Hochschild (Co)homology of D-modules on rigid analytic spaces I
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915785436299264 |
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| author | Vázquez, Fernando Peña |
| author_facet | Vázquez, Fernando Peña |
| contents | We introduce a formalism of Hochschild (co)-homology for $\mathcal{D}$-cap modules on smooth rigid analytic spaces based on the homological tools of Ind-Banach $\mathcal{D}$-cap modules. We introduce several categories of $\mathcal{D}$-cap bimodules for which this theory is well-behaved. Among these, the most important example is the category of diagonal $\mathcal{C}$-complexes. We give an explicit calculation of the Hochschild complex for diagonal $\mathcal{C}$-complexes, and show that the Hochschild complex of $\mathcal{D}$-cap is canonically isomorphic to the de Rham complex of $X$. In particular, we obtain a Hodge-de Rham spectral sequence converging to the Hochschild cohomology groups of $\mathcal{D}$-cap. We obtain explicit formulas relating the Hochschild cohomology and homology of a given diagonal $\mathcal{C}$-complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hochschild (Co)homology of D-modules on rigid analytic spaces I Vázquez, Fernando Peña Number Theory Algebraic Geometry Rings and Algebras Representation Theory We introduce a formalism of Hochschild (co)-homology for $\mathcal{D}$-cap modules on smooth rigid analytic spaces based on the homological tools of Ind-Banach $\mathcal{D}$-cap modules. We introduce several categories of $\mathcal{D}$-cap bimodules for which this theory is well-behaved. Among these, the most important example is the category of diagonal $\mathcal{C}$-complexes. We give an explicit calculation of the Hochschild complex for diagonal $\mathcal{C}$-complexes, and show that the Hochschild complex of $\mathcal{D}$-cap is canonically isomorphic to the de Rham complex of $X$. In particular, we obtain a Hodge-de Rham spectral sequence converging to the Hochschild cohomology groups of $\mathcal{D}$-cap. We obtain explicit formulas relating the Hochschild cohomology and homology of a given diagonal $\mathcal{C}$-complex. |
| title | Hochschild (Co)homology of D-modules on rigid analytic spaces I |
| topic | Number Theory Algebraic Geometry Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2504.16707 |