Lemma 37.52.8. Let $f : X \to S$ be a morphism of schemes. Assume

1. $f$ is proper, flat, and of finite presentation, and

2. the geometric fibres of $f$ are reduced.

Then the function $n_{X/S} : S \to \mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ is locally constant.

Proof. By Lemma 37.52.7 the function $n_{X/S}$ is lower semincontinuous. For $s \in S$ consider the $\kappa (s)$-algebra

$A = H^0(X_ s, \mathcal{O}_{X_ s})$

By Varieties, Lemma 33.9.3 and the fact that $X_ s$ is geometrically reduced $A$ is finite product of finite separable extensions of $\kappa (s)$. Hence $A \otimes _{\kappa (s)} \kappa (\overline{s})$ is a product of $\beta _0(s) = \dim _{\kappa (s)} H^0(E \otimes ^\mathbf {L} \kappa (s))$ copies of $\kappa (\overline{s})$. Thus $X_{\overline{s}}$ has $\beta _0(s) = \dim _{\kappa (s)} A$ connected components. In other words, we have $n_{X/S} = \beta _0$ as functions on $S$. Thus $n_{X/S}$ is upper semi-continuous by Derived Categories of Schemes, Lemma 36.32.1. This finishes the proof. $\square$

Comment #2798 by BB on

The term "semicontinuous" is spelled incorrectly in the first sentence of the proof. Also, unless I'm mistaken, this term more frequently appears as hyphenated as "semi-continuous" elsewhere in the SP.

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