The Stacks project

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. $\square$

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