Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

TL;DR

This paper develops a framework for understanding extensions of finite algebraic groups by smooth groups using nonabelian 2-cohomology, providing bounds and finiteness results especially over perfect fields.

Contribution

It introduces the concept of F-kernels and links extensions of algebraic groups to nonabelian 2-cohomology, with explicit bounds and finiteness theorems.

Findings

Extensions of finite algebraic groups are derived from finite group extensions over perfect fields.

Explicit bounds on the order of finite groups involved in extensions when the group is linear.

Finiteness results for the sets of nonabelian 2-cohomology classes.

Abstract

For a finite smooth algebraic group $F$ over a field $k$ and a smooth algebraic group $\bar G$ over the separable closure of $k$, we define the notion of $F$-kernel in $\bar G$ and we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of $F$ by an arbitrary smooth $k$-group $G$. We show in particular that any such extension comes from an extension of finite $k$-groups when $k$ is perfect and we give explicit bounds on the order of these finite groups when $G$ is linear. We prove moreover some finiteness results on these sets.