Lemma 13.18.2 (013J)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.18: Injective resolutions
Lemma 13.18.2 (cite)

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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.

Proof. Omitted. For the second statement use $L^\bullet = \tau _{\geq n}K^\bullet $ for some $n \ll 0$. See Homology, Section 12.15 for the definition of the truncation $\tau _{\geq n}$. $\square$

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