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

TL;DR

This paper proves Grothendieck's conjecture that schemes over finitely generated fields can be reconstructed from their étale topoi, extending previous results to infinite fields of any characteristic.

Contribution

It establishes the full faithfulness of the functor from schemes to their étale topoi over infinite fields, generalizing prior work to arbitrary characteristic.

Findings

Reconstruction of schemes from étale topoi in characteristic 0.

Reconstruction of perfections of schemes in positive characteristic.

Extension of Grothendieck's conjecture to infinite fields.

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.