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Rational $p$-adic Hodge theory for rigid-analytic varieties

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arxiv 2306.06100 v1 pith:XIAMSAIQ submitted 2023-06-09 math.AG math.NT

Guido Bosco This is my paper · ORCID
classification math.AGmath.NT
keywords cohomologyadicrationalrigid-analyticvarietiescomparisonetalepro-
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abstract

We study a cohomology theory for rigid-analytic varieties over $\mathbb{C}_p$, without properness or smoothness assumptions, taking values in filtered quasi-coherent complexes over the Fargues-Fontaine curve, which compares to other rational $p$-adic cohomology theories for rigid-analytic varieties $-$ namely, the rational $p$-adic pro-\'etale cohomology, the Hyodo-Kato cohomology, and the infinitesimal cohomology over the positive de Rham period ring. In particular, this proves a conjecture of Le Bras. Such comparison results are made possible thanks to the systematic use of the condensed and solid formalisms developed by Clausen-Scholze. As applications, we deduce some general comparison theorems that describe the rational $p$-adic pro-\'etale cohomology in terms of de Rham data, thereby recovering and extending results of Colmez-Niziol.

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Cited by 7 Pith papers

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    Adjoining formal p-adic logarithms of p and 2pi i to the Fargues-Fontaine period ring B makes its Galois cohomology vanish in degrees >=1.

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