Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916850320801792 |
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| author | Gabber, Ofer Zavyalov, Bogdan |
| author_facet | Gabber, Ofer Zavyalov, Bogdan |
| contents | First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean geometry. The result says that we can always algebraize a geometrically reduced affinoid rigid-analytic space in "one direction" in an appropriate sense. As an application of this result, we show the remaining cases of the Artin-Grothendieck Vanishing for affinoid algebras, which were previously conjectured in [BM21, §7]. This allows us to deduce a stronger version of the rigid-analytic Artin-Grothendieck Vanishing Conjecture (see [Han20, Conj. 1.2]) over a field of characteristic 0. Using a completely different set of ideas, we also obtain a weaker version of this conjecture over a field of characteristic p>0. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing Gabber, Ofer Zavyalov, Bogdan Algebraic Geometry Commutative Algebra Number Theory First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean geometry. The result says that we can always algebraize a geometrically reduced affinoid rigid-analytic space in "one direction" in an appropriate sense. As an application of this result, we show the remaining cases of the Artin-Grothendieck Vanishing for affinoid algebras, which were previously conjectured in [BM21, §7]. This allows us to deduce a stronger version of the rigid-analytic Artin-Grothendieck Vanishing Conjecture (see [Han20, Conj. 1.2]) over a field of characteristic 0. Using a completely different set of ideas, we also obtain a weaker version of this conjecture over a field of characteristic p>0. |
| title | Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing |
| topic | Algebraic Geometry Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2402.08741 |