If $f:X \rightarrow S$ is locally of finite type, there is unique largest open subset $U$ in $X$ such that $fU$ is etale.
Suppose $f$ is finite and $U$ is nonempty. Is it true that $fU$ is finite etale?
Thanks in advance.
If $f:X \rightarrow S$ is locally of finite type, there is unique largest open subset $U$ in $X$ such that $fU$ is etale. Suppose $f$ is finite and $U$ is nonempty. Is it true that $fU$ is finite etale? Thanks in advance. 


No, because $U\to S$ finite implies that $U\to X$ is finite (at least when $X\to S$ is separated), so $U$ would be closed in $X$. If you want an example, take a nontrivial morphism from a projective smooth curve $X$ to the projective line over $\mathbb C$. You might ask whether $U\to f(U)$ (if the latter is open in $S$) is finite, but this is not true either. Consider for example $S=\mathrm{Spec}\mathbb Z$ and $X=\mathrm{Spec}\mathbb Z[t]$ with $t^3+t^2+2t+2=0$. In the fiber above $2$, there is one étale point and one ramified point. 


No. If $f^{1}(f(U))\neq U$, then $f_U$ is not proper and hence not finite. This can easily happen if there are unramified points mapping to a branch point. (In other words if there are unramified and ramified points mapping to the same image). 


Here is an example of a finite morphism with no finite etale restriction. Let $X = S = \mathbb{P}^1_k$ and $f : X \rightarrow S$ is defined by sending $x$ to $x^2$. Then this map is ramified at $0, \infty$. so $U = \mathbb{P}^1_k \{0,\infty\}$. Then restriction of $f$ can't be finite and etale. It should be open and closed so, surjective. But $\mathbb{P}^1_k$ has no nontrivial finite etale cover. 

