The Stacks project

Lemma 37.74.6. Let $f : X \to Y$ be a separated, locally quasi-finite, and universally open morphism of schemes. Let $n_{X/Y}$ be as in Lemma 37.74.5. If $n_{X/Y}(y) \geq d$ for some $y \in Y$ and $d \geq 0$, then $n_{X/Y} \geq d$ in an open neighbourhood of $y$.

Proof. The question is local on $Y$ hence we may assume $Y$ affine. Let $K$ be an algebraic closure of the residue field $\kappa (y)$. Our assumption is that $(X_ y)_ K$ has $\geq d$ connected components. Then for a suitable quasi-compact open $X' \subset X$ the scheme $(X'_ y)_ K$ has $\geq d$ connected components; details omitted. After replacing $X$ by $X'$ we may assume $X$ is quasi-compact. Then $f$ is quasi-finite. Let $x_1, \ldots , x_ n$ be the points of $X$ lying over $y$. Apply Lemma 37.41.5 to get an ├ętale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

\[ U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j} \]

as in locus citatus. Observe that $n_{X/Y}(y) = \sum _ i m_ i$ in this situation; some details omitted. Since $f$ is universally open, we see that $V_{i, j} \to U$ is open for all $i, j$. Hence after shrinking $U$ we may assume $V_{i, j} \to U$ is surjective for all $i, j$. This proves that $n_{U \times _ Y X/U} \geq \sum _ i m_ i = n_{X/Y}(y) \geq d$. Since the construction of $n_{X/Y}$ is compatible with base change the proof is complete. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F36. Beware of the difference between the letter 'O' and the digit '0'.