Lemma 76.36.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume
$f$ is proper, flat, and of finite presentation, and
the geometric fibres of $f$ are reduced.
Then the function $n_{X/S} : |Y| \to \mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ (Lemma 76.30.1) is locally constant.
Proof. By Lemma 76.36.7 the function $n_{X/Y}$ is lower semincontinuous. Thus it suffices to show it is upper semi-continuous. To do this we may work étale locally on $Y$, hence we may assume $Y$ is an affine scheme. For $y \in Y$ consider the $\kappa (y)$-algebra
By Spaces over Fields, Lemma 72.14.3 and the fact that $X_ y$ is geometrically reduced $A$ is finite product of finite separable extensions of $\kappa (y)$. Hence $A \otimes _{\kappa (y)} \kappa (\overline{y})$ is a product of $\beta _0(y) = \dim _{\kappa (y)} A$ copies of $\kappa (\overline{y})$. Thus $X_{\overline{y}}$ has $\beta _0(y)$ connected components. In other words, we have $n_{X/S} = \beta _0$ as functions on $Y$. Thus $n_{X/Y}$ is upper semi-continuous by Derived Categories of Spaces, Lemma 75.26.2. This finishes the proof. $\square$
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