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

Abstract

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold forétale 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-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proetale 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.

The -cohomology in the semistable case

“…Our final goal is to deduce the semistable comparison isomorphism for suitable proper, ‘semistable’ formal schemes (see Theorem 9.5). This extends [BMS18, 1.1(i)], which treated the good reduction case (see also [TT15, 1.2] for a result ‘with coefficients’ over an absolutely unramified base), and is similar to the semistable comparison established by Colmez–Niziol [CN17, 5.26]. More precisely, [CN17, 5.26] also includes cases in which the log structures are not ‘vertical’.…”

On uniqueness ofp-adic period morphisms, II

“…There are by now four main different approaches to the construction of these period morphisms: syntomic: Fontaine and Messing [FM87], Hyodo and Kato [HK94], Kato [Kat94a], Tsuji [Tsu99a], Yamashita [Yam11], Colmez and Nizioł [CN17]; almost étale: Faltings [Fal89, Fal02], Scholze [Sch13], Li and Pan [LP19], Diao et al . [DLLZ19], Tan and Tong [TT19], Bhatt, Morrow and Scholze [BMS18, BMS19], Česnavičius and Koshikawa [ČK19]; motivic: Nizioł [Niz98, Niz08]; -cohomology: Beilinson [Bei12, Bei13], Bhatt [Bha12]. …”

“…almost étale: Faltings [Fal89, Fal02], Scholze [Sch13], Li and Pan [LP19], Diao et al . [DLLZ19], Tan and Tong [TT19], Bhatt, Morrow and Scholze [BMS18, BMS19], Česnavičius and Koshikawa [ČK19];…”

“…More work followed: Li and Pan [LP19] extended Scholze's de Rham comparison for trivial coefficients to the open case (with a nice compactification), Diao et al . [DLLZ19] added a treatment of nontrivial coefficients; from another angle, Tan and Tong [TT19] extended Scholze's proof to the case of good reduction (over an unramified base) proving the crystalline conjecture in this setting.…”

A note on p-adic Simpson correspondence

“…Theorem 1.1 answers a question raised by Liu-Zhu [LZ17] about the relation of their p-adic Simpson correspondence and Faltings' p-adic Simpson correspondence. It should be regarded as a general form of a thoerem proven by Tan-Tong [TT19]. If a filtered de Rham bundle (V, ∇, Fil) underlies a Fontaine-Faltings module, then the crystalline representation corresponding to (V, ∇, Fil) restricted to the geometric fundamental group coincides with the generalized representation corresponding to the graded Higgs bundle Gr Fil (V, ∇).…”