Depth Two and the Galois Coring

Abstract

We study the cyclic module ${}_{S}R$ for a ring extension $A\Vert B$ with centralizer $R$ and bimodule endomorphism ring $S=End{}_B{A}_B$. We show that if $A\Vert B$ is an H-separable Hopf subalgebra, then $B$ is a normal Hopf subalgebra of $A$. We observe from math.RA/0107064 and math.RA/0108067 depth two in the role of noncommutative normality (as in field theory) in a depth two separable Frobenius characterization of irreducible semisimple-Hopf-Galois extensions. We prove that a depth two extension has a Galois $A$-coring structure on $A{ø}_{R}T$ where $T$ is the right $R$-bialgebroid dual to $S$.

9 pages