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The Stacks project

Lemma 84.4.8. 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 84.4.6. The second one is a special case of Lemma 84.4.7. \square


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