Let $f:X\to Z$ be an etale morphism of integral schemes with $Z$ normal, and let $k(Z)\subseteq L\subseteq k(X)$ be an intermediate field between the associated function fields. Can we find a scheme $Y$ with $k(Y)=L$ so that $f$ factorises into an etale tower $X\to Y\to Z$? Can anything be said if $f$ is only assumed to be smooth?



`Normalisation' in etale towers?
Yes, you can take $Y$ to be the image of $X$ in the normalization of $Z$ in $L$. You need to show that if in the composite $X \to Y \to Z$ all schemes are normal, $X\to Y$ is surjective and $X \to Z$ is étale, then $X \to Y$ is also étale. By base-changing along $X\to Z$ you can assume that $X \to Z$ is an isomorphism, in which case the statement is pretty obvious.


[Edit]: why is $Y$ open? Because a quasi-finite dominant morphism $X \to Y$ between normal schemes is open. First, reduce to the finite case by embedding $X$ into the normalization of $Y$ in $k(X)$ and using Zariski's main theorem. Then by taking a Galois closure one can assume that $X \to Y$ is Galois, and this case is easy.


[Another edit]: The statement is false in the smooth case. Suppose that $X$ is a smooth variety over a field $k$, with a finite group $G$ acting on it, such that the quotient $X/G$ is singular. Take $Z = \mathop{\rm Spec}k$ and $L = k(X/G)$.


