The Endomorphism Ring Theorem for Galois and D2 extensions

2005/03/10 by Lars Kadison, Kadison, Lars · 2 citations
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.QA #msc:13B05 #msc:16S40 #msc:20L05 #msc:81R50

paper · pdf · doi:10.48550/arxiv.math/0503194

20 pp, some additional material including a converse endomorphism ring theorem for certain Frobenius extensions, which yields a complete answer to question 1 in math.RA/0107064

arxiv created 2005/04/01 · arxiv updated 2009/12/01

Abstract

Let S be the left bialgebroid \End _BA_B over the centralizer R of a right 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 A^{\rm op}. As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension (cf. math.QA/0502188 and math.QA/0409589). We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.

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