Duality for $p$-adic geometric pro-étale cohomology
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908578491662336 |
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| author | Colmez, Pierre Gilles, Sally Nizioł, Wiesława |
| author_facet | Colmez, Pierre Gilles, Sally Nizioł, Wiesława |
| contents | We prove that $p$-adic geometric pro-étale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincaré duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizioł, from a Poincaré duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincaré duality for ${\mathbf B}^+_{\rm st}$-twisted Hyodo-Kato and filtered $\mathbf{B}^+_{\rm dr}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Duality for $p$-adic geometric pro-étale cohomology Colmez, Pierre Gilles, Sally Nizioł, Wiesława Algebraic Geometry Number Theory We prove that $p$-adic geometric pro-étale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincaré duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizioł, from a Poincaré duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincaré duality for ${\mathbf B}^+_{\rm st}$-twisted Hyodo-Kato and filtered $\mathbf{B}^+_{\rm dr}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result. |
| title | Duality for $p$-adic geometric pro-étale cohomology |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2411.12163 |