Lemma 13.19.2 (0645)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.19: Projective resolutions
Lemma 13.19.2 (cite)

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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.

Proof. Dual to Lemma 13.18.2. $\square$

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