Non-Archimedean analytic geometry as relative algebraic geometry

Abstract

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of Toën--Vaquié--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry and in this way we also get definitions of (higher) non-Archimedean analytic stacks. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. We also include a treatment of quasi-coherent sheaf theory in analytic geometry. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

added material on quasi-coherent modules, connection to derived analytic geometry, corrected mistakes

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Cited by in corpus (21)

• Dagger Geometry As Banach Algebraic Geometry
• Stein Domains in Banach Algebraic Geometry
• Derived complex analytic geometry I: GAGA theorems
• Analytic geometry over F_1 and the Fargues-Fontaine curve
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• Fréchet Modules and Descent
• Bounded linear endomorphisms of rigid analytic functions
• The global derived period map
• On analytic analogues of quantum groups
• A Tannakian Reconstruction Theorem for IndBanach Spaces
• Analytic cyclic homology in positive characteristic
• Quasi-coherent sheaves in differential geometry
• Analytification, localization and homotopy epimorphisms
• D-modules on rigid analytic spaces I
• A Beilinson-Bernstein Theorem for analytic quantum groups
• Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties
• A note on relative homological epimorphisms of topological algebras
• On Liu morphisms in non-Archimedean geometry
• Dense quasi-free subalgebras of the Toeplitz algebra
• A note on deriving unbounded functors of exact categories, with applications to Ind- and Pro- functors
• Blow-ups and normal bundles in connective and nonconnective derived geometries