hochschild constructions for green functors

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Hochschild Constructions for Green FunctorsSerge Bouc a ba Institut de Mathématiques de Jussieu , Université Paris , Paris, Franceb Institut de Mathématiques de Jussieu , Université Paris , 7-Denis Diderot, 75251, Paris,Cedex 05, FrancePublished online: 01 Feb 2007.

To cite this article: Serge Bouc (2003) Hochschild Constructions for Green Functors, Communications in Algebra, 31:1,403-436, DOI: 10.1081/AGB-120016767

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Hochschild Constructions for Green Functors

Serge Bouc*

Institut de Mathematiques de Jussieu, Universite Paris, Paris, France

ABSTRACT

Let G be a finite group, and R be a commutative ring. If A is a Greenfunctor for G over R, and G is a crossed G-monoid, then the Mackeyfunctor AG obtained by the Dress construction has a natural struc-ture of Green functor, and its evaluation AG(G ) is an R-algebra. Thisframework involves as special cases the construction of the Hochs-child cohomology algebra of the group algebra from the ordinarycohomology functor, and the construction of the crossed Burnsidering from the ordinary Burnside functor. This article presents someproperties of those Green functors AG, and the functorial relationsbetween the corresponding categories of modules. As a consequence,a general product formula for the algebra AG(G) is stated.

Key Words: Green functor;Burnside functor;Hochschildcohomology.

AMS Classification: 16E40; 19A22.

*Correspondence: Serge Bouc, Institut de Mathematiques de Jussieu, UniversiteParis, 7-Denis Diderot, 75251, Paris, Cedex 05, France; E-mail: [email protected].

COMMUNICATIONS IN ALGEBRA�

Vol. 31, No. 1, pp. 403–436, 2003

403

DOI: 10.1081/AGB-120016767 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

Let G be a finite group, and R be a commutative ring. Thispaper provides a possible generalization to any Green functor for Gover R of the construction of the Hochschild cohomology ringHH�(G, R) of the group algebra RG from the ordinary cohomology func-tor H�(�, R). This construction also involves as a special case the con-struction of the crossed Burnside ring of G from the ordinary Burnsidefunctor.

The general abstract setting is the following: let A be a Green functorfor the group G. Let Gc denote the group G, on which G acts by conjuga-tion. Suppose G is a crossed G-monoid, i.e., that G is a G-monoid over theG-group Gc. Then the Mackey functor AG obtained from A by Dressconstruction has a natural structure of Green functor. In particularAG(G) is a ring.

In the case where G is the crossed G-monoid Gc, and A is thecohomology functor (with trivial coefficients R), the ring AG(G) is theHochschild cohomology ring of G over R. If A is the Burnside functorfor G over R, then the ring AG(G) is the crossed Burnside ring of Gover R.

This article presents some properties of those Green functors AG, andthe functorial relations between the corresponding categories of modules.Instead of exposing first the general framework, and since the Greenfunctor setting is rather abstract, the first sections only deal with thetwo above special cases.

In particular, they provide a new proof for a result of Siegel andWitherspoon (1999), which was conjectured by Cibils (1997) and Cibilsand Solotar (1997). This result describes the multiplicative structure ofthe Hochschild cohomology ring of a finite group in terms of cup pro-ducts, transfers and restrictions on the ordinary cohomology.

The case of the crossed Burnside ring can be viewed as a conceptualreason for the existence of a ring homomorphism from the crossed Burn-side ring to the center of the Mackey algebra, which was used in Bouc(preprint) to determine the p-blocks of the Mackey algebra.

The next sections present the general theoretic setting: first the defini-tion of crossed G-monoids, and then the construction of the Green func-tor structure on AG, for a Green functor A and a crossed G-monoid G.A general formula for the product in the ring AG(G) is stated. Here onemore example is built: it is possible to attach to any normal subgroupN of a finite group G a kind of ‘‘relative’’ Hochschild cohomology ringHH�(G, N, R), which is the ordinary cohomology ring for N¼ 1, andthe usual Hochschild cohomology ring for N¼G.

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The next section is an exposition of the functorial relations betweencategories of A-modules and AG-modules. These relations explain in par-ticular the existence of a natural ring homomorphism from the crossedBurnside ring to the center of the Mackey algebra.

2. GREEN FUNCTORS AND G-SETS

When dealing with Mackey and Green functors, two different (butequivalent) points of view are possible: in the first one, a Mackey functorfor G over R is a collection of R-modules M(H), indexed by thesubgroups H of G, together with transfer maps tKH :M(H)!M(K ), withrestrictions maps rKH : MðKÞ !MðHÞ, and conjugation maps cx,H :M(H )!M(xH ), whenever H�K are subgroups of G, and x2G. Thesemaps are subject to a list of conditions: transitivity, commutation, trivi-ality, and Mackey formula.

In the second point of view, a Mackey functor for G over R is abivariant functor from the category of finite G-sets to the category ofR-modules, which transforms disjoint unions into direct sums, and hassome compatibility property with cartesian squares (see Bouc (1997,1.1.2) for details).

Similarly, a Green functor A is a Mackey functor ‘‘with a compatiblering structure’’: each evaluation A(H) has a ring structure (with unit), fora product (a, b) 7! a.b, restrictions maps and conjugation maps are homo-morphisms of rings (with unit), and the Frobenius relations on productsand transfers hold. There is also a definition of Green functors in terms ofG-sets (Bouc, 1997, 2.2):

Definition 2.1. Let R be a commutative ring. A Green functor A (over R)for the group G is a Mackey functor (over R) endowed for any G-sets X andY with bilinear maps

AðXÞ � AðY Þ ! AðX � YÞ

denoted by (a, b) 7! a� b which are bifunctorial, associative, and unitary, inthe following sense:

� (Bifunctoriality) If f : X!X 0 and g :Y!Y 0 are morphisms ofG-sets, then the squares

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AðX Þ � AðYÞ ��!� AðX � YÞ

A�ð f Þ�A�ðgÞj#

j#A�ð f�gÞ

AðX 0Þ � AðY 0Þ ��!� AðX 0 � Y 0Þ

AðX Þ � AðYÞ ��!� AðX � YÞ

A�ð f Þ�A�ðgÞ"j "jA�ð f�gÞ

AðX 0Þ � AðY 0Þ ��!� AðX 0 � Y 0Þ

are commutative.� (Associativity) If X, Y and Z are G-sets, then the square

AðXÞ � AðY Þ � AðZÞ ��!IdAðXÞ�ð�ÞAðXÞ � AðY � ZÞ

ð�Þ�IdAðZÞj#

j#�

AðX � Y Þ � AðZÞ ��!� AðX � Y � ZÞ

is commutative, up to identifications (X�Y )�ZffiX�Y�Z’X� (Y�Z).

� (Unitarity) If � denotes the G-set with one element, there existsan element eA2A(�) such that for any G-set X and for any a2A(X)

A�ðpX Þða� eAÞ ¼ a ¼ A�ðqX ÞðeA � aÞ

denoting by pX (resp. qX) the (bijective) projection from X�� (resp.from ��X) to X.

If A and B are Green functors for the group G, a morphism f (of Greenfunctors) from A to B is a morphism of Mackey functors such that forany G-sets X and Y, the square

AðX Þ � AðYÞ ��!� AðX � Y ÞfX�fYj#

j#fX�Y

BðX Þ � BðYÞ ��!� BðX � YÞ

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is commutative. The composition of morphisms of Green functors is definedin the obvious way. The category of Green functors for G over R is denotedby GreenR(G).

Remark 2.2. One can pass from the second definition of Green functorsto the first one by the following procedure: suppose that A is a Greenfunctor in the second sense. Then setting A(H)¼A(G=H), for a subgroupH of G, one can define the maps tKH and rKH , for H�K by

tKH ¼ A�ðpKHÞ rKH ¼ A�ðpKHÞ

where pKH is the natural projection. If x2G, then the conjugation mapcx,H is defined by

cx;H ¼ A�ðgx;HÞ

where gx,H :G=H!G=xH maps gH to gx�1 xH. One can also define aproduct (a, b) 7! a.b on A(H) by

a:b ¼ A�ðdG=HÞða� bÞ

where dG=H is the diagonal inclusion from G=H to (G=H)� (G=H).

Remark 2.3. One can pass from the first definition to the second by thefollowing construction (Bouc, 1997, 2.3): suppose that A is a Green func-tor in the first sense. Then setting A(G=H)¼A(H) leads by linearity to adefinition of A(X) for any finite G-set X: one can set

AðXÞ ¼Mx2X

AðGxÞ !G

where the exponent denotes fixed points under the natural action of G onLx2XA(Gx) by permutation of the components, and Gx is the stabilizer

of x in G. If f :X!Y is a map of G-sets, then the maps A�( f ) andA�( f ) are defined by

A�ð f ÞðuÞy ¼X

x2½Gyn f �1ðyÞ�tGyGx ðuxÞ for u 2 AðX Þ and y 2 Y

A�ð f ÞðvÞx ¼ rGf ðxÞGx

vf ðxÞ for v 2 AðY Þ and x 2 X


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where Gy is the stabilizer of y in G, and [Gy n f�1(y)] is a set of represen-tatives of the orbits of Gy on f�1(y). The summation in the formula doesnot depend on the choice of such a set of representatives.

The product of elements a2A(X) and b2A(Y ) is defined as the ele-ment of A(X�Y ) whose component in A(G(x, y)), for (x, y)2X�Y, isequal to

ða� bÞx;y ¼ rGxGðx;yÞðaxÞ:rGyGðx;yÞðbyÞ:

One can also express the previous formulae after choosing sets of repre-sentatives of orbits of G on X, Y, and X�Y. For instance, there is anisomorphism

AðX Þ ffiM

x2½GnX �AðGxÞ

where [GnX] is a set of representatives of orbits of G on X. Now if H andK are subgroups of G, then there is an isomorphism of G-sets

ðG=HÞ � ðG=KÞ ffiG

x2½HnG=K �G=ðH \x KÞ

hence an isomorphism

A ðG=HÞ � ðG=KÞð Þ ffiM

x2½HnG=K�AðH \x KÞ:

and the product a� b of a2A(G=H) and b2A(G=K ) is the elementMx2½HnG=K �

rHH\xKa : rxKH\xK

xb 2M

x2½HnG=K �AðH \ xKÞ

where xb¼ cx,K(b).

Remark 2.4. There is an alternative procedure to pass from the first defi-nition of Green functors to the second one, which is often easier to handlefor computations. It uses co-invariants rather than invariants: if A is aGreen functor for G in the first sense, and X is a finite G-set, then onecan set

AðX Þ ¼Mx2X

AðGxÞ !

G

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where the subscript denotes co-invariants (i.e., the largest quotientwith trivial G-action). If x2X and u2A(Gx), denote by [x, u]G theimage inA(X) of the element u ofA(Gx). Then if f :X!Y is amap ofG-sets

A�ð f Þð½x; u�GÞ ¼ ½ f ðxÞ; tGf ðxÞGx

u�G for x 2 X and u 2 AðGxÞA�ð f Þð½y; v�GÞ ¼

Px2Gyn f �1ðyÞ

½x; rGy

Gxv�G for y 2 Y and v 2 AðGyÞ:

The product of the elements [x, u]G and [y, v]G is given by

½x; u�G � ½y; v�G ¼X

w2½GxnG=Gy�

�ðx;wyÞ; rGx

Gx\Gwyu:r

Gwy

Gx\Gwy

wv�G:

Example 2.5. Let X and Y be finite G-sets, and set

H�ðG;RX Þ ¼M1n¼0

HnðG;RX Þ:

This is usually denoted by H�(G, RX), but this notation could be confus-ing here with the Mackey functor formalism. Then the cup-product oncohomology gives maps

H�ðG;RX Þ �H�ðG;RY Þ ! H�ðG;RX R RY Þand identifying RXRRY with R(X�Y ), this gives cross productmaps

H�ðG;RX Þ �H�ðG;RY Þ ! H�ðG;RðX � Y ÞÞ:It is easy to check that this gives H�(G, �) a Green functor structure, andthat if K is a subgroup of G, the induced ring structure on

H�ðG;RðG=KÞÞ ffi H�ðK ;RÞcoincides with the ordinary ring structure of H�(K, R) for cup-products.

Example 2.6. Let B denote the Burnside functor. If X is a finite G-set,then B(X) is the Grothendieck group of the category of G-sets over X.The obvious product

Z T# ; #X Y

0@

1A 7! Z � T

#X � Y


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extends linearly to a cross product B(X)�B(Y )!B(X�Y ), which givesB its structure of Green functor (Bouc, 1997, 2.4).

3. THE DRESS CONSTRUCTION

The Dress construction is a fundamental endofunctor of the categoryof Mackey functors for G over R, defined as follows. Let G be a fixedfinite G-set. If M is a Mackey functor for G over R, then the Mackeyfunctor MG is the bivariant functor defined on the finite G-set Y by

MGðY Þ ¼MðY � GÞ:

If f :Y!Z is a map of G-sets, then

ðMGÞ�ð f Þ ¼M�ð f � IdGÞ ðMGÞ�ð f Þ ¼M�ð f � IdGÞ:

One checks easily (Bouc, 1997, 1.2) that MG is a Mackey functor for Gover R.

It follows from the definitions that the evaluation of MG at the trivialG-set �¼G=G is equal to

MGð�Þ ¼MðGÞ:

Suppose now that G is a finite G-monoid, i.e., that G is a finite monoid onwhich G acts by monoid automorphisms. Let A be a Green functor for Gover R. Apart from the product on A(G) defined by the Green functorstructure, defined by

ða; bÞ 2 AðGÞ 7! a:b ¼ A�g#g; g

0@

1Aða� bÞ;

where� g#g;g

�denotes the diagonal inclusion of G into G�G, there is

another natural product on A(G), defined by

ða; bÞ 2 AðGÞ 7! a�G b ¼ A�g1; g2#

g1g2

0@

1Aða� bÞ

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where A�� g1;g2#

g1g2

�denotes the image by A� of the multiplication G�G!G.

Let� �#1G

�denote the map from � to G sending the unique element of � to

the unit of G. Set eAG¼A�

� �#1G

�(eA).

Lemma 3.1. The above product gives A(G) a ring structure, with unit eAG.

Proof. This will follow from Theorem 5.1 below, since A(G) will be theevaluation at the trivial G-set of the Green functor AGu. &

Notation 3.2. In the sequel, for a Green functor A for G over R, and aG-monoid G, the ring A(G) will always be understood as the R-moduleA(G), equipped with the product (a, b) 7! a�G b.

Example 3.3. Let A denote the cohomology functor H�(G, �), andG¼Gc denote the group G, on which G acts by conjugation. Then

AðGÞ ¼ AðGcÞ ¼ H�ðG;RGcÞ

is isomorphic to the Hochschild cohomology HH�(G, R), as an R-module. But Proposition 3.1 of Siegel and Witherspoon (1999) shows thatthis isomorphism is actually a ring isomorphism.

Example 3.4. Let B denote the Burnside functor for G, and G¼Gc

denote the group G, on which G acts by conjugation. Then

AðGÞ ¼ BðGcÞ

is isomorphic to the crossed Burnside ring Bc(G) of G, as a Z-module. It isclear moreover from the definitions of the ring structure on Bc(G) (Bouc,preprint, 2.1) that this is actually a ring isomorphism.

Example 3.5. Let k be a commutative ring, and let Mk(G) denote theGrothendieck group of the category of finitely generated kG-modules,for relations given by direct sum decompositions. The usual operationsof induction and restriction of modules endow Mk with a structure ofMackey functor, and the tensor product of modules (over k) gives a struc-ture of Green functor on Mk (for G, over Z). The ring Mk(G) is usuallycalled the Green ring of kG-modules.

Let G¼Gc denote the group G, on which G acts by conjugation, asin the previous example. Then the ring Mk(G) is isomorphic to theGrothendieck ring of Hopf bimodules for the Hopf algebra kG. This


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was originally a question of Cibils, and follows easily from his Theorems2.1 and 3.1 in Cibils (1997) using the product formula of Theorem 6.1.

4. CROSSED G-MONOIDS

Definition 4.1. Let G be a finite group. A crossed G-monoid (G, j) is apair consisting of a finite monoid G with a left action of G by monoid auto-morphisms (denoted by (g, g) 7! gg or (g, g) 7! gg, for g2G and g2G), and amap of G-monoids j from G to Gc (i.e., a map j which is both a map ofmonoids and a map of G-sets). A morphism of crossed G-monoids from(G, j) to (G0, j0) is a map of G-monoids y :G!G0 such that j0 y¼j.

A crossed G-group (G, j) is a crossed G-monoid for which G is a group.

Remark 4.2. Generally the map j :G!Gc will be clear from context,and will be understood in the notation.

Proposition 4.3. Let (G, j) be a crossed G-monoid. If X is any G-set, thereis a natural monoid action of G on X, denoted by

ðg; xÞ 2 G� X 7! g � x 2 X

and defined by

g � x ¼ jðgÞx:This action of G has the following properties:

1. If X is a G-set, if g2G, if g2G and x2X, then

gðg � xÞ ¼ gg � gx:

2. If f : X!Y is a morphism of G-sets, then for all g2G and all x2X

f ðg � xÞ ¼ g � f ðxÞ:

Moreover the square

G� X ��!IdG�f G� Yj#

j#

X ��!f

Y

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is cartesian, where the vertical arrows are given by the actions of Gon X and Y.

3. If X is a G-set and if y : (G, j)! (G0, j0) is a morphism of crossedG-monoids, then for all g2G and all x2X

yðgÞ � x ¼ g � x:

Proof. The only non-obvious point concerns the cartesian square in 2).Suppose x2X and (g, y)2G�Y are such that f(x)¼ g � y. Then settingz¼j(g)�1x gives an element (g, z)2G�X such that (IdG� f )(g, z)¼(g, y) and g � z¼ x. And if (g0, z0) is another pair in G�X satisfyingthese two conditions, then g¼ g0 and g � z¼ g0.z, thus j(g)z¼j(g)z0, andz¼ z0. &

Remark 4.4. Property 1) is equivalent to saying that the action (g, x) 7!g � x of G on X is a map of G-sets from G�X to X. Property 2) implies inparticular that if X and Y are two G-sets, if (x, y)2X�Y and g2G, theng � (x, y)¼ (g � x, g � y).

Example 4.5. Let H be a normal subgroup of G, and j be the inclusionhomomorphism from H to G. Then Hc¼ (H, j) is a crossed G-group.

Example 4.6. Let G be any G-monoid (i.e., any monoid with a left actionof G by monoid automorphisms). Let u be the trivial monoid homo-morphism from G to G. Then Gu¼ (G, u) is a crossed G-monoid.

Example 4.7. Let (G, j) be a crossed G-monoid. Then j(G) is a normalsubgroup of G, and j�1(1) is a G-submonoid of G. There is a naturalinclusion of crossed G-monoids from j�1(1)u to (G, j), and a natural sur-jection from (G, j) to j(G)c.

Example 4.8. Let E be a group of cardinality 1, with trivial G-action.Let u :E!Gc be the map sending the unique element of E to the unitof G. Then (E, u) is an initial object in the category of crossed G-monoids.On the other hand the crossed G-monoid Gc¼ (G, IdG) is a final object inthe category of crossed G-monoids.

5. THE GREEN FUNCTOR STRUCTURE ON AG

Let R be a commutative ring, and G be a crossed G-monoid. If A is aGreen functor for G over R, then the Dress construction gives a Mackey


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functor AG defined on the G-set X by

AGðX Þ ¼ AðX � GÞ:If X and Y are finite G-sets, define maps

AGðX Þ R AGðY Þ ! AGðX � YÞ : a b 7! a�G b

by

a�G b ¼ A�x; g1; y; g2#

x; g1 � y; g1g2

0@

1Aða� bÞ:

The notation A�� x;g1;y;g2

#x;g1�y;g1g2

�means A�( f ), where f ¼

�#

x;g1;y;g2

x;g1�y;g1g2

�is the map

from X�G�Y�G to X�Y�G mapping (x, g1, y, g2) to (x, g1�y, g1g2).This definition makes sense, since the map f is a map of G-sets if G is

a crossed G-monoid. Moreover if a2A(X�G) and b2A(Y�G), thena� b2A(X�G�Y�G), hence a�G b2A (X�Y�G) ¼AG(X�Y ).

Let moreover eAGdenote the element A�

� �#1G

�(eA) of A(G)¼AG(�).

Theorem 5.1. The functor AG is a Green functor for G over R, with uniteAG

. Moreover the correspondence A 7!AG is an endo-functor of the cate-gory GreenR(G).

Proof. First AG is a Mackey functor for G over R. Moreover, the pro-duct �G is bifunctorial : suppose that f :X!X 0 and g :Y!Y 0 are mapsof G-sets. I must check that for any a2AG(X) and b2AG(Y )

ðAGÞ�ð f ÞðaÞ �G ðAGÞ�ðgÞðbÞ ¼ ðAGÞ�ð f � gÞða�G bÞ:

This is equivalent to

A�ð f � IdGÞðaÞ�G A�ðg� IdGÞðbÞ ¼A�ð f � g� IdGÞða�G bÞ ð5:2ÞThe right hand side of this equation is equal to

A�ð f � g� IdGÞA�x;g1;y;g2#

x;g1 � y;g1g2

0@

1Aða� bÞ

¼A�

x;g1;y;g2#

f ðxÞ;gðg1 � yÞ;g1g2

0@

1Aða� bÞ:

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The left hand side of Eq. (5.2) is equal to

A�x0;g1;y0;g2#

x0;g1 � y0;g1g2

0@

1A A�ð f � IdGÞðaÞ�A�ðg� IdY ÞðbÞð Þ:

Since A is a Green functor, this is equal to

A�

x0;g1;y0;g2#

x0;g1 � y0;g1g2

0B@

1CAA�

x;g1;y;g2#

f ðxÞ;g1;gðyÞ;g2

0B@

1CAða� bÞ ¼ � � �

� � � ¼A�

x;g1;y;g2#

f ðxÞ;g1 � gðyÞ;g1g2

0B@

1CAða� bÞ:

It follows that Eq. (5.2) holds, since g1�g(y)¼ g (g1�y) for any g12G andy2Y.

Similarly, I must check that for any a0 2AG(X0) and b0 2AG(Y

0)

A�ðf �IdGÞða0Þ�GA�ðg�IdGÞðb0Þ¼A�ðf �g�IdGÞða0 �G b

0Þ ð5:3Þ

The left hand side of this equation is equal to

A�x;g1;y;g2#

x;g1 �y;g1g2

0@

1A A�ðf �IdGÞða0Þ�A�ðg�IdGÞðb0Þð Þ:

Since A is a Green functor, this is equal to

A�x;g1;y;g2#

x;g1 �y;g1g2

0@

1AA�

x;g1;y;g2#

f ðxÞ;g1;gðyÞ;g2

0@

1Aða0 �b0Þ ð5:4Þ

The right hand side of Eq. (5.3) is equal to

A�ðf �g�IdGÞA�x0;g1;y0;g2#

x0;g1 �y0;h01h02

0@

1Aða0 �b0Þ ð5:5Þ


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Now the square

XG ��!x;g1;y;g2

#f ðxÞ;g1;gðyÞ;g2

0@

1AX 0GY 0G

x;g1;y;g2

#x;g1�y;g1g2

0@

1A

x0;g1;y0;g2#

x0;g1�y0;g1g2

0@

1A

XYG��!f�g�IdG

X 0Y 0G

is cartesian if G is a crossed G-monoid. It follows that expression (5.5) isequal to expression (5.4), and Eq. (5.3) holds.

Next I have to check that the product on AG is associative. In otherwords, if X, Y, and Z are finite G-sets, if a2AG(X), if b2AG(Y ) andc2AG(Z), I must show that

a�G ðb�G cÞ ¼ ða�G bÞ �G c ð5:6Þ

in AG(X�Y�Z). The left hand side is equal to

A�x; g1; y; z; g2

#x; g1 � y; g1 � z; g1g2

0@

1A a� A�

y; g1; z; g2#

y; g1 � z; g1g2

0@

1Aðb� cÞ

0@

1A:

Since A is a Green functor, this is also equal to

A�x; g1; y; z; g2#

x; g1 � y; g1 � z; g1g2

!A�

x; g1; y; g2; z; g3#x; g1; y; g2:z; g2g3

!ða� b� cÞ

which is also equal to

A�x; g1; y; g2; z; g3#

x; g1 � y; g1 � ðg2:zÞ; g1ðg2g3Þ

!ða� b� cÞ: ð5:7Þ


A�x; y; g1; z; g2#

x; y; g1 � z; g1g2

!A�

x; g1; y; g2#x; g1 � y; g1g2

!ða� bÞ � c

!:

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Since A is a Green functor, this is also

A�x; y; g1; z; g2#

x; y; g1 � z; g1g2

!A�

x; g1; y; g2; z; g3#x; g1 � y; g1g2; z; g3

!ða� b� cÞ


A�x; g1; y; g2; z; g3#

x; g1 � y; ðg1g2Þ � z; ðg1g2Þg3

!ða� b� cÞ:

This is equal to (5.7). Hence AG is associative.Finally I have to check that eAG

is a unit for AG. Let X be a finiteG-set, and let a2AG(X). Then

eAG �G a ¼ A�

g1; x; g2#

g1 � x; g1g2

0B@

1CA A�

�#1G

0B@

1CAðeAÞ � a

0B@

1CA

¼ A�

g1; x; g2#

g1 � x; g1g2

0B@

1CAA�

x; g

#1G; x; g

0B@

1CAðaÞ

¼ A�

x; g

#1G � x; 1Gg

0B@

1CAðaÞ ¼ a:

One checks similarly that a�G eAG¼ a.

To complete the proof of Theorem 5.1, it remains to check that theconstruction A 7!AG is an endo-functor of GreenR(G). So let y :A!A0

be a morphism of Green functors. Then y is in particular a morphismof Mackey functors, and it induces a morphism of Mackey functors yGfrom AG!A0G, whose evaluation at the finite G-set X is the map

ðyGÞX ¼ yX�G : AGðXÞ ¼ AðX � GÞ ! A0ðX � GÞ ¼ AG0 ðX Þ:

It is clear moreover that with obvious notation (y y0)G¼ yG y0G,and that if y is the identity endomorphism of A, then yG is the identityof AG.

It remains to check that yG is a morphism of Green functors, i.e., thatit is compatible with the product. So let X and Y be finite G-sets, let


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a2AG(X) and b2AG(Y ). Then

ðyGÞX�Y ða�G bÞ ¼ yX�Y�G A�

x; g1; y; g2#

x; g1 � y; g1g2

0B@

1CAða� bÞ

0B@

1CA

¼ A0�

x; g1; y; g2#

x; g1 � y; g1g2

0B@

1CA yX�G�Y�Gða� bÞð Þ

ðbecause y : A! A0 is a morphism of

Mackey functorsÞ

¼ A0�

x; g1; y; g2#

x; g1 � y; g1g2

0B@

1CA yX�GðaÞ � yY�GðbÞð Þ

ðbecause y is a morphism of Green functorsÞ¼ yX�GðaÞ �G yY�GðbÞ¼ ðyGÞX ðaÞ �G ðyGÞY ðbÞ:

This shows that yG is a morphism of Green functors from AG to A0G, andcompletes the proof of Theorem 5.1. &

Remark 5.8. The evaluation at the trivial G-set of the Green functorAG is

AGð�Þ ¼ Að� � GÞ ffi AðGÞ

and with this identification the product on A(G) is given by

ða; bÞ 2 AðGÞ � AðGÞ 7!A�g1; g2#

g1g2

0@

1Aða� bÞ:

This product coincides with the product defined in Sec. 3, and justifiesLemma 3.1.

Proposition 5.9. Let f : (G, j)! (G0, j0) be a morphism of crossedG-monoids. For any G-set X, denote by Af,X the map A�(IdX� f ) fromAG(X) to AG0(X). Then these maps Af, X define a morphism of Green func-tors Af from AG to AG0. Moreover, if f is injective, then Af is a split injectionof Mackey functors.

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Proof. The maps Af,X define a morphism of Mackey functors from AG

to AG0 (Bouc, 1997, 1.2). Moreover clearly

Af ;�ðeAGÞ ¼ A�ð f ÞA��#1G

0@

1AðeAÞ ¼ A�

�#1G0

0@

1AðeAÞ ¼ eAG0 :

Moreover if X and Y are finite G-sets, if a2AG(X) and b2AG(Y ), then

A�x; y; g#

x; y; f ðgÞ

0@

1Aða�G bÞ ¼ A�

x; g#

x; f ðgÞ

0@

1AðaÞ �G0 A�

y; g#

y; f ðgÞ

0@

1AðbÞ:

Indeed, the left hand side is equal to

A�

x; y; g

#x; y; f ðgÞ

0B@

1CAA�

x; g1; y; g2#

x; g1 � y; g1g2

0B@

1CAða� bÞ

¼ A�

x; g1; y; g2#

x; g1 � y; f ðg1g2Þ

0B@

1CAða� bÞ

whereas the right hand side is equal to

A�x; g01; y; g

02

#x; g01:y; g

01g02

0@

1A A�

x; g#

x; f ðgÞ

0@

1AðaÞ � A�

y; g#

x; f ðgÞ

0@

1AðbÞ

0@

1A

or

A�x; g01; y; g

02

#x; g01 � y; g01g02

0@

1AA�

x; g1; y; g2#

x; f ðg1Þ; y; f ðg2Þ

0@

1Aða� bÞ ¼ � � �

� � � ¼ A�x; g1; y; g2#

x; f ðg1Þ � y; f ðg1Þf ðg2Þ

0@

1Aða� bÞ

Now f(g1) � y¼ g1 � y and f(g1)f(g2)¼ f(g1g2) for all y2Y and g1, g22G,since f is a morphism of crossed G-monoids.

Finally if f is injective, then themapsAfX ¼A�(IdX� f ) define amorph-

ism of Mackey functors Af from AG0 to AG (Bouc, 1997, 1.2), which is asection to Af, i.e., such that Af Af ¼ IdAG

. The proposition follows. &


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Corollary 5.10. Let G be a crossed G-monoid. For any finite G-set X,

denote by iX the map A�

� x#

x;1G

�from A(X ) to A(X�G)¼AG(X ). Then

iX is a split injective map of R-modules, and the maps iX define an injectivemorphism of Green functors from A to AG.

Proof. This is the special case of the proposition, where f is the (unique)morphism of crossed G-monoids from (E, u) to (G, j) (see example 4.8),sending the unique element of E to 1G. &

6. THE PRODUCT FORMULA

This section states a product formula for the ring AG(G), which gen-eralizes Theorem 5.1 of Siegel and Witherspoon (1999).

Theorem 6.1. Let A be a Green functor for G over R, and G be a crossedG-monoid. Then

AGðGÞ ¼ AðGÞ ¼Mg2G

AðGgÞ !G

and for g2G, the g-component of the product of the elements a and b ofA(G) is given by

ða�G bÞg ¼X

ða;bÞ2GgnðG�GÞab¼g

tGg

Gða;bÞ rGaGða;bÞaa:r

Gb

Gða;bÞbb

� �:

Corollary 6.2. Taking first sets of orbit representatives, there is an iso-morphism of R-modules

AðGÞ ffiM

g2½GnG�AðGgÞ

where [GnG] is a set of representatives of the orbits of G in G. With thisnotation, the product of a2A(Gg) and b2A(Gd) is equal toM

e2½GnG�

Mw2½GgnG=Gd�

tGeGgðw;eÞg\Ggðw;eÞwd

gðw;eÞðrGg

Gg\Gwda � rGwd

Gg\Gwd

wbÞ

where g(w, e) is an element of the unique class Geg in GenG such thatg(gwd)= e.

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Proof. This follows from the formulae given in Remark 2.3: the producta� b is the element of A(G�G) whose component (a, b) is equal to

rGaGða;bÞ

aa � rGb

Gða;bÞbb

and a�G b is the image of a� b by A�(m), where m :G!G is the productin G. The formula of the theorem follows.

The formula in the corollary is a translation using sets of representa-tives, and is a generalization of the product formula of (Siegel and With-erspoon, 1999) Theorem 5.1. &

Remark 6.3. There is a similar formula for the product in A(G), if oneuses co-invariants rather than invariants, as in Remark 2.4. There is anisomorphism of R-modules

AðGÞ ffiMg2G

AðGgÞ !

G

and denoting by [g, a]G the image in A(GG) of the element a of A(g),for g2G, the product of [g, a]G and [d, b]G (for d2G and b2A(Gd)) isequal to

½g; a�G �G ½d; b�G ¼X

w2½GgnG=Gd�gwd; tGgwd

Gg\GwdrGg

Gg\Gwda:rGwd

Gg\Gwd

wbÞ

� �h iG:

This kind of formula was used in Bouc (Preprint).

Corollary 6.4. Let H be a normal subgroup of G. Suppose that A is a(graded) commutative Green functor. If for any subgroup K of G, thegroup H\CG(K ) acts trivially on A(K ), then the ring A(Hc) is (graded)commutative.

Proof. Let g be an element ofH, and consider the Gg-set Pg of pairs (a, b)of elements of H such that ab¼ g. The correspondence

s : ða; bÞ 7! ðb; abÞ

is a permutation of P, which commutes with the action of Gg.Moreover

Ga \ Gb ¼ Gb \ Gab :


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Let a and b be elements of A(Hc). If A is graded, suppose moreover thata and b are homogeneous. The g-component of the product of a and bis equal to

ða�G bÞg ¼X

ða;bÞ2GgnPg

tGg

Gða;bÞðrGa

Gða;bÞaa � rGb

Gða;bÞbbÞ:

Since A is (graded) commutative, this is also equal (up to sign if A isgraded) to

ða�G bÞg ¼X

ða;bÞ2GgnPg

tGg

Gða;bÞrGb

Gða;bÞbb � rGa

Gða;bÞaa

� �:

Changing the order of summation via the permutation s gives

ða�G bÞg ¼X

ðb;aÞ2GgnPg

tGg

Gðb;aÞ

�rGb

Gðb;aÞbb � rbGa

Gða;bÞaba

�

¼X

ðb;aÞ2GgnPg

tGg

Gðb;aÞ

�rGb

Gðb;aÞbb �b rGa

Gða;bÞaa

�:

Now for (b, a)2Pg, set K¼Gb, a¼CG(hb, ai). Then b2H\CG(K ),thus b acts trivially on A(K ). Hence brGa

Gða;bÞaa ¼ rGa

Gða;bÞaa and

ða�G bÞg ¼X

ðb;aÞ2GgnPg

tGg

Gðb;aÞrGb

Gðb;aÞbb � rGa

Gða;bÞaa

� �¼ ðb�G aÞg

and A(Hc) is (graded) commutative. &

Remark 6.5. Corollary 6.4 shows in particular that the crossed Burnsidering of G is commutative. Similarly, the Hochschild cohomology ringof G is graded commutative. This was first proved by Gerstenhaber(1963).

Remark 6.6. Let N be a normal subgroup of G, viewed as a crossedG-monoid via the inclusion morphism from N to G. Then taking for Athe cohomology functor H�(�, R) leads to a ring AN(G). This ring isisomorphic to the cohomology ring H�(G, R) if N is trivial, andto the Hochschild cohomology ring HH�(G, R) if N¼G. It is akind of ‘‘relative’’ Hochschild cohomology, and could be denoted by

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HH� (G, N, R). As an R-module, it is isomorphic to

HH�ðG; N; RÞ ffiMn2Nmod:G

H�ðCGðnÞ; RÞ

In other wordsHH�(G, N, R)ffiH�(G, RN), where the action of G on RNcomes from the conjugation action of G on N. Corollary 6.4 shows thatHH�(G, N, R) is graded commutative.

7. SEMI-DIRECT PRODUCTS OF

CROSSED G-MONOIDS

Theorem 5.1 shows that the correspondence A 7!AG is an endo-functor of GreenR(G). It is natural to compose those endo-functors,and this leads to the notion of semi-direct product of crossed G-monoids.

Proposition 7.1. Let (G, j) and (G0, j0) be crossed G-monoids. Let G00

denote the direct product G0 �G, with diagonal G-action. Define the follow-ing multiplication on G00:

ðg01; g1Þðg02; g2Þ ¼ ðg01ðg1 � g02Þ; g1g2Þ 8g1; g2 2 G; 8g01; g02 2 G0:

Define j00 :G00 !Gc by j00(g0, g)¼j0(g0)j(g) for all g2G and g0 2G0. Then(G00, j00) is a crossed G-monoid, with unit (1G0, 1G).

Proof. This is a series of straightforward verifications. &

Definition 7.2. The crossed G-monoid (G00,j00) of Proposition 7.1 is calledthe semi-direct product of the crossed G-monoids (G0,j0) and (G,j), and itis denoted by (G0, j0)e (G, j), or G0 eG for short.

Proposition 7.3. Let A be a Green functor for G over R. If G and G0 arecrossed G-monoids, then the Green functor (AG)G0 is naturally isomorphic toAG0 e G.

Proof. Let X be a finite G-set. Then

ðAGÞG0 ðXÞ ¼ AGðX � G0Þ ¼ AðX � G0 � GÞ


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and this induces clearly an isomorphism of Mackey functors

ðAGÞG0 ffi AG0�G

where G0 �G is the direct product of G0 and G, with diagonal G-action,i.e., the underlying G-set of G0 eG.

Now let X and Y be finite G-sets, let a2A(X�G0 �G), viewed as(AG)G0(X), and b2A(Y�G0 �G), viewed as (AG)G0(Y ). Then the productof a and b in (AG)G0(X�Y ) is equal to

a�G;G0 b ¼ AG�

x; g01; y; g02

#x; g01 � y; g01g02

0B@

1CAða�G bÞ

¼ A�

x; g01; y; g02; g

#x; g01 � y; g01g02; g

0B@

1CAA�

x; g01; g1; y; g02; g2

#x; g01; g1 � y; g1 � g02; g1g2

0B@

1CAða� bÞ

¼ A�

x; g01; g1; y; g02; g2

#x; g01 � ðg1 � yÞ; g01ðg1 � g02Þ; g1g2

0B@

1CAða� bÞ

¼ a�G0 eG b

since moreover g01�(g1�y)¼j0(g0)j(g)y¼j00(g0, g)y¼ (g0, g) � y.It follows that the previous isomorphism of Mackey functors is

compatible with the product. Moreover, the unit of (AG)G0 is by definitionequal to the following element of A(G0 �G):

eðAGÞG0 ¼ AG��#1G0

0@

1AðeAGÞ

¼ A�g#

1G0 ; g

0@

1AA�

�#1G

!ðeAÞ

¼ A��#

1G0 ; 1G

0@

1AðeAÞ

¼ eAG0 eG

and the proposition follows. &

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8. FROM A-MODULES TO AG-MODULES

It follows in particular from Corollary 5.10 that there is a functor ofrestriction rG along the Green functor homomorphism i :A!AG, fromthe category AG-Mod of AG-modules to the category A-Mod. This sectiondescribes a functor iG from A-Mod to AG-Mod.

Notation 8.1. Let A be a Green functor for G over R, and M be anA-module. If X and Y are finite G-sets, if a2AG(X) and m2M(Y ), denoteby a�Gm the element of M(X�Y ) defined by

a�G m ¼M�x; g; y#

x; g � y

0@

1Aða�mÞ 2MðX � YÞ:

Theorem 8.2. Let G be a crossed G-monoid. If A is a Green functor for Gover R, and if M is an A-module, then the product

ða;mÞ 2 AGðX Þ �MðYÞ 7! a�G m 2MðX � Y Þ

endows M with a structure of AG-module, denoted by iG(M). If f :M!N isa morphism of A-modules, then the maps fX :M(X)!N(X) define amorphism iG( f ) of AG-modules from iG(M) to iG(N).

Proof. Again I have to check that this product is bifunctorial, associa-tive, and that eAG

is a left unit. Let f :X!X 0 and g: Y!Y 0 be maps ofG-sets. Let a2AG(X) and m2M(Y ). I must check that

A�ð f ÞðaÞ �G M�ðgÞðmÞ ¼M�ð f � gÞða�G mÞ:

From the definitions, this amounts to checking that

M�

x0; g; y0

#x0; g � y0

0B@

1CAðA�ð f ÞðaÞ �M�ðgÞðmÞÞ

¼M�ð f � gÞM�x; g; y

#x; g � y

0B@

1CAða�mÞ


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or equivalently that

M�

x0; g; y0

#x0; g � y0

0B@

1CAM�

x; g; y

#f ðxÞ; g; gðyÞ

0B@

1CAða�mÞ ¼ � � �

� � � ¼M�

x; y

#f ðxÞ; gðyÞ

0B@

1CAM�

x; g; y

#x; g � y

0B@

1CAða�mÞ:

This holds, since both sides are equal to

M�x; g; y#

f ðxÞ; g � gðyÞ

0@

1Aða�mÞ:

I must also check that for a0 2AG(X0) and m0 2M(Y 0)

A�ð f Þða0Þ �G M�ðgÞðm0Þ ¼M�ð f � gÞða0 �G m0Þ:By similar transformations, this amounts to checking that

M�

x; g; y

#x; g � y

0B@

1CAM�

x; g; y

#f ðxÞ; g; gðyÞ

0B@

1CAða0 �m0Þ ¼ � � �

� � � ¼M�x; y

#f ðxÞ; gðyÞ

0B@

1CAM�

x0; g; y0

#x0; g � y0

0B@

1CAða0 �m0Þ:

This holds because M is a Mackey functor, and because the square

XGY ��!x;g;y

#f ðxÞ;g;gðyÞ

0@

1A

X 0GY 0x;g;y

#x;g�y

0@

1Aj# j

#x0;g;y0

#x0;g�y0

0@

1A

XY ��!x;y

#f ðxÞ;gðyÞ

0@

1A

X 0Y 0

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is a cartesian square of G-sets if G is a crossed G-monoid. This shows thatthe product (a, m) 7! a�G m is bifunctorial.

It is moreover associative: let X, Y, Z be finite G-sets, and leta2AG(X), b2AG(Y ), and m2M(Z). Then I must check that

a�G ðb�G mÞ ¼ ða�G bÞ �G m ð8:3Þ

in MG(X�Y�Z). The left hand side is equal to

M�x; g; y; z#

x; g � y; g � z

0@

1A a�M�

y; g; z#

y; g � z

0@

1Aðb�mÞ

0@

1A:

Since M is a module for the Green functor A, this is also equal to

M�x; g; y; z#

x; g � y; g � z

0@

1AM�

x; g1;y; g2; z#

x; g1; y; g2 � z

0@

1Aða� b�mÞ


M�x; g1;y; g2; z

#x; g1 � y; g1 � ðg2:zÞ

0@

1Aða� b�mÞ: ð8:4Þ


M�x; y; g; z#

x; y; g � z

0@

1A A�

x; g1; y; g2#

x; g1 � y; g1g2

0@

1Aða� bÞ �m

0@

1A:

Since M is a module for the Green functor A, this is also

M�x; y; g; z#

x; y; g � z

0@

1AM�

x; g1; y; g2; z#

x; g1 � y; g1g2; z

0@

1Aða� b�mÞ:

This is equal to

M�x; g1; y; g2; z

#x; g1 � y; ðg1g2Þ � z

0@

1Aða� b�mÞ

which is equal to (8.4). Hence the product is associative.


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Finally, if X is any G-set and m2M(X), then

eAG �G m ¼M�

g; x

#g � x

0B@

1CA A�

�#1G

0B@

1CAðeAÞ �m

0B@

1CA

¼M�

g; x

#g � x

0B@

1CAM�

x

#1G; x

0B@

1CAðmÞ

¼M�

x

#1G � x

0B@

1CAðmÞ ¼ m:

This shows that iG(M) is an AG-module.Now if f :M!N is a morphism of A-modules, if X and Y are finite

G-sets, if a2AG(X) and m2M(Y ), then

fX�Y ða�G mÞ ¼ fX�YM�x; g; y#

x; g � y

!ða�mÞ

¼ N�x; g; y#

x; g � y

!fX�G�Y ða�mÞ

ðbecause f is a morphism of Mackey functorsÞ

¼ N�x; g; y#

x; g � y

!ða� fY ðmÞÞ

ðbecause f is a morphism of A-modulesÞ¼ a�G fY ðmÞ:

Hence f defines a morphism of AG-modules from iG(M) to iG(N), and theproof of the theorem is complete. &

Proposition 8.5. Let G be a crossed G-monoid, and A be a Green functorfor G over R. If M is an A-module, then the composition rG iG is iso-morphic to the identity functor of A-Mod.

Proof. Let M be an A-module. If X is a finite G-set, then

rG iGðMÞðXÞ ¼MðXÞ

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thus rG iG(M) is isomorphic to M as a Mackey functor. Moreover iff :M!N is a morphism of A-modules, then rG iG( f )¼ f. It remains tocheck that rG iG(M) is isomorphic to M as an A-module, i.e. that if Xand Y are finite G-sets, if a2A(X) and m2M(Y ), then

a�m ¼ iX ðaÞ �G m:

This is equivalent to

a�m ¼M�

x; g; y

#x; g � y

0B@

1CA A�

x

#x; 1G

0B@

1CAðaÞ �m

0B@

1CA

¼M�

x; g; y

#x; g � y

0B@

1CAM�

x; y

#x; 1G; y

0B@

1CAða�mÞ

¼M�

x; y

#x; 1G � y

0B@

1CAða�mÞ

¼ a�m

which proves the proposition. &

Corollary 8.6. The functor iG is a full embedding of A-Mod intoAG-Mod.

Proof. Since rG iG is isomorphic to the identity functor, the functoriG is faithful. Moreover, the functor rG is clearly faithful: if f :M!Nis a morphism of AG-modules, then for any finite G-set X, the maprG( f )X :M(X)!N(X) is equal to the map fX. Now the functors iG andrG induce maps

HomAðM;NÞ �!iG HomAGðiGðMÞ; iGðNÞÞ �!rG

HomAðM;NÞ

whose composition is the identity. Since the map rG is moreoverinjective, it follows that both maps are bijections, and the corollaryfollows. &


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9. CENTRES AND CENTRALIZERS

Recall the following definitions from Bouc (1997) 6.5 and 12.2:

Definition 9.1. Let A be a Green functor for G over R. If X and Y arefinite G-sets, if a2A(X) and b2A(Y ), set

a�op b ¼ A�y; x#x; y

0@

1Aðb� aÞ 2 AðX � YÞ:

If M is a Mackey subfunctor of A, define the commutant of M in A by

CAðMÞðX Þ ¼ fa 2 AðX Þj8Y ; 8m 2MðY Þ; a�m ¼ a�op mg

Also define zA(X) as the set of natural transformations from the identityfunctor I of A-Mod to the endofunctor IX of A-Mod given by the Dressconstruction associated to X.

The commutant of M in A is a Green subfunctor of A (Bouc, 1997,6.5.3). In Sec. 12.2 of Bouc (1997), it is shown that zA has a natural struc-ture of Green functor. Its evaluation at the trivial G-set is the center ofthe category A-Mod, i.e., the set of natural transformations from theidentity functor of A-Mod to itself.

Theorem 9.2. Let G be a crossed G-monoid, and A be a Green functor. LetC(A, G) denote the commutant of i(A) in AG. If X and Y are finite G-sets, ifM is an A-module, and if a2C(A, G)(X), define a map zX(a)M, Y :M(Y )!M(Y�X) by

zX ðaÞM;Y ðmÞ ¼M�x; y#y; x

0@

1Aða�G mÞ:

Then:

1. For given X, a and M, the maps zX(a)M,Y define a morphism ofA-modules zX(a)M from M to MX.

2. For given X and a, these morphisms zX(a)M define an element zX(a)of zA(X).

3. The maps zX define a morphism of Green functors z from C(A, G)to zA.

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Proof. Let f :Y!Z be a map of G-sets. Then for m2M(Y )

zX ðaÞM;Z M�ð f ÞðmÞð Þ ¼M�x; z

#z; x

0@

1A a�G M�ð f ÞðmÞð Þ

¼M�x; z

#z; x

0@

1AM�

x; y#

x; f ðyÞ

!ða�G mÞ

ðbecause iGðMÞ is an AG-moduleÞ

¼M�y; x

#f ðyÞ; x

0@

1AM�

x; y#y; x

!ða�G mÞ

¼ ðMX Þ�ð f ÞðzX ðaÞM;Y ðmÞÞ:

A similar computation, using the fact that M�x; y#

y; x

!¼M� y; x#

x; y

!,

shows that

zX ðaÞM;Y M�ð f Þ ¼ ðMX Þ�ð f Þ zX ðaÞM;Z

hence that the maps zX (a)M,Y define a morphism of Mackey functorszX(a)M from M to MX.

Now let Y and Z be finite G-sets, let a2A(Z) and m2M(Y ). Then

a� zX ðaÞM;Y ðmÞ ¼ a�M�

x; y

#y; x

0B@

1CAða�G mÞ

¼M�

z; x; y

#z; y; x

0B@

1CA a� ða�G mÞð Þ

ðbecause M is an A-moduleÞ

¼M�

z; x; y

#z; y; x

0B@

1CA iZðaÞ �G ða�G mÞð Þ

ðby Proposition 8:5Þ


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¼M�z; x; y#

z; y; x

!ðiZðaÞ �G aÞ �G mð Þ

¼M�z; x; y#

z; y; x

!ðiZðaÞ �op

G aÞ �G m�

ðbecause a 2 CAGðiðAÞÞðX ÞÞ

¼M�z; x; y#

z; y; x

!ðAGÞ�

x; z#z; x

!ða�G iZðaÞÞ �G m

!

¼M�z; x; y#

z; y; x

!M�

x; z; y#

z; x; y

!ða�G iZðaÞÞ �G mð Þ


¼M�z; x; y#

z; y; x

!M�

x; z; y#

z; x; y

!a�G ðiZðaÞ �G mð ÞÞ

¼M�x; z; y#

z; y; x

!a�G ða�mÞð Þ

ðby proposition 8:5Þ

¼ zX ðaÞM;Z�Y ða�mÞ

which proves assertion 1).To prove assertion 2), I must check that if f :M!N is a morphism of

A-modules, then for any finite G-set Y, the square

MðYÞ ��!fYNðY Þ

zX ðaÞM;Y zX ðaÞN;Y

MðY � XÞ ��!fY�X

NðY � X Þ

is commutative. But for m2M(Y )

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fY�X zX ðaÞM;Y ðmÞ ¼ fY�XM�

x;y

#y;x

0B@

1CAða�G mÞ

¼N�

x;y

#y;x

0B@

1CAfX�Y ða�G mÞ

ðbecause f is a morphism of Mackey functorsÞ

¼N�

x;y

#y;x

0B@

1CAða�G fY ðmÞÞ

ðbecause iGð f Þ is a morphism of AG-modulesÞ¼ zX ðaÞN;Y fY ðmÞ

as was to be shown.For assertion 3), first recall the Mackey functor structure on zA

(Bouc, 1997, 12.2.1) : if X is a finite G-set and y is a natural transforma-tion from the identity functor I of A-Mod to IX, then y is characterizedby maps of A-modules yM :M!MX, for each A-module M, i.e., by mapsof R-modules yM,Y :M(Y )!M(Y�X), for any finite G-set Y. Now iff :X!Z is a map of finite G-sets, then (zA)�( f )(y) is the natural transfor-mation from I to IZ characterized by the maps

ðzAÞ�ð f ÞðyÞM;Y ¼M�ðIdY � f Þ yM;Y :

Similarly, if y0 2 zA(Z), then zA� ( f )(y0) is the element of zA(X) defined by

z�Að f Þðy0Þ ¼M�ðIdY � f Þ y0M;Y :

Let C denote the Green functor C(A, G). To show that the mapszX :C(X)! zA(X) define a morphism of Mackey functors, I must checkthat for any m2M(Y )

M�ðIdY � f Þ zX ðaÞM;Y ðmÞ ¼ zZ C�ð f ÞðaÞð ÞM;Y ðmÞ ð9:3Þ

and similarly

M�ðIdY � f Þ zZðaÞM;Y ðmÞ ¼ zX C�ð f ÞðaÞð ÞM;Y ðmÞ: ð9:4Þ


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The left hand side of Eq. (9.3) is equal to

M�ðIdY � f Þ zX ðaÞM;Y ðmÞ ¼M�ðIdY � f ÞM�x; y

#y; x

0B@

1CAða�G mÞ

¼M�

x; y

#y; f ðxÞ

0B@

1CAða�G mÞ

¼M�

z; y

#y; z

0B@

1CAM�

x; y

#f ðxÞ; y

0B@

1CAða�G mÞ

¼M�

z; y

#y; z

0B@

1CA ðAGÞ�ð f ÞðaÞ �G m�

ðbecause iGðMÞ is an AG-moduleÞ¼ zZ C�ð f ÞðaÞð ÞðmÞ

as was to be shown, since C�( f )¼ (AG)�( f ), because C is a subfunctorof AG. The proof of Eq. (9.4) is similar, using the fact that

M�� x; y#y; x

�¼M�

� y; x#x; y

�.

It remains to check that the maps zX define a morphism of Greenfunctors from C¼C(A, G) to zA. Recall from Bouc (1997, 12.2.1) thatif X and Y are finite G-sets, if y2 zA(X) and c2 zA(Y ), then the producty�c is the element of zA(X�Y ) determined by the maps

ðy� cÞM;Z ¼M�z; y; x#

z; x; y

0@

1A yM;Z�Y cM;Z

for any A-module M and any finite G-set Z.So let a2C(X) and b2C(Y ). Then for m2M(Z)

zX ðaÞ�zY ðbÞð ÞM;ZðmÞ¼M�

z; y; x

#z; x; y

0@

1AzX ðaÞM;Z�Y zY ðbÞM;ZðmÞ¼� � �

� � � ¼M�

z; y; x

#z; x; y

0@

1AM�

x; z; y

#z; y; x

0@

1A

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� a�G M�

y; z

#z; y

0B@

1CAðb�G mÞ

264

375

¼M�

x; z; y

#z; x; y

0B@

1CA

� M�

x; y; z

#x; z; y

0B@

1CAða�G b�G mÞ

264

375


¼M�

x; y; z

#z; x; y

0B@

1CAða�G b�G mÞ

¼ zX�Y ða�G bÞM;ZðmÞ

This shows that zX(a)� zY(b)¼ zX�Y(a� Gb). The last verification con-cerns units: for any m2M(Z)

z�ðeAGÞM;ZðmÞ ¼M�

�; y#y; �

0B@

1CAðeAGÞ �G mÞ

¼ m

since eAGis the unit of AG. Hence z�(eAG

) is the identity transformation ofthe identity functor of A-Mod. This completes the proof of thetheorem. &

Remark 9.5. Theorem 9.2 provides in particular a natural ring homo-morphism from C(A, G)(�) to the center of the category A-Mod. Onecan check from the definitions that

CðA;GÞð�Þ ¼ a 2 AðGÞ j 8X ; 8a 2 AðXÞ; a� a

8><>:¼ A�

g; x#

g � x; g

0@

1Aða� aÞ

9=;


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as a subring of A(G). If A is the Burnside functor B, and G¼Gc, thenactually C(A, G)¼AG, and the previous ring homomorphism is the nat-ural morphism from the crossed Burnside ring of G over R to the centerof the Mackey algebra of G over R. This morphism leads in particular toa description of the block idempotents of the Mackey algebra (Bouc,Preprint).

REFERENCES

Bouc, S. (2001). The p-blocks of the Mackey algebra. Preprint, to appearin Algebras and Representation Theory.

Bouc, S. (1997). Green-functors and G-sets. In: Lecture Notes in Mathe-matics. Vol. 1671. Springer.

Cibils, C. (1997). Tensor product of Hopf bimodules over a groupalgebra. Proc. Amer. Math. Soc. 125:1315–1321.

Cibils, C., Solotar, A. (1997). Hochschild cohomology algebra of abeliangroups. Arch. Math. 68:17–21.

Gerstenhaber, M. (1963). The cohomology structure of an associativering. Ann. Math. 78:267–288.

Siegel, S. F., Witherspoon, S. J. (1999). The Hochschild cohomology ringof a group algebra. Proc. London Math. Soc. 79:131–157.

Received September 2001Revised February 2002

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hochschild constructions for green functors

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