The Stacks project

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

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

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