Finite Spaces and Schemes

TL;DR

This paper explores the concept of ringed finite spaces, establishing their relation to schemes and finite topological spaces, and introduces a functor connecting schematic finite spaces to schemes.

Contribution

It introduces schematic finite spaces and morphisms, and constructs a functor linking these to quasi-compact, quasi-separated schemes.

Findings

Category of ringed finite spaces includes finite topological spaces and affine schemes

Schematic finite spaces behave like schemes with respect to quasi-coherence

A functor from schematic finite spaces to schemes is fully faithful and essentially surjective

Abstract

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a ringed finite space. We introduce the notions of schematic finite space and schematic morphism, showing that they behave, with respect to quasi-coherence, like schemes and morphisms of schemes do. Finally, we construct a fully faithful and essentially surjective functor from a localization of a full subcategory of the category of schematic finite spaces and schematic morphisms to the category of quasi-compact and quasi-separated schemes.