(also nonabelian homological algebra)

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

An abelian category has **property (sup)** if:

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

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

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 $R$, and the category of sheaves of abelian groups on a fixed topological space $X$.

According to the Appendix B of the Thomason–Trobaugh contribution to Grothendieck Festschrift, it is not known whether the category $Qcoh_Y$ of quasicoherent sheaves over an arbitrary scheme $Y$ 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.