Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields
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| Format: | Preprint |
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2026
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| _version_ | 1866916035306717184 |
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| author | Merici, Alberto Rülling, Kay Saito, Shuji |
| author_facet | Merici, Alberto Rülling, Kay Saito, Shuji |
| contents | We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field $K$ of characteristic zero, can be refined to an $\mathbf{A}^1$-invariant cohomology theory of smooth (not necessarily proper) schemes over $K$ with values in $\mathcal{O}_K$-lattices, and the same holds for $K$ of positive characteristic in dimensions at most $3$. As one application, we obtain that the automorphism group of the function field of a proper smooth variety $X$ of dimension at most 3 over a field of positive characteristic acts quasi-unipotently on the cohomology of the structure sheaf of $X$. The construction of the lattices relies on a variant of the tame cohomology of Hübner--Schmidt with coefficients in a twisted version of the tame structure sheaf and uses results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22046 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields Merici, Alberto Rülling, Kay Saito, Shuji Algebraic Geometry 14F06, 14G22, 14J50 We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field $K$ of characteristic zero, can be refined to an $\mathbf{A}^1$-invariant cohomology theory of smooth (not necessarily proper) schemes over $K$ with values in $\mathcal{O}_K$-lattices, and the same holds for $K$ of positive characteristic in dimensions at most $3$. As one application, we obtain that the automorphism group of the function field of a proper smooth variety $X$ of dimension at most 3 over a field of positive characteristic acts quasi-unipotently on the cohomology of the structure sheaf of $X$. The construction of the lattices relies on a variant of the tame cohomology of Hübner--Schmidt with coefficients in a twisted version of the tame structure sheaf and uses results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put. |
| title | Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields |
| topic | Algebraic Geometry 14F06, 14G22, 14J50 |
| url | https://arxiv.org/abs/2605.22046 |