Master Class on New Developments in Finite Generation of Cohomology – Abstracts
Lecture Series by Paul Sobaje
Lecture 1 – A primer on affine group schemes
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
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
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
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
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 GLn.
Lecture 3 – (CFG) for Noetherian algebras over a field - Part I
We introduce Grosshans filtrations, and we prove that the cohomology of GLn with Grosshans graded coefficients is finitely generated.
Lecture 4 – (CFG) for Noetherian algebras over a field - Part II
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
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
We explain how geometry helps to construct certain finite resolutions of GLn-representations, which were instrumental in the previous lectures.
Lecture by Peter Symonds
Group scheme actions on rings and the Cech complex
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.