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arXiv · 1710.06145

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

Kestutis Cesnavicius·Teruhisa Koshikawa
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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.

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Kestutis Cesnavicius, Teruhisa Koshikawa. 2017-10-17. The $A_{inf}$-cohomology in the semistable case. https://doi.org/10.1112/s0010437x1800790x

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