The Stacks project

Proof. By Lemma 76.29.3 formation of $E$ commutes with base change. To check a subset of $|Y|$ is open, we may replace $Y$ by the members of an étale covering. Thus we may assume $Y$ is affine. Then $Y$ is a cofiltered limit of affine schemes of finite type over $\mathbf{Z}$. Hence we can assume $X \to Y$ is the base change of $X_0 \to Y_0$ where $Y_0$ is the spectrum of a finite type $\mathbf{Z}$-algebra and $X_0 \to Y_0$ is flat and proper. See Limits of Spaces, Lemma 70.7.1, 70.6.12, and 70.6.13. Since the formation of $E$ commutes with base change (see above), we may assume the base is Noetherian.

Assume $Y$ is Noetherian. The set is constructible by Lemma 76.29.4. Hence it suffices to show the set is stable under generalization (Topology, Lemma 5.19.10). By Properties, Lemma 28.5.10 we reduce to the case where $Y = \mathop{\mathrm{Spec}}(R)$, $R$ is a discrete valuation ring, and the closed fibre $X_ y$ is geometrically reduced. To show: the generic fibre $X_\eta $ is geometrically reduced.

If not then there exists a finite extension $L$ of the fraction field of $R$ such that $X_ L$ is not reduced, see Spaces over Fields, Lemmas 72.11.4 (characteristic zero) and 72.11.5 (positive characteristic). There exists a discrete valuation ring $R' \subset L$ with fraction field $L$ dominating $R$, see Algebra, Lemma 10.120.18. After replacing $R$ by $R'$ we reduce to Lemma 76.29.5. $\square$