Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a connected (1-dimensional) Dedekind scheme.


Suppose that the morphism $g\circ f$ from $X$ to $Z$ is etale.


**Question.** Is the morphism $g:Y\longrightarrow Z$ etale?


**Remark.** The hypothesis on the dimension is not necessary probably.


**Remark.** One may assume $Y$ to be integral.


**Remark.** To assure that $f$ is finite one may suppose that $Y$ is excellent.


Interesting cases one may consider are $Z\subset \mathrm{Spec} \mathbf{Z}$ or $Z\subset \mathbf{P}^1\_k$ non-empty open.



Is the following morphism etale
If $g$ is étale, then $Y$ is regular because $Z$ is regular. Thus $X=Y$. So the anwser to your question is yes if and only if $X=Y$. 


