So this question is probably not “research level”, although, for what it is worth, it is coming up in a research paper I am presently writing.

Let $X,Y$ be irreducible affine varieties over $mathbb{C}$. Suppose $f:Xto Y$ is a bijective birational morphism. Assume that $X$ is normal, but $Y$ is not and that both $X,Y$ are singular (not smooth).

**Question 1**: Is $f$ étale?

I know that $f$ is the normalization map, and that if $Y$ is normal then $f$ is an isomorphism in this setting. I also know that the Grothendeick classes of $X$ and $Y$ are equal (induced by $f$). Also, at corresponding smooth points there is a local analytic isomorphism and so at such points the map is étale.

I suspect the answer to this question is no, and that there are easy counter-examples (that I have probably seen before and am just forgetting now).

So let me ask a second question that might be more useful to me than the expected counter-example.

**Question 2**: Are there easy to check (think computational) conditions on $X$, $Y$, or $f$ that will guarantee that $f$ is étale?