Reconstruction of schemes from their étale topoi
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910545628626944 |
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| author | Carlson, Magnus Haine, Peter J. Wolf, Sebastian |
| author_facet | Carlson, Magnus Haine, Peter J. Wolf, Sebastian |
| contents | Let $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its étale topos defines a fully faithful functor from the localization of the category of finite type $k$-schemes at the universal homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for infinite fields of arbitrary characteristic. In characteristic $0$, this shows that seminormal finite type $k$-schemes can be reconstructed from their étale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their étale topoi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstruction of schemes from their étale topoi Carlson, Magnus Haine, Peter J. Wolf, Sebastian Algebraic Geometry Category Theory Let $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its étale topos defines a fully faithful functor from the localization of the category of finite type $k$-schemes at the universal homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for infinite fields of arbitrary characteristic. In characteristic $0$, this shows that seminormal finite type $k$-schemes can be reconstructed from their étale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their étale topoi. |
| title | Reconstruction of schemes from their étale topoi |
| topic | Algebraic Geometry Category Theory |
| url | https://arxiv.org/abs/2407.19920 |