SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

摘要

Let H = (H_L, H_R, S) be a Hopf algebroid, and A a left H_L-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids， which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra 100 (1986) 363-379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math. 179 (2003) 75-121], we construct a Morita context connecting A#H and A~(H_L) the invariant subalgebra of H_L on A.