1. Introduction

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The theory of Mackey functors for a finite group G over a ring R provides

a single framework for the various representation theories of G and its subgroups.

So it looks like an extension of the notion of jRG-module. The usual notions of

induction, restriction, inflation, ... for modules, have their analogues for Mackey

functors. I will describe here a missing item in that list: tensor induction.

The first part of this paper is actually more general, and not specific to Mackey

functors. It introduces an extension of the notion of right exact functor between

abelian categories to non-additive functors: the usual definition of right exactness

actually implies additivity, so it has to be modified in order to be extended. The

main theorem of this general setting concerns the extension of a (non necessarily

additive) functor from a suitable sub-category V of an abelian category A to an

abelian category i3, to a right exact functor from A to B.

Next I apply those results to various constructions of tensor induction. In

all those cases, I will consider two finite groups G and H, a finite set U with

a H x Gop-action (or H-set-G, or biset), and I will define a (generalized) tensor

induction associated to U', which will be a functor between categories C{G) and

C(H) naturally attached to G and H.

In the first case, the category C(G) is the category of Mackey functors for G. I

will apply the extension theorem to the subcategory of "permutation functors", and

this leads to a generalized tensor induction functor Ty from Mackey functors for

G to Mackey functors for if, associated to a finite biset U. This tensor induction

behaves well with respect to composition of functors, tensor product of Mackey

functors, and disjoint unions of bisets 2. There is also a kind of binomial formula

for the tensor induction of a direct sum.

Next I consider the relations between tensor induction and other functors be-

tween categories of Mackey functors, such as induction, restriction, inflation, ... I

also define a reasonable notion of direct product of Mackey functors, and study its

relations with tensor induction. Finally, I extend those notions to the case of Green

functors.

The second case deals with the category C(G) of cohomological Mackey func-

tors for G over a commutative ring R, and uses the subcategory of "permutation

cohomological Mackey functors". There is a generalized tensor induction functor

associated to finite biset U, whenever U is "free enough" with respect to R. This

cohomological tensor induction is closely related to the tensor induction for Mackey

functors.

It leads to the definition of a generalized tensor induction for p-permutation

modules and p-permutation algebras: this was the very starting point of that work,

in a conversation with Jacques Thevenaz, who asked me about the possibility of

such a generalized construction, giving a suitable functorial structure for the Dade

group. In our joint recent preprint [BT98], we give an independent exposition of

this generalized tensor induction for permutations algebras for p-groups, and use it

to solve some open questions about the Dade group.

The third case is the case of the category C(G) of RG modules, using the subcat-

egory of free RG-modules. This leads to a generalized tensor induction associated

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Received by the editor November 20, 1997, and in revised form November 16, 1998.

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The construction of a tensor induction for Mackey functors with those properties was a

question of T. Yoshida at the Seattle AMS conference (Problem 37 in [ACPW98])

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