nLab
property sup

Context

Category theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

The property (sup)

Idea

The property (sup) is an optional property of an abelian category, introduced in (Gabriel).

Definition

An abelian category has property (sup) if:

Definition (sup)

For any ascending chain Ω\Omega of subobjects of a fixed object MM, the supremum of Ω\Omega exists; and, for any subobject LML\hookrightarrow M, the canonical morphism

sup{LPPΩ}(supΩ)L sup\{L\cap P | P\in \Omega\}\to (sup \Omega) \cap L

is an isomorphism.

Examples

Gabriel’s property (sup) is satisfied by any Grothendieck category (in some expositions it is listed as a part of the definition), e.g. the category of all modules over a fixed ring RR, and the category of sheaves of abelian groups on a fixed topological space XX.

According to the Appendix B of the Thomason–Trobaugh contribution to Grothendieck Festschrift, it is not known whether the category Qcoh YQcoh_Y of quasicoherent sheaves over an arbitrary scheme YY is a Grothendieck category (although this is known for a large class), but it is an elementary fact that it does satisfy Gabriel’s property (sup). If an abelian category is noetherian it clearly satisfies the property (sup).

References

Revised on December 30, 2010 12:51:23 by Urs Schreiber (89.204.153.71)