The coherent cohomology ring of an algebraic group

TL;DR

This paper characterizes the structure of the cohomology ring of an algebraic group scheme, showing it is a free module and an exterior algebra, and determines its Hopf algebra structure when the group is connected.

Contribution

It provides a detailed description of the algebraic and Hopf algebra structures of the cohomology ring of algebraic group schemes, including the case of connected groups.

Findings

$H^*(G)$ is a free module over $H^0(G)$ of finite rank.

$H^*(G)$ is the exterior algebra of its degree 1 component.

The Hopf algebra structure of $H^*(G)$ is explicitly determined for connected groups.

Abstract

Let $G$ be a group scheme of finite type over a field, and consider the cohomology ring $H^*(G)$ with coefficients in the structure sheaf. We show that $H^*(G)$ is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When $G$ is connected, we determine the Hopf algebra structure of $H^*(G)$.