Rational p-adic Hodge theory for rigid-analytic varieties

2023/06/09 by Bosco, Guido · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.06100

Abstract

We study a cohomology theory for rigid-analytic varieties over 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-étale 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-étale cohomology in terms of de Rham data, thereby recovering and extending results of Colmez-Niziol.

Cited by

• A p-adic Simpson correspondence for singular rigid-analytic varieties
• Une conjecture C_{\rm st} pour la cohomologie à support compact

Related

• On p-adic comparison theorems for rigid analytic varieties, I
• Rational analytic syntomic cohomology
• On the cohomology of p-adic analytic spaces, I: The basic comparison theorem
• Compactly supported p-adic pro-étale cohomology of analytic varieties
• Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve