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 36.36.9 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F8H. Beware of the difference between the letter 'O' and the digit '0'.