On the algebra of some group schemes

Abstract

The algebra of a finite group over a field $k$ of characteristic zero is known to be a projective separable $k$-algebra; but these separable algebras are of a very special type, characterized by Brauer and Witt.

In contrast with that, we prove that any projective separable $k$-algebra is a quotient of the group algebra of a suitable group scheme, finite étale over $k$. In particular, any finite separable field extension $K\subset L$, even a noncyclotomic one, may be generated by a finite étale $K$-group scheme.