A tower condition characterizing normality

Abstract

We define left relative H-separable tower of rings and continue a study of these begun by Sugano. It is proven that a progenerator extension has right depth two if and only if the ring extension together with its right endomorphism ring is a left relative H-separable tower. In particular, this applies to twisted or ordinary Frobenius extensions with surjective Frobenius homomorphism. For example, normality for Hopf subalgebras of finite-dimensional Hopf algebras is also characterized in terms of this tower condition.

19 pages. Added a module condition characterizing H-separable extension, a Frobenius extension condition so that left and right tower conditions are equivalent, and a going up and down proposition between the tower conditions here and in http://arxiv.org/abs/0707.3756