nLab
noetherian category

Contents

Definition

A category CC is noetherian if it is svelte and every object in CC is a noetherian object.

An abelian category is called locally noetherian if it satisfies axiom (AB5) and has a small generating family of noetherian objects.

The full subcategory of noetherian objects in any locally noetherian abelian category is itself a noetherian abelian category. A category RModR Mod of modules over a noetherian commutative unital ring RR, and more generally, the category Qcoh(X)Qcoh(X) of quasicoherent sheaves of 𝒪\mathcal{O}-modules over any noetherian scheme XX, is a locally noetherian abelian category; moreover the full subcategory of noetherian objects in Qcoh(X)Qcoh(X) in this case is the category of coherent sheaves of 𝒪\mathcal{O}-modules over XX.

A filtered colimit of injective objects in any locally noetherian abelian category is injective.

Revised on December 21, 2012 18:28:15 by Urs Schreiber (131.174.40.34)