nLab property sup

The property (sup)


Category theory

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


The property (sup)


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


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

is an isomorphism.


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).


Last revised on December 30, 2010 at 12:51:23. See the history of this page for a list of all contributions to it.