Cohomology of ample groupoids
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918110860148736 |
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| author | Deaconu, Valentin Ionescu, Marius |
| author_facet | Deaconu, Valentin Ionescu, Marius |
| contents | We introduce a cochain complex for ample groupoids $\mathcal G$ using a flat resolution defining their homology with coefficients in $\mathbb Z$. We prove that the cohomology of this cochain complex with values in a $\mathcal G$-module $M$ coincides with the previously introduced continuous cocycle cohomology of $\mathcal G$. In particular, this groupoid cohomology is invariant under Morita equivalence. We derive an exact sequence for the cohomology of skew products by a $\mathbb Z$-valued cocycle. We indicate how to compute the cohomology with coefficients in a $\mathcal G$-module $M$ for $AF$-groupoids and for certain action groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00166 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomology of ample groupoids Deaconu, Valentin Ionescu, Marius Operator Algebras Algebraic Topology We introduce a cochain complex for ample groupoids $\mathcal G$ using a flat resolution defining their homology with coefficients in $\mathbb Z$. We prove that the cohomology of this cochain complex with values in a $\mathcal G$-module $M$ coincides with the previously introduced continuous cocycle cohomology of $\mathcal G$. In particular, this groupoid cohomology is invariant under Morita equivalence. We derive an exact sequence for the cohomology of skew products by a $\mathbb Z$-valued cocycle. We indicate how to compute the cohomology with coefficients in a $\mathcal G$-module $M$ for $AF$-groupoids and for certain action groupoids. |
| title | Cohomology of ample groupoids |
| topic | Operator Algebras Algebraic Topology |
| url | https://arxiv.org/abs/2501.00166 |