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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