Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911416782422016 |
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| author | Gabber, Ofer Lodh, Rémi |
| author_facet | Gabber, Ofer Lodh, Rémi |
| contents | We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's numerical criteria, in both the variant for the multiplicity of the special fibre and that for the index of the generic fibre. The second is a logarithmic criterion: we show that, over a log regular base, a proper flat fs log smooth morphism is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelian varieties with good reduction, providing necessary and sufficient conditions for the log smoothness of their regular models over arbitrary discrete valuation rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria Gabber, Ofer Lodh, Rémi Algebraic Geometry Number Theory 14A20, 14A21, 14F06, 14C20, 14C22, 14G17, 14G20, 14H20, 14K15, 14K30, 11G20, 11S15 We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's numerical criteria, in both the variant for the multiplicity of the special fibre and that for the index of the generic fibre. The second is a logarithmic criterion: we show that, over a log regular base, a proper flat fs log smooth morphism is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelian varieties with good reduction, providing necessary and sufficient conditions for the log smoothness of their regular models over arbitrary discrete valuation rings. |
| title | Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria |
| topic | Algebraic Geometry Number Theory 14A20, 14A21, 14F06, 14C20, 14C22, 14G17, 14G20, 14H20, 14K15, 14K30, 11G20, 11S15 |
| url | https://arxiv.org/abs/2403.01831 |