Subgroup Depth and Twisted Coefficients

Abstract

Danz computes the depth of certain twisted group algebra extensions in [10], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8]. In this article, 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 ⊗ 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 the subgroup depth behaves exactly like the combinatorial depth with respect to the core of a subgroup, and extend results in [22] to coideal subalgebras of finite dimension.