Representations and Cohomology of finite group schemes

TL;DR

This paper reviews recent advances in the representation theory of finite group schemes, focusing on cohomology, support varieties, and bundle constructions, highlighting developments over the past fifteen years.

Contribution

It synthesizes key theories and results in finite group scheme representations, including cohomology, support spaces, and bundle constructions, emphasizing recent progress and generalizations.

Findings

Finite generation of cohomology for finite group schemes

Development of support varieties and $ au$-points theory

Construction of vector bundles from cohomology rings

Abstract

The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the $\pi$-points and $\Pi$-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on projective varieties associated with cohomology ring of an infinitesimal group scheme $G$. In the last section we discuss varieties of elementary subalgebras of modular Lie algebras, generalizations of modules of constant Jordan type, and new constructions of bundles on projective varieties associated to a modular Lie algebra.