Integral p $p$ -adic Hodge theory

The -cohomology in the semistable case

“…The base changes (1.2.1) also allow us to extend the cohomology specialization results obtained in the good reduction case in [BMS18]. Qualitatively, in Proposition 7.7 we show that is torsion free if and only if is torsion free, in which case is torsion free.…”

“…One of the goals of the present paper is to show that the answer is positive if the logarithmic de Rham cohomology of the models and is torsion free (see (8.6.2) and Theorem 8.7): in this case, both and agree with the -lattice in that is functorially determined by . The good reduction case of this result may be derived from the work of Bhatt–Morrow–Scholze [BMS18] on integral -adic Hodge theory, and our approach, as well as the bulk of this paper, is concerned with extending the framework of [BMS18] to the semistable case.…”

“…To approach the question above, we set , let be the basic period ring of Fontaine, and, for a semistable -model of , similarly to the smooth case treated in [BMS18], construct the -cohomology object that is quasi-isomorphic to a bounded complex of finite free -modules and has finitely presented cohomology . We show that base changes of recover other cohomology theories: see §7.2; here denotes the logarithmic crystalline (that is, Hyodo–Kato) cohomology, (respectively, ) carries the log structure associated to (respectively, ), and is endowed with the base change of the standard log structure of .…”

“…The analysis of , specifically, its relation to the -adic étale and the logarithmic crystalline cohomologies, permits us to reprove in Theorem 9.5 the semistable conjecture of Fontaine–Jansen [Kat94a, Conjecture 1.1]: Other proofs of this conjecture have been given in [Tsu99], [Fal02], [Niz08], [Bha12], [Bei13a], and [CN17], whereas [BMS18] used to reprove the crystalline conjecture. Similarly to [CN17], we establish (1.3.1) for -adic formal -schemes that are proper, flat, and ‘semistable’.…”

“…A key result that leads to (1.3.1) is the absolute crystalline comparison isomorphism of Corollary 5.43, whose construction in §5 forms the technical core of this paper. This construction is based on an ‘all possible coordinates’ technique that is a variant of its analogue used to establish (1.3.2) in the smooth case in [BMS18, §12]. The presence of singularities and log structures creates additional complications that do not appear in the smooth case and are examined in §5.…”