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I got myself in a confusing situation the other week while trying to read a bit of algebraic geometry. I'm hoping someone can pull me out.

Suppose $k$ is a field, and $V$ a homogeneous variety with generic point $(x)$ over $k$. Denote by $Z$ the algebraic set of zeroes in $k^a$ of a homogeneous ideal in $k[X]$ generated by forms $f_1,\dots,f_r\in k[X]$. It is a theorem of Hilbert and Zariski that $V\cap Z$ has only the trivial zero if and only if each $x_i$ is integral over $k[f_1(x),\dots,f_r(x)]$.

I searched for a proof of this fact and played with it over the course of last week, and came up with nothing. I would like to know, why does $V\cap Z$ have only the trivial zero if and only if each $x_i$ is integral over $k[f_1(x),\dots,f_r(x)]$?

Thank you for your expertise.

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1 Answer 1

up vote 5 down vote accepted

The proof of this theorem is on page 42 of Lang's Introduction to Algebraic Geometry.
This answers your question technically , however:

Admonition
Your question is phrased in the language of Weil's Foundations of algebraic Geometry .
This language has been killed by Grothendieck's scheme theory, developed in the treatise Eléments de Géométrie Algébrique (=EGA), about 55 years ago.
Practically nobody nowadays understands Weil's obsolete language or is interested in it: I am no exception and although I gave you a reference to a proof in that language I wouldn't dream of reading one line of it.
Let me insist that if you express yourself in that language, you will probably never meet anybody who understands you and your chances of publishing a paper in that style are exactly zero.

To end on a cheerful note, you are lucky to live in a time where there are a huge number of excellent books on algebraic geometry written from the point of view of scheme theory, or at least compatible with it, from Hulek's very elementary one to Qing Liu's quite advanced one.
You will find a long list of such books with helpful comments and explanations in this link to our sister site MathOverflow

Edit
Here is Brian Conrad's opinion from his homepage:
" With all due respect to the role of Andre Weil in the development of algebraic geometry, nobody should ever again have to read Weil's "Foundations of algebraic geometry": EGA must be an adequate logical starting point for the subject."

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Thank you Georges for the reference! :) –  Vika Feb 6 '12 at 20:17
    
Dear Georges, I wanted to ask one more thing before I accept. At the end of page 42, Lang says, "we see than an equation similar to the above holds with coefficients which do not have constant terms." Can you spell out what Lang is saying here more explicitly? Thank you. –  Vika Feb 8 '12 at 3:44
    
Dear @Vika, each coefficient $a_i(f)$ is obtained by taking a polynomial $a_i(Y_1,...,Y_s)$ in $s $ variables over $k$ and replacing each variable $Y_j$ by a polynomial $f_j(X_1,...,X_n)$ depending on a set of other variables $X_1,...,X_n$. In the end you obtain a polynomial in these variables $X_1,...,X_n$ and Lang asserts that this polynomial has constant term zero. –  Georges Elencwajg Feb 8 '12 at 7:32
    
@GeorgesElencwajg Would you be willing to include a citation for where this result appears in EGA? –  vgty6h7uij Mar 8 '13 at 13:41

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