Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras

TL;DR

This paper develops spectral sequences to compute Hochschild (co)homology of crossed products by inverse monoid actions, with applications to Steinberg algebras of ample groupoids, revealing new connections between algebraic and topological structures.

Contribution

It introduces new (co)homology spectral sequences for crossed products by inverse monoids, linking Hochschild (co)homology to inverse semigroup (co)homology, and applies these to Steinberg algebras of ample groupoids.

Findings

Spectral sequences converge to Hochschild (co)homology of crossed products.

Spectral sequences simplify when certain flatness conditions are met.

Hochschild homology of Steinberg algebras relates to inverse semigroup homology.

Abstract

Given a unital action $\theta $ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_\theta S$ with values in a bimodule over $A\rtimes_\theta S$. The spectral sequences involve a new kind of (co)homology of the inverse monoid $S,$ which is based on $KS$-modules. The spectral sequences take especially nice form, when $(A\rtimes_\theta S)^e $ is flat as a left (homology case) or right (cohomology case) $A^e$-module, involving also the Hochschild (co)homology of $A.$ Same nice spectral sequences are also obtained if $K$ is a commutative ring, over which $A$ is projective, and $S$ is $E$-unitary. We apply our results to the Steinberg algebra $A_K(\mathscr{G})$ over a field $K$ of an ample groupoid $\mathscr{G},$ whose unit space $\mathscr{G} ^{(0)}$ is compact. In the homology case our spectral sequence collapses on the $p$-axis, resulting in an isomorphism between the Hochschild homology of $A_K(\mathscr{G})$ with values in an $A_K(\mathscr{G})$-bimodule $M$ and the homology of the inverse semigroup of the compact open bisections of $\mathscr{G}$ with values in the invariant submodule of $M.$