Localization and coalescence of condensed ringed spaces

TL;DR

This paper uses condensed mathematics to embed the category of locally ringed spaces into a new framework, establishing a connection between simple objects and adic spectra of Huber pairs.

Contribution

It introduces a condensed mathematics approach to locally ringed spaces, extending Gillam's embedding with a coreflection mapping to adic spectra.

Findings

Established a coreflective embedding into a category related to adic spectra

Mapped simple objects to spectra of rings and Huber pairs

Extended Gillam's results using condensed mathematics

Abstract

Gillam proved that the category of locally ringed spaces admits a fully faithful embedding into a certain category, which has a right adjoint that maps some simple objects to the spectra of rings. In this paper, we use condensed mathematics to define an analogous category and its coreflective full subcategories, and prove that one of their coreflections maps certain simple objects to the adic spectra of certain Huber pairs.