The tempered disk and the tempered cohomology
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| Format: | Preprint |
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2024
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| author | Bambozzi, Federico Chiarellotto, Bruno Vanni, Pietro |
| author_facet | Bambozzi, Federico Chiarellotto, Bruno Vanni, Pietro |
| contents | Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic geometry (then associating a "spectrum" to each ind-Banach algebra). We want to look at the behaviour of this spectrum from a differential point of view. In such a spectrum, for example, there exist open subsets having functions with log-growth as sections for the structural sheaf. In this framework, a transfer theorem for the log-growth of solutions of p-adic differential equations can be interpreted as a continuity theorem (analogue to the transfer theorem for their radii of convergence in the Berkovich spaces). As a dividend of such a theory, we define a new cohomology theory in terms of the Hodge-completed derived de Rham cohomology of the ind-Banach derived analytic space associated to a smooth k-scheme, X_k (k residual field of V), via the use of "tempered tubes".
We finally compare our tempered de Rham cohomology with crystalline cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_09473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The tempered disk and the tempered cohomology Bambozzi, Federico Chiarellotto, Bruno Vanni, Pietro Algebraic Geometry Functional Analysis Number Theory Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic geometry (then associating a "spectrum" to each ind-Banach algebra). We want to look at the behaviour of this spectrum from a differential point of view. In such a spectrum, for example, there exist open subsets having functions with log-growth as sections for the structural sheaf. In this framework, a transfer theorem for the log-growth of solutions of p-adic differential equations can be interpreted as a continuity theorem (analogue to the transfer theorem for their radii of convergence in the Berkovich spaces). As a dividend of such a theory, we define a new cohomology theory in terms of the Hodge-completed derived de Rham cohomology of the ind-Banach derived analytic space associated to a smooth k-scheme, X_k (k residual field of V), via the use of "tempered tubes". We finally compare our tempered de Rham cohomology with crystalline cohomology. |
| title | The tempered disk and the tempered cohomology |
| topic | Algebraic Geometry Functional Analysis Number Theory |
| url | https://arxiv.org/abs/2410.09473 |