Vol. 13, No. 7, 2019

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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

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.

$p$-adic Hodge theory, proétale topos, crystalline cohomology
Mathematical Subject Classification 2010
Primary: 14F30
Secondary: 11G25
Received: 12 March 2016
Revised: 2 May 2019
Accepted: 2 June 2019
Published: 21 September 2019
Fucheng Tan
Research Institute for Mathematical Sciences
Kyoto University
Jilong Tong
School of Mathematical Sciences
Capital Normal University