Depth three towers and Jacobson-Bourbaki correspondence

TL;DR

This paper introduces the concept of depth three towers of rings as a generalization of depth two extensions, exploring their properties, connections to Galois theory, and a Jacobson-Bourbaki correspondence in this context.

Contribution

It defines depth three towers of rings, establishes their relation to Frobenius extensions, and develops a pre-Galois theory and a Jacobson-Bourbaki correspondence for these structures.

Findings

Depth three towers generalize depth two extensions.

A pre-Galois theory for certain endomorphism rings is established.

A Jacobson-Bourbaki correspondence is extended to depth three intermediate rings.

Abstract

We introduce a notion of depth three tower of three rings C < B < A as a useful generalization of depth two ring extension. If A = End B_C and B | C is a Frobenius extension, this also captures the notion of depth three for a Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C is depth three, then A | C is depth two (cf. math.QA/0001020). 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 \otimes_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in the last two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.