# The Stacks Project

## Tag 0E0N

Lemma 36.45.8. 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 36.45.7 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. $\square$

The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 13029–13039 (see updates for more information).

\begin{lemma}
\label{lemma-proper-flat-geom-red}
Let $f : X \to S$ be a morphism of schemes. Assume
\begin{enumerate}
\item $f$ is proper, flat, and of finite presentation, and
\item the geometric fibres of $f$ are reduced.
\end{enumerate}
Then the function $n_{X/S} : S \to \mathbf{Z}$
counting the numbers of geometric connected components
of fibres of $f$ is locally constant.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}
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
\ref{varieties-lemma-proper-geometrically-reduced-global-sections}
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 \ref{perfect-lemma-jump-loci-geometric}.
This finishes the proof.
\end{proof}

Comment #2798 by BB on September 7, 2017 a 2:01 pm UTC

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.

Comment #2901 by Johan (site) on October 7, 2017 a 3:26 pm UTC

THanks, fixed here.

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