The Gabriel-Popescu theorem states that every Grothendieck category is a localization of a module
category. More explicitly, let 
be a Grothendieck category
with a generator 
. This means that 
is a faithful functor.
Let 
be its endomorphism ring. Then the functor
 |
is a fully faithful functor and has a left adjoint that is an exact functor. Fully faithful
means that the functor gives a bijection
between every Hom-set in 
and the corresponding Hom-set
in 
,
the module category whose objects are right R and whose morphisms are module
homomorphisms. Consequently, 
is equivalent to the quotient of 
by a localizing subcategory.
Such a subcategory is closed under subobjects (objects
embedded by monomorphisms), quotient objects (cokernels of subobject inclusions), extensions (objects
assembled in short exact sequences), and coproducts.
