The Stacks project

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. $\square$


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