The Stacks project

Lemma 12.19.14. Let $\mathcal{A}$ be an abelian category. Let $A \to B \to C$ be a complex of filtered objects of $\mathcal{A}$. Assume $\alpha : A \to B$ and $\beta : B \to C$ are strict morphisms of filtered objects. Then $\text{gr}(\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )) = \mathop{\mathrm{Ker}}(\text{gr}(\beta ))/\mathop{\mathrm{Im}}(\text{gr}(\alpha )))$.

Proof. This follows formally from Lemma 12.19.12 and the fact that $\mathop{\mathrm{Coim}}(\alpha ) \cong \mathop{\mathrm{Im}}(\alpha )$ and $\mathop{\mathrm{Coim}}(\beta ) \cong \mathop{\mathrm{Im}}(\beta )$ by Lemma 12.19.4. $\square$

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