Fully bounded grothendieck categories part I: locally noetherian categories

Abstract

In this paper we aim to generalize some well-known properties of left noetherian fully left bounded rings to locally noetherian Grothendieck categories. For technical reasons the general theory will only be developed for locally noetherian categories having a noetherian generator. However, to show that the theory works in much more general situations as well, similar results will be proved to hold in other categories, e.g. the category of graded left modules over a graded noetherian ring, which will be dealt with in Part II of this paper. In the first paragraphs we recall some properties of locally noetherian categories and localization in Grothendieck categories. Next, we will introduce prime kernel functors in Grothendieck categories. Once the technical machinery developed we generalize results of Krause and Gabriel to Grothendieck categories with a noetherian generator. We conclude by giving some relations between symmetric kernel functors and fully bounded Grothendieck categories.