Non-Archimedean analytic geometry as relative algebraic geometry

Ben-Bassat, Oren; Kremnizer, Kobi

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 49-126.

Nous montrons que la géométrie analytique non-archimédienne peut être considérée comme la géométrie algébrique relative, au sens de Toën-Vaquié, au-dessus de la catégorie des espaces de Banach non-archimédiens. Pour toute catégorie symétrique monoïdale fermée quasi-abélienne nous définissons une topologie sur certaines sous-catégories de la catégorie des schémas affines (relatifs). Dans le cas où la catégorie monoïdale est celle des groupes abéliens, la topologie coïncide avec la topologie de Zariski usuelle. En examinant cette topologie pour la catégorie des espaces de Banach, nous retrouvons la G-topologie faible ou encore la topologie des sous-ensembles admissibles sur un affinoïde utilisée en géométrie rigide. Cela donne une approche de type foncteur des points à la géométrie analytique non-archimédienne. Nous démontrons que la catégorie des espaces analytiques de Berkovich (et aussi des espaces analytiques rigides) se plonge de manière pleinement fidèle dans la catégorie des schémas relatifs. Nous définissons une notion de faisceau quasi-cohérent sur les espaces analytiques que nous utilisons pour caractériser les familles couvrantes. En chemin nous utilisons l’algèbre homologique dans les catégories quasi-abéliennes développée par Schneiders.

We show that non-Archimedean 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 define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

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