The -cohomology in the semistable case

Abstract

For a proper, smooth scheme X over a p-adic field K, we show that any proper, flat, semistable OK-model X of X whose logarithmic de Rham cohomology is torsion free determines the same OK -lattice inside H i dR (X/K) and, moreover, that this lattice is functorial in X. For this, we extend the results of Bhatt-Morrow-Scholze on the construction and the analysis of an A inf -valued cohomology theory of p-adic formal, proper, smooth O K -schemes X to the semistable case. The relation of the A inf -cohomology to the p-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine-Jannsen.

Breuil–Kisin modules via crystalline cohomology

“…In the case of low ramification ei < p − 1, it should be straightforward to prove that this is indeed the case using work of Caruso [15] (generalizing earlier work of Breuil [10] in the case e = 1), which provides the essential integral comparison isomorphisms needed to adapt the arguments of §5 to this setting. To get results without restriction on the ramification of K requires the generalization of [6, Theorem 1.8] to the case of semistable reduction, which has recently been established by Cesnavicius and Koshikawa [17].…”

p-Adic Geometry

“…All of these results were recently extended to the case of semistable reduction byČesnavičius-Koshikawa,[CK17]. In that situation, part (iii) implies that also H í et (X, Zp) is p-torsion free.Corollary 8.2.…”

Hodge theory of $p$-adic varieties: a survey

“…From the beginning of p-adic Hodge theory we knew that we can do integral computations for large primes p (Fontaine-Lafaille theory) and also that the denominators that appear in general can be universally bounded (in terms of p and the dimension of the varieties). However, perhaps no expert expected the result proved recently by Bhatt-Morrow-Scholze [9], [10] (see Bhatt-Scholze [11] for a different treatment via prismatic cohomology and Česnavičius-Koshikawa [16] for generalizations): it is enough to "twist" by the element µ ∈ B cr (a lift of ζ p − 1, for a primitive pth root of unity ζ p , from C p ) to obtain optimal integral p-adic comparison theorems.…”

“…In [64], [65], I proved this to be indeed the case for the syntomic, the old almost étale, and the Beilinson approaches; the case of the (rational) comparison morphism of Bhatt-Morrow-Scholze and Česnavičius-Koshikawa was treated in the PhD thesis of Sally Gilles [47]. The proof in [47] is direct, using the fact that the comparison morphism constructed in [47] (a global geometric version of the morphism constructed by Colmez-Nizioł [22]) is very similar ( 7 ) to that of [9] and [16] and can be shown directly to be the same as the comparison morphism of Fontaine-Messing. My proofs exploit the motivic approach (A3) in [64] and the h-topology approach (B1) in [65].…”

“…(3) There is a potentially semistable comparison theorem for smooth and ("almost") proper varieties obtained by Colmez-Nizioł via the syntomic technique that works surprisingly well in the analytic context [22], [25]. (4) There are integral comparison theorems of Bhatt-Morrow-Scholze, Česnavičius-Koshikawa, and Bhatt-Scholze [9], [10], [16], [11] that introduce a new cohomology (A inf -chomology or prismatic cohomology) as a gobetween for étale and de Rham cohomologies.…”