A subcategory of an abelian category is a localizing subcategory (French: sous-catégorie localisante) if there exists an exact localization functor having a right adjoint (which is automatically then fully faithful) and for which i.e. the full subcategory of generated by objects such that .
One sometimes says that is the localizing subcategory associated with quotient (or localized) category (which is then equivalent to the Serre quotient category ).
A localizing subcategory determines up to an equivalence of categories commuting with the localization functors; it is the quotient functor to the Serre quotient category . The right adjoint to is usually called the section functor. Denote the unit of the adjunction . Then for , is the maximal subobject of contained in , called the -torsion part of . An object is -torsionfree if the -torsion part of is , i.e. is isomorphism, and is -closed (local object with respect to morphisms inverting under ) if is an isomorphism. The section functor realizes the equivalence of categories between and the full subcategory of generated by -closed objects.
A thick subcategory (in strong sense) is localizing iff every object in has the largest subobject among the subobjects from and if the only subobject from is a zero object then there is a monomorphism from to a -closed object.
Localizing subcategories are precisely those which are topologizing, closed under extensions and closed under all colimits which exist in . In other words, and are in iff any given extension of by is in ; and it is closed under colimits existing in .
A strictly full subcategory is localizing iff the class of all for which and is precisely the class of all morphisms inverted by some left exact localization admiting right adjoint.
If admit colimits and has a set of generators, then any localizing subcategory ,and the Serre quotient , admit colimits and has a set of generators (Gabriel, Prop. 9) and the quotient functor preserves colimits (in the same Grothendieck universe if we work with universes). The generators of are the images of the generators in under the quotient functor . If is locally noetherian abelian category then any localizing subcategory and the quotient category are locally noetherian (Gabriel, Cor. 1). (If is locally finitely presented, and are locally finitely presented.?) If is locally noetherian and is the full subcategory of noetherian objects in , then the assignment which to any localizing subcategory assigns the full subcategory of noetherian objects in is the bijection between the localizing subcategories in and (strongly) thick subcategories in (Gabriel Prop. 10).
In this setup, there is a bijective correspondence between hereditary torsion theories, localizing subcategories and exact localizations having right adjoint.
(i) is localizing
(ii) is closed under coproducts
(iii) is cocomplete (closed under arbitrary colimits)
(iv) any colimit of objects in in belongs to
(v) the corresponding localizing functor preserves colimits
The notion is introduced by Gabriel:
A comprehensive (and very reliable) source is