The Stacks project

Lemma 83.8.3. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above:

  1. $H^ q(X_{\acute{e}tale}, \mathcal{F}) = H^ q_{ph}(X, a_ X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,

  2. $H^ q(X_{\acute{e}tale}, K) = H^ q_{ph}(X, a_ X^{-1}K)$ for $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves

Example: if $A$ is a torsion abelian group, then $H^ q_{\acute{e}tale}(X, \underline{A}) = H^ q_{ph}(X, \underline{A})$.

Proof. This follows from Lemma 83.8.2 by Cohomology on Sites, Remark 21.14.4. $\square$

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