Lemma 18.42.3. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a coherent ring. Let $M$ be a flat $\Lambda $-module. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the module $\underline{M}(U)$ is a flat $\Lambda $-module.

**Proof.**
Let $I \subset \Lambda $ be a finitely generated ideal. By Algebra, Lemma 10.39.5 it suffices to show that $\underline{M}(U) \otimes _\Lambda I \to \underline{M}(U)$ is injective. As $\Lambda $ is coherent $I$ is finitely presented as a $\Lambda $-module. By Lemma 18.42.2 we see that $\underline{M}(U) \otimes I = \underline{M \otimes I}$. Since $M$ is flat the map $M \otimes I \to M$ is injective, whence $\underline{M \otimes I} \to \underline{M}$ is injective.
$\square$

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