Abstract
Let H be a finite‐dimensional Hopf algebra over a field k, B a left H‐module algebra, and H∗ the dual Hopf algebra of H. For an H∗‐Azumaya Galois extension B with center C, it is shown that B is an H∗‐DeMeyer‐Kanzaki Galois extension if and only if C is a maximal commutative separable subalgebra of the smash product B#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.
Highlights
Let H be a finite-dimensional Hopf algebra over a field k, B a left H-module algebra, and H∗ the dual Hopf algebra of H
An H∗-Azumaya Galois extension B was characterized in terms of the smash product B#H see [7, Theorem 3.4]
Observing that the commutator VB(BH ) of BH in B is an H∗-Azumaya Galois extension, in the present paper, we will give a characterization of an H∗-Azumaya Galois extension B in terms of VB(BH )
Summary
Let H be a finite-dimensional Hopf algebra over a field k, B a left H-module algebra, and H∗ the dual Hopf algebra of H. Let C be a commutative separable CH -algebra containing CH as a direct summand as a CH -module. C is a separable CH -algebra, so C is a splitting ring for the Azumaya CH -algebra C#H such that C ⊗CH (C#H) HomC (C#H, C#H) (see the proof of [4, Theorem 5.5, page 64]).
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