Depth two, normality and a trace ideal condition for Frobenius extensions

Details Depth two, normality and a trace ideal condition for Frobenius extensions Depth two, normality and a trace ideal condition for Frobenius extensions

2004-09-20

arxiv.org/abs/math/0409346v2

Mathematics

Group Theory

final version, 19 pages. to appear: Communications in Algebra

Scientific paper

We review the depth two and Hopf algebroid-Galois theory in math.RA/0108067 and specialize to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse we observe that normal Hopf subalgebras over a field are depth two extensions. We introduce a generalized Miyashita-Ulbrich action on the centralizer of a ring extension, and apply it to a study of depth two and separable extensions, providing new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.

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