Locally Stable Grothendieck Categories: Applications

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications 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.