The Stacks project

Lemma 18.28.9. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let

\[ 0 \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to 0 \]

be a short exact sequence of presheaves of $\mathcal{O}$-modules.

  1. If $\mathcal{F}_2$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_1$.

  2. If $\mathcal{F}_1$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_2$.

If $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings then the same result holds in $\textit{Mod}(\mathcal{O})$.

Proof. Let $\mathcal{G}^\bullet $ be an arbitrary exact complex of presheaves of $\mathcal{O}$-modules. Assume that $\mathcal{F}_0$ is flat. By Lemma 18.28.8 we see that

\[ 0 \to \mathcal{G}^\bullet \otimes _{p, \mathcal{O}} \mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{p, \mathcal{O}} \mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{p, \mathcal{O}} \mathcal{F}_0 \to 0 \]

is a short exact sequence of complexes of presheaves of $\mathcal{O}$-modules. Hence (1) and (2) follow from the snake lemma. The case of sheaves of modules is proved in the same way. $\square$


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