Two criteria for locally Noetherian Grothendieck categories (Q2079659)

The author generalizes two characterizations of noetherian rings to locally noetherian Grothendieck categories. Thus, a Grothendieck category is locally noetherian if and only if every object satisfying the CS-condition is a direct sum of uniform objects. Also, a locally finitely generated Grothendieck category is locally noetherian if and only if every finitely injective object is injective if and only if every finitely injective object is a direct sum of injectives. The following result is also deduced during the process: if \(M\) is a quasi-continuous object of a locally finitely generated Grothendieck category, then \(M\) is a direct sum of indecomposable objects if and only if \(M\) has a direct sum decomposition that complements summands if and only if each local summand of \(M\) is a direct summand.