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Publications mathématiques de l'IHÉS

, Volume 127, Issue 1, pp 71–93 | Cite as

La conjecture du facteur direct

Article

Résumé

M. Hochster a conjecturé que pour toute extension finie \(S\) d’un anneau commutatif régulier \(R\), la suite exacte de \(R\)-modules \(0\to R \to S \to S/R\to0\) est scindée. En nous appuyant sur sa réduction au cas d’un anneau local régulier \(R\) complet non ramifié d’inégale caractéristique, nous proposons une démonstration de cette conjecture dans le contexte de la théorie perfectoïde de P. Scholze. Les deux ingrédients-clé sont le « lemme d’Abhyankar perfectoïde » et l’analyse des extensions kummériennes de \(R\) par une technique d’épaississement sur des voisinages tubulaires.

Nous montrons par les mêmes techniques l’existence d’algèbres de Cohen-Macaulay pour les anneaux locaux d’inégale caractéristique. Il s’ensuit que les revêtements finis d’anneaux réguliers sont dominés par des plats.

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Copyright information

© IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuParisFrance

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