A remark on the structure of torsors under an affine group scheme

TL;DR

This paper discusses conditions under which torsors under affine group schemes are trivial, extending known results from algebraic groups over algebraically closed fields to more general cases, with applications in Tannakian categories.

Contribution

It generalizes the triviality of torsors to affine group schemes not necessarily of finite type, broadening the scope of existing theorems.

Findings

Torsors under affine algebraic groups are trivial over algebraically closed fields.

Under certain conditions, torsors under more general affine group schemes are also trivial.

Application to isomorphisms of fibre functors in neutral Tannakian categories.

Abstract

It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories.