Realization of spaces of commutative rings

TL;DR

This paper explores the topological structure of spaces of subrings within a commutative ring, introducing new concepts like patch bundles, presheaves, and algebras to analyze their properties and relationships.

Contribution

It introduces the notions of patch bundles, presheaves, and algebras to study spaces of subrings, providing new tools for understanding their topological and algebraic structure.

Findings

Patch bundles approximate spaces with Stone spaces.

Patch presheaves encode spaces into stalks over Boolean algebras.

Patch algebras realize rings as factor rings or localizations.

Abstract

Motivated by recent work on the use of topological methods to study collections of rings between an integral domain and its quotient field, we examine spaces of subrings of a commutative ring, where these spaces are endowed with the Zariski or patch topologies. We introduce three notions to study such a space $X$: patch bundles, patch presheaves and patch algebras. When $X$ is compact and Hausdorff, patch bundles give a way to approximate $X$ with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space $X$ into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying $X$. To both objects, a patch bundle and a patch presheaf, we associate what we call a patch algebra, a commutative ring that efficiently realizes the rings in $X$ as factor rings, or even localizations, and whose structure reflects various properties of the rings in $X$.