The Stacks project

Proof. The question is étale local on $Y$, hence we may and do assume $Y$ is an affine scheme. Let $y \in Y$. Set $n = n_{X/S}(y)$. Note that $n < \infty $ as the geometric fibre of $X \to Y$ at $y$ is a proper algebraic space over a field, hence Noetherian, hence has a finite number of connected components. We have to find an open neighbourhood $V$ of $y$ such that $n_{X/S}|_ V \geq n$. Let $X \to Y' \to Y$ be the Stein factorization as in Theorem 76.36.5. By Lemma 76.36.2 there are finitely many points $y'_1, \ldots , y'_ m \in Y'$ lying over $y$ and the extensions $\kappa (y'_ i)/\kappa (y)$ are finite. More on Morphisms, Lemma 37.42.1 tells us that after replacing $Y$ by an étale neighbourhood of $y$ we may assume $Y' = V_1 \amalg \ldots \amalg V_ m$ as a scheme with $y'_ i \in V_ i$ and $\kappa (y'_ i)/\kappa (y)$ purely inseparable. Then the algebraic spaces $X_{y_ i'}$ are geometrically connected over $\kappa (y)$, hence $m = n$. The algebraic spaces $X_ i = (f')^{-1}(V_ i)$, $i = 1, \ldots , n$ are flat and of finite presentation over $Y$. Hence the image of $X_ i \to Y$ is open (Morphisms of Spaces, Lemma 67.30.6). Thus in a neighbourhood of $y$ we see that $n_{X/Y}$ is at least $n$. $\square$