Odd H-depth and H-separable extensions

TL;DR

This paper introduces the concept of H-depth for subring pairs, relating it to Hochschild homology, matrix invariants, and Morita theory, providing new characterizations for various classes of ring extensions.

Contribution

It defines H-depth for subring pairs and derives formulas for semisimple and QF extensions, connecting H-depth to matrix invariants and Morita theory.

Findings

H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix.

For QF extensions, H-depth is derived from the odd depth of the endomorphism ring extension.

In certain categories, the minimum depth and H-depth are always finite.

Abstract

A subring pair B < A has right depth 2n if the n+1'st relative Hochschild bar resolution group is isomorphic to a direct summand of a multiple of the n'th relative Hochschild bar resolution group as A-B-bimodules; depth 2n+1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n-1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix, and for QF extensions it is derived from the odd depth of the endomorphism ring extension. For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory. In certain nice categories of bimodules the minimum depth and H-depth of certain types of ring extensions are always finite.