Ideal depth of QF extensions

TL;DR

This paper introduces the concept of ideal depth for ring extensions, providing bounds and relations with existing depth notions, especially for Frobenius and QF extensions, along with structural formulas and embedding results.

Contribution

It defines ideal depth for ring homomorphisms, relates it to existing depth measures, and establishes embedding and structural results for Frobenius and QF extensions.

Findings

Ideal depth bounds classical subring depth in certain algebra extensions.

In QF extensions, minimum left and right even depths coincide.

Depth 3 QF extensions can embed into depth 2 extensions.

Abstract

A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra pair or semisimple complex algebra extensions. If R | S is a QF extension of finite depth, minimum left and right even depth are shown to coincide. If R < S is moreover a Frobenius extension with R a right S-generator, its subring depth is shown to coincide with its tower depth. In the process formulas for the ring, module, Frobenius and Temperley-Lieb structures are provided for the tensor product tower above a Frobenius extension. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depth n extensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.