Curves are algebraic $K(π,1)$: theoretical and practical aspects
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914955435966464 |
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| author | Levrat, Christophe |
| author_facet | Levrat, Christophe |
| contents | We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(π,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $π_1(X)$-module. To this end, we explicitly construct some Galois coverings of $X$ corresponding to Galois coverings of the normalisation of its irreducible components. When $k$ is finite or separably closed, we explicitly describe finite quotients of $π_1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products $H^1\times H^1\to H^2$ and $H^1\times H^2\to H^3$ in terms of finite group cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curves are algebraic $K(π,1)$: theoretical and practical aspects Levrat, Christophe Algebraic Geometry Number Theory 14F20, 14H30, 11G20 We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(π,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $π_1(X)$-module. To this end, we explicitly construct some Galois coverings of $X$ corresponding to Galois coverings of the normalisation of its irreducible components. When $k$ is finite or separably closed, we explicitly describe finite quotients of $π_1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products $H^1\times H^1\to H^2$ and $H^1\times H^2\to H^3$ in terms of finite group cohomology. |
| title | Curves are algebraic $K(π,1)$: theoretical and practical aspects |
| topic | Algebraic Geometry Number Theory 14F20, 14H30, 11G20 |
| url | https://arxiv.org/abs/2306.03295 |