Lemma 21.52.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume every object of $\mathcal{C}$ has a covering by quasi-compact objects. Then every compact object of $D(\mathcal{O})$ is a direct summand in $D(\mathcal{O})$ of a finite complex whose terms are finite direct sums of $\mathcal{O}$-modules of the form $j_!\mathcal{O}_ U$ where $U$ is a quasi-compact object of $\mathcal{C}$.
Proof. Apply Lemma 21.52.1 where $S \subset \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}))$ is the set of modules of the form $j_!\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ quasi-compact. Assumption (1) holds by Modules on Sites, Lemma 18.28.8 and the assumption that every $U$ can be covered by quasi-compact objects. Assumption (2) follows as
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U) \]
which commutes with direct sums by Sites, Lemma 7.17.7. $\square$
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Your email address will not be published.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)