2007/07/25 by Lars Kadison, Kadison, Lars · 1 citation
Mathematics · #13B05 #16W30 (Primary) 46L37 #81R15 (Secondary) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math.OA #math.QA #msc:13B05 #msc:16W30 #msc:46L37 #msc:81R15
paper · pdf · doi:10.48550/arxiv.0707.3756
26 pp., depth three towers with new section on finite depth, and corrections
arxiv created 2007/07/25 · openalex publication_date 2007/07/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a notion of depth three tower of three rings C < B < A with depth two ring extension A | B recovered when B = C. If A = \End BC and B | C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension in arXiv:math/0107064 and arXiv:math/0108067, such that if B | C is depth three, then A | C is depth two (a phenomenon of finite depth subfactors, see arXiv:math/0006057). We provide a similar definition of finite depth Frobenius extension with embedding theorem utilizing a depth three subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure KG contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring \End BAC and coring (A øB A)C involving Morita context bimodules and left coideal subrings. This is applied in two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.