Subgroup depth and twisted coefficients

TL;DR

This paper investigates the relationships between subgroup depth, h-depth, and coring depth in algebra extensions, providing new bounds and theoretical insights into their structures, especially in the context of Hopf algebras and Galois corings.

Contribution

It establishes that the h-depth of group crossed product algebra extensions is bounded by the h-depth of the corresponding group algebra extension, and extends depth results to coideal subalgebras.

Findings

h-depth of crossed product algebra ≤ h-depth of group algebra extension

Corings associated with A tensor C have a depth extending h-depth

Subgroup depth matches combinatorial depth relative to subgroup core

Abstract

Danz computes the depth of certain twisted group algebra extensions in Comm. Alg. (2011), which are less than the values of the depths of the corresponding untwisted group algebra extensions in Burciu et al, I.E.J.A. (2011). In this paper, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right $H$-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A tensor C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R < H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that subgroup depth behaves exactly like combinatorial depth with respect to the core of a subgroup, and extend results in Kadison J.Pure Appl.Alg. (2014) to coideal subalgebras of finite dimension.