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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?

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  • $\begingroup$ 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 $\endgroup$ – HHH Aug 6 '15 at 18:08
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According to Exercise 4 in Section 5.8 in Abelian Categories with Applications to Rings and Modules by N. Popescu (Academic Press, 1973) the answer to your first question is yes. (I have not checked the details, i.e., solved the exercise.)

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