nLab subscheme of an Abelian category

Motivation

By the Gabriel-Rosenberg reconstruction theorem a scheme $X$ can be reconstructed from the category $Qcoh_X$ of quasicoherent sheaves on $X$. If $i : Y\hookrightarrow X$ is a subscheme of $X$ then we can identify the category quasicoherent sheaves on $Y$ with certain abelian subcategory of quasicoherent sheaves on $X$. One attempts to abstractly deefine the properties of such a subcategory which guarantee that it corresponds to a subcategory of that kind.

Definition

(Alexander Rosenberg in his 1996 book and MPI preprints with Lunts from 1996)

A subscheme of an abelian category $A$ is a coreflective topologizing subcategory of $A$.

If a subscheme is also reflective then we call it Zariski closed.

Properties

If an abelian category $A$ has the Gabriel’s property (sup) then every

(a) The intersection of any set of subschemes of $A$ is a subscheme.

(b) The intersection of any set of Zariski closcd suhschemes of $A$ is a Zariski closed subscheme.

Literature

• V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf