Lemma 12.12.9. Let $\mathcal{A}$ be an abelian category.

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

A morphism of cochain complexes $f : A^\bullet \to B^\bullet $ is injective if and only if each $f^ n : A^ n \to B^ n$ is injective.

A morphism of cochain complexes $f : A^\bullet \to B^\bullet $ is surjective if and only if each $f^ n : A^ n \to B^ n$ is surjective.

A sequence of cochain 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 $B^ i$.

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