The Stacks project

Lemma 37.68.6. 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 37.68.5. 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 37.37.5 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. $\square$


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