  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 math.RA/0108067. First we extend results on the equivalence of
certain properties of Hopf-Galois extensions with corresponding properties of
the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring
theory math.RA/0002105. Next we show that $T^{\rm op}$ is a Hopf algebroid over
the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 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 \cite[Schauenburg]{Sch} and the equivalence of
Yetter-Drinfel'd modules with modules over Drinfel'd double \cite[Majid]{Maj}.
In our last section, an exposition of results of Sugano \cite{Su82,Su87} 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 $A
\supseteq B$.
