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 étale 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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