Categorical construction of Schemes
Arvid Siqveland
TL;DR
The paper develops a categorical framework for schemes in the associative setting by introducing $a\mathrm{Spec}\,A$, built from a deformation-theoretic localization $A_P$ in a finite collection of simple modules and aprime $A$-modules. It proves that for commutative rings, particularly domains and algebras finitely generated over a field, $a\mathrm{Spec}\,A$ recovers the classical $\mathrm{Spec}\,A$, and shows that affine schemes can be constructed via projective limits of localizations $A_{\mathfrak m}$ (or $A_f$), with a sheaf $\mathcal{O}_X$ arising from these limits. This yields a natural, functorial framework that aligns with Hartshorne's definition while enabling a systematic extension to associative rings. The approach leverages deformation theory to algebraize local moduli, providing a streamlined pathway to abstract scheme theory beyond commutative rings.
Abstract
In the authors book, Associative Algebraic Geometry, 2023, and the following article Shemes of Associative Algebras,\\ https://doi.org/10.48550/arXiv.2410.17703,2024, we use an algebraization of the semi-local formal moduli of simple modules to construct associative schemes. Here, we consider a commutative ring for which we can use the localization in maximal ideals as local moduli. This gives a categorical definition of schemes that is equivalent to the definition in Hartshorne's book, Algebraic Geometry, 1977. The definition includes a construction of the sheaf associated to a presheaf using projective limits, and this makes the basic results in scheme theory more natural.