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?

  • 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. $\endgroup$ Dec 16, 2013 at 20:23

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.