The Stacks project

Remark 35.8.6. Remark 35.8.4 and Lemma 35.8.5 have the following applications:

  1. Let $S$ be a scheme. The construction $\mathcal{F} \mapsto \mathcal{F}^ a$ is the pullback under the morphism of ringed sites $\text{id}_{\tau , Zar} : ((\mathit{Sch}/S)_\tau , \mathcal{O}) \to (S_{Zar}, \mathcal{O})$ or the morphism $\text{id}_{small, {\acute{e}tale}, Zar} : (S_{\acute{e}tale}, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$.

  2. Let $f : X \to Y$ be a morphism of schemes. For any of the morphisms $f_{sites}$ of ringed sites of Remark 35.8.4 we have

    \[ (f^*\mathcal{F})^ a = f_{sites}^*\mathcal{F}^ a. \]

    This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma 18.13.3.

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