Q:

Descending étaleness

Let $f:X\rightarrow Y$ be a finite morphism of reduced schemes, and let $Y^n\rightarrow Y$ be the normalization morphism. Assume that $X\times_Y Y^n\rightarrow Y^n$ becomes finite \'etale, is it true that $f:X\rightarrow Y$ is finite étale?

A:

The "Main theorem" in "Mark S. McCormick, Etaleness and Normality, Journal of Algebra 219, 1999, 437-465" almost answers your question. 
Sorry, this should be a comment not an answer, but I have few reputation points and the page does not allow me to make comments.

