Locally stable Grothendieck categories: applications (Q1942034)

Let \(\mathcal A\) be a Grothendieck category. The authors introduce the notion of stable localizing subcategory, as a localizing subcategory that is an intersection of localizing subcategories defined by indecomposable injectives. Then \(\mathcal A\) is called a locally stable category if every localizing subcategory is stable. It is shown that the locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally Noetherian categories are examples of locally stable categories. The main result states that if \(\mathcal C\) is a localizing subcategory of \(\mathcal A\), then \(\mathcal A\) is locally stable if and only if \(\mathcal C\) and the quotient category \(\mathcal A/\mathcal C\) are locally stable. Applications are given to the category of modules over a left fully bounded Noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.