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The Stacks project

Lemma 84.6.3. 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:

  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 84.6.2 by Cohomology on Sites, Remark 21.14.4. \square


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