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

Mikhailo Dokuchaev, Mykola Khrypchenko, Juan Jacobo Simón

Abstract

Given a unital action $θ$ 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 $A\rtimes_θS$ with values in a bimodule over $A\rtimes_θS$. 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_θS)^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 $M.$

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

Abstract

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

Paper Structure

This paper contains 15 sections, 57 theorems, 296 equations.

Table of Contents

  1. Preliminaries
  2. Compatible actions of inverse semigroups and partial actions of groups
  3. From compatible actions of $S$ to partial actions of $\mathcal{G}(S)$
  4. The crossed product $A\rtimes_\theta S$
  5. $K S$ as skew group algebra for $E$-unitary $S$
  6. (Co)homology of an inverse monoid $S$ with values in a $K S$-module
  7. Homology of $A\rtimes_\theta S$
  8. Preparatory results
  9. Homology of $A\rtimes_\theta S$ over a field $K$
  10. Homology of $A\rtimes_\theta S$ with $E$-unitary $S$
  11. Cohomology of $A\rtimes_\theta S$
  12. Preliminary results
  13. Cohomology of $A\rtimes_\theta S$ over a field $K$
  14. Cohomology of $A\rtimes_\theta S$ with $E$-unitary $S$
  15. Application to Steinberg algebras

Key Result

Lemma 2.2

Equality tl-0_g(a) defines a bijection from $\mathcal{D}_{g{}^{-1}}$ to $\mathcal{D}_g$ whose inverse is $\tilde{\theta}_{g{}^{-1}}$.

Theorems & Definitions (123)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 113 more