Realization of spaces of commutative rings
Laura Cossu, Bruce Olberding
TL;DR
The paper addresses spaces of subrings between a domain $D$ and its quotient field $F$, analyzed under the Zariski and patch topologies. It introduces patch spaces, patch presheaves, and patch algebras as three complementary realizations, showing how patch bundles approximate spaces by Stone spaces, how patch presheaves encode spaces via stalks, and how patch algebras realize the rings in the space as quotients or localizations. A category equivalence between patch presheaves and patch bundles is established, and the patch algebra construction ties these viewpoints together by recovering the rings of the space as quotients modulo minimal primes or as localizations at maximal ideals. In indecomposable or domain settings, the Pierce spectrum of patch algebras aligns with the underlying Stone or spectral spaces, enabling concrete reconstructions (and yielding Rickart/Gelfand-type properties for patch algebras) and paving the way for applications to Zariski-Riemann spaces and spectral spaces.
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 $X$.