Affine manifolds are rigid analytic spaces in characteristic one, II

TL;DR

This paper extends rigid analytic geometry to algebraic geometry over monoids in characteristic one, establishing new notions of morphisms and embedding affine manifolds within this framework.

Contribution

It introduces a novel framework linking affine manifolds with rigid analytic spaces over monoids, including criteria for morphisms and methods to recover affine manifolds.

Findings

Affine manifolds embed as a subcategory with algebraic and topological criteria.

Affine manifolds can be recovered from Novikov field points or as universal Hausdorff quotients.

Base change yields toric analytic spaces fibering over affine manifolds.

Abstract

I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a subcategory defined by simple algebraic (normal) and topological (overconvergent) criteria. The affine manifold of a rigid space can be recovered either as a set of `Novikov field' points or as a universal Hausdorff quotient. After base change to any topological field, one obtains a `toric' analytic space that fibres over the affine manifold.