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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