Universal classes for algebraic groups

TL;DR

This paper constructs specific cohomology classes for algebraic groups, confirming long-standing predictions and contributing to understanding the finite generation of their cohomology algebras.

Contribution

It explicitly constructs cocycles representing key cohomology classes in algebraic groups, advancing the proof of finite generation of their cohomology algebras.

Findings

Constructed cocycles for classes in the rational cohomology of GL

Confirmed predictions by van der Kallen about these classes

Contributed to the proof that reductive groups have finitely generated cohomology

Abstract

We exhibit cocycles representing certain classes in the rational cohomology of of the general linear group with coefficients in the divided powers of a Frobenius twist of the adjoint representation. These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras.