Lemma 13.19.3. Let \mathcal{A} be an abelian category. Assume \mathcal{A} has enough projectives.
Any object of \mathcal{A} has a projective resolution.
If H^ n(K^\bullet ) = 0 for all n \gg 0 then K^\bullet has a projective resolution.
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.
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