The Stacks project

Lemma 31.14.5. Let $f : S' \to S$ be a morphism of schemes. Let $D$ be a effective Cartier divisors on $S$. If the pullback of $D$ is defined then $f^*\mathcal{O}_ S(D) = \mathcal{O}_{S'}(f^*D)$ and the canonical section $1_ D$ pulls back to the canonical section $1_{f^*D}$.

Proof. Omitted. $\square$

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