The Stacks project

Lemma 17.10.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a quasi-coherent $\mathcal{O}_ Y$-module is quasi-coherent.

Proof. Arguing as in the proof of Lemma 17.8.2 we may assume $\mathcal{G}$ has a global presentation by direct sums of copies of $\mathcal{O}_ Y$. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have an exact sequence

\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ Y \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ Y \longrightarrow \mathcal{G} \longrightarrow 0 \]

then upon applying $f^*$ we obtain the exact sequence

\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \longrightarrow f^*\mathcal{G} \longrightarrow 0. \]

This implies the lemma. $\square$

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