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

TL;DR

This paper constructs crystalline comparison isomorphisms in p-adic Hodge theory for proper smooth formal schemes over an unramified base, extending the theory to nontrivial coefficients and relative settings using pro-étale techniques.

Contribution

It develops the crystalline comparison isomorphisms in the absolutely unramified case, incorporating nontrivial coefficients and relative cases via pro-étale methods.

Findings

Established crystalline comparison isomorphisms for proper smooth formal schemes

Proved the Poincaré lemma for crystalline period sheaves

Demonstrated geometric acyclicity of crystalline period sheaves

Abstract

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale 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 pro-\'etale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-\'etale site. Moreover, we need to prove the Poincar\'e 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.