Schemes of Objects in Abelian Categories
Arvid Siqveland

TL;DR
This paper generalizes the concept of schemes from ring categories to arbitrary locally small categories, introducing a framework for constructing schemes of objects in these broader contexts.
Contribution
It extends the definition of schemes by replacing the category of rings with a general category and introduces a method to construct schemes of objects within this setting.
Findings
Generalizes schemes to arbitrary locally small categories.
Provides a construction method for schemes of objects.
Builds on previous categorical scheme constructions.
Abstract
In the article Categorical Construction of Schemes, arXiv:2511.03433 we gave a natural definition of ordinary schemes based on the fact that the localization of a ring in a maximal ideal is a local representation of the corresponding function field. In this text, we replace the category of rings with a general locally small category , we consider a subcategory of base-points, and assume that each that contains i.e. there is a morphism there exists a local representing object Assuming that coproducts exists, we can use the construction of ordinary schemes to construct schemes of objects in any such category.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras · advanced mathematical theories
