Lemma 13.19.10 (064D)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.19: Projective resolutions
Lemma 13.19.10 (cite)

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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

$(\tau _{\leq n - 1}K^\bullet , K^\bullet , \tau _{\geq n}K^\bullet , f, g, h)$

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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