# The cardinality of fibers of quasi-finite morphism is locally constant (The open-ness of proper locus)

Mathematics Asked by zom on October 19, 2020

Let $$f : X to Y$$ be an etale surjective separated quasi-compact morphism of schemes.
Then does there exist an integer $$d > 0$$ such that there exists the maximal open subset over which $$f$$ is finite etale of degree $$d$$?

Since the degree function of finite locally free morphism is locally constant, it suffices to see that, there exists the maximal open subset over which $$f$$ is finite etale.

Or, by Deligne-Rapoport’s II.1.19 of "Les schemas…", it suffices to see that there exists an integer $$d > 0$$ such that the subset $${ y in Y | # f^{-1}(bar{y}) = d }$$ (where $$bar{y}$$ is a geometric point over $$y$$) is open.

(Or by the surjectivity, it’s sufficient to show the open-ness of $${ y in Y | # f^{-1}(bar{y}) < d }$$ for any $$d$$.
If $$f$$ is proper, then this is just a semi-continuity theorem.
But now $$f$$ is not proper.)

How can I show it?

Please give me a proof or references.

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