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Publication# Sur quelques foncteurs de bi-ensembles

Résumé

This thesis is in the context of representation theory of finite groups. More specifically, it studies biset functors. In this thesis, I focus on two biset functors: the Burnside functor and the functor of p-permutation modules. For the Burnside functor we first give a result that characterize some B-groups; B-groups being the essential ingredient in the classification of composition factors of the Burnside functor. The second result compares the Burnside functor and the functor of free modules. Note that the functor of free modules is not a biset functor since the inflation of a free module is not necessarily free. To compare those functors we will work on an adjunction between the category of biset functors and the category of functors that do not have inflation. An aspect of the work done on the functor of p-permutation module is to compare the functor of p-permutation modules and the functor of ordinary representations. On the other hand, because of the classification of p-permutation modules, we try to express the functor o p-permutation modules in terms of the functor of projective modules (which is not a biset functor). We will use an adjunction between the category of biset functors and a category that contains the functor of projective modules.

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En mathématiques, un groupe fini es

Théorie des représentations

La théorie des représentations est une branche des mathématiques qui étudie les structures algébriques abstraites en représentant leurs éléments comme des transformations linéaires d'espaces vectori

Foncteur

Dans la théorie des catégories, un foncteur est une construction transformant les objets et morphismes d'une catégorie en ceux d'une autre catégorie, d'une façon compatible. On parle alors d'une cons

This dissertation is concerned with modular representation theory of finite groups, and more precisely, with the study of classes of representations, which we shall term relative endotrivial modules. Given a prime number p, a finite group G of order divisible by p, we shall say that a kG-module M is endotrivial relatively to the kG-module V if its endomorphism algebra Endk(M) is isomorphic, as a kG-module, to a direct sum of a trivial module and another module which is projective relatively to V , i.e. in short Endk(M) ≅ k ⊕ (V – projective). More accurately, in the first part of the text projectivity relative to kG-modules is used to define groups of relative endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. However, in order to achieve this goal we first need to develop the theory of projectivity relative to modules, in particular with respect to standard group operations such as induction, restriction and inflation. Then, for finite groups having a cyclic Sylow p-subgroup, using the structure of the group T(G) of endotrivial modules described in [MT07], we give a complete classification of the groups of relative endotrivial modules. We also study the case of groups that have a Sylow p-subgroup isomorphic to a Klein group C2 × C2, as well as the case of p-nilpotent groups. In a second part of the text, it is shown how our new groups of relative endotrivial modules provide a natural context to generalise the Dade group of a p-group P to an arbitrary finite group. The classification of endo-permutation modules and the complete description of the structure of the Dade group D(P) was completed in 2004 by S. Bouc with [Bou06]. This adventure had started about 25 years earlier with the first papers and results by E. Dade in [Dad78a] and [Dad78b] in 1978, and the final classification was in fact achieved through the non-effortless combined work of several (co)-authors between 1998 and 2004, including S. Bouc, J. Carlson, N. Mazza and J. Thévenaz. It is most interesting to note that crucial building pieces for this classification are indeed the endotrivial modules, which are particular cases of endo-permutation modules. Yet, for an arbitrary finite group G, no satisfying equivalent group structure to the Dade group on a class of kG-modules has been defined so far. With the goal to fill this gap, we turn the problem upside down, in some sense, and show how one can regard an endo-permutation module as an endotrivial module, of course not in the ordinary sense, but in the relative sense. This shall enable us to endow a set of isomorphism classes of endo-p-permutation modules with a group structure, similar to that of the Dade group. We shall call this new group, the generalised Dade group of the group G, explicitly compute its structure and show how it is closely related to that of the G-stable points of the Dade group of a Sylow p-subgroup of G.

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].