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The Stacks project

Lemma 37.74.7. 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} attains a maximum d < \infty , then the set

Y_ d = \{ y \in Y \mid n_{X/Y}(y) = d\}

is open in Y and the morphism f^{-1}(Y_ d) \to Y_ d is finite.

Proof. The openness of Y_ d is immediate from Lemma 37.74.6. To prove finiteness over Y_ d we redo the argument of the proof of that lemma. Namely, let y \in Y_ d. Then there are at most d points of X lying over y. Say x_1, \ldots , x_ n are 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 d = 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 and we may assume U maps into W. This proves that n_{U \times _ Y X/U} \geq \sum _ i m_ i = d. Since the construction of n_{X/Y} is compatible with base change we know that n_{U \times _ Y X/U} = d. This means that W has to be empty and we conclude that U \times _ Y X \to U is finite. By Descent, Lemma 35.23.23 this implies that X \to Y is finite over the image of the open morphism U \to Y. In other words, we see that f is finite over an open neighbourhood of y as desired. \square


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