Cohomological Descent for Faltings Ringed Topos
Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000264/type/journal_article |
| _version_ | 1797229751730962432 |
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| author | Tongmu He |
| author_facet | Tongmu He |
| author_sort | Tongmu He |
| collection | DOAJ |
| description | Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings’ approach to any integral model, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings. |
| first_indexed | 2024-04-24T15:17:34Z |
| format | Article |
| id | doaj.art-d3b5488e46b34502a36241aa14722c76 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-24T15:17:34Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-d3b5488e46b34502a36241aa14722c762024-04-02T09:10:59ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.26Cohomological Descent for Faltings Ringed ToposTongmu He0https://orcid.org/0000-0002-7387-7968Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, FranceFaltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings’ approach to any integral model, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings.https://www.cambridge.org/core/product/identifier/S2050509424000264/type/journal_article14F30 |
| spellingShingle | Tongmu He Cohomological Descent for Faltings Ringed Topos Forum of Mathematics, Sigma 14F30 |
| title | Cohomological Descent for Faltings Ringed Topos |
| title_full | Cohomological Descent for Faltings Ringed Topos |
| title_fullStr | Cohomological Descent for Faltings Ringed Topos |
| title_full_unstemmed | Cohomological Descent for Faltings Ringed Topos |
| title_short | Cohomological Descent for Faltings Ringed Topos |
| title_sort | cohomological descent for faltings ringed topos |
| topic | 14F30 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000264/type/journal_article |
| work_keys_str_mv | AT tongmuhe cohomologicaldescentforfaltingsringedtopos |