Hopf algebroids and Galois extensions

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To a finite Hopf-Galois extension A | B we associate dual bialgebroids S:= and T:= (A øB 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 KT,Doi to depth two extensions using coring theory math.RA/0002105. Next we show that T 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 [Schauenburg]Sch and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [Majid]Maj. In our last section, an exposition of results of Sugano 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 B.