Hopf algebroids and H-separable extensions

Abstract

Since an H-separable extension A|B is of depth two, we associate to it dual bialgebroids S:= End B A B and T:= (A ⊗B A) B over the centralizer R as in Kadison-Szlachanyi. We show that S has an antipode τ and is a Hopf algebroid. T 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|B, we show that End A A ⊗ B A ≅ T α End B A.