Lemma 59.91.2 (0A0C)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.91: The proper base change theorem
Lemma 59.91.2 (cite)

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Lemma 59.91.2. Let $(A, I)$ be a henselian pair. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $i : Z \to X$ be the closed immersion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$ into $X$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, i_{small}^{-1}\mathcal{F})$.

Proof. This follows from Lemma 59.82.2 and 59.91.1 and the fact that any scheme finite over $X$ is proper over $\mathop{\mathrm{Spec}}(A)$. $\square$

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