Endotrivial modules for finite groups schemes II

TL;DR

This paper extends the understanding of endotrivial modules from finite groups to finite group schemes, proving finiteness of isomorphism classes of modules of fixed dimension and supporting the conjecture of their finite generation.

Contribution

It proves that for finite group schemes, only finitely many endotrivial modules of a given dimension exist, advancing the theory beyond finite groups.

Findings

Finiteness of isomorphism classes of endotrivial modules of fixed dimension for finite group schemes

Supports the conjecture that the group of endotrivial modules is finitely generated

Applications to lifting and twisting structures in infinitesimal group schemes

Abstract

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we prove that for an arbitrary finite group scheme G, and for any fixed integer n > 0, there are only finitely many isomorphism classes of endotrivial modules of dimension n. This provides evidence to support the speculation that the group of endotrivial modules for a finite group scheme is always finitely generated. The result also has some applications to questions about lifting and twisting the structure of endotrivial modules in the case that G is an infinitesimal group scheme associated to an algebraic group.