The Stacks project

Lemma 82.6.6. In Lemma 82.6.4 if $f$ is finite, then $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_ X^{-1}K)$ for $K$ in $D^+(X_{\acute{e}tale})$.

Proof. Let $V \to Y$ be a surjective étale morphism where $V$ is a scheme. It suffices to prove the base change map is an isomorphism after restricting to $V$. Hence we may assume that $Y$ is a scheme. As the morphism is finite, hence representable, we conclude that we may assume both $X$ and $Y$ are schemes. In this case the result follows from the case of schemes (Étale Cohomology, Lemma 58.94.6 part (2)) using the comparison of topoi discussed in Section 82.3 and in particular given in Lemma 82.3.1. Some details omitted. $\square$


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