Finite depth and Jacobson-Bourbaki correspondence

TL;DR

This paper introduces a new notion of depth three ring towers, explores their properties, and extends the Jacobson-Bourbaki correspondence to depth two extensions with depth three intermediate rings.

Contribution

It defines depth three towers of rings, relates them to Frobenius extensions, and generalizes the Jacobson-Bourbaki correspondence in this context.

Findings

Depth three towers relate to subgroup normal closures in group algebra examples.

A pre-Galois theory for endomorphism rings in depth three towers is developed.

The Jacobson-Bourbaki correspondence is extended to depth two extensions with depth three intermediate rings.

Abstract

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 B_C 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 K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring \End {}_BA_C and coring (A \o_B 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.