Lemma 18.42.2 (093K)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.42: Constant sheaves
Lemma 18.42.2 (cite)

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Lemma 18.42.2. Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring and let $M$ and $Q$ be $\Lambda$ -modules. If $Q$ is a finitely presented $\Lambda$ -module, then we have $\underline{M \otimes _\Lambda Q}(U) = \underline{M}(U) \otimes _\Lambda Q$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ .

Proof. Choose a presentation $\Lambda ^{\oplus m} \to \Lambda ^{\oplus n} \to Q \to 0$ . This gives an exact sequence $M^{\oplus m} \to M^{\oplus n} \to M \otimes Q \to 0$ . By Lemma 18.42.1 we obtain an exact sequence

$\underline{M}(U)^{\oplus m} \to \underline{M}(U)^{\oplus n} \to \underline{M \otimes Q}(U) \to 0$

which proves the lemma. (Note that taking sections over $U$ always commutes with finite direct sums, but not arbitrary direct sums.) $\square$

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