Homology of Epsilon-Strongly Graded Algebras

TL;DR

This paper develops spectral sequences to compute the Hochschild (co)homology of epsilon-strongly graded algebras, linking it to partial group (co)homology and conjugacy class decompositions.

Contribution

It introduces spectral sequences for Hochschild (co)homology of epsilon-strongly graded algebras, connecting them to partial group (co)homology and conjugacy class structures.

Findings

Spectral sequences converge to Hochschild (co)homology of the algebra.

Decomposition of homology spectral sequence by conjugacy classes.

Identification of the $E^2$-page with ordinary group homology of centralizers.

Abstract

Let $G$ be a group and $S$ a unital epsilon-strongly $G$-graded algebra. We construct spectral sequences converging to the Hochschild (co)homology of $S$. Each spectral sequence is expressed in terms of the partial group (co)homology of $G$ with coefficients in the Hochschild (co)homology of the degree-one component of $S$. Moreover, we show that the homology spectral sequence decomposes according to the conjugacy classes of $G$, and, by means of the globalization functor, its $E^2$-page can be identified with the ordinary group homology of suitable centralizers.