The Stacks project

Lemma 59.106.1. With notation as above, if $K$ is in the essential image of $R\epsilon _*$, then the maps $c^ K_{X, Z, X', E}$ of Cohomology on Sites, Lemma 21.26.1 are quasi-isomorphisms.

Proof. Denote ${}^\# $ sheafification in the h topology. We have seen in More on Flatness, Lemma 38.37.7 that $h_ X^\# = h_ Z^\# \amalg _{h_ E^\# } h_{X'}^\# $. On the other hand, the map $h_ E^\# \to h_{X'}^\# $ is injective as $E \to X'$ is a monomorphism. Thus this lemma is a special case of Cohomology on Sites, Lemma 21.29.3 (which itself is a formal consequence of Cohomology on Sites, Lemma 21.26.3). $\square$

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