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Let $f: X \rightarrow Y$ be a finite injective dominant unramified morphism, where $X$ is a normal and $Y$ is a regular scheme over a field $k$ (which I am not assuming to be algebraically closed or of characteristic 0).

Is it true that $f$ is birational?

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    $\begingroup$ Welcome, new contributor. The answer is no: take $\mathrm{Spec}(\mathbb{C})\to \mathrm{Spec}(\mathbb{R})$. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jul 11, 2021 at 9:30
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    $\begingroup$ For varieties over $\mathbb{C}$, the birationality should be a consequence of the following result: If $f:X \to Y$ is a morphism of varieties and either $f$ is finite or $X$ and $Y$ are projective, then $f$ is an isomorphism if and only if $f$ is bijective and $\mathrm{d} f: T_x(X) \to T_{f(x)}(Y)$ is injective for all $x \in X$. $\endgroup$
    – Francesco Polizzi
    Commented Jul 11, 2021 at 9:51
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    $\begingroup$ See mathoverflow.net/questions/164937/… $\endgroup$
    – Francesco Polizzi
    Commented Jul 11, 2021 at 9:52

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