The Stacks project

Lemma 59.100.8. For a scheme $X$ and $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above:

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

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

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

Proof. This follows from Lemma 59.100.7 by Cohomology on Sites, Remark 21.14.4. $\square$

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