
Lemma 12.9.2. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C}$ be a subcategory of $\mathcal{A}$. Then $\mathcal{C}$ is a Serre subcategory if and only if the following conditions are satisfied:

1. $0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

2. $\mathcal{C}$ is a strictly full subcategory of $\mathcal{A}$,

3. any subobject or quotient of an object of $\mathcal{C}$ is an object of $\mathcal{C}$,

4. if $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is an extension of objects of $\mathcal{C}$ then also $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Moreover, a Serre subcategory is an abelian category and the inclusion functor is exact.

Proof. Omitted. $\square$

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