Serre subcategory

Remarks

Equivalently, $\mathcal{C}$ is nonempty and closed under subobjects, quotient objects and extensions. It then contains a zero object and is closed under isomorphisms, and it is itself an abelian category with exact inclusion. In a Grothendieck category, a Serre subcategory closed under arbitrary direct sums is called a localizing subcategory.

Example: the torsion abelian groups form a Serre subcategory of the category of abelian groups. Subgroups and quotients of torsion groups are torsion. If $0 \to A' \to A \to A'' \to 0$ is exact with $A'$ and $A''$ torsion and $a \in A$, then $na$ maps to $0$ in $A''$ for some $n \ge 1$, so $na \in A'$ and $mna = 0$ for some $m \ge 1$.

Non-example: the free abelian groups do not form a Serre subcategory. The sequence $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ is exact with $\Z$ free, but the quotient $\Z/2\Z$ is not free.