Lemma 13.29.3. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{I} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset. Assume $\mathcal{I}$ contains $0$, is closed under (finite) products, and every object of $\mathcal{A}$ is a subobject of an element of $\mathcal{I}$. Let $K^\bullet $ be a complex. There exists a commutative diagram

in the category of complexes such that

the vertical arrows are quasi-isomorphisms and termwise injective,

$I_ n^\bullet $ is a bounded below complex with terms in $\mathcal{I}$,

the arrows $I_{n + 1}^\bullet \to I_ n^\bullet $ are termwise split surjections and $\mathop{\mathrm{Ker}}(I^ i_{n + 1} \to I^ i_ n)$ is an element of $\mathcal{I}$.

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