The $A_{inf}$-cohomology in the semistable case

TL;DR

This paper extends $A_{inf}$-cohomology theory to semistable schemes over $p$-adic fields, establishing functorial lattices in de Rham cohomology and proving the semistable conjecture of Fontaine--Jannsen.

Contribution

It generalizes Bhatt--Morrow--Scholze's $A_{inf}$-cohomology to the semistable case and links it to classical $p$-adic cohomologies, confirming the semistable conjecture.

Findings

Unified $A_{inf}$-cohomology for semistable schemes.

Established functorial lattices in de Rham cohomology.

Reproved the semistable conjecture of Fontaine--Jannsen.

Abstract

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-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 $\mathcal{O}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-cohomology to the $p$-adic \'{e}tale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine--Jannsen.