Quasi-Finite + Affine -> Finite?

Question

If $f:X\to Y$ is an affine morphism of schemes, say with $Y$ irreducible, that is quasi-finite - all of the fibers, including the generic fiber, are finite - is it true that $f$ is finite?

If not, what is a good counterexample.

EDIT: I also am requiring that $f$ be closed.

Answer 1

Let $U = \mathbb{P}^1 - \{1\}$ and let $f : U \to \mathbb{P}^1$ be the squaring map $z\mapsto z^2$. Then $f$ is a counterexample. (Note that it is still surjective.)

The point is that "closed" is not sufficient. The property you want is "universally closed", that is, remains closed after any base change.

It is true that quasi-finite + affine + universally closed $\Rightarrow$ finite. (The precise statement is that finite = quasi-finite + proper.)

Answer 2

Let $$f\colon \textrm{Spec}\left(k[x,y]/(xy-1)\right) \to \textrm{Spec} \left(k[x]\right)$$ be the projection of the hyperbola onto the affine line. Then, $f$ is quasi-finite, but it is not finite.