Imre Bálint, Kornél Szlachányi
Finitary Galois Extensions over Noncommutative Bases
Preprint series: ESI preprints

MSC:

16W30 Coalgebras, bialgebras, Hopf algebras, See also {57T05,

13B05 Galois theory (commutative rings)

16D90 Module categories, See also {16Exx, 16Gxx, 16S90}; module theory in a category-theoretic context; Morita equivalence and duality

Abstract: We study Galois extensions $M^{(\co\text{-})H}\subset M$ 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-Take\-uchi, Doi-Takeuchi and Cohen-Fischman-Montgomery.
We find that the Galois extensions $N\subset M$ 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 Brzezi\'nski-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 $E=\End\,_NM_N$ into $_N\M_N^\op$.

Keywords: Hopf-Galois theory, quantum groupoid, double algebra, depth 2 extensions, Frobenius extensions, Yetter-Drinfeld modules