Schemes of Objects in Abelian Categories

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 $\cat C$, we consider a subcategory $\cat B\subset C$ of base-points, and assume that each $X\in\ob\cat C$ that contains $P\in\ob\cat B,$ i.e. there is a morphism $P\rightarrow X,$ there exists a local representing object $X_P.$ Assuming that coproducts exists, we can use the construction of ordinary schemes to construct schemes of objects in any such category.