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

• Magnus Carlson ^[1] ; Peter J. Haine ^[2] ; Sebastian Wolf ^[3]
  1. [1] Goethe University Frankfurt

     Goethe University Frankfurt

     Frankfurt, Alemania

     
  2. [2] University of California System

     University of California System

     Estados Unidos

     
  3. [3] University of Regensburg

     University of Regensburg

     Kreisfreie Stadt Regensburg, Alemania

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01054-5
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• Resumen

  • 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 kschemes 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.

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