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The Stacks project

Lemma 13.19.3. Let \mathcal{A} be an abelian category. Assume \mathcal{A} has enough projectives.

  1. Any object of \mathcal{A} has a projective resolution.

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

  3. If K^\bullet is a complex with K^ n = 0 for n > a, then there exists a projective resolution \alpha : P^\bullet \to K^\bullet with P^ n = 0 for n > a such that each \alpha ^ n : P^ n \to K^ n is surjective.

Proof. Dual to Lemma 13.18.3. \square


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