Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

TL;DR

This paper explores the relationship between Hochschild homology of Hopf-Galois extensions and their coinvariants, focusing on coactions, comodules, and descent properties in algebraic structures.

Contribution

It introduces a framework connecting cotensor products of Hochschild homology with coinvariants in Hopf-Galois extensions, extending prior descent results.

Findings

HH.(A,M) is a right comodule over C

The cotensor product N relates HH.(A,M) and HH.(B,N)

Extension of descent results for Hochschild homology

Abstract

Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the action of the Galois group, and HH.(B) are shown to be isomorphic.