Depth Two,Normality, and a Trace Ideal Condition for Frobenius Extensions |
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Authors: | Lars Kadison Burkhard Külshammer |
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Institution: | 1. Department of Mathematics and Statistics , University of New Hampshire , Durham, New Hampshire, USA kadison@math.unh.edu;3. Mathematics Institute, Friedrich Schiller University , Jena, Germany |
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Abstract: | A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules A A B and B A A . In this case the structures (A ? B A) B and End B A B are bialgebroids over the centralizer C A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two 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. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields 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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Keywords: | Depth two Generalized Miyashita–Ulbrich action Normal subgroup Picard group Separable extension |
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