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Morphismes de périodes et cohomologie syntomique

Sally Gilles

Vol. 17 (2023), No. 3, 603–666
Abstract

On commence par donner la version géométrique d’un résultat de Colmez et Nizioł établissant un théorème de comparaison entre les cycles proches p-adiques arithmétiques et la cohomologie des faisceaux syntomiques. La construction locale de cet isomorphisme utilise la théorie des (φ,Γ)-modules et s’obtient en réduisant l’isomorphisme de périodes à un théorème de comparaison entre des cohomologies d’algèbres de Lie. En appliquant ensuite la méthode des « coordonnées plus générales » utilisée par Bhatt, Morrow et Scholze, on construit un isomorphisme global. On peut notamment déduire de ce théorème la conjecture semi-stable de Fontaine et Jannsen. Ce résultat a également été prouvé par (entre autres) Tsuji, via l’application de Fontaine et Messing, et par Česnavičius et Koshikawa, qui généralisent la preuve de la conjecture cristalline de Bhatt, Morrow et Scholze. On utilise l’application construite précédemment pour montrer que le morphisme de périodes de Tsuji est égal à celui de Česnavičius et Koshikawa.

We start by giving the geometric version of a result of Colmez and Nizioł establishing a comparison theorem between p-adic arithmetic nearby cycles and syntomic sheaf cohomology. The local construction of this isomorphism uses (φ,Γ)-modules and is obtained by reducing the period isomorphism to a comparison theorem between cohomologies of Lie algebras. Then, applying Bhatt, Morrow and Scholze’s “more general coordinates” method, we construct a global isomorphism. We can deduce from this theorem the semistable conjecture of Fontaine and Jannsen. This result was also proved, among others, by Tsuji, using Fontaine and Messing’s map, and by Česnavičius and Koshikawa, who generalize the proof of Bhatt, Morrow and Scholze’s crystalline conjecture. We use the previously constructed mapping to show that Tsuji’s period morphism is equal to the one of Česnavičius and Koshikawa.

Keywords
syntomic cohomology, period morphism, $p$-adic cohomology, $p$-adic Hodge theory
Mathematical Subject Classification
Primary: 11S25, 14F20, 14F30, 14F40
Secondary: 14G22
Milestones
Received: 26 January 2021
Revised: 22 March 2022
Accepted: 10 May 2022
Published: 12 April 2023
Authors
Sally Gilles
Huxley Building, South Kensington Campus
Imperial College London
London
United Kingdom

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