My question refers to following previous thread:
Geometric interpretation of the Noether normalization lemma
Obviously the core problem is: If $X= Spec(A), Y= Spec(R)$ are affine varieties and $\phi: X \to Y$ is a finite surjective morphism (so equivalently $A$ is a finite $R$-module)
why $\phi$ has finite fibers, so if $P \in Y$ why is $\phi^{-1}(P)$ finite?
Using again the language of commutative algebra the problem boils down to following:
Let $R$ wlog local with maximal ideal $p$ (in general case: localize $R$ by at $p$). We then get a morphism $R/p \to A/pA$.
I want to know why is $A/pA$ is finite dimensional $R/p$-vector space.
Following the thread The fibers of a finite morphism of affine varieties are all finite
it suffice to prove that. But I don't see why it holds, so why is $A/pA$ is finite dimensional $R/p$-vector space?
Best Answer
Since $A$ is a finite $R$-module, it is generated as an $R$-module by some elements $a_1, \dots, a_n$. Then $A/pA$ is spanned as an $R/p$-vector space by the images of the $a_i$.