nLab property sup

category theory

Applications

Homological algebra

homological algebra

Introduction

diagram chasing

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 $\Omega$ 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.

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 $R$, 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).

References

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