Lemma 21.17.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. If $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}$-modules, then $\mathcal{K}^\bullet |_ U$ is a K-flat complex of $\mathcal{O}_ U$-modules.
Proof. Let $\mathcal{G}^\bullet $ be an exact complex of $\mathcal{O}_ U$-modules. Since $j_{U!}$ is exact (Modules on Sites, Lemma 18.19.3) and $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}$-modules we see that the complex
\[ j_{U!}(\text{Tot}(\mathcal{G}^\bullet \otimes _{\mathcal{O}_ U} \mathcal{K}^\bullet |_ U)) = \text{Tot}(j_{U!}\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \]
is exact. Here the equality comes from Modules on Sites, Lemma 18.27.9 and the fact that $j_{U!}$ commutes with direct sums (as a left adjoint). We conclude because $j_{U!}$ reflects exactness by Modules on Sites, Lemma 18.19.4. $\square$
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Your email address will not be published.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)