Realization of spaces of commutative rings

Cossu, Laura

;Olberding, Bruce

2025-01-01

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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, endowed with the Zariski or patch topologies. We introduce three notions to study such a space (Formula presented.) : patch bundles, patch presheaves and patch algebras. When (Formula presented.) is compact and Hausdorff, patch bundles give a way to approximate (Formula presented.) with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space (Formula presented.) into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying (Formula presented.). 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 (Formula presented.) as factor rings, or even localizations, and whose structure reflects various properties of the rings in (Formula presented.).