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arXiv · 2511.04191

Schemes of Objects in Abelian Categories

Arvid Siqveland
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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.

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Arvid Siqveland. 2025-11-06. Schemes of Objects in Abelian Categories. https://arxiv.org/abs/2511.04191

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