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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.

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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

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There is a whole theory of localisation within the context of ring theory that uses Gabriel's notion of quotient category. A starting point might be Popescu's book: Abelian categories with applications to rings and modules, if you can get hold of a copy. The theory has an immense literature so I will not attempt to give a list (do a Maths Review search with a long search time frame!!!!) A related area is that of Torsion Theory, so search on that as well.

You do not make precise what areas of application might be of interest to you? Perhaps looking at the book by Gabriel and Zisman would indicate another (non-Abelian) perspective. That leads on to localisations in Quillen model category theory and beyond to triangulated categories, and ....

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

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Regarding the first paragraph: I always advertise Stenström's Rings of Quotients in this context. Also, "Almost Ring Theory" as developed by Gabber/Ramero relies on the same setting, and from here one gets to Faltings' work in p-adic Hodge theory. And/or to Scholze's Perfectoid Spaces. –  Torsten Schoeneberg May 12 '13 at 12:10
    
I know and like Stenström's earlier lecture notes, but do not know his book. –  Tim Porter May 12 '13 at 19:14

Recent work of Nick Katz (see his book "Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms", available on www.math.princeton.edu/~nmk) depends quite crucially on the properties of a quotient abelian category (of perverse sheaves on the multiplicative group) defined and studied previously by Gabber and Loeser; the quotient has the effect of allowing a certain convolution operation to be used to define a Tannakian structure, as Katz explains in Chapter 2 of his book.

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Have you considered registering one of your five accounts? –  S. Carnahan May 12 '13 at 12:14

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