Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras
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| Format: | Preprint |
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2025
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| author | Dokuchaev, Mikhailo Khrypchenko, Mykola Simón, Juan Jacobo |
| author_facet | Dokuchaev, Mikhailo Khrypchenko, Mykola Simón, Juan Jacobo |
| contents | 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 $M.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras Dokuchaev, Mikhailo Khrypchenko, Mykola Simón, Juan Jacobo Rings and Algebras Group Theory K-Theory and Homology Operator Algebras Primary 20M18, Secondary 16S35, 16S99, 16W22, 18G40, 22A22 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 $M.$ |
| title | Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras |
| topic | Rings and Algebras Group Theory K-Theory and Homology Operator Algebras Primary 20M18, Secondary 16S35, 16S99, 16W22, 18G40, 22A22 |
| url | https://arxiv.org/abs/2506.20321 |