The Stacks project

Lemma 59.102.7. For a scheme $X$ and $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ 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 $H^ q_{\acute{e}tale}(X, \underline{A}) = H^ q_{ph}(X, \underline{A})$.