nLab noetherian category

Definition

A category $C$ is noetherian if it is svelte and every object in $C$ 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.

Examples and properties

The full subcategory of noetherian objects in any locally noetherian abelian category is itself a noetherian abelian category.

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

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

Related concepts

• Noetherian ring

• Noetherian poset