The Stacks project

Lemma 12.16.2. Let $\mathcal{A}$ be an abelian category. The category of graded objects $\text{Gr}(\mathcal{A})$ is abelian.

Proof. Let $f : A = (A^ i) \to B = (B^ i)$ be a morphism of graded objects of $\mathcal{A}$ given by morphisms $f^ i : A^ i \to B^ i$ of $\mathcal{A}$. Then we have $\mathop{\mathrm{Ker}}(f) = (\mathop{\mathrm{Ker}}(f^ i))$ and $\mathop{\mathrm{Coker}}(f) = (\mathop{\mathrm{Coker}}(f^ i))$ in the category $\text{Gr}(\mathcal{A})$. Since we have $\mathop{\mathrm{Im}}= \mathop{\mathrm{Coim}}$ in $\mathcal{A}$ we see the same thing holds in $\text{Gr}(\mathcal{A})$. $\square$

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