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The
$A_{\text{inf}}$-cohomology in the semistable case
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 11 / November 2019
- Published online by Cambridge University Press:
- 09 September 2019, pp. 2039-2128
- Print publication:
- November 2019
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- Article
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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 
$H_{\text{dR}}^{i}(X/K)$ 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 
$A_{\text{inf}}$-valued cohomology theory of 
$p$-adic formal, proper, smooth 
${\mathcal{O}}_{\overline{K}}$-schemes 
$\mathfrak{X}$ to the semistable case. The relation of the 
$A_{\text{inf}}$-cohomology to the 
$p$-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.
