# Non-Archimedean analytic geometry as relative algebraic geometry

Ben-Bassat, O
Kremnizer, K

1 January 2017

## Journal:

Annales de la faculté des sciences de Toulouse Mathématiques

## Last Updated:

2020-05-16T21:08:57.3+01:00

## 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.

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