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Derived non-archimedean analytic spaces

Porta, Mauro and Yu, Tony Yue (2018) Derived non-archimedean analytic spaces. Selecta Mathematica - New Series, 24 (2). pp. 609-665. ISSN 1022-1824. doi:10.1007/s00029-017-0310-1.

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We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition turns out to be insufficient. In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie. We prove the following three fundamental properties of derived non-archimedean analytic spaces: (1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the ∞-category of derived non-archimedean analytic spaces. (2) The ∞-category of derived non-archimedean analytic spaces admits fiber products. (3) The ∞-category of higher non-archimedean analytic Deligne–Mumford stacks embeds fully faithfully into the ∞-category of derived non-archimedean analytic spaces. The essential image of this embedding is spanned by n-localic discrete derived non-archimedean analytic spaces. We will further develop the theory of derived non-archimedean analytic geometry in our subsequent works. Our motivations mainly come from intersection theory, enumerative geometry and mirror symmetry.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Porta, Mauro0000-0002-1239-3409
Yu, Tony Yue0000-0002-6019-8552
Additional Information:© 2018 Springer. Published 09 February 2017. Issue Date: April 2018. We are grateful to Vladimir Berkovich, Antoine Chambert-Loir, Brian Conrad, Antoine Ducros, Bruno Klingler, Maxim Kontsevich, Jacob Lurie, Marco Robalo, Matthieu Romagny, Pierre Schapira, Carlos Simpson, Michael Temkin, Bertrand Toën and Gabriele Vezzosi for valuable discussions. The authors would also like to thank each other for the joint effort. This research was partially conducted during the period Mauro Porta was supported by Simons Foundation grant number 347070 and the group GNSAGA, and Tony Yue Yu served as a Clay Research Fellow.
Funding AgencyGrant Number
Simons Foundation347070
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA)UNSPECIFIED
Clay Mathematics InstituteUNSPECIFIED
Subject Keywords:Derived geometry · Rigid analytic geometry · Non-archimedean geometry · Berkovich space · Analytic stack · Higher stack · Pregeometry · Structured topos
Issue or Number:2
Classification Code:Mathematics Subject Classification: Primary 14G22; Secondary 14A20 18B25 18F99
Record Number:CaltechAUTHORS:20210914-164412737
Persistent URL:
Official Citation:Porta, M., Yu, T.Y. Derived non-archimedean analytic spaces. Sel. Math. New Ser. 24, 609–665 (2018).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110830
Deposited By: George Porter
Deposited On:14 Sep 2021 21:21
Last Modified:14 Sep 2021 21:21

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