Lemma 13.19.11. Let $\mathcal{A}$ be an abelian category. Let $\beta : P^\bullet \to L^\bullet$ and $\alpha : E^\bullet \to L^\bullet$ be maps of complexes. Let $n \in \mathbf{Z}$. Assume

1. $P^\bullet$ is a bounded complex of projectives and $P^ i = 0$ for $i < n$,

2. $H^ i(\alpha )$ is an isomorphism for $i > n$ and surjective for $i = n$.

Then there exists a map of complexes $\gamma : P^\bullet \to E^\bullet$ such that $\alpha \circ \gamma$ and $\beta$ are homotopic.

Proof. Consider the cone $C^\bullet = C(\alpha )^\bullet$ with map $i : L^\bullet \to C^\bullet$. Note that $i \circ \beta$ is zero by Lemma 13.19.10. Hence we can lift $\beta$ to $E^\bullet$ by Lemma 13.4.2. $\square$

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