Affine manifolds are rigid analytic spaces in characteristic one, I

TL;DR

This paper extends the framework of schemes relative to monoids with zero to formal schemes, enabling new models for degenerating Abelian varieties and providing combinatorial classifications and criteria within formal geometry.

Contribution

It introduces a formal scheme framework for monoid-based schemes, including models for degenerating Abelian varieties, and offers combinatorial classification and algebraisation criteria.

Findings

Normal formal monoid schemes classified by cone complexes

Provided algebraisation criterion for formal schemes

Reformulated separated and proper morphisms in formal geometry

Abstract

I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian varieties. Following the usual toric paradigm, normal formal monoid schemes can be classified in terms of certain cone complexes, and their properties understood in combinatorial terms. I use this to give a simple algebraisation criterion. I also reformulate the traditional notions of separated and proper morphism in a manner amenable to the context of relative formal geometry, and give characterisations in terms of the topology of cone complexes.