Lemma 82.6.3. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above:

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

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

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

## Comments (0)