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.

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

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

Related concepts

• Noetherian ring

• Noetherian poset