The Stacks project

Lemma 31.23.8. Let $f : X \to Y$ be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition 31.23.4).

  1. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. There is a canonical pullback map $f^* : \Gamma (Y, \mathcal{K}_ Y(\mathcal{F})) \to \Gamma (X, \mathcal{K}_ X(f^*\mathcal{F}))$ for meromorphic sections of $\mathcal{F}$.

  2. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. A regular meromorphic section $s$ of $\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\mathcal{L}$.

Proof. Omitted. $\square$

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