Lemma 12.16.12. Let $\mathcal{A}$ be an abelian category.

Let $A$ be a filtered object and $X \subset A$. Then for each $p$ the sequence

\[ 0 \to \text{gr}^ p(X) \to \text{gr}^ p(A) \to \text{gr}^ p(A/X) \to 0 \]is exact (with induced filtration on $X$ and quotient filtration on $A/X$).

Let $f : A \to B$ be a morphism of filtered objects of $\mathcal{A}$. Then for each $p$ the sequences

\[ 0 \to \text{gr}^ p(\mathop{\mathrm{Ker}}(f)) \to \text{gr}^ p(A) \to \text{gr}^ p(\mathop{\mathrm{Coim}}(f)) \to 0 \]and

\[ 0 \to \text{gr}^ p(\mathop{\mathrm{Im}}(f)) \to \text{gr}^ p(B) \to \text{gr}^ p(\mathop{\mathrm{Coker}}(f)) \to 0 \]are exact.

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