Finitary Galois Extensions over Noncommutative Bases

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-FischmanMontgomery. 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 ME of the endomorphism Hopf algebroid E = End N MN into NM op N .