Lemma 84.4.3 (0DFZ)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 84: More on Cohomology of Spaces
Section 84.4: Proper base change
Lemma 84.4.3 (cite)

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Lemma 84.4.3. Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \to \mathop{\mathrm{Spec}}(A)$ is a proper morphism. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (X_0, \mathcal{F}|_{X_0})$ .

Proof. This is a special case of Lemma 84.4.2. $\square$

Comments (2)

Comment #7738 by Laurent Moret-Bailly on September 07, 2022 at 09:15

End of second sentence: I think it should be "is a proper morphism".

Comment #7987 by Stacks Project on January 14, 2023 at 11:45

Thanks and fixed here.

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