Non-Archimedean analytic geometry as relative algebraic geometry

Abstract

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 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. ContentsOREN BEN-BASSAT, KOBI KREMNIZER A.5. Enough projectives and injectives 47 A.6. The closed structure in the category of Banach spaces 51 A.7. Completion 51 A.8. Banach algebras and modules 52 Appendix B. Category theory background 54 References 55

Dagger geometry as Banach algebraic geometry

“…This characterization is analogous to the one given in [7] for the weak G-topology of classical affinoid spaces over non-Archimedean base fields. In this section we deal with the category Comm(CBorn k ) instead of Comm(Ind(Ban k )).…”

“…The flatness follows from the fact that the two non-expanding categories Ban A,≤1 R and Ban nA,≤1 R are closed symmetric monoidal categories with the same monoidal structure and internal hom spaces as Ban A R and Ban nA R respectively, and that both P A (M) and P nA (M) are coproducts of flat objects and hence flat. The projectivity of these objects is proven exactly along the lines of the analogous facts when R is a complete valuation field (see [7]). …”

“…For the quasi-abelian structure, a simple adaptation of arguments of Proposition 3.14 works, we omit the details (see also [7] in the case that R is a complete non-Archimedean valuation field).…”

“…In this article we mainly focus on the first condition, looking at the particular cases such as C = Ind(Ban A Z ) and C = Born k . In [7] we looked at the case C = Ban nA k . The conceptual idea is to show that on sub-categories of Comm(C) op ⊂ Comm(sC) op the abstractly defined topologies on Comm(sC) op restrict to well known topologies in analytic geometry.…”

“…The focus of the current article is understanding the meaning of these conditions in cases relevant to our new approach to analytic geometry. This idea of this approach is due to Kobi Kremnizer and this article is one in a series of papers including [7], [8] and others to appear which will develop it. The main symmetric monoidal categories that are used in this article are the category of complete Bornological vector spaces of convex type over a complete, non-trivially valued field and a generalization of this: the categories of Ind-Banach modules over a complete normed ring.…”