The Stacks project

Lemma 13.11.5. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a complex.

1. If $H^ n(K^\bullet ) = 0$ for all $n \ll 0$, then there exists a quasi-isomorphism $K^\bullet \to L^\bullet $ with $L^\bullet $ bounded below.

2. If $H^ n(K^\bullet ) = 0$ for all $n \gg 0$, then there exists a quasi-isomorphism $M^\bullet \to K^\bullet $ with $M^\bullet $ bounded above.

3. If $H^ n(K^\bullet ) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes

   \[ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } \]

   where all the arrows are quasi-isomorphisms, $L^\bullet $ bounded below, $M^\bullet $ bounded above, and $N^\bullet $ a bounded complex.