Hochschild (Co)homology of D-modules on rigid analytic spaces II
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913935520694272 |
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| author | Vázquez, Fernando Peña |
| author_facet | Vázquez, Fernando Peña |
| contents | Let $X$ be a smooth $p$-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite order differential operators $\mathcal{D}_X$-cap. In particular, we show that the Hochschild cohomology complex of $\mathcal{D}_X$-cap is a strict complex of nuclear Fréchet spaces which is quasi-isomorphic to the de Rham complex of $X$. We then use this to compare the first Hochschild cohomology group of $\mathcal{D}_X$-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of $\mathcal{D}_X$-cap with the deformation theory of $\mathcal{D}_X(X)$-cap. Assuming some finiteness conditions on the de Rham cohomology of $X$, we define explicit isomorphisms between the first Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of bounded outer derivations of $\mathcal{D}_X(X)$-cap, and between the second Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of infinitesimal deformations of $\mathcal{D}_X(X)$-cap. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hochschild (Co)homology of D-modules on rigid analytic spaces II Vázquez, Fernando Peña Number Theory Algebraic Geometry Rings and Algebras Representation Theory Let $X$ be a smooth $p$-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite order differential operators $\mathcal{D}_X$-cap. In particular, we show that the Hochschild cohomology complex of $\mathcal{D}_X$-cap is a strict complex of nuclear Fréchet spaces which is quasi-isomorphic to the de Rham complex of $X$. We then use this to compare the first Hochschild cohomology group of $\mathcal{D}_X$-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of $\mathcal{D}_X$-cap with the deformation theory of $\mathcal{D}_X(X)$-cap. Assuming some finiteness conditions on the de Rham cohomology of $X$, we define explicit isomorphisms between the first Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of bounded outer derivations of $\mathcal{D}_X(X)$-cap, and between the second Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of infinitesimal deformations of $\mathcal{D}_X(X)$-cap. |
| title | Hochschild (Co)homology of D-modules on rigid analytic spaces II |
| topic | Number Theory Algebraic Geometry Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2504.17167 |