\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

Lemma 36.46.8. 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 36.46.7 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 32.9.3 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 35.28.1. This finishes the proof. $\square$

Comments (2)

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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