On composition of torsors

TL;DR

This paper investigates the conditions under which torsors over schemes can be composed into a single torsor under an extended group, providing both positive and negative results based on group properties.

Contribution

It introduces criteria for composing torsors into a single torsor under an extended group scheme, highlighting when such compositions are possible or not.

Findings

Positive results for certain group schemes

Negative results demonstrating limitations

Conditions depend on the nature of the groups involved

Abstract

Let $K$ be a field, let $X$ be a connected smooth $K$-scheme and let $G,H$ be two smooth connected $K$-group schemes. Given $Y \to X$ a $G$-torsor and $Z \to Y$ an $H$-torsor, we study whether one can find an extension $E$ of $G$ by $H$ so that the composite $Z \to X$ is an $E$-torsor. We give both positive and negative results, depending on the nature of the groups $G$ and $H$.