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DERIVED NON-ARCHIMEDEAN ANALYTIC HILBERT SPACE

Part of: Categories and geometry Arithmetic problems. Diophantine geometry Algebraic geometry: Foundations Special categories

Published online by Cambridge University Press:  26 August 2020

Jorge António Open the ORCID record for Jorge António [Opens in a new window]  and
Mauro Porta
Jorge António
Affiliation:
Institut de Mathématiques de Toulouse, 118 Rue de Narbonne, 31400Toulouse, France ([email protected])
Mauro Porta
Affiliation:
Institut de Recherche Mathématique Avancée, 7 Rue René Descartes,67000Strasbourg, France ([email protected])
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Abstract

In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space $\mathbf{R}\text{Hilb}(X)$ for a separated $k$-analytic space $X$. Such representability results rely on a localization theorem stating that if $\mathfrak{X}$ is a quasi-compact and quasi-separated formal scheme, then the $\infty$-category $\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$ of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the $\infty$-category $\text{Coh}^{-}(\mathfrak{X})$. Along the way, we prove several results concerning the $\infty$-categories of formal models for almost perfect modules on derived $k$-analytic spaces.

Keywords

derived geometryrigid geometryHilbert schemeformal modelsderived generic fiber

MSC classification

Primary: 14G22: Rigid analytic geometry
Secondary: 14A20: Generalizations (algebraic spaces, stacks) 18B25: Topoi 18F99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 335 - 366
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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