  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 $H^*(G)$.
