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

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

2. 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.

Proof. Dual to Lemma 13.18.2. $\square$

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