{m}\mathrm{o}\mathrm{r}1^{3\mathrm{h}\mathrm{i}\mathrm{S}\mathrm{m}}$$e_{M}$:, $ $\cong RB$$M\otimes DM\wedgearrow \mathrm{H}(\mathrm{R}\mathrm{B}$$)$.

2.2. Lemma :The Mackey functor M is dualizable if and only if M is finitely
generated ancl projective.
at the trivial C-set
Proof: (Sketch) Suppose that M is dualizable. Evaluating
gives an isom orphism$\overline{J}M$$(DM\otimes M)(\cdot)\wedgearrow \mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)}(M)$

Choosing an element in the left hand side mapping to the identity of M shows that
: M
andthere exist a positive integer n, finite -sets , and morphisms, for i {1,: RBn} such that for any G-set$G$$b^{(i)}$$arrow M_{Y_{\mathrm{t}}}$

(2.3)

$\in$

$a^{[i\rangle}$$\mathrm{Y}_{l}$$arrow RB_{Y_{\mathrm{t}}}$$Z$\ldots ,$\mathrm{I}\mathrm{d}_{k\mathrm{I}(Z\}}=\sum_{i=1}^{n}M_{*}(_{z}^{zy}1^{\mathrm{t}}\cdot)M^{*}(zy_{l}1)zy_{\mathrm{i}}y_{\mathrm{t}}\circ b_{ZY_{*}}^{\{i)}\circ a_{Z}^{(i)}$Using the adjunction
phismsgive morphisms
is equivalent to, the mor: $RB_{Y_{i}}arrow M$ , and one can check that equality 2.3$\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{h}1_{\partial \mathrm{C}}\mathrm{k}_{R}(G\}}(RB, M_{Y_{t}})\cong \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{N}1\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\rangle(RB_{Y_{\mathrm{a}}},M)}$$b^{[i)}$$\tilde{b}^{(i)}$$\mathrm{I}\mathrm{d}_{M}=\sum_{=1}^{n}\tilde{b}^{\langle i)}\circ a^{(\overline{t})}$, this shows that $M$ is a direct summand of RBx- Since $RB_{X}$ is
Settingfinitely generated and projective, so is $M$ .
$X=i=1\mathrm{U}Y_{i}n$111

Conversely, if $M$ is finitely generated and projective, then $M$ is a direct summandof some functor $RB_{X}$ , ) 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 iseasy, because$\mathrm{f}\mathrm{i}$$\mathrm{r}$$’\mu(RB_{X},RB)\otimes RB_{X}\cong\prime H\wedge(RB_{X}, RB)_{X}\cong H(RBX, RBx)$

Here the first isomorphism is a consequence of Lemma 7.2.3 of [2], which implies thatfor any Mackey functors $M$ and $N$ , and any finite G-set $X$ , one has thatand from the fact that $RB$ is a unit for the tensor product . The
0second isomorphism follows easily from the definitions of .
$(M\otimes N)_{X}\wedge$$\cong$$\otimes\wedge$$M_{X}\otimes N\wedge\cong M\otimes N_{X\tau}\wedge$$\mathcal{H}$2.4. Burnside trace and dimension. Let M be a dualizable (i.e. finitely generated. There is a sequence ofand projective) Mackey functor, and let fmorphisms$\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\}}(M)$$RBarrow \mathcal{H}(\tilde{\mathrm{r}}_{M}\lambda f, M)arrow DM\otimes Marrow M\otimes DMarrow M\sigma_{M}\wedge\otimes DM\wedge e_{M}arrow RBj_{M}^{-1}f\otimes \mathrm{I}\mathrm{d}\wedge$M) :is the unique morphism of Green functors from % toHere the morpbism
2.1.1(Proposition
functor
M)
is
a
Green
there is a unique such morphism, becauseof [2]), and RB is an initial object in the category of Green functors for G over $R$
comes from the commutativity of O.
(Proposition 2.4.4 of (2]). The morphism
Mackeyfunctors is an endomorphism of RB,ofmorphismsofThe composition these$RB(\cdot)$ at the trivial G-set,evaluation
theisomorphic
to
is
Since
$RB(G)$
gives
an element denoted by
R,
this
over
of
G
algebrato the Burnside$RB(G)$.
ofof $RB(G)$ ill be called the2.5. Definition and Notation :This element,itwill be called the Burnside
endomorphism
When
theBurnside trace offf.
and denoted by Bdim(M).
dimension of$\mathcal{H}(M,$$i_{M}$$\mathcal{H}(M,$$\sigma_{ff}$$\mathrm{i}.\mathrm{e}_{\backslash }$$\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)}(RB)$$\mathrm{B}\mathrm{t}\mathrm{r}(f)$$\mathrm{B}\mathrm{t}\mathrm{r}(f)$$=\mathrm{I}\mathrm{d}$$M_{\rangle}$

2.6. Proposition :Let M be a dualizable Mackey functor for G over R, let X bea finite G set, and let p : $RB_{X}arrow M$ and s : M $arrow RBx$ be morphis ms of Mackey
the element of $RB(X^{2})$ corresponding to the. Letfunctors such that, then :diagonal inclusion xof X into , ij f$p\circ s=\mathrm{I}\mathrm{d}_{M}$$\epsilon_{X}$$X^{2}.$$\mapsto(x_{1}x)$

Btr(f)$\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{h}1\mathrm{a}\mathrm{c}\mathrm{k}_{\mathrm{R}}(G)}$$=RB_{*}(\downarrow.)xRB^{*}(_{xx}^{x}\downarrow)s\mathrm{x}fxPx\langle\epsilon \mathrm{x}$).

is the map sending
Proof: First of all, the isomorphism$f.(\cdot)\in
RB(\cdot)$
of $\in RB(\cdot)$ in
image
$RB$
the
.
Now
to
the endomorphism of$M$ . The hypotheses imply thatmap
identityofis theof the componentof the identity map of $M$ is the element
the image by$Xarrow$in the direct sum$\mathrm{E}\mathrm{n}\mathrm{d}\mu \mathrm{a}\mathrm{t}\mathrm{k}_{R}\{G\}(RB)\cong RB(’)$

$f$

.

.

$H(M, M)(\cdot)=\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{A}4\mathrm{a}\mathrm{c}\mathrm{k}_{R}\{G)}(M)$$(j_{M}^{-1})$$s\otimes\tilde{p}$$\otimes$$\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}\{G)(RB, M_{X})$$\bigoplus_{Xarrow}.\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\}}(M, RBx)$.

defining $(DM\hat{\Phi}M)(\cdot)$ , where is the morphism $RBarrow M_{X}$ obtained by adjunctionis thebyfrom the morphism : $RB_{X}arrow M$ . The image of this element$Xarrow$um
directtheinof the component
element$\tilde{p}$$p$$\tilde{P}\otimes$$s$.

$s\otimes\tilde{p}$$\mathrm{s}$$\mathrm{H}\circ \mathrm{m}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}[G\}(M,RB_{X})}$$\bigoplus_{Xarrow}.\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{h}\mathfrak{i}\mathrm{a}\mathrm{c}\mathrm{k}_{R}[G)}(RB,M_{X})\otimes$$(\sigma M)$

112

is sot to, this element. By the map
defining
in the same component of the direct sum. And finally, by the map $(e_{M}.).$ , this is sentof $RB$, where : $M_{X}arrow RB$ is the morphism deduced byto the endomorphism
adjunction from : $Marrow RBX$
It follow that$(M\hat{\otimes}DM)(\cdot)$$\overline{p}\otimes$$(f\otimes \mathrm{I}\mathrm{d}).$$fx\tilde{P}\otimes$ $s$$s$$\tilde{s}$$\tilde{s}fx\tilde{p}$$s$$\mathrm{s}$So it is the image ofNow the map.

$\overline{p}$.

$=(\overline{s}f\mathrm{x}\tilde{p}).(\cdot)$$\mathrm{B}\mathrm{t}r(f)$.

$\in RB(\cdot)$by the map.

$RB(\cdot)\overline{p}arrow$ $M(X)arrow M(X)arrow RB(\cdot)f_{X}\tilde{\Leftrightarrow}$is the map$\mathrm{R}\mathrm{B}\{\mathrm{X}).RB(X)RB(X^{2})arrow M(X)\underline{RB^{*}(1)x}\underline{RB_{\sim}(_{xx}^{x}\downarrow 1}\mathrm{P}X$

and$RB_{*}(_{xx}^{x}\downarrow)RB^{*}(x!)$$(\cdot)=\mathrm{e}\mathrm{x}-$Moreover the map$\tilde{s}$.

,is the map$\mathrm{M}(\mathrm{X})arrow RB(X^{2})RB(X)arrow.RB(\cdot)\underline{RB^{\cdot}(_{xx}^{x}\downarrow)}RB.(x\downarrow)$

It follow that Btr(f) $=RB_{*}(x!)RB^{*}(_{xx}^{x}1)sx$ fxpx(\epsilon x), as was to be Shown.

0$\mathrm{s}$The following is the special case $M=RB_{X}$ : then

2.7. Corollary : Let X and Z be finitefrom Z to X. Let$G$$\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\rangle}(M)$$\cong RB(X^{2})$:-sets, and let a and b be maps of G-sets

$z$

$f=$

X

$X$

be the co responding element of $RB(X^{2})_{f}$ viewed as an endomorphism of RBx- Then
Btr(/) $=\{z \in Z|a(z)=b(z)\}$) $=X$ .
In particular Bdim(Rcase,
Proof: In thisone can suppose that the maps and are the identity map.
Btr(/)The result forfollows fiiom a straightforward computation, and the result for
Bdim(R $Bx$ ) is the special case $Z=X$ and0$B_{X}$

$p$

$s$

$)$

$a=b=\mathrm{I}\mathrm{d}\mathrm{x}-$is isomorphic to
2.8. Example :Suppose that X $=G/1$ . Then En$arrow
RB((G/1)^{2})$, maps, denoted by x
the group algebra RG : this isomorphism RG$\inG$
the element gof the canonical basis of RG to the element$\mathrm{d}_{hl_{\partial \mathrm{C}\mathrm{k}_{R}\{G)(RB_{X})}}$$\mapsto\hat{x}$113

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

whereis a Kronecker symbol, so in general Btr(x^)
map on the group algebra $RG$ .
is the usual$\delta_{g,1}$$=\mathrm{t}\mathrm{r}_{RG}(x)$

.

$G/1$

, where$\mathrm{t}\mathrm{r}_{RG}(x)$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$2.9. Remark :Through the equivalence 1.3, the category of Mackey functors is)-modules, there is the Hattoriequivalent to the category of $\mu_{R}(G)$ -modules. For(see [6] and [8]), which associates to any endomorphisin $f$
Stallings trace map
in the zero-thof a finitely generated projective $\mu_{R}(G)$ -module, an element), i.e.
Hochschild homology group of$\mu_{R}\langle G$$\mathrm{b}_{\mu R\{G)}$$\mathrm{b}_{\mu_{R}(G)}(f)$$\mu_{R}\langle G$$r\mathrm{b}_{\mu_{R}(G)}(f)\in HH_{0}(\mu_{R}(G))=\mu_{R}(G)/[\mu_{R}(G),\mu_{R}(G)]$One can show easily that with this equivalence, the Burnside trace Btr(/) is the image
by the mapof$\prime \mathrm{R}_{\mu R(G)}(f)$$HH_{0}(\mu_{R}(G))arrow RB(G)$induced by the “equalizer map” from $\mu_{R}(G)=RB(\Omega_{G}^{2}.)$ to $RB(G)$ , sending the element$z$

to the equalizer $\{z\in Z|a(z)=b(z)\}$ , 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$U\circ X=\{(u,x)\in U$

x$X|\forall g\in G,$ u. g$=u\Rightarrow g.$x $=x\}$and denote by $U\circ_{G}X$ the quotient of $U\circ X$ by the right action of G given by(u, x) .g $=(u$ .g,$g^{-1}.$x), $\forall(u, x,g)\in U$ x Xx $G$

X from $B(G)$ to $B(H)$ .
This construction extends to a map X
from O-set to H-set, which preserves
isafunctor
X
The construction
disjoint unions and pullback squares. Conversely, any functor G-set H -set with thesetwo properties is isomorphic to a functor , for some finite (H,$G)$ -biset U (see [1] for$\mapsto U\circ G$$\mapsto U\circ_{G}X$$\gamma_{U}$$arrow$$\gamma_{U}$$\det$ails).

By composition, the functor$\Gamma_{U}$$\gamma u$

induces a functor: MackH(G)defined by Yg $(N)=N\circ\gamma U$ , for any$arrow \mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G)$$\mathrm{V}[perp] \mathrm{I}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{y}$,functor $N$ for $H$ over

$R$

.

114

have left and right adjoints, respectively
3.2. Adjoint functors. These functors: an explicit, but rather complicated description of the functors
anddenoted by
was given in Chapter 9 of [2]. A simpler description ([4]) can be obtained a follows,
using the equivalence 1.3 : consider$\Gamma_{U}$$\mathcal{R}_{U}$$\mathcal{L}_{U}$$\mathcal{L}u$$RB_{U}=RB(\Omega_{H}\mathrm{x}\{U\circ c\Omega_{G}))$

This is a $(\mu_{R}(H),\mu_{R}(G))$ -bimodule, for the actions extending linearly the following(X, (a, b)) is an G-set over
, and that (Z, (e, f)) is an G-set overproducts : suppose that

G-set overfollowing diagram$\Omega_{G}\mathrm{x}\Omega_{G}$$\Omega_{H}\mathrm{x}\Omega_{H}$$\Omega_{H}$x, that$(\mathrm{Y}_{9}(c, d))$$(U\circ G\Omega G)$

is a. Build the$\int_{G}^{k}\backslash _{D}^{l}B$$X\swarrow^{/}g\backslash ^{h}\swarrow\backslash U\circ_{G}’ \mathrm{Y}\mathrm{h}j$$\int_{\Omega_{H}}a\backslash ^{b}ae\int_{\Omega_{H}}\backslash _{j}^{U\mathrm{o}_{H^{C}}}\int_{\Omega_{G}U\circ c}\mathrm{h}_{\mathrm{o}_{G}}^{U\mathrm{o}_{H}d}U\Omega_{G}$where all the squares are pull-back squares. Then the left and right actions on $RB_{U}$ aredefined by( ,$X$ $(a, b)$).

, $(c, d))=(E$ , $(agk, (U\circ_{H}d)jl\dot{)})$$(Z,$ $(e, f)).(\mathrm{Y}$It is easy to this from this definition that there is an isomorphism of left$\mu R(H)\sim \mathrm{n}\mathrm{l}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}$$RB_{U}\cong RB_{U\mathrm{o}_{G}\Omega_{G}}(\Omega_{H})$In particular $RB_{U}$ is projective and finitely generated as $\mu_{R}(H)$ -module. Moreover, one
can show that if $N$ is a Mackey functor for $H$ over $R$ , then the natural isomorphism of
R-modules$(N\circ U)(\Omega_{G})=N$($U\circ_{G}$Qc)$\cong \mathrm{H}\mathrm{o}\mathrm{m}_{\mu_{R}(H)}(RB_{U},N(\Omega_{H}))$is an isomorphism of-modules. Thus if $M$ is a Mackey functor for
gives by standard arguments$\mu_{R}(G)$$G$

over , this$R$$\mathcal{L}_{U}(M)(\Omega_{H})\cong RBrt$ $\otimes_{\mu n(G)}M(\Omega_{G})$More generally, if $Z$ is a finite if-set, the same argum ent shows that
(3.3)

where the rightabove$Lu(M)(Z)\cong RB(Z\mathrm{x} U\circ_{G}\Omega_{G})\otimes_{\mu_{R}\{G)}M(\Omega_{G})$$\mu_{R}${ $G)-$ module structure on$RB(Z\mathrm{x}U\circ_{G}\Omega_{G})$,is given by pullback as

115

3.4. Remark : A similar argument (see [4]), considering the $(\mu R(G),\mu R(H))$ -bimodtde$RB_{U}^{\#}=\mathrm{H}\mathrm{o}\mathrm{m}_{\mu R\{H)}(RBu,\mu_{R}(H))$

gives the description of the right adjoint$R_{U}(M)(\Omega_{H})$$\mathcal{R}u$, by$\cong \mathrm{H}\mathrm{o}\mathrm{m}_{\mu_{\mathrm{R}}\langle G)}(RB_{U}^{\mathrm{Q}},M(\Omega_{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
)

$=$

$(M\otimes N)\wedge$$=$

Bdim(M

Bdim

Bdim(

$\oplus N$$\mathrm{M}$$=$

Bdim $M+$ Bdim $N$
BdimM BdimM
BdimM$\cdot$0

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,biset. Let M be a finitely generated projective Mackey functor for G over R, and let. Thenf$G)\sim$$\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{h}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(G\}}(M)$$\mathrm{B}\mathrm{t}\mathrm{r}(\mathcal{L}_{U}(f))=U\circ c$

Proof;

Btr(f)

Since $M$ is finitely generated and projective, there exist a finite O-set $X$ andmorphisms : $RB_{X}$ $arrow M$ and $s:Marrow RB_{X}$ such thatthe notation of Proposition 2.6$p$

Btr(f)

Applying the functor$\mathcal{L}_{U}(p)$:$\mathcal{L}_{U}(p)$$\circ \mathcal{L}_{U}(s)$fxPx. In this case, with$(\epsilon x)$

gives morphisms$\mathcal{L}_{U}(s)$$\mathcal{L}_{U}(RB_{X})arrow \mathcal{L}_{U}(M)$that{see Lemma 5.4 of [3]$\mathrm{s}^{\tau}\iota \mathrm{z}\mathrm{c}\mathrm{h}$$\mathcal{L}u$$=RB_{*}(\downarrow.)xRB^{*}\mathrm{t}_{xx}^{x}\downarrow)sx$$p\circ s=\mathrm{I}\mathrm{d}_{\mathrm{M}}$$=\mathrm{I}\mathrm{d}_{L\sigma\{M\}}$:$\mathcal{L}_{U}(M)arrow \mathcal{L}_{U}(RB_{X})$. Now there is an isomorphism of Mackey functors$)$$\mathcal{L}_{U}(RB_{X})\cong RB_{U\mathrm{o}_{G}X}$

which cau be seen as follow : if$\mathrm{s}$$\pi_{Z,X}$:Cu (RBx)(Z)$Z$

is a finite$\cong \mathrm{R}\mathrm{B}(Z \mathrm{x} U\mathrm{o}G\Omega G)$$\mathrm{H}$-set, then there is a map$\otimes_{\mu_{R}(G)}RB(\Omega_{G}\mathrm{x} X)arrow RB(Z\mathrm{x}U\mathrm{o}_{G}X)$

is some H-set over $Z\mathrm{x}Uoc\Omega c$ and is some G-set over
. To check that this map, to the pullback product of and 717(T) over
$X=\Omega_{G}$ , where it is trivial, and thenis an isomorphism, first consider the case
. The general case follows, becaus
the case where $X$ is a disjoint union of copies ofsending$\Omega_{G}\mathrm{x}X$$S$(& $T$ , where$T$$\mathrm{S}$$U\circ_{G}\Omega_{G}$$S$$\pi z,x$$\Omega_{G}$11$\mathrm{G}$any G-set is a subset of such a disjoint union, and the corresponding map
the retract of an isomorphism.
Denote by the if-set $U\circ cX$ . By definition now$\mathrm{t}\mathrm{t}\mathrm{z},\mathrm{x}$is then$T$

Btr$(\mathcal{L}_{U}(f))=RB_{*}(\mathrm{s}\downarrow.)RB^{*}(_{\mathrm{t}t}^{t}\downarrow)\mathcal{L}_{U}(s)_{T}\mathcal{L}_{U}(f)\tau \mathcal{L}u(p)\mathrm{r}(\epsilon\tau)$by the above isoTo compute this, the first thing to do is to find a preimage of,is an inclusion. For:: suppose thatmorphism$\epsilon\tau$$\tilde{J}\in\{1, \ldots, n\}$$Xarrow j=1\coprod^{n}\Omega_{G}$$\omega$$\pi\tau,x$. Denote bycomponent of
the inverse \’image by of theinto $X$ . Then
, and by the inclusion of
the restriction of todenote by$X_{j}$$\omega$$X_{j}$$\omega$$\tilde{J}^{- \mathrm{t}\mathrm{h}}$$\coprod_{\overline{J}^{=1}}^{n}\Omega_{G}$$X_{j}$$\mathrm{i}_{j}$$\pi_{T,X}^{-1}(\epsilon_{T})=\sum_{\mathrm{j}=1}^{n}$

The image of this elem ent by$\mathcal{L}u(s)\tau \mathcal{L}_{U}(f)\tau L_{U}$ $\langle p)\tau$is equal to$\sum_{j=1}^{n}$and the image of this by the isomorphism$\pi\tau,x$

is equal to the element$S$ $= \sum n$$i=1$

of $RB\{T^{2}$ ), where$\mathrm{x}_{U\mathrm{o}_{G}\Omega_{G}}$denotes the pullback product over$\mathrm{B}\mathrm{t}\mathrm{r}(\mathcal{L}y(f))=RB_{*}(t\downarrow.)RB^{*}(_{tt}^{\mathrm{f}}\iota)(S)$

Since the functor$\gamma_{U}$preserves pullbacks, this is equal to$U \circ_{G}RB_{*}(\downarrow.)xRB^{*}(_{x}x\downarrow_{x})[\sum_{j=1}^{n}$$U\circ_{G}\Omega_{G}$. Then$\omega j$

117

where

$\mathrm{x}_{\Omega_{G}}$denotes the pullback product overand since , $f$ , and
$s$

$p$

$\Omega_{G}$. Now

are morphisms of Mackey functors, it follows that$=RB_{X*}(\omega_{j})RB_{X}^{*}(\overline{\iota}_{j})s_{X}fxP\mathrm{x}(\epsilon_{X})$,and this is also equal to$\cross xs_{X}$$\Omega_{G}$where$\mathrm{x}_{X}$fxPx(\epsilon x),$X$

is the pullback over $X$ . Finally$\mathrm{B}\mathrm{t}\mathrm{r}(\mathcal{L}\mathrm{u} (f))$

is equal to$U\circ_{G}RB_{*}(x\downarrow.)RB^{*}$

and by associativity of pullback products, this is equal to$\mathrm{x}_{X}s_{X}fxPx(\epsilon_{X})$Now$=\epsilon x$

and$\epsilon x$

is the identity element for$\mathrm{X}\chi$. HenceBtr(/) $(f)\}=U\mathrm{o}GRB_{*}(x!)RB^{*}(_{x^{1}x}^{x})sxf_{X}px(\epsilon x)=U\mathrm{o}_{G}$ Btr(/)

as vas to be shown.

,$\square$118

3.7. -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 psubgroups.
of G, and isomorphism classes of indecomposable trivial source
the ProThus if V is such an indecomposable trivial source module, denote byjective Mackey functor for G over k such that $P_{V}(1)=V$ . It is natural to look at the
Burnside dimension of Py.
Proposition 3.6 involves the special case of restriction to a subgroup :if H is a
subgroup of G, and if U $=G$ , view ed as an (if, G)-biset by left and right multiplication,: Mack#(G)\rightarrowis the restriction functor
then the corresponding functor
is also the
H-set
:
Gset
The
functor[2]).
(see Section 9.9.1 ofrestriction functor.
is a permutation
Suppose that Q is a -subgroup of G. Then the module$kQ$ -module. So there is a finite Q-set Xq, such that$k’G\sim \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathrm{s}$$P_{V}$$\mathrm{R}\mathrm{a}\mathrm{e}_{H}^{G}$$\mathcal{L}u$$arrow$$\gamma_{U}$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{R}(H)$${\rm Res}_{Q}^{G}V$$\mathrm{R}\epsilon \mathrm{s}_{Q}^{G}V\cong k\lambda_{Q}^{r}$,does not depend on the choice of such a Q-stableand up to isomorphism, the Q-set
basis. In particular, this gives a well defined element $X_{Q}\in kB(Q)$ . Then obviously, if$X_{Q}$$Q’\subseteq Q$${\rm Res}_{Q^{t}}^{Q}X_{Q}=X_{Q’}$and if x $\in P$ , then xXQ $=XoeQ$ .
Hence the sequence, indexed by the setof G, is an element of (. The map$\beta_{V}=(XQ)_{9\in\underline{s}_{\mathrm{p}}(G)}$$\lambda_{G}$:$arrow\lim_{Q\in_{-}s_{\varphi}\{G)},kB(Q))^{G}$$\underline{s}_{p}(G)$of all p-subgroups$(u_{Q}) \in(\varliminf_{Q\in\underline{\epsilon}_{\mathrm{p}}(G\}}kB(Q))^{G}\mapsto-\sum_{Q\in_{=^{\mathrm{Q}}\mathrm{p}}(G)/G}\frac{\tilde{\chi}]Q,.\mathrm{k}(c)}{|N_{G}(Q)\cdot Q|}.\mathrm{I}\mathrm{n}\mathrm{d}_{Q}^{G}u_{Q}\in kB(G)$where ] Q, .
is the reduced Euler-Poincare’ characteristics of the poset ofpsubgroupsof G containing Q as a proper subgroup, is injective, and right inverse to the map$\tilde{\chi}$$.\mathrm{g}_{\{G)}$$\beta G$:X$\in kB(G)\mapsto({\rm Res}_{Q}^{G}X)_{Q\in\underline{s}_{\mathrm{p}}(G\}}$and evaluation at the: since
Now back to the projective Mackey functor
trivial subgroup commute, the evaluation at the trivial subgroup of the Mackey functor
is isomorphic to kXq. Thus
$P_{V}$${\rm Res}_{Q}^{G}$${\rm Res}_{Q}^{G}Pv$${\rm Res}_{Q}^{G}P_{V}\cong kB_{X_{\mathrm{Q}}}$

It follows that, and this gives finaly$\mathrm{R}\mathrm{a}\mathrm{e}_{Q}^{G}\mathrm{B}\dim(P_{V})=Xq$$p_{G}(\mathrm{B}\dim(P_{V}))=\beta_{V}$3.8. Remark :It is natural to ask if the stronger result$\mathrm{B}\dim\langle P_{V})=\lambda_{G}(\beta_{V})$holds. One can show it is the case if
characteristic 0 (e.g. if is perfect)
$k$

$k$

is the residue field of a discrete valuation ring of

119

3.9. Tensor induction. In this section, the ground riq R is the ring of integers.
If G and H axe finite groups, if U is a finite {H, $G)$ -biset, one can define another fun ctor
, called tensor induction, associated to U (see [3] for: Macks(G)details). This functor is not additive, but rather multiplicative (i.e. it commutes withthe tensor product of Mackey functors).
, defined on the subcatIt is defined by extending the functor
additive)functor defined
(non
exact
right
functors,toaMackeyegory of permutation. It follows in particular from this construction thaton$\mathbb{Z}$$T_{U}$$arrow \mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}(\mathrm{i}\mathrm{T})$$B_{X}\mapsto B_{\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{n}_{G}(U^{\mathrm{o}p},X\rangle}$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{k}_{\mathrm{Z}}(G)$$T_{U}(B_{X})=B_{\mathrm{H}\mathrm{o}\mathrm{m}_{G}(U^{\sigma \mathrm{p}},X)}$

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

Proposition: Let G and H be finite groups, and let U be a finite ( H,G)-bi$et
Let moreover M be a finitely generated projective Mackey functor for G over Z. If, thenf
3,10.

$\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{M}\mathrm{a}\mathrm{d}\mathrm{c}\mathrm{z}(G\}}(M)$

Btr $(T_{U}(f))=$ Homo ($U^{o\mathrm{p}}$, Btr( $f)$ ): $B(G)arrow B(H)$
3.11. Remark:Here in the right hand side, the map-sets. Thismap
ondefinednaturalofthering
Burnsideis the extension to the
3 of [3], by
maps,
or
as
in
Sectionpolynomialconsideringbyextension can be achieved
considering finite -posets and associated Lefschetz invariants.
$\mathrm{H}\mathrm{o}\mathrm{m}_{G}(U^{op}, -)$

$G$

$G$

References[1] S. Bouc. Construction de foncteurs entre cat\’egories de183(0239);737-825J 1996.

$G$-ensembles. J. of Algebr\^a[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 CJ.R.M (Lum iny)
http:$//\mathrm{w}\mathrm{w}$. math . j ussieu .

$\mathrm{f}\mathrm{r}/\sim \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{c}/\mathrm{b}\mathrm{i}$set. pdf, 2002.

[5] A. Dress. Contributions to the theorry of induced representations, volume 342 of
Lecture Notes in Mathematics, pages 183-240. Springer-Verlag, 1973.
[6] A. Hattori. Rank element of a projective module. Nagoya J. Maths, 25:113-120,
1965,[7] J. P. May, Picard groups, Grothendieck rings, and Burnside rings of categories.
Advances in Mathematics, 163:1-16, 2000.

120[8] J. R. Stailings. Centerless groups- an algebraic formulation of Gottlieb’s theorem.
Topology, 4:129-134, 1965.

[9] J. Th\’evenaz and P. Webb. The structure of Mackey functors. Trans. Amer. Math.
Sac., 347(6):1865-1961, June 1965.