The Stacks project

Lemma 18.36.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites. Let $p$ be a point of $\mathcal{C}$ or $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and set $q = f \circ p$. Then

\[ (f^*\mathcal{F})_ p = \mathcal{F}_ q \otimes _{\mathcal{O}_{\mathcal{D}, q}} \mathcal{O}_{\mathcal{C}, p} \]

for any $\mathcal{O}_\mathcal {D}$-module $\mathcal{F}$.

Proof. We have

\[ f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_\mathcal {C} \]

by definition. Since taking stalks at $p$ (i.e., applying $p^{-1}$) commutes with $\otimes $ by Lemma 18.26.2 we win by the relation between the stalk of pullbacks at $p$ and stalks at $q$ explained in Sites, Lemma 7.34.2 or Sites, Lemma 7.34.3. $\square$

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