# Applications and examples of quotient categories of abelian categories

I want to get motivated to learn more about quotient categories of abelian categories by a Serre subcategory or even by a localizing category as they are described in Pierre Gabriel's thesis "Des catégories abéliennes".

So I am looking for some applications of this theory or theorems, which can be proven nicely by using quotient categories.

• I believe that Serre's "mod $\mathcal{C}$" theory is a nice application of these ideas in topology. – Akhil Mathew May 11 '13 at 17:58

Serre's classes also have applications in profinite group theory (pro-$\mathcal{C}$-groups), but that uses the localisation less directly.