Algebra depth in tensor categories

TL;DR

This paper explores the concept of algebra depth within tensor categories, linking it to Hopf algebra structures, separability, and Morita invariants, and introduces a tensor categorical perspective on depth.

Contribution

It introduces a tensor categorical definition of depth and relates algebraic properties like semisimplicity and Frobenius extensions to depth and Morita invariants.

Findings

Depth is linearly related to the length of the annihilator chain in semisimple Hopf algebras.

A tensor categorical definition of depth is proposed and connected to previous results.

Depth, Bratteli diagram, and cyclic homology are Morita invariants.

Abstract

Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown in a last section that the depth, Bratteli diagram and relative cyclic homology of algebra extensions are Morita invariants.