Bounded linear endomorphisms of rigid analytic functions
Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of th...
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| Format: | Journal article |
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2016
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| _version_ | 1797091204802805760 |
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| author | Ardakov, K Ben-Bassat, O |
| author_facet | Ardakov, K Ben-Bassat, O |
| author_sort | Ardakov, K |
| collection | OXFORD |
| description | Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of the structure sheaf. In the case of complex manifolds, Ishimura proved that the analogous sheaves are isomorphic. In the rigid analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to \mathcal{E}$ is an isomorphism if and only if the ground field $K$ is algebraically closed and its residue field is uncountable. |
| first_indexed | 2024-03-07T03:29:38Z |
| format | Journal article |
| id | oxford-uuid:ba428b39-eb60-4070-a261-1c55f7dc6a2c |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:29:38Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | oxford-uuid:ba428b39-eb60-4070-a261-1c55f7dc6a2c2022-03-27T05:08:34ZBounded linear endomorphisms of rigid analytic functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ba428b39-eb60-4070-a261-1c55f7dc6a2cSymplectic Elements at Oxford2016Ardakov, KBen-Bassat, OLet $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of the structure sheaf. In the case of complex manifolds, Ishimura proved that the analogous sheaves are isomorphic. In the rigid analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to \mathcal{E}$ is an isomorphism if and only if the ground field $K$ is algebraically closed and its residue field is uncountable. |
| spellingShingle | Ardakov, K Ben-Bassat, O Bounded linear endomorphisms of rigid analytic functions |
| title | Bounded linear endomorphisms of rigid analytic functions |
| title_full | Bounded linear endomorphisms of rigid analytic functions |
| title_fullStr | Bounded linear endomorphisms of rigid analytic functions |
| title_full_unstemmed | Bounded linear endomorphisms of rigid analytic functions |
| title_short | Bounded linear endomorphisms of rigid analytic functions |
| title_sort | bounded linear endomorphisms of rigid analytic functions |
| work_keys_str_mv | AT ardakovk boundedlinearendomorphismsofrigidanalyticfunctions AT benbassato boundedlinearendomorphismsofrigidanalyticfunctions |