The Stacks project

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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