Finitary Galois extensions over noncommutative bases

Abstract

We study Galois extensions M ( co - ) H ⊂ M for H-(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf–Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer–Takeuchi, Doi–Takeuchi and Cohen–Fischman–Montgomery. We find that the Galois extensions N ⊂ M over some Frobenius Hopf algebroid are precisely the balanced depth 2 Frobenius extensions. We prove that the Yetter–Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized Brzeziński–Militaru theorem. Contravariant “fiber functors” are used to prove an analogue of Ulbrich's theorem and to get a monoidal embedding of the module category M E of the endomorphism Hopf algebroid E = End M N N into M N op N .