I am currently reading Basic Algebraic Geometry by Shafarevich. I have two questions about finite maps on affine and quasi-projective varieites.
For a finite map $f:X\rightarrow Y$ between affine varieties $X$ and $Y$, we say that $f$ is finite if $f(X)$ is dense in $Y$ and $k[X]$ is integral over $k[Y]$ (where $k[Y]$ has been identified with a subring of $k[X]$).
Question 1
The first question is about the proof of Theorem 1.13. The statement of the theorem is as follows.
If $f:X\rightarrow Y$ is a regular map of affine varieties, and every $x\in Y$ has an affine neighbourhood $U\ni x$ such that $V = f^{-1}(U)$ is affine and $f:V\rightarrow U$ is finite, then $f$ itself is finite.
The proof itself is not so hard, but I find myself unable to understand the first step, which is to assume that each neighbourhood $U$ in the statement of the theorem above is a principal open set.
As far as I understand it, the neighbourhood $U$ of $x$ may be decomposed into a finite union of principal open sets $U_1,\ldots, U_r$ and for each $U_i$, the set $V_i := f^{-1}(U_i)$ is also a principal open set in $X$. However, I fail to understand why $f$ must be finite when restricted to any of these sets $V_i$.
Question 2
The definition of finite maps on quasi-projective varieties follows the "local definition" above. More precisely,
A map $f: X \rightarrow Y$ of quasi-projective varieties is finite if any point $y\in Y$ has an affine neighbourhood $U$ such that $V = f^{-1}(U)$ is affine and $f:V \rightarrow U$ is a finite map between affine varieties.
This definition makes sense given the theorem above, but it got me wondering regarding the following fairly easy fact about finite maps between affine varieties.
Fact: If $X,Y$ are affine varieties and $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are finite maps, then so is their composition $g\circ f$.
It's not clear to me if the above holds for finite maps between quasiprojective varieties as defined above. The "obvious" proof again runs into a similar obstacle where one needs to show that a restriction of a finite map to some open affine subset is still finite.
Possibly I am missing some simple clean fact about finite maps. If so, I would like to know it.
Best Answer
For question (1), what you are trying to show is that if you have a finite ring map $A \to B$ and some element $f \in A$ then $A_f \to B_f$ is finite (meaning invert the multiplicative subset of powers of $f$ in both rings). That is, we have some $A$-module surjection $A^n \twoheadrightarrow B$. Since localization is exact, and it commutes with direct sums, we get a surjection $A_f^n \twoheadrightarrow B_f$, as desired.
In general you're coming upon the idea of properties of morphisms that are affine-local on the target, where their truth can be verified either by checking every affine open on the target, or by checking it on a cover of the target by affine opens. This idea is discussed very cleanly in Vakil's Rising Sea notes section 5.3.