The Stacks project

Lemma 36.37.3. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{E}^\bullet $ and $\mathcal{F}^\bullet $ be finite complexes of finite locally free $\mathcal{O}_ X$-modules. Then any $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ can be represented by a diagram

\[ \mathcal{E}^\bullet \leftarrow \mathcal{G}^\bullet \to \mathcal{F}^\bullet \]

where $\mathcal{G}^\bullet $ is a bounded complex of finite locally free $\mathcal{O}_ X$-modules and where $\mathcal{G}^\bullet \to \mathcal{E}^\bullet $ is a quasi-isomorphism.

Proof. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 we can represent $\alpha $ by a diagram

\[ \mathcal{E}^\bullet \leftarrow \mathcal{H}^\bullet \to \mathcal{F}^\bullet \]

where $\mathcal{H}^\bullet $ is a complex of quasi-coherent $\mathcal{O}_ X$-modules and where $\mathcal{H}^\bullet \to \mathcal{E}^\bullet $ is a quasi-isomorphism. For $n \ll 0$ the maps $\mathcal{H}^\bullet \to \mathcal{E}^\bullet $ and $\mathcal{H}^\bullet \to \mathcal{F}^\bullet $ factor through the quasi-isomorphism $\mathcal{H}^\bullet \to \tau _{\geq n}\mathcal{H}^\bullet $ simply because $\mathcal{E}^\bullet $ and $\mathcal{F}^\bullet $ are bounded complexes. Thus we may replace $\mathcal{H}^\bullet $ by $\tau _{\geq n}\mathcal{H}^\bullet $ and assume that $\mathcal{H}^\bullet $ is bounded below. Then we may apply Lemma 36.37.1 to conclude. $\square$

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