Ringed Finite Spaces

TL;DR

This paper introduces the concept of ringed finite spaces, exploring their homotopy, sheaves, and morphisms, and demonstrates how they relate to schemes, providing a new framework for understanding finite ringed spaces and their properties.

Contribution

It develops the theory of ringed finite spaces, extending homotopy and sheaf concepts, and embeds schemes into this framework, bridging finite topological spaces and algebraic geometry.

Findings

Extended homotopy theory for ringed finite spaces

Analyzed quasi-coherent sheaves on these spaces

Embedded schemes into the category of finite ringed spaces

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 make a study of the homotopy of ringed finite spaces (that extends the homotopy of finite topological spaces) and a study of the quasi-coherent sheaves on a ringed finite space. This leads to set and develop basic notions on ringed finite spaces and morphisms, as being affine, schematic, semi-separated, etc, focusing on its cohomological properties. Finally, we see how to embed the category of quasi-compact and quasi-separated schemes in a localization of the category of finite ringed spaces.