Hopf algebroids and H-separable extensions

Since an H-separable extension $A \vert B$ is of depth two, we associate to it dual bialgebroids $S := \operatorname{End}{}_BA_B$ and $T := (A \otimes_B A)^B$over the centralizer $R$ as in Kadison-Szlachányi. We show that $S$ has an antipode $\tau$ and is a Hopf algebroid. $T^{\operatorname{op}}$ is also Hopf algebroid under the condition that the centralizer $R$ is an Azumaya algebra over the center $Z$ of $A$. For depth two extension $A \vert B$, we show that $\operatorname{End}{}_AA\otimes_B A \cong T \ltimes \operatorname{End}{}_BA$.