Lemma 69.4.2 (0DK3)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.4: Finite morphisms
Lemma 69.4.2 (cite)

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Lemma 69.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$ . Let $\overline{y}$ be a geometric point of $Y$ with lifts $\overline{x}_1, \ldots , \overline{x}_ n$ in $X$ . Then

$(f_*\mathcal{F})_{\overline{y}} = \prod \nolimits _{i = 1, \ldots , n} \mathcal{F}_{\overline{x}_ i}$

for any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ .

Proof. Choose an étale neighbourhood $(V, \overline{v})$ of $\overline{y}$ . Then the stalk $(f_*\mathcal{F})_{\overline{y}}$ is the stalk of $f_*\mathcal{F}|_ V$ at $\overline{v}$ . By Properties of Spaces, Lemma 66.18.12 we may replace $Y$ by $V$ and $X$ by $X \times _ Y V$ . Then $Z \to X$ is a finite morphism of schemes and the result is Étale Cohomology, Proposition 59.55.2. $\square$

Comments (2)

Comment #5891 by WH on January 18, 2021 at 16:22

Should the direct sum here be a product, since this is for any (not necessarily abelian) sheaf?

Comment #6094 by Johan on April 29, 2021 at 19:11

Yes indeed. Thanks WH! See changes here.

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