The Stacks project

Lemma 12.13.3. Let $\mathcal{A}$ be an abelian category.

1. The category of chain complexes in $\mathcal{A}$ is abelian.

2. A morphism of complexes $f : A_\bullet \to B_\bullet $ is injective if and only if each $f_ n : A_ n \to B_ n$ is injective.

3. A morphism of complexes $f : A_\bullet \to B_\bullet $ is surjective if and only if each $f_ n : A_ n \to B_ n$ is surjective.

4. A sequence of chain complexes

   \[ A_\bullet \xrightarrow {f} B_\bullet \xrightarrow {g} C_\bullet \]

   is exact at $B_\bullet $ if and only if each sequence

   \[ A_ i \xrightarrow {f_ i} B_ i \xrightarrow {g_ i} C_ i \]

   is exact at $B_ i$.