Lemma 17.20.2. Let $f : X \to Y$ be a flat morphism of ringed spaces. Then the pullback functor $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is exact.

Proof. The functor $f^*$ is the composition of the exact functor $f^{-1} : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(f^{-1}\mathcal{O}_ Y)$ and the change of rings functor

$\textit{Mod}(f^{-1}\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X), \quad \mathcal{F} \longmapsto \mathcal{F} \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{O}_ X.$

Thus the result follows from the discussion following Definition 17.20.1. $\square$

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