Vol. 16, No. 3, 2022

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Enveloppe étale de morphismes plats

Daniel Ferrand

Appendix: Ofer Gabber

Vol. 16 (2022), No. 3, 521–565
Abstract

Pour un morphisme T S, plat et de présentation finie, l’enveloppe étale — un avatar du π0(TS) — peut ne pas exister  ; par contre l’enveloppe étale séparée, i.e., celle qui est universelle pour les schémas étales et séparés sur S, existe dès que S est localement connexe. On la note T πs(TS)  ; c’est le quotient de T par la relation d’équivalence minimale à graphe ouvert et fermé dans T ×ST  ; cette construction permet de démontrer qu’un morphisme fpqc de changement de base S S induit un isomorphisme πs(S×STS) S×Sπs(TS) si ses fibres géométriques sont connexes. Par ailleurs, lorsque S est normal, disons intègre, et que le morphisme T S est normal, on dispose d’un isomorphisme πs(Tξξ) πs(TS)ξ  ; il permet d’étendre à des morphismes normaux des résultats connus sur un corps de base. Toujours sous des hypothèses de normalité, le morphisme canonique π0(TS) πs(TS) fait apparaître le schéma πs comme l’enveloppe séparée de l’espace algébrique π0.

For a flat morphism T S of finite presentation, the étale envelope — an avatar of π0(TS) — may fail to exist; but, the separated étale envelope, i.e., the one which is universal for the separated étale S-schemes only, is shown to exist as soon as S is locally connected; denoted πs(TS), it is the quotient of T by the equivalence relation which is minimal among those with clopen graph in T ×ST; from that we prove that, for a fpqc base change S S, the morphism πs(S×STS) S×Sπs(TS) is an isomorphism if the geometric fibres of S S are connected. When S is integral with generic point ξ, and if both S and the morphism T S are normal, then one gets the isomorphism πs(Tξξ) πs(TS)ξ, on the strength of which one can often extend to S results previously proven when the base is a field.

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Keywords
étale envelope, connected component, clopen subsets of a scheme
Mathematical Subject Classification 2010
Primary: 14A15
Secondary: 14A20
Milestones
Received: 27 January 2019
Revised: 20 March 2021
Accepted: 18 July 2021
Published: 9 July 2022
Authors
Daniel Ferrand
Institut Mathématique de Jussieu
Sorbonne Université
Paris
France
Ofer Gabber
IHES
Bures-sur-Yvette
France