Lemma 13.19.10. Let $\mathcal{A}$ be an abelian category. Let $P^\bullet $, $K^\bullet $ be complexes. Let $n \in \mathbf{Z}$. Assume that
$P^\bullet $ is a bounded complex consisting of projective objects,
$P^ i = 0$ for $i < n$, and
$H^ i(K^\bullet ) = 0$ for $i \geq n$.
Then $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(P^\bullet , K^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(P^\bullet , K^\bullet ) = 0$.
Proof. The first equality follows from Lemma 13.19.8. Note that there is a distinguished triangle
by Remark 13.12.4. Hence, by Lemma 13.4.2 it suffices to prove $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(P^\bullet , \tau _{\leq n - 1}K^\bullet ) = 0$ and $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(P^\bullet , \tau _{\geq n} K^\bullet ) = 0$. The first vanishing is trivial and the second is Lemma 13.19.4. $\square$
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