Morita Equivalences Induced by Bimodules over Hopf-galois Extensions

Let H be a Hopf algebra, and A, B be H-Galois extensions. We investigate the category AM H B of relative Hopf bimodules, and the Morita equivalences between A and B induced by them. Introduction This paper is a contribution to the representation theory of Hopf-Galois extensions, as originated by Schneider in [15]. More specifically, we consider the following questions. Let H be a Hopf algebra, and A, B right H-comodule algebras. Moreover, assume that A and B are faithfully flat H-Galois extensions. (1) If A and B are Morita equivalent, does it follow that AcoH and BcoH are also Morita equivalent? (2) Conversely, if AcoH and BcoH are Morita equivalent, when does it follow that A and B are Morita equivalent? These questions have been considered in [10] in the context of strongly group graded algebras, the motivation coming from problems raised in the modular representation theory of finite groups. The results of the present paper generalize the results of [10, Sections 2 and 3]. Given a right H-comodule algebra A, and a left H-comodule algebra B, we consider (A ⊗ B,H)-Hopf modules. These are at the same time left A ⊗ B-modules and right H-comodules, with a suitable compatibility condition. There are various ways to look at these Hopf modules: they are Doi-Hopf modules (see [8]) over a certain Doi-Hopf datum (with two possible descriptions of the underlying module coalgebra), and they can also be viewed as comodules over a coring (see Section 3). The main result of Section 2, and also the main tool used during the rest of the paper, is a structure Theorem for (A⊗B,H)-Hopf modules, stating that the category of (A ⊗ B,H)-Hopf modules is equivalent to the category of left modules over the cotensor product A HB, under the condition that A is a faithfully flat H-Galois extension. The results from Section 2 can be applied to relative Hopf bimodules: let A and B be right H-comodule algebras, and consider (A,B)-bimodules with a right H-coaction, satisfying a certain compatibility condition. The category 2000 Mathematics Subject Classification. 16W30, 16D90.