Lemma 59.91.13 (0DDF)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.91: The proper base change theorem
Lemma 59.91.13 (cite)

previous lemma

Lemma 59.91.13. Let $f : X \to Y$ be a proper morphism of schemes. Let $\overline{y} \to Y$ be a geometric point.

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

Comments (2)

Comment #6991 by Marco on January 26, 2022 at 09:48

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)?

Comment #6992 by Johan on January 26, 2022 at 12:42

Yes

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