# Why this map is birational?

Mathematics Asked on January 7, 2022

Let $$Y$$ be a connected normal Noetherian scheme, $$f: Xto Y$$ is an etale morphism of finite type. We assume $$X$$ is also connected. I have proved that in this case, $$X$$ is also normal. Denote the functional field of $$Y$$ and $$X$$ by $$K(Y)$$ and $$K(X)$$. Then $$K(X)$$ is a finite separable extension of $$K(Y)$$. Let $$widetilde{Y}$$ be the normalization of $$Y$$ in $$K(X)$$. Since $$X$$ is normal and f is dominant, by universal property of normalization, we could get a map $$g:Xto widetilde{Y}$$. My question is

Why g is birational?

In general, $$g$$ is not birational. So I guess, in this case, $$g$$ is birational because $$f$$ is etale. But I do not know how to prove it. Could you tell me how to prove it or where I could find a proof?

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