The endomorphism ring theorem for Galois and depth two extensions.
type: Item
instance of: scholarly article
title: The endomorphism ring theorem for Galois and depth two extensions.
zbMATH Open document ID: 1115.16019
author: Lars Kadison
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: 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 \(D2\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 5077756
zbMATH Keywords: depth two extensions
zbMATH Keywords: bialgebroids
zbMATH Keywords: Hopf Galois extensions
zbMATH Keywords: endomorphism algebras
zbMATH Keywords: Frobenius extensions
reviewed by: Margaret Beattie
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.01.013
OpenAlex ID: W2049812586
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DOI: 10.1016/J.JALGEBRA.2006.01.013
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description: scientific article; zbMATH DE number 5077756