Cohomology of classical algebraic groups from the functorial viewpoint

TL;DR

This paper establishes a functorial approach to compute the rational cohomology of classical algebraic groups using extension groups in polynomial functor categories, extending known results beyond general linear groups.

Contribution

It generalizes the computation of rational cohomology to all classical algebraic groups via functor categories, providing new tools and applications.

Findings

Extension groups compute cohomology of classical groups.

Cohomological stabilization property established.

Existence of Hopf algebra structures on stable cohomology.

Abstract

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.