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arXiv · 2407.19920

Reconstruction of schemes from their \'{e}tale topoi

Magnus Carlson·Peter J. Haine·Sebastian Wolf
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Abstract

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 \'{e}tale 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 \'{e}tale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their \'{e}tale topoi.

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Magnus Carlson, Peter J. Haine, Sebastian Wolf. 2024-07-29. Reconstruction of schemes from their \'{e}tale topoi. https://arxiv.org/abs/2407.19920

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