# What is going on with the map $X \mapsto X^2$

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 $a^2$...

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am I missing something? Why is it topologically identity? $(x-a)$ pulls back to $(x-a^2)$. And this is also how you can think about it geometrically: it's the map $z \to z^2$ from $\mathbb{A}^1$ to $\mathbb{A}^1$. – Soarer Sep 1 '11 at 1:34
you are absolutely right.... – DBr Sep 1 '11 at 2:00
@Soarer: Perhaps you'd like to post that as an answer? – Zev Chonoles Sep 26 '11 at 4:48
@Zev, done. (random characters) – Soarer Sep 29 '11 at 5:12

$(x−a)$ pulls back to $(x−a^2)$. This is also how you can think about it geometrically: it's the map $z \to z^2$ from $\mathbb{A}^1$ to $\mathbb{A}^1$.