## Tag `0E0N`

Chapter 36: More on Morphisms > Section 36.45: Stein factorization

Lemma 36.45.8. Let $f : X \to S$ be a morphism of schemes. Assume

- $f$ is proper, flat, and of finite presentation, and
- 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}
```

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