Lemma 63.10.6. Let $f' : X' \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $i : X \to X'$ be a thickening and denote $f = f' \circ i$. Let $\Lambda $ be a ring. For $K'$ in $D^+_{tors}(X'_{\acute{e}tale}, \Lambda )$ or $K' \in D(X'_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have $Rf_!i^{-1}K' = Rf'_!K'$.
Proof. This is true because $i^{-1}$ and $i_* = i_!$ inverse equivalences of categories by the topological invariance of the small étale topos (Étale Cohomology, Theorem 59.45.2) and we can apply Lemma 63.9.2. $\square$
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