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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"?

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  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. $\endgroup$ – dantopa Jun 19 at 2:51
  • $\begingroup$ An open subset of a variety is naturally equipped with the structure of a variety. Thus when one speaks of an affine neighborhood, it is an open subset that contains the point in question which is (isomorphic to) an affine variety. An affine neighborhood is a full fledged variety - if it satisfies the hypotheses of any theorems, the conclusions are true. $\endgroup$ – RghtHndSd Jun 19 at 3:11
  • $\begingroup$ When one says "affine", it is short for "affine variety". $\endgroup$ – RghtHndSd Jun 19 at 3:14
  • $\begingroup$ When $X$ is several affine varieties glued together (for example $X= P^1(\Bbb{C})$ is $\Bbb{C}[z]$ glued with $\Bbb{C}[z^{-1}]$) there are no non-constant regular functions on $X$ anymore, so you need to remove a few hypersurfaces $S_j$ from $X$ to get a ring of regular function whose fraction field is the whole function field. $X-\cup S_j$ is said an affine open dense subset. $\endgroup$ – reuns Jun 19 at 4:05
  • $\begingroup$ @RghtHndSd: Thanks for the reply! This makes sense to me now. The particular case of principal open subsets are a special case of this. I'm happy to accept your comment as an answer to my question. $\endgroup$ – stupidqnasker Jun 20 at 15:54
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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.

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  • $\begingroup$ This is something which I still find a bit confusing. The reason for this for me is that I tend to think that an affine variety is necessarily Zariski closed. The only way I have to convince myself that this is not true is the fact that a principal open set of $ X $ at $ a \in A $ is affine since it is isomorphic to $ \text{Spec}(A_{(a)}). $ $\endgroup$ – Math N00b Aug 28 at 1:39
  • $\begingroup$ @MathN00b: You should never think of an affine variety as closed! An affine variety is always missing points - which is why it's not a projective variety. Projective space is closed, affine space is an open inside the projective space. This loose language is especially applicable when one thinks of an abstract variety as being locally a bunch of affine varieties that are patched together. $\endgroup$ – RghtHndSd Aug 29 at 0:30

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