Homotopy of ringed finite spaces

TL;DR

This paper extends the homotopy classification of finite topological spaces to ringed finite spaces and shows that the category of quasi-coherent modules on such spaces is a homotopy invariant.

Contribution

It introduces a homotopy theory for ringed finite spaces and demonstrates that quasi-coherent modules form a homotopy invariant category.

Findings

Extended Stong's homotopy classification to ringed finite spaces

Proved the category of quasi-coherent modules is a homotopy invariant

Connected the category of ringed finite spaces with affine schemes and finite topological 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 study the homotopy of ringed finite spaces, extending Stong's homotopy classification of finite topological spaces to ringed finite spaces. We also prove that the category of quasi-coherent modules on a ringed finite space is a homotopy invariant.