Lemma 12.16.9 (0129)—The Stacks project

Lemma 12.16.9. Let $\mathcal{A}$ be an abelian category. Let $(A, F)$ be a filtered object of $\mathcal{A}$. Let $X \subset Y \subset A$ be subobjects of $A$. On the object

\[ Y/X = \mathop{\mathrm{Ker}}(A/X \to A/Y) \]

the quotient filtration coming from the induced filtration on $Y$ and the induced filtration coming from the quotient filtration on $A/X$ agree. Any of the morphisms $X \to Y$, $X \to A$, $Y \to A$, $Y \to A/X$, $Y \to Y/X$, $Y/X \to A/X$ are strict (with induced/quotient filtrations).

Proof. The quotient filtration $Y/X$ is given by $F^ p(Y/X) = F^ pY/(X \cap F^ pY) = F^ pY/F^ pX$ because $F^ pY = Y \cap F^ pA$ and $F^ pX = X \cap F^ pA$. The induced filtration from the injection $Y/X \to A/X$ is given by

\begin{align*} F^ p(Y/X) & = Y/X \cap F^ p(A/X) \\ & = Y/X \cap (F^ pA + X)/X \\ & = (Y \cap F^ pA)/(X \cap F^ pA) \\ & = F^ pY/F^ pX. \end{align*}

Hence the first statement of the lemma. The proof of the other cases is similar. $\square$

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