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
\xymatrix{ \ldots \ar[r] & \tau _{\geq -2}K^\bullet \ar[r] \ar[d] & \tau _{\geq -1}K^\bullet \ar[d] \\ \ldots \ar[r] & I_2^\bullet \ar[r] & I_1^\bullet }
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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