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DERIVED NON-ARCHIMEDEAN ANALYTIC HILBERT SPACE
Part of:
Categories and geometry
Algebraic geometry: Foundations
Special categories
Arithmetic problems. Diophantine geometry
Published online by Cambridge University Press: 26 August 2020
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.
MSC classification
Primary:
14G22: Rigid analytic geometry
- 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
References
António, J., Moduli of
$p$
-adic representations of a profinite group, preprint, 2017, arXiv:1709.04275.Google Scholar
$p$
-adic representations of a profinite group, preprint, 2017, arXiv:1709.04275.Google ScholarAntónio, J.,
$p$
-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302.Google Scholar
$p$
-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302.Google ScholarAntónio, J., Moduli of
$\ell$
-adic pro-étale local systems for smooth non-proper schemes, preprint, 2019, arXiv:1904.08001.Google Scholar
$\ell$
-adic pro-étale local systems for smooth non-proper schemes, preprint, 2019, arXiv:1904.08001.Google ScholarBosch, S., Lectures on Formal and Rigid Geometry, Lecture Notes in Mathematics, (Springer International Publishing, 2014). ISBN: 978-3-319-04416-3; 978-3-319-04417-0.CrossRefGoogle Scholar
Bosch, S. and Lütkebohmert, W., Formal and rigid geometry. II. Flattening techniques, Math. Ann.
296(3) (1993), 403–429.CrossRefGoogle Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math.
119(2) (1995), 361–398.CrossRefGoogle Scholar
Hennion, B., Porta, M. and Vezzosi, G., Formal gluing along non-linear flags, preprint, 2016, arXiv:1607.04503.Google Scholar
Lurie, J., Spectral algebraic geometry, available at http://www.math.harvard.edu/∼lurie/papers/SAG-rootfile.pdf, 2018.Google Scholar
Lurie, J., Higher Topos Theory, Annals of Mathematics Studies, Volume 170 (Princeton University Press, Princeton, NJ, 2009).CrossRefGoogle Scholar
Lurie, J., DAG IX: Closed immersions, preprint, 2011.Google Scholar
Lurie, J., DAG V: Structured spaces, preprint, 2011.Google Scholar
Lurie, J., DAG VII: Spectral schemes, preprint, 2011.Google Scholar
Porta, M. and Yu, T. Y., Derived non-archimedean analytic spaces, preprint, 2016, arXiv:1601.00859. To appear in Selecta Mathematica:MR 3782411.Google Scholar
Porta, M. and Yu, T. Y., Higher analytic stacks and GAGA theorems, Adv. Math.
302 (2016), 351–409.CrossRefGoogle Scholar
Porta, M. and Yu., T. Y., Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683. To appear in J. Eur. Math. Soc.Google Scholar
Porta, M. and Yu, T. Y., Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730.Google Scholar