Lemma 82.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

The rule

\[ \mathcal{F}^ a : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow \textit{Ab},\quad (f : Y \to X) \longmapsto \Gamma (Y, f^*\mathcal{F}) \]satisfies the sheaf condition for fpqc and a fortiori fppf and étale coverings,

$\mathcal{F}^ a = \pi _ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{\acute{e}tale}$,

$\mathcal{F}^ a = a_ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{fppf}$,

the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O})$,

the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{fppf}, \mathcal{O})$,

we have $\epsilon _{X, *}a_ X^*\mathcal{F} = \pi _ X^*\mathcal{F}$ and $a_{X, *}a_ X^*\mathcal{F} = \mathcal{F}$,

we have $R^ i\epsilon _{X, *}(a_ X^*\mathcal{F}) = 0$ and $R^ ia_{X, *}(a_ X^*\mathcal{F}) = 0$ for $i > 0$.

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