Lemma 18.28.7 (03EV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.28: Flat modules
Lemma 18.28.7 (cite)

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Lemma 18.28.7. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\mathcal{C}$ . Consider the functor $j_ U : \mathcal{C}/U \to \mathcal{C}$ .

The presheaf of $\mathcal{O}$ -modules $j_{U!}\mathcal{O}_ U$ (see Remark 18.19.7) is flat.
If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, $j_{U!}\mathcal{O}_ U$ is a flat sheaf of $\mathcal{O}$ -modules.

Proof. Proof of (1). By the discussion in Remark 18.19.7 we see that

$j_{U!}\mathcal{O}_ U(V) = \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{O}(V)$

which is a flat $\mathcal{O}(V)$ -module. Hence (1) follows from Lemma 18.28.2. Then (2) follows as $j_{U!}\mathcal{O}_ U = (j_{U!}\mathcal{O}_ U)^\#$ (the first $j_{U!}$ on sheaves, the second on presheaves) and Lemma 18.28.3. $\square$

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