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Selecta Mathematica

, Volume 24, Issue 2, pp 609–665 | Cite as

Derived non-archimedean analytic spaces

  • Mauro Porta
  • Tony Yue Yu
Article
  • 81 Downloads

Abstract

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. (1)

    The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the \(\infty \)-category of derived non-archimedean analytic spaces.

     
  2. (2)

    The \(\infty \)-category of derived non-archimedean analytic spaces admits fiber products.

     
  3. (3)

    The \(\infty \)-category of higher non-archimedean analytic Deligne–Mumford stacks embeds fully faithfully into the \(\infty \)-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.

Keywords

Derived geometry Rigid analytic geometry Non-archimedean geometry Berkovich space Analytic stack Higher stack Pregeometry Structured topos 

Mathematics Subject Classification

Primary 14G22 Secondary 14A20 18B25 18F99 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.David Rittenhouse LaboratoryUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

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