Cohomology and Support in Representation Theory

Descent techniques in modular representation theory

Paul Balmer (UCLA)

This series of lectures will present some techniques imported from algebraic geometry and descent theory into modular representation theory. Such tools can be useful to the modern mathematician, beyond the specific case presented here.

Lecture I. Introduction and motivation

We shall explain that restriction of representations from a group G to a subgroup H can be revamped as an extension-of-scalars. The parallel with étale topology in algebraic geometry will be emphasized. In this viewpoint, the problem of extending representations from H to G naturally becomes a descent problem. We shall explain the basics of descent theory and discuss the technical difficulties that arise in the context of representation theory.

Lecture II. G-set representations and the sipp topology

We shall present a Grothendieck topology on the category of finite G-sets, called the sipp topology. We shall also discuss categories of (plain) representations, derived categories and stable categories.

Lecture III. Descent and stacks

We shall prove that the plain, derived and stable categories all form stacks with respect to the sipp topology, in the sense of Lecture II. We do not assume that participants know what stacks are and we shall explain this notion. We shall then translate the above theorem in purely representation-theoretic terms and see how it sheds light on the question of extending modular representations from a Sylow subgroup to the ambient group.

Lecture IV. Applications to endotrivial modules

We shall illustrate the techniques of the first lectures with the problem of describing the kernel and the image of the restriction homomorphism T(G) → T(P), where G is an arbitrary finite group, P is a Sylow p-subgroup of G and T(-) denotes the Picard group of isomorphism classes of tensor-invertible objects in the stable category (also unfortunately known as endotrivial modules).

Reference: This series of lectures is based on an article entitled Stacks of Group Representations, available online on the lecturer's publication page.

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