Lemma 37.53.7. Let $X \to S$ be a flat proper morphism of finite presentation. Let $n_{X/S}$ be the function on $S$ counting the numbers of geometric connected components of fibres of $f$ introduced in Lemma 37.28.3. Then $n_{X/S}$ is lower semi-continuous.

**Proof.**
Let $s \in S$. Set $n = n_{X/S}(s)$. Note that $n < \infty $ as the geometric fibre of $X \to S$ at $s$ is a proper scheme over a field, hence Noetherian, hence has a finite number of connected components. We have to find an open neighbourhood $V$ of $s$ such that $n_{X/S}|_ V \geq n$. Let $X \to S' \to S$ be the Stein factorization as in Theorem 37.53.5. By Lemma 37.53.2 there are finitely many points $s'_1, \ldots , s'_ m \in S'$ lying over $s$ and the extensions $\kappa (s'_ i)/\kappa (s)$ are finite. Then Lemma 37.42.1 tells us that after replacing $S$ by an étale neighbourhood of $s$ we may assume $S' = V_1 \amalg \ldots \amalg V_ m$ as a scheme with $s'_ i \in V_ i$ and $\kappa (s'_ i)/\kappa (s)$ purely inseparable. Then the schemes $X_{s_ i'}$ are geometrically connected over $\kappa (s)$, hence $m = n$. The schemes $X_ i = (f')^{-1}(V_ i)$, $i = 1, \ldots , n$ are flat and of finite presentation over $S$. Hence the image of $X_ i \to S$ is open (Morphisms, Lemma 29.25.10). Thus in a neighbourhood of $s$ we see that $n_{X/S}$ is at least $n$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: