Let $$f\colon X \to Y$$ be a morphism of affine normal algebraic varieties over $\mathbb{C}$. Assume that $f$ is birational and bijective on closed points. Does normality imply that $f$ is an isomorphism? Does it follow from Stein's factorization?


(I am especially interested in the surface case, but I do not see why this should help.)



About normalization
Please confer Zariski's Main Theorem, pp. 288-289 of Mumford's "Red Book of Varieties and Schemes". Using the "original form", $f$ is an open immersion. Using the hypothesis about bijectivity on closed points, $f$ is an isomorphism.


