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.
Categories: math.NT math.AG