Uniqueness of six-functor formalisms
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913889802780672 |
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| author | Dauser, Adam Kuijper, Josefien |
| author_facet | Dauser, Adam Kuijper, Josefien |
| contents | We present an alternative formulation of Scholze's notions of cohomologically proper and cohomologically étale with respect to an abstract six-functor formalism. These conditions guarantee canonical isomorphisms between the direct and exceptional direct images for certain "proper" morphisms, and between the inverse and exceptional inverse images for certain "étale" morphisms. Using this framework, we prove Scholze's conjecture, showing that a six-functor formalism with sufficiently many cohomologically proper and étale morphisms is uniquely determined by the tensor product and inverse image functors, and can be obtained by a construction of Liu-Zheng and Mann. Additionally, we show that a generalisation of the conjecture fails, and propose a measure of this failure in terms of K-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of six-functor formalisms Dauser, Adam Kuijper, Josefien Algebraic Geometry K-Theory and Homology 14F08 (primary), 18B10, 18G90 (secondary) We present an alternative formulation of Scholze's notions of cohomologically proper and cohomologically étale with respect to an abstract six-functor formalism. These conditions guarantee canonical isomorphisms between the direct and exceptional direct images for certain "proper" morphisms, and between the inverse and exceptional inverse images for certain "étale" morphisms. Using this framework, we prove Scholze's conjecture, showing that a six-functor formalism with sufficiently many cohomologically proper and étale morphisms is uniquely determined by the tensor product and inverse image functors, and can be obtained by a construction of Liu-Zheng and Mann. Additionally, we show that a generalisation of the conjecture fails, and propose a measure of this failure in terms of K-theory. |
| title | Uniqueness of six-functor formalisms |
| topic | Algebraic Geometry K-Theory and Homology 14F08 (primary), 18B10, 18G90 (secondary) |
| url | https://arxiv.org/abs/2412.15780 |