Let $X$ and $Y$ be **affine** varieties over $\mathbb C$, and consider a
morphism $f:X\to Y$ and the induced homomorhism

$$ \varphi=f^*:B=\mathbb C[Y]\to A=\mathbb C[X]. $$

It is very easy to see that if $\varphi$ is surjective then $f$ is injective. The converse is not true, as the inclusion $\mathbb C\setminus\{0\}\to\mathbb C$ shows.

My question is:

Assume $f$ is a

finiteinjective morphism. Is it true that $\varphi$ must be surjective?

Here *finite* means that $A$ is integral over $B$ or, equivalently, that $A$ is a fintely generated $B$-module. Further, we are allowed to assume that $X$ and $Y$ are irreducible, and that $A$ and $B$ are free polynomial rings, if that helps.

One idea was to localize at an arbitrary maximal ideal $\mathfrak m$ of $B$, but I do not know if this works.