The Stacks project

Remark 13.18.5. Let $\mathcal{A}$ be an abelian category. Using the fact that $K(\mathcal{A})$ is a triangulated category we may use Lemma 13.18.4 to obtain proofs of some of the lemmas below which are usually proved by chasing through diagrams. Namely, suppose that $\alpha : K^\bullet \to L^\bullet $ is a quasi-isomorphism of complexes. Then

is a distinguished triangle in $K(\mathcal{A})$ (Lemma 13.9.14) and $C(\alpha )^\bullet $ is an acyclic complex (Lemma 13.11.2). Next, let $I^\bullet $ be a bounded below complex of injective objects. Then

is an exact sequence of abelian groups, see Lemma 13.4.2. At this point Lemma 13.18.4 guarantees that the outer two groups are zero and hence $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(L^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(K^\bullet , I^\bullet )$.