The Stacks project

Lemma 83.4.8. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces. Let $\overline{y} \to Y$ be a geometric point.

  1. For a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $(R^ nf_*\mathcal{F})_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$.

  2. For $E \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $(R^ nf_*E)_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, E_{\overline{y}})$.

Proof. In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to $X_{\overline{y}} = \overline{y} \times _ Y X$. Since pulling back by $\overline{y} \to Y$ produces the stalk of $\mathcal{F}$, the first statement of the lemma is a special case of Theorem 83.4.6. The second one is a special case of Lemma 83.4.7. $\square$

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