Consider the homomorphism $f : \mathbb{C}[X] \to \mathbb{C}[X]$, $X \mapsto X^2$. It induced a morphism of affine schemes $\operatorname{spec} f : \mathbb{A} \to \mathbb{A}$ which topologically is the identity function. How am I meant to think about $\operatorname{spec} f$ geometrically?
EDIT: As pointed out in the comments, $spec \; f$ is not infact topologically the identity, it does exactly what it should. It sends $a$ to $a^2$...