In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.
Serre subcategories Let A {\displaystyle {\mathcal {A}}} be an abelian category. A non-empty full subcategory C {\displaystyle {\mathcal {C}}} is called a Serre subcategory (or also a dense subcategory), if for every short exact sequence 0 → A ′ → A → A ″ → 0 {\displaystyle 0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0} in A {\displaystyle {\mathcal {A}}} the object A {\displaystyle A} is in C {\displaystyle {\mathcal {C}}} if and only if the objects A ′ {\displaystyle A'}
and A ″ {\displaystyle A''} belong to C {\displaystyle {\mathcal {C}}} . In words: C {\displaystyle {\mathcal {C}}} is closed under subobjects, quotient objects and extensions. Each Serre subcategory C {\displaystyle {\mathcal {C}}} of A {\displaystyle {\mathcal {A}}} is itself an abelian category, and the inclusion functor C → A {\displaystyle {\mathcal {C}}\to {\mathcal {A}}} is exact. The importance of this notion stems from the fact that kernels of exact functors between abelian categories are Serre subcategories, and that one can build (for locally small A {\displaystyle {\mathcal {A}}} ) the quotient category (in the sense of Gabriel, Grothendieck, Serre) A / C {\displaystyle {\mathcal {A}}/{\mathcal {C}}} , which has the same objects as A {\displaystyle {\mathcal {A}}} , is abelian, and comes with an exact functor (called the quotient functor) T : A → A / C {\displaystyle T\colon {\mathcal {A}}\rightarrow {\mathcal {A}}/{\mathcal {C}}} whose kernel is C {\displaystyle {\mathcal {C}}} .
Localizing subcategories Let A {\displaystyle {\mathcal {A}}} be locally small. The Serre subcategory C {\displaystyle {\mathcal {C}}} is called localizing if the quotient functor
T : A → A / C {\displaystyle T\colon {\mathcal {A}}\rightarrow {\mathcal {A}}/{\mathcal {C}}} has a right adjoint
S : A / C → A {\displaystyle S\colon {\mathcal {A}}/{\mathcal {C}}\rightarrow {\mathcal {A}}} . Since then T {\displaystyle T} , as a left adjoint, preserves colimits, each localizing subcategory is closed under colimits. The functor T {\displaystyle T} (or sometimes S T {\displaystyle ST} ) is also called the localization functor, and S {\displaystyle S} the section functor. The section functor is left-exact and fully faithful. If the abelian category A {\displaystyle {\mathcal {A}}} is moreover cocomplete and has injective hulls (e.g. if it is a Grothendieck category), then a Serre subcategory C {\displaystyle {\mathcal {C}}} is localizing if and only if
C {\displaystyle {\mathcal {C}}} is closed under arbitrary coproducts (a.k.a. direct sums). Hence the notion of a localizing subcategory is equivalent to the notion of a hereditary torsion class. If A {\displaystyle {\mathcal {A}}} is a Grothendieck category and C {\displaystyle {\mathcal {C}}} a localizing subcategory, then C {\displaystyle {\mathcal {C}}} and the quotient category
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