Centers of perfectoid purity

Anne Fayolle

doi:10.5802/jep.313

Centers of perfectoid purity
[Centres de pureté perfectoïde]

Anne Fayolle ¹

¹ Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1381-1415.

Abstract (VO)
Résumé

We introduce a mixed characteristic analog of log canonical centers in characteristic $0$ and centers of $F$-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of the ring, analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under étale morphisms.

Nous introduisons un objet analogue en caractéristique mixte des centres log canoniques en caractéristique $0$ et des centres de pureté de Frobenius en caractéristique positive. Nous le nommons centre de pureté perfectoïde. Nous prouvons qu’un anneau intégralement clos n’admet pas de centre non trivial. Nous démontrons également l’existence d’un centre de pureté perfectoïde spécial qui détecte la pureté perfectoide de l’anneau, de manière analogue à l’idéal premier de scindage d’Aberbach et Enescu. Enfin, nous étudions le comportement de ces centre sous les morphismes étales.

Reçu le : 2025-05-10
Accepté le : 2025-09-03
Publié le : 2025-09-15

DOI : 10.5802/jep.313

Classification : 14G45, 14F18, 14B05, 13A35
Keywords: $F$-singularities, centers of $F$-purity, perfectoid singularities, log canonical center
Mots-clés : Singularités de Frobenius, centre de pureté de Frobenius, singularités perfectoïde, centre log canonique

Affiliations des auteurs :

Anne Fayolle ¹

¹ Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Anne Fayolle. Centers of perfectoid purity. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1381-1415. doi : 10.5802/jep.313. https://jep.centre-mersenne.org/articles/10.5802/jep.313/

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