Categorical construction of Schemes

TL;DR

This paper presents a categorical approach to defining schemes using algebraization of local moduli, providing an alternative, more natural construction aligned with classical algebraic geometry.

Contribution

It introduces a new categorical construction of schemes based on local moduli at maximal ideals, aligning with Hartshorne's classical definition.

Findings

Categorical scheme definition equivalent to Hartshorne's

Sheaf construction via projective limits clarified

Provides a more natural framework for scheme theory

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.