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2019 Crystalline comparison isomorphisms in $p$-adic Hodge theory:the absolutely unramified case
Fucheng Tan, Jilong Tong
Algebra Number Theory 13(7): 1509-1581 (2019). DOI: 10.2140/ant.2019.13.1509

Abstract

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

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Fucheng Tan. Jilong Tong. "Crystalline comparison isomorphisms in $p$-adic Hodge theory:the absolutely unramified case." Algebra Number Theory 13 (7) 1509 - 1581, 2019. https://doi.org/10.2140/ant.2019.13.1509

Information

Received: 12 March 2016; Revised: 2 May 2019; Accepted: 2 June 2019; Published: 2019
First available in Project Euclid: 16 January 2020

zbMATH: 07110515
MathSciNet: MR4009670
Digital Object Identifier: 10.2140/ant.2019.13.1509

Subjects:
Primary: 14F30
Secondary: 11G25

Keywords: $p$-adic Hodge theory , Crystalline cohomology , proétale topos

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.13 • No. 7 • 2019
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