Dagger geometry as Banach algebraic geometry

Abstract

Abstract. In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Klönne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together.

Analytic geometry over F1 and the Fargues-Fontaine curve

“…One can prove that U is not a forgetful functor, i.e. it is not faithful, using the same reasoning of Remark 3.33 of [3]. The next proposition shows one of the advantages of using the category of essential monomorphic objects instead of IndpˆSetsq.…”

“…In this way we can easily introduce the homotopy Zariski topology (as introduced by Toën-Vezzosi in any HAG context) on the category of affine derived analytic spaces over F 1 (see Definition 5.9). It has been proved in previous works of the authors (see [3], [4] and [5]) that the homotopy Zariski topology on the category of simplicial Banach/bornological algebras over complete valued field restricts to the usual topology when it is considered over "affine" (i.e. affinoid, Stein or Stein-like) analytic spaces.…”

Dagger completions and bornological torsion-freeness

“…It is somewhat similar to the proof of (4). Next we prove (2). Assume that M is complete, that M is torsion-free, and that f (M ) is not closed in M .…”

Bounded linear endomorphisms of rigid analytic functions

“…That is to say, he identified this sheaf of continuous endomorphisms with the sheaf of formal differential operators, whose total symbol in a small enough local chart around any point, can be used to define a holomorphic function on . Some recent work: , suggests defining a type of analytic geometry that would work in the same way over a general valuation field (Archimedean or not) or even a Banach ring. In this type of geometry, one constructs various Grothendieck topologies on the ‘affine’ objects.…”

“…The resulting bornology is called the von Neumann bornology. We will not give more details here, and instead refer the interested reader to and the references given therein, but we will explain the natural bornological structure on the set .…”