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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