The Stacks project

Lemma 59.103.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. The rule

\[ (\mathit{Sch}/S)_ h \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma (X, f_{small}^{-1}\mathcal{F}) \]

is a sheaf and a fortiori a sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. In fact this sheaf is equal to $\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ and $\epsilon _ S^{-1}\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_ h$.

Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the h topology. However, in Lemma 59.102.1 we have shown that $\pi _ S^{-1}\mathcal{F}$ is a sheaf even in the stronger ph topology. $\square$

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