The Stacks project

Lemma 18.32.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. The pullback $f^*\mathcal{L}$ of an invertible $\mathcal{O}_\mathcal {D}$-module is invertible.

Proof. By Lemma 18.32.2 there exists an $\mathcal{O}_\mathcal {D}$-module $\mathcal{N}$ such that $\mathcal{L} \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{N} \cong \mathcal{O}_\mathcal {D}$. Pulling back we get $f^*\mathcal{L} \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{N} \cong \mathcal{O}_\mathcal {C}$ by Lemma 18.26.2. Thus $f^*\mathcal{L}$ is invertible by Lemma 18.32.2. $\square$

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