A category is noetherian if it is svelte and every object in 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 of modules over a noetherian commutative unital ring , and more generally, the category of quasicoherent sheaves of -modules over any noetherian scheme , is a locally noetherian abelian category; moreover the full subcategory of noetherian objects in in this case is the category of coherent sheaves of -modules over .
A filtered colimit of injective objects in any locally noetherian abelian category is injective.
Last revised on December 21, 2012 at 18:28:15. See the history of this page for a list of all contributions to it.