The Stacks project

Lemma 59.87.4. Consider a cartesian diagram of schemes

Assume that

1. $f$ is flat and open,

2. the residue fields of $S$ are separably algebraically closed,

3. given an étale morphism $U \to X$ with $U$ affine we can write $U$ as a finite disjoint union of open subschemes of $X$ (for example if $X$ is a normal integral scheme with separably closed function field),

4. any nonempty open of a fibre $X_ s$ of $f$ is connected (for example if $X_ s$ is irreducible or empty).

Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.