Lemma 59.87.3 (0EZY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.87: Base change for pushforward
Lemma 59.87.3 (cite)

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Lemma 59.87.3. Consider the cartesian diagrams of schemes

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

Assume that $S$ is the spectrum of a separably closed field. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$ .

Proof. We may work locally on $X$ . Hence we may assume $X$ is affine. Then we can write $X$ as a cofiltered limit of affine schemes of finite type over $S$ . By Lemma 59.86.3 we may assume that $X$ is of finite type over $S$ . Then Lemma 59.87.1 applies because any scheme of finite type over a separably closed field is a finite disjoint union of connected and geometrically connected schemes (see Varieties, Lemma 33.7.6). $\square$

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