nLab
Serre subcategory

Definition

A (nonempty) full subcategory $T$ of an abelian category ${}_A\mathrm{Mod}$ of (say left) modules over a ring $A$ is a Serre subcategory if for any exact sequence

$$0\to M\to M\prime \to M″\to 0$$0\to M\to M'\to M''\to 0

$M\prime$ is in $T$ iff $M$ and $M″$ are in $T$.

Following Serre, one then defines the category $A/T$ whose objects are the objects of $A$ and where the morphisms in $A/T$ are defined by

$${\mathrm{Hom}}_{A/T}(X,Y):=\mathrm{colim}{\mathrm{Hom}}_A(X\prime ,Y/Y\prime ),$$\mathrm{Hom}_{A/T}(X,Y) := \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),

where the colimit is over all $X\prime ,Y\prime$ in $A$ such that $Y\prime$ and $X/X\prime$ are in $T$. The quotient category $A/T$ is abelian.

Very often the same definition of Serre subcategory is used in an arbitrary abelian category $A$ (we will say in that case weakly Serre subcategory); but in fact, at least when the abelian category is not a Grothendieck category, it is more appropriate to ask for an additional condition in the definition of Serre subcategory, so that the standard theorems on correspondences with other canonical data in localization theory remain valid.

To this aim, for any subcategory of an arbitrary abelian category $A$ one denotes by $\overline{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) subquotient of $N$ in $T$ has a (nonzero) subobject from $T$. This becomes an idempotent operation on the class of subcategories of $A$ with $T\subset \overline{T}$ iff $T$ is topologizing. Moreover $\overline{T}$ is always thick in the stronger sense (that is, thick and topologizing).

Serre subcategories in the strong sense are those nonempty full subcategories which are stable under the operation $T\mapsto \overline{T}$.