Lemma 13.19.2. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a complex of $\mathcal{A}$.

If $K^\bullet $ has a projective resolution then $H^ n(K^\bullet ) = 0$ for $n \gg 0$.

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

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