Crystalline comparison isomorphisms in p-adic Hodge theory :the absolutely unramified case

Tan, Fucheng; Tong, Jilong

doi:10.2140/ant.2019.13.1509

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Crystalline comparison isomorphisms in $p$-adic Hodge theory:the absolutely unramified case

Fucheng Tan and Jilong Tong

Vol. 13 (2019), No. 7, 1509–1581

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.

Keywords

$p$-adic Hodge theory, proétale topos, crystalline cohomology

Mathematical Subject Classification 2010

Primary: 14F30

Secondary: 11G25

Milestones

Received: 12 March 2016

Revised: 2 May 2019

Accepted: 2 June 2019

Published: 21 September 2019

Authors

Fucheng Tan
Research Institute for Mathematical Sciences Kyoto University Japan

Jilong Tong
School of Mathematical Sciences Capital Normal University Beijing China

Vol. 13, No. 7, 2019