2005/04/30 by Lars Kadison, Kadison, Lars
Mathematics · #16D90 #18E05 #20D25 #22D30 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.CT #math.RA #msc:16D90 #msc:18E05 #msc:20D25 #msc:22D30
paper · pdf · doi:10.48550/arxiv.math/0505004
17 pp, additional section discussing Morita equivalence with generalizations applied to the problem in the main body, depth two bimodules, functorial characterizations of left D2 extensions and prebraided commutativity of the centralizer
openalex publication_date 2005/04/30 · arxiv created 2005/06/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = \End BAB and sub-tensor-square ring T = (A øB A)B. Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A ‖ B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which replaces the key module AAe for an Azumaya algebra A. After establishing these characterizations, we characterize left D2 extensions in terms of the module TR, and ask whether a weak generator condition on RT might characterize left D2 extensions as well, possibly a problem in σ(M)-categories or its generalizations. We also show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, its normality yields a Hopf subalgebra analogue of a factoid for subgroups and their centralizers, and a special case of a conjecture that D2 Hopf subalgebras are normal.