Hopf algebroids and Galois extensions

Abstract

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer $R$ using the depth two theory in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra [21,8] to depth two extensions using coring theory [3]. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem \cite[23,5.1] for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules [28] and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [24]. In our last section, an exposition of results of Sugano [29,30] leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.