Lemma 36.37.2. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of $D(\mathcal{O}_ X)$ can be represented by a bounded complex of finite locally free $\mathcal{O}_ X$-modules.
Proof. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 we can represent $E$ by a complex $\mathcal{F}^\bullet $ of quasi-coherent $\mathcal{O}_ X$-modules. Observe that $E$ is in $D^ b(\mathcal{O}_ X)$ because $X$ is quasi-compact. Hence $\tau _{\geq n}\mathcal{F}^\bullet $ is a bounded below complex of quasi-coherent $\mathcal{O}_ X$-modules which represents $E$ if $n \ll 0$. Thus we may apply Lemma 36.37.1 to conclude. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)