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The Stacks project

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

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

  2. 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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