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<p>Let <i>k</i> be a field that is finitely generated over its prime field. In Grothendieck’s anabelian letter to Faltings, he conjectured that sending a <i>k</i>-scheme to its étale topos defines a fully faithful functor from the localization of the category of finite type <i>k</i>-schemes at the universal homeomorphisms to a category of topoi. By extending results of Voevodsky, we prove Grothendieck’s conjecture for infinite finitely generated fields of arbitrary characteristic. In characteristic 0, this shows that seminormal finite type <i>k</i>-schemes can be reconstructed from their étale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type <i>k</i>-schemes can be reconstructed from their étale topoi.</p>

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A generalization of Voevodsky’s reconstruction theorem of schemes from their étale topoi

  • 期刊论文

  • 发表日期：

    2025年7月10日

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  • Magnus Carlson,
  • Peter J. Haine,
  • Sebastian Wolf

摘要

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. By extending results of Voevodsky, we prove Grothendieck’s conjecture for infinite finitely generated 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.