Realization of spaces of commutative rings

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.).