Is projection $\mathbb{A}^2 \to \mathbb{A}$ finite?

Question

I am misunderstanding something very basic here. We know a regular map $\varphi : W \to V$ is finite iff $k[W]$ is a finite $k[V]$-algebra. We also know that finite morphisms have finite fibers. Projection onto the first coordinate $\mathbb{A}^2 \to \mathbb{A}^1$ seems to be a finite morphism since $k[x,y]$ is a finite $k[x]$-algebra. But it surely doesn't have finite fibers. The pull back of a point gives $\mathbb{A}^1$. I clearly have misunderstood something basic. Please help me fix it.

Answer 1

$k[x,y]$ is not a finite $k[x]$-algebra. It is a finitely generated $k[x]$-algebra.

This terminology is confusing but standard. Finite always refers to finiteness as a module. To take a slightly simpler example: $k[x]$ is a finitely generated $k$-algebra, but is definitely not finite as a $k$-vector space.