The Stacks project

Lemma 17.8.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ is locally generated by sections if $\mathcal{G}$ is locally generated by sections.

Proof. Given an open subspace $V$ of $Y$ we may consider the commutative diagram of ringed spaces

\[ \xymatrix{ (f^{-1}V, \mathcal{O}_{f^{-1}V}) \ar[r]_{j'} \ar[d]_{f'} & (X, \mathcal{O}_ X) \ar[d]^ f \\ (V, \mathcal{O}_ V) \ar[r]^ j & (Y, \mathcal{O}_ Y) } \]

We know that $f^*\mathcal{G}|_{f^{-1}V} \cong (f')^*(\mathcal{G}|_ V)$, see Sheaves, Lemma 6.26.3. Thus we may assume that $\mathcal{G}$ is globally generated.

We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have a surjection

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

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

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

This implies the lemma. $\square$

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