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

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

2. 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.

3. 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.

4. 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 $B^ i$.

Proof. Omitted. $\square$

There are also:

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