Degree 2 cohomological invariants of linear algebraic groups

TL;DR

This paper characterizes degree 2 cohomological invariants of smooth, connected linear algebraic groups over any field, using étale cohomology, including non-reductive groups over imperfect fields.

Contribution

It provides a comprehensive description of degree 2 invariants for all smooth, connected linear algebraic groups, expanding understanding beyond reductive cases and over arbitrary fields.

Findings

Describes degree 2 invariants for all smooth, connected linear groups.

Utilizes étale cohomology of sheaves on simplicial schemes.

Includes non-reductive groups over imperfect fields.

Abstract

The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants taking values in the Brauer group. Our main tool is the \'etale cohomology of sheaves on simplicial schemes. We get a description of these invariants for \emph{every} smooth and connected linear groups, in particular for non reductive groups over an imperfect field.