HOPF-GALOIS EXTENSIONS AND H -MORITA CONTEXTS Andrei Marcus Faculty of Mathematics and Computer Science, Babes ¸-Bolyai University, Str. Mihail Kog˘ alniceanu 1, RO-400084 Cluj-Napoca, Romania e-mail: marcus@math.ubbcluj.ro Abstract This article is an extended version of the talk given at the conference on “New techniques in Hopf algebras and graded ring theory”, held in Brussels, September 19–23, 2006. We con- sider a Hopf algebra H and two H -Galois extensions A and B. We investigate the category A M H B of relative Hopf bimodules, and the Morita equivalences between A and B induced by them. The talk is based on the paper [3], so no proofs are given, but we discuss the motivation which lies behind this study, and also how the general results apply to group graded algebras. INTRODUCTION Various questions in the modular representation theory of finite groups, most notably Brou´ e’s Abelian Defect Group Conjecture, lead to the problem of constructing Morita or derived equiva- lences between two algebras A and B strongly graded by the finite group G, under the hypothesis that their identity components A 1 and B 1 are Morita or derived equivalent. This situation occurs as follows. Let k be an algebraically closed field of characteristic p > 0. If K is a normal subgroup of the finite group H , with G = H /K, and b is a G-invariant block idempotent with defect group D of the group algebra kK, then the Brauer correspondent c of b in kN K (D) is a G-invariant block of idempotent kN K (D); under the assumption that D is abelian, Brou´ e’s abelian defect group conjecture predicts that there is a Rickard equivalence between the block algebras A 1 = kKb and B 1 = kN K (D)c, that is, the bounded homotopy categories H b (A) and H b (B) are equivalent as triangulated categories; moreover, such an equivalence should be compatible with p ′ -extensions, that is, if p does not divide the order of G, then the equiva- lence can be extended to a Rickard equivalence between the G-graded k-algebras A = kHb and B = kN H (D)c induced by a bounded complex of G-graded (A, B)-bimodules. The methods de- veloped in [9] and [10] allow the reduction of Brou´ e’s conjecture for the principal blocks to the case of simple groups, and are useful in the verification of the conjecture for various classes of groups and blocks (see [13]). Another instance where graded Morita equivalences occur natu- rally is the case of blocks with normal defect groups (see [7]). The paper [3] is a contribution to the representation theory of Hopf-Galois extensions, as origi- nated by Schneider in [14], and it generalizes the results of [10, Sections 2 and 3]. We consider the following questions. Let H be a Hopf algebra, and A, B right H -comodule algebras. More- over, assume that A and B are faithfully flat H -Galois extensions. 1. If A and B are Morita equivalent, does it follow that A coH and B coH are also Morita equiv- alent? 127