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 FrobeniusHopf 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 N M N into N M op N .

Corrigendum to Cleft Extensions of Hopf Algebroids

“…= f (h (1) (1) ) j c (h (1) (2) ) j(h (2) ) = f (h (1) ) [0] j c ( f (h (1) ) [1] ) j(h (2) ). The proof is then completed as in the paper.…”

“…(Left H L -colinearity of ϑ is needed to see that the range of the map (5.6) is in B, as stated.) If B ⊆ A is a cleft extension by a Hopf algebroid H with a bijective antipode, then similarly to (6), any morphism g ∈ Hom HR,− (H, A) is checked to satisfy, for h ∈ H, g(h) = j c (h (1) ) j S −1 (g(h (2) ) [1] ) g(h (2) ) [0] . Therefore …”

Scalar Extension of Bicoalgebroids

“…For a coalgebra C, ∆, ε , the coproduct ∆ : C → C ⊗ C on elements is denoted ∆(c) = c (1) ⊗ c (2) , with an implicit finite summation understood, i.e. c (1) ⊗ c (2) = i c (1) i ⊗ c (2) i . Quite similarly, a right C-coaction ρ M : M → M ⊗ C will be denoted ρ M (m) = m [0] ⊗ m [1] and a left C-coaction λ N : N → C ⊗ N will be denoted λ N (n) = n [−1] ⊗ n [0] .…”

“…Incidentally, the construction that was shown in [1] to be a non-commutative version of scalar extension was defined (for Hopf-algebras) in [2] -alongside with bicoalgebroids.…”

“…It is proved that the Yetter-Drinfel'd category is monoidal and pre-braided just as in the case of bialgebroids, and is embedded into the one-sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of [2] and [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given.…”