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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
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@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

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up vote 3 down vote accepted

$(x−a)$ 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 $\mathbb{A}^1$.

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