Lemma 18.42.3 (093L)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.42: Constant sheaves
Lemma 18.42.3 (cite)

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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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