# Master Class on New Developments in Finite Generation of Cohomology – Abstracts

## Lecture Series by Paul Sobaje

Lecture 1 – A primer on affine group schemes

Paul Sobaje

We introduce and motivate the basic theory of affine group schemes and their representations. We will define and discuss the significance of ground rings, the importance of flatness, and examples of finite group schemes.

Lecture 2 – Representations of reductive groups

Paul Sobaje

We survey the representation theory of a reductive group G over a field, with primary attention given to fields of positive characteristic. We will discuss highest weight theory, good filtrations and tilting modules, the Frobenius morphism and its consequences.

Lecture 3 – Open problems in the representation theory of reductive groups

Paul Sobaje

We conclude with several open problems in the modular representation theory of reductive groups, including those related to an application of CFG (support variety theory).

## Lecture Series by Vincent Franjou and Antoine Touzé

Lecture 1 – Power reductivity and finite generation of invariants

Vincent Franjou

The notion of power reductivity is introduced and it is used to prove finite generation of invariants (FG) over an arbitrary base ring. Power reductive group schemes include Chevalley groups and finite group schemes.

Lecture 2 – Cohomological finite generation (CFG) and van der Kallen's conjecture

Antoine Touzé

We introduce the cohomological finite generation property (CFG). We give a short historical overview of the problem and we state van der Kallen's conjecture. We explain that how the proof of (CFG) reduces to the case of GL_{n}.

Lecture 3 – (CFG) for Noetherian algebras over a field - Part I

Antoine Touzé

We introduce Grosshans filtrations, and we prove that the cohomology of GL_{n} with Grosshans graded coefficients is finitely generated.

Lecture 4 – (CFG) for Noetherian algebras over a field - Part II

Antoine Touzé

We prove van der Kallen conjecture when the ground ring is a Noetherian algebra over a field.

Lecture 5 – Power reductivity strikes again: CFG over a Noetherian ring

Vincent Franjou

We explain that CFG over a Noetherian ring is equivalent to uniformly bounded abelian torsion for the cohomology groups. We deduce the generalization of Friedlander and Suslin's finite generation result to Noetherian rings (i.e. CFG for finite group schemes over a Noetherian ring).

Lecture 6 – Resolutions of the diagonals

Vincent Franjou

We explain how geometry helps to construct certain finite resolutions of GL_{n}-representations, which were instrumental in the previous lectures.

## Lecture by Peter Symonds

Group scheme actions on rings and the Cech complex

Peter Symonds

I will talk about some recent results on Castelnuovo-Mumford regularity of rings of invariants for actions of finite group schemes, generalizing what is known for finite groups.