Lemma 59.91.13. Let $f : X \to Y$ be a proper morphism of schemes. 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 $(R^ nf_*E)_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, E|_{X_{\overline{y}}})$.

Proof. In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to the scheme theoretic fibre $X_{\overline{y}} = \overline{y} \times _ Y X$. 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 59.91.11. The second one is a special case of Lemma 59.91.12. $\square$

Comment #6991 by Marco on

Are you sure that we don't need that $\mathcal{F}$ has no $p$-torsion where $p$ is the characteristic of the residue field over the point $y$ in order to conclude (1)?

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