The Stacks project

Lemma 17.9.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a finite type $\mathcal{O}_ Y$-module is a finite type $\mathcal{O}_ X$-module.

Proof. Arguing as in the proof of Lemma 17.8.2 we may assume $\mathcal{G}$ is globally generated by finitely many sections. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have a surjection

\[ \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ Y \to \mathcal{G} \to 0 \]

then upon applying $f^*$ we obtain the surjection

\[ \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X \to f^*\mathcal{G} \to 0. \]

This implies the lemma. $\square$

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