The endomorphism ring theorem for Galois and depth two extensions

Let S be the left bialgebroid over the centralizer R of a right depth two (D2) algebra extension A|B, which is to say that its tensor-square is isomorphic as A–B-bimodules to a direct summand of a finite direct sum of A with itself. We prove that its left endomorphism algebra is a left S-Galois extension of Aop. As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension. We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.