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?
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4$\begingroup$ Yes, at least if you assume moreover that $f$ is locally of finite presentation. Indeed, first $X\times_Y Y^n\to Y^n$ flat implies $X\to Y$ flat by EGA4, part 3, prop. 11.5.5. Second, being unramified is a property that depends only on the fibres, so $X\times_Y Y'\to Y'$ unramified always implies $X\to Y$ unramified for $Y'\to Y$ surjective. $\endgroup$– Matthieu RomagnyCommented Dec 16, 2013 at 20:23
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The "Main theorem" in "Mark S. McCormick, Etaleness and Normality, Journal of Algebra 219, 1999, 437-465" almost answers your question.
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