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Given a quasi-finite (the each fiber is a finite set) morphism between two affine varieties (in the sense of the zero set of polynomials): $\phi:X\to Y$.

What can we say about the induced ring homomorphism $\phi^*:A(Y)\to A(X)$ as well as the relation between $A(Y),A(X)$? More precisely, I know if $\phi$ is finite (quasi-finite+proper) iff $\phi^*: A(Y)\to A(X)$ is finite, can we say something like this when $\phi$ is quasi-finite?

Moreover, if $\phi:X\to Y$ is a morphism between two affine varieties with the same dimension, is $\phi$ quasi-finite? Or what additional condition do we need to add so that $\phi$ is quasi-finite?

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    $\begingroup$ To answer your last question: no, blowups are not quasi-finite. You can make these affine if you want by removing some stuff, e.g. the map $\mathbf A^2 \to \mathbf A^2$ given by $(x,y) \mapsto (x,xy)$ is not quasi-finite. For your first question, you might be interested in this version of Zariski's main theorem. $\endgroup$ Nov 25, 2019 at 3:32
  • $\begingroup$ @R.vanDobbendeBruyn the Zariski's main theorem mentioned is for schemes not for the affine varieties in my sense. Also I want to know how the induced ring homomorphism as well as the coordinate ring behave when the morphism is just quasi-finite, I don't think Zariski's main theorem answers my question. $\endgroup$
    – 6666
    Nov 25, 2019 at 3:49
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    $\begingroup$ It tells you that you should think of quasi-finite morphisms as finite morphism plus a localisation, although it is slightly more general than that (localisation corresponds to the case where $Y \to Z$ is a standard affine open immersion $D(f) \subseteq \operatorname{Spec} C$, but it can also be another type of open immersion). $\endgroup$ Nov 25, 2019 at 4:36

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The first part of the answer is an extension of the comment by R. van Dobben de Bruyn.

We have

$$\textit{ the class of open immersions of schemes } = \textit{ the class of étale monomorphism of schemes} $$

Hence we deduce (by Zariski's main theorem) that $\phi^*:A[Y] \rightarrow A[X]$ is quasi-finite if and only if there exists an affine $k$-scheme $Z$ and a factorization $\phi^* = i^*\cdot p^*$ such that $p^*:A[Y] \rightarrow A[Z]$ is a finite morphism and $i^*:A[Z] \rightarrow A[X]$ is an étale epimorphism of $k$-algebras. Now étale can be characterized as formally étale and (in case of finitely generated $k$-algebras) of finite type. For characterizing epimorphisms of rings you may be interested in the following MO question.

For your second question it suffices to assume (in addition to $\mathrm{dim}(X) = \mathrm{dim}(Y)$) that $\phi:X\rightarrow Y$ is flat. By flatness all fibers have the same dimension. Moreover, flat morphism locally of finite presentation are open, thus $\phi$ is dominant and hence the generic fiber of $\phi$ is finite. Hence all fibers are finite and $\phi$ is quasi-finite (clearly it is of finite type).

Moreover, if $\phi:X\rightarrow Y$ is a dominant morphism of varieties such that $\mathrm{dim}(X) = \mathrm{dim}(Y)$, then there exists open dense subset $V\subseteq Y$ such that $\phi^{-1}(V)\rightarrow V$ is finite, but in general not all fibers are finite as it was pointed by R. van Dobben de Bruyn in the comments.

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