I am trying to read the book Basic Algebraic Geometry Chapter 1 by Shafarevich, but am not able to follow the meanings of some basic statements.
In Section 5.3 of Chapter 1, Shafarevich defines finite maps $f:X \rightarrow Y$ between affine varieties $X$ and $Y$ as follows.
Definition: $f$ (a regular map from $X$ to $Y$ with $f(X)$ dense in $Y$) is a finite map if $k[X]$ is integral over $k[Y]$.
Here $k$ is the underlying field, and $k[X]$ is the coordinate ring of $X$.
A couple of interesting theorems are proved about finite maps: (a) Finite maps are surjective, and (b) A finite map takes closed sets to closed sets.
After this, Shafarevich states the following theorem.
Theorem: If $f:X\rightarrow Y$ is a regular map of affine varieties and every point $x\in Y$ has an affine neighborhood $U\ni x$ such that $V = f^{-1}(Y)$ is affine and $f:V\rightarrow U$ is finite, then $f$ itself is finite.
Here, I do not understand what an "affine neighbourhood" means.
Is it a set that is isomorphic to an open set in some affine space? In this case, what does it mean for $f:V\rightarrow U$ to be finite? Do we take the same definition? If so, do the intermediate theorems also hold in this setting?
In the proof of this theorem, Shafarevich restricts the open sets to be principal open sets, which I understand better as they are isomorphic to affine varieties. But I am not sure if this is what he means by an affine neighbourhood.
Finally, there is the following definition.
Definition: A regular map $f:X \rightarrow Y$ of quasiprojective varieties is finite if any $y\in Y$ has an affine neighbourhood $V$ such that the set $U = f^{-1}(V)$ is affine and $f:U\rightarrow V$ is an finite map between affine varieties.
Again, I have the same questions as above. Additionally, what does it mean for $U$ to be "affine"? Is he referring to an "affine neighbourhood" or an "affine variety"?
Best Answer
As indicated in the comments, the primary confusion was over the use of "affine". In algebraic geometry, "affine" is short for "affine variety". In particular, an affine neighborhood is an open subset of a variety (which nauturally carries the structure of a variety) containing the point in question which is an affine variety.