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Let $\phi : X \rightarrow Y$ be a morphism of affine varieties and let $\phi^\ast : k[Y] \rightarrow k[X]$ be the induced map on coordinate rings. My text says that if $\phi^\ast$ is surjective then $\phi(X)$ is closed in $Y$; the "proof" given is that $\phi(X) = V(\ker \phi^\ast)$. Clearly $\phi(X) \subseteq V(\ker \phi^\ast)$ in general, but how do I get the other direction?

Here's what I have so far:

Consider the maps $\varepsilon_i \in k[X]$ with $\varepsilon_i(x_1, x_2, \ldots x_n) = x_i$. Since $\phi^\ast$ is surjective, there are maps $\gamma_i \in k[Y]$ such that $\varepsilon_i = \gamma_i \circ \phi$. If we let $\gamma = (\gamma_1, \gamma_2, \ldots \gamma_n)$ then $\gamma \circ \phi$ is the identity on $X$, but it's not even clear that the image of $\gamma$ lies in $X$, so I don't know what to do with this.

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up vote 6 down vote accepted

The general result in commutative algebra is:

Theorem Let $\phi:A\to B$ be a homomorphism of commutative rings and let $f:\text{Spec}(B)\to \text{Spec}(A)$ be the corresponding continuous map between spectra. If $\phi$ is surjective, then $f(\text{Spec}(B))$ is a closed subset of $\text{Spec}(A)$.

Proof. We assert that $f(\text{Spec}(B))=V(\text{Ker}(\phi))$. If $p\in\text{Spec}(B)$, then $f(p)=\phi^{-1}(p)\supseteq \text{Ker}(\phi)$. Conversely, if $q\in V(\text{Ker}(\phi))$, then we wish to prove that $q=\phi^{-1}(p)=f(p)$ for some $p\in \text{Spec}(B)$. In fact, the following exercise completes the proof:

Exercise: Prove that $p=\phi(q)$ is a prime ideal of $B$ and that $f(p)=\phi^{-1}(p)=q$. (Hint: you will need to use the surjectivity of $\phi$ to prove that $p$ is an ideal of B. However, you will need to use the surjectivity of $\phi$ and the fact that $q\supseteq \text{Ker}(\phi)$ to show that $p$ is a prime ideal of $B$.)

Therefore, $f(\text{Spec}(B))=V(\text{Ker}(\phi))$ is a closed subset of $\text{Spec}(A)$. Q.E.D.

I hope this helps!

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