MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X^\nu,Y^\nu$ be normalizations of affine varieties $X$ and $Y.$ If a morphism $f:X\to Y$ is a bijection, does it imply that its lift $f^\nu: X^\nu\to Y^\nu$ is an isomorphism?

share|cite|improve this question
up vote 4 down vote accepted

We should be able to construct a counterexample as follows:

Let $Y$ be an affine curve which is smooth away from a single node. We obtain $X$ from the normalization of $Y$ by removing one of the points mapping to the node of $Y$. The map from $X$ to $Y$ is a bijection, but the map $X^\nu\to Y^\nu$ is not an isomorphism.

share|cite|improve this answer
I guess the right assumption that would make things work is that $f$ is proper and separable. – Piotr Achinger Oct 28 '12 at 21:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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