The Stacks project

Lemma 59.101.2. Let $S$ be a scheme. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$ the maps

\[ \pi _ S^*\mathcal{F} \longrightarrow R\epsilon _{S, *}(a_ S^*\mathcal{F}) \quad \text{and}\quad \mathcal{F} \longrightarrow Ra_{S, *}(a_ S^*\mathcal{F}) \]

are isomorphisms with $a_ S : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$ as above.

Proof. This is an immediate consequence of parts (6) and (7) of Lemma 59.101.1. $\square$

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