Lemma 59.79.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module and denote $\mathcal{F}^ a$ the associated quasi-coherent sheaf on the small étale site of $X$ (Proposition 59.17.1). Then

$H^ q_ Z(X, \mathcal{F})$ agrees with $H^ q_ Z(X_{\acute{e}tale}, \mathcal{F}^ a)$,

if the complement of $Z$ is retrocompact in $X$, then $i_*\mathcal{H}^ q_ Z(\mathcal{F}^ a)$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules equal to $(i_*\mathcal{H}^ q_ Z(\mathcal{F}))^ a$.

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