The Stacks project

Lemma 20.42.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect.

  1. For any element $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is given by a morphism of complexes $\alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{F}^\bullet |_{U_ i}$.

  2. Given a morphism of complexes $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet $ whose image in the group $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is zero, there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is homotopic to zero.

Proof. Proof of (1). By the construction of the derived category we can find a quasi-isomorphism $f : \mathcal{F}^\bullet \to \mathcal{G}^\bullet $ and a map of complexes $\beta : \mathcal{E}^\bullet \to \mathcal{G}^\bullet $ such that $\alpha = f^{-1}\beta $. Thus the result follows from Lemma 20.42.7. We omit the proof of (2). $\square$


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Your email address will not be published.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 08C9. Beware of the difference between the letter 'O' and the digit '0'.