Lemma 13.18.2. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a complex of $\mathcal{A}$.

If $K^\bullet $ has an injective resolution then $H^ n(K^\bullet ) = 0$ for $n \ll 0$.

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.

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