# Properties of quasi finite morphism of affine varieties

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?

• 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. Nov 25, 2019 at 3:32
• @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.
– 6666
Nov 25, 2019 at 3:49
• 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). Nov 25, 2019 at 4:36

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