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

Abstract. A ring extension A |B is depth two if its tensor-square sat-isfies a projectivity condition w.r.t. the bimodules AAB and BAA. In this case the structures (A⊗B A)B and EndBAB are bialgebroids over the centralizer CA(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 charac-ter 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 gen-eralized Miyashita-Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable exten-sions, 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