Semisimple extensions and elements of trace 1

Abstract

Let H be a Hopf algebra which is Frobenius over a commutative ring k. Let A be an H-module algebra, A H its ring of invariants under the H-action, and A # H the associated semidirect, or smash, product. This setup has been shown by the authors and S. Montgomery [CF, CFM], to give rise to a Morita context [AH, A, A, A # H] with structure maps[,]:A@AHA-+A#H,and(,):A~,.,A+AH.Thecomodule theoretic concept of H-Galois extensions was shown [CFM] under these circumstance to be equivalent to surjectivity of [ , 1. In this paper we start out by showing that another important comodule theoretic concept, that of a total integral [Dl], can be translated to a module theoretic concept as well. We show that existence of a total integral is equivalent to the surjectivity of ( , ), which we also termed suggestively: existence of an element c E A of trace 1. Furthermore, if this element c centralizes AH, then AH is an AH-bimodule direct summand of A. We apply these to the Morita context associated with the H*-module algebra A # H, to prove the main theorem of this paper (Theorem 1.8). This theorem deals with semisimple extensions (sometimes coined “Maschke type” theorems), or better yet, with separable extensions. We show that when A/AH is H*-Galois,