Duplicial functors, descent categories and generalized Hopf modules

TL;DR

This paper explores the conditions under which cyclic homology invariants can be described using modules and comodules over bialgebras, particularly focusing on Hopf algebras with bijective antipodes and generalized Hopf modules.

Contribution

It formulates conditions enabling a dual description of coalgebras in the context of cyclic homology, extending the Hopf module theorem to broader settings.

Findings

Describes when left and right coalgebras can be simultaneously represented as modules and comodules.

Identifies Hopf algebras with bijective antipodes as key examples.

Connects generalized Hopf modules to cyclic homology invariants.

Abstract

B\"ohm and \c{S}tefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^\chi_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $\chi$ between two comonads. For the key example associated to a bialgebra $H$, right $\chi$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $\chi$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples.