On automorphisms of separable algebras

Abstract

This paper presents a Galois theory for separable algebras over a (not necessarily Noetherian) semi-local ring.Our theory is patterned after, but independent of the Galois theory of G. Hochschild; over a perfect field Hochschild's theory and ours coincide.In order to present this theory we first present some results concerning separable algebras over semilocal rings.The most important of these is a generalization of the Skolem-Noether theorem.Our theorem states that any algebra monomorphism of a separable subalgebra without central idempotents into a central separable algebra whose center is semilocal extends to an inner automorphism.