The endomorphism ring theorem for Galois and depth two extensions. (Q854896)

This paper continues a series of papers by the author and his collaborators on depth two extensions. Let \(A|B\) be a right D2 (depth two) algebra extension. The right D2 condition means that \(A\otimes_BA\) is centrally projective w.r.t the bimodule \(_AA_B\), i.e., \(_A(A\otimes_BA)_B\) is isomorphic to a direct summand of \(\bigoplus^n{_AA_B}\). Let \(R\) denote \(C_A(B)\), the centralizer of \(B\) in \(A\), let \(S\) denote the bimodule endomorphism algebra \(\text{End}{_BA_B}\), and let \(T\) denote \((A\otimes_BA)^B\), the \(B\)-centralized elements of \(A\otimes_BA\). Then it was shown in a paper by the author and \textit{K. Szlachányi}, [Adv. Math. 179, No. 1, 75-121 (2003; Zbl 1049.16022)], that \(S\) is a left bialgebroid over \(R\) and \(T\) is a right bialgebroid over \(R\). As well, the algebra \(A\) is embedded in \(\text{End}{_BA}\) by the map \(\rho\) taking \(a\) to right multiplication by \(a\). The main theorem states that in this situation \(\text{End}{_BA}\) is a left \(S\)-comodule algebra which is a Galois extension of \(\rho(A)\). The author recalls from a previous paper [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 51, 209-231 (2005; Zbl 1134.16016)] that Galois extensions are one-sided D2 and balanced extensions. Here he shows the converse result that if \(A|B\) is a right generator Frobenius extension, and \(\text{End\,}A_B|A\) is D2, then \(A|B\) is \(D2\).