The Category of Modules of Quotients

Abstract

In this chapter we will prove that every quotient category Mod-(A, ℑ) is a Grothendieck category. The proof uses arguments of a rather general nature, and it can also be used to show that the category of abelian sheaves on a topological space is a Grothendieck category. The second main result of this chapter is the Popescu-Gabriel theorem, which states that every Grothendieck category actually is a quotient category of a module category.