Lemma 59.86.5 (0EZU)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.86: Preliminaries on base change
Lemma 59.86.5 (cite)

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Lemma 59.86.5. Consider a cartesian diagram of schemes

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

where $g : T \to S$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be an abelian sheaf on $T_{\acute{e}tale}$ . Let $q \geq 0$ . The following are equivalent

For every geometric point $\overline{x}$ of $X$ with image $\overline{s} = f(\overline{x})$ we have

$H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S T, \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S T, \mathcal{F})$
$f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is an isomorphism.

Proof. Since $Y = X \times _ S T$ we have $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ X Y = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S T$ . Thus the map in (1) is the map of stalks at $\overline{x}$ for the map in (2) by Theorem 59.53.1 (and Lemma 59.36.2). Thus the result by Theorem 59.29.10. $\square$

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