The Stacks project

Lemma 12.19.12. Let $\mathcal{A}$ be an abelian category.

  1. 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 $A/X$).

  2. 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.

Proof. We have $F^{p + 1}X = X \cap F^{p + 1}A$, hence map $\text{gr}^ p(X) \to \text{gr}^ p(A)$ is injective. Dually the map $\text{gr}^ p(A) \to \text{gr}^ p(A/X)$ is surjective. The kernel of $F^ pA/F^{p + 1}A \to A/X + F^{p + 1}A$ is clearly $F^{p + 1}A + X \cap F^ pA/F^{p + 1}A = F^ pX/F^{p + 1}X$ hence exactness in the middle. The two short exact sequence of (2) are special cases of the short exact sequence of (1). $\square$

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