The Stacks project

Lemma 36.37.4. 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. Let $\alpha ^\bullet , \beta ^\bullet :\mathcal{E}^\bullet \to \mathcal{F}^\bullet $ be two maps of complexes defining the same map in $D(\mathcal{O}_ X)$. Then there exists a quasi-isomorphism $\gamma ^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet $ where $\mathcal{G}^\bullet $ is a bounded complex of finite locally free $\mathcal{O}_ X$-modules such that $\alpha ^\bullet \circ \gamma ^\bullet $ and $\beta ^\bullet \circ \gamma ^\bullet $ are homotopic maps of complexes.

Proof. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 (and the definition of the derived category) there exists a quasi-isomorphism $\gamma ^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet $ where $\mathcal{G}^\bullet $ is a complex of quasi-coherent $\mathcal{O}_ X$-modules such that $\alpha ^\bullet \circ \gamma ^\bullet $ and $\beta ^\bullet \circ \gamma ^\bullet $ are homotopic maps of complexes. Choose a homotopy $h^ i : \mathcal{G}^ i \to \mathcal{F}^{i - 1}$ witnessing this fact. Choose $n \ll 0$. Then the map $\gamma ^\bullet $ factors canonically over the quotient map $\mathcal{G}^\bullet \to \tau _{\geq n}\mathcal{G}^\bullet $ as $\mathcal{E}^\bullet $ is bounded below. For the exact same reason the maps $h^ i$ will factor over the surjections $\mathcal{G}^ i \to (\tau _{\geq n}\mathcal{G})^ i$. Hence we see that we may replace $\mathcal{G}^\bullet $ by $\tau _{\geq n}\mathcal{G}^\bullet $. Then we may apply Lemma 36.37.1 to conclude. $\square$

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