Lemma 17.20.4. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module flat over $Y$. Then the functor

is exact.

Lemma 17.20.4. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module flat over $Y$. Then the functor

\[ \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X),\quad \mathcal{G} \longmapsto f^*\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} \]

is exact.

**Proof.**
This is true because $f^*\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} = f^{-1}\mathcal{G} \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{F}$, the functor $f^{-1}$ is exact, and $\mathcal{F}$ is a flat $f^{-1}\mathcal{O}_ Y$-module.
$\square$

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