# locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective if and only if every injective $R$-module is a direct sum of indecomposable injective $R$-modules.

1) Is it true that a Grothendieck category $\mathcal{C}$ is locally noetherian if and only if every injective object is a direct sum of indecomposable injective objects?

2) Is it true that a scheme $X$ in algebraic geometry is (locally) noetherian if and only if every injective quasi-coherent sheaf is a direct sum of indecomposable injective quasi-coherent sheaves?

• In algebraic geometry every injective quasi-coherent sheaf over a locally noetherian scheme can be written as a diret sum of some indecomposable injective qc sheves but an example is appeared in Hartshorne (Residues and duality) states that the category of qc sheves over a locally noetherian scheme may not be locally noetherian
– HHH
Aug 6, 2015 at 18:08