Notes on schematic finite spaces

TL;DR

This paper explores schematic finite spaces, providing characterizations and demonstrating their strong relationship with quasi-compact, quasi-separated schemes, thereby advancing the understanding of finite ringed spaces in algebraic geometry.

Contribution

It introduces various characterizations of schematic finite spaces and their morphisms, linking them closely to well-studied schemes in algebraic geometry.

Findings

Schematic finite spaces can be characterized in multiple ways.

They are strongly related to quasi-compact, quasi-separated schemes.

The theory of quasi-coherent modules extends naturally to these spaces.

Abstract

The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show that schematic finite spaces are strongly related to quasi-compact quasi-separated schemes.