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Non-Archimedean analytic geometry as relative algebraic geometry
- Publication Year :
- 2013
-
Abstract
- We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--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.<br />Comment: added material on quasi-coherent modules, connection to derived analytic geometry, corrected mistakes
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1312.0338
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.5802/afst.1526