Gelfand-Fuks cohomology of vector fields on algebraic varieties
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914293798141952 |
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| author | Billig, Yuly Dykes, Kathlyn |
| author_facet | Billig, Yuly Dykes, Kathlyn |
| contents | For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gelfand-Fuks cohomology of vector fields on algebraic varieties Billig, Yuly Dykes, Kathlyn Representation Theory Algebraic Geometry 17B56 (Primary) 17B66 (Secondary) For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras. |
| title | Gelfand-Fuks cohomology of vector fields on algebraic varieties |
| topic | Representation Theory Algebraic Geometry 17B56 (Primary) 17B66 (Secondary) |
| url | https://arxiv.org/abs/2410.20316 |