Localization of ringed spaces

TL;DR

This paper introduces a localization construction for ringed spaces with prime ideal data, enabling the formation of inverse limits and a general spectrum functor, with applications to relative schemes.

Contribution

It defines a localization of ringed spaces with prime ideal data, establishing universal properties and enabling new constructions like inverse limits and a broad spectrum functor.

Findings

The category of locally ringed spaces has all inverse limits.

The localization construction generalizes the spectrum functor.

Applications to the theory of relative schemes.

Abstract

Let $X$ be a ringed space together with the data $M$ of a set $M_x$ of prime ideals of $\O_{X,x}$ for each point $x \in X$. We introduce the localization of $(X,M)$, which is a locally ringed space $Y$ and a map of ringed spaces $Y \to X$ enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general $\Spec$ functor. We conclude with a discussion of relative schemes.