The Ainf-cohomology in the semistable case

For a proper, smooth scheme X over a p-adic fi eld K, we show that any proper, flat, semistable O-K-model X of X whose logarithmic de Rham cohomology is torsion free determines the same O-K-lattice inside H-dR(i) (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 Ainf-valued cohomology theory of p-adic formal, proper, smooth O (K) over bar -schemes X to the semistable case. The relation of the A(inf)-cohomology to the p-adic etale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine-Jannsen.