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?



Properties of quasi finite morphism of affine varieties
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](https://en.wikipedia.org/wiki/Formally_%C3%A9tale_morphism) 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](https://mathoverflow.net/questions/109/what-do-epimorphisms-of-commutative-rings-look-like).


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.


