Geometry of Spaces of Subrings

Abstract

Motivated by examples such as the local rings of projective integral schemes and the Zariski-Riemann space of a field, we develop a geometry for continuously indexed families of subrings of a commutative ring $R$. The indexing spaces are Stone spaces, and continuity is taken with respect to the Zariski topology on the space of subrings. To each such family we associate a subsheaf of the constant sheaf with value $R$ whose stalks are the indexed subrings. This gives a patch ringed space. Its global sections are locally constant $R$-valued functions whose value at each point belongs to the corresponding subring. The ring of global sections, with this representation, is the associated patch algebra. Our main result establishes an algebra-geometry correspondence between these continuously indexed families of subrings and patch ringed spaces that can be expressed as a category equivalence, with each of the two categories dually equivalent to the category of patch algebras. Thus the indexing space, its sheaf, and the indexed subrings can be recovered functorially from the associated patch algebra, which serves as a coordinate ring for this geometry. We describe kernels and factorizations of patch algebra morphisms in geometric terms and characterize their injectivity and surjectivity. We include applications to Pierce duality and Bhatt-Scholze $w$-localizations of integral separated quasi-compact schemes.