nLab
subscheme of an Abelian category

Motivation

By the Gabriel-Rosenberg reconstruction theorem a scheme XX can be reconstructed from the category Qcoh XQcoh_X of quasicoherent sheaves on XX. If i:YXi : Y\hookrightarrow X is a subscheme of XX then we can identify the category quasicoherent sheaves on YY with certain abelian subcategory of quasicoherent sheaves on XX. 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 AA is a coreflective topologizing subcategory of AA.

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

Properties

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

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

(b) The intersection of any set of Zariski closcd suhschemes of AA 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

Created on June 15, 2010 18:28:30 by Zoran Škoda (161.53.130.104)