Schemes of Objects in Abelian Categories
Arvid Siqveland
TL;DR
The paper generalizes the construction of schemes by replacing rings with a locally small category $\mathbf{C}$ and introducing base-points $\mathbf{B}$ with local representations $X_P$. It formalizes a universal localization $X_x$ for base-points, and builds a global object $\mathcal{O}(X)$ from localized pieces via coproducts (covariant) or products (contravariant), enabling the notion of affine objects when $\mathcal{O}(X)\cong X$ and schemes of objects via open covers where $\mathcal{O}(U)\cong U$. The framework further extends to multi-pointed (associative) schemes by considering $r$-pointed localizations $X_M$, defining $O(X)$ and $\mathcal{O}(X)$ in that context, and highlighting the potential mismatch between $\prod X_x$ and the product of point-sets. Through multiple examples across algebra, geometry, and categories, the work provides a broad, systematic approach to constructing scheme-like structures in diverse categorical settings.
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.