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The Stacks project

Lemma 12.19.9. 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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