Lemma 76.29.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is quasi-compact and locally of finite presentation. Then the set
is étale locally constructible.
Proof. Choose an affine scheme $V$ and an étale morphism $V \to Y$. The meaning of the statement is that the inverse image of $E$ in $|V|$ is constructible. By Lemma 76.29.3 we may replace $Y$ by $V$, i.e., we may assume that $Y$ is an affine scheme. Then $X$ is quasi-compact. Choose an affine scheme $U$ and a surjective étale morphism $U \to X$. For a morphism $\mathop{\mathrm{Spec}}(k) \to Y$ the morphism between fibres $U_ k \to X_ k$ is surjective étale. Hence $U_ k$ is geometrically reduced over $k$ if and only if $X_ k$ is geometrically reduced over $k$, see Spaces over Fields, Lemma 72.11.7. Thus the set $E$ for $X \to Y$ is the same as the set $E$ for $U \to Y$. In this way we see that the lemma follows from the case of schemes, see More on Morphisms, Lemma 37.26.5. $\square$
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