0
$\begingroup$

Specifically, how to show that an affine variety over complex number is never compact in Euclidean topology unless it is a single point. I got a hint on this qiestion: Given an affine variety X, show that the image of X under the projection map onto the first coordinate is either a point or an open subset (in the Zariski topology).

  • 0
    Does someo$n$e help me?2011-02-23

3 Answers 3

3

As Plop states, the hint follows from Chevalley's theorem. However, in this context one shouldn't need to appeal to the full strength of that theorem.

In fact, Chevalley's theorem is a variation on Noether normalization (and both are variations on the Nullstellensatz --- see this MO answer), but Noether normalization is usually taught at an ealier stage than Chevalley's theorem, so you might consider using it instead. (Regard this as an alternative hint.)

  • 0
    @charm: Dear Charm, Firstly, do you know the statement of Noether normalization? Regards,2011-02-24
2

The hint is a consequence of Chevalley's theorem on constructible sets: http://en.wikipedia.org/wiki/Constructible_set_%28topology%29

A non-empty Zariski-open susbset of the affine line is clearly not compact, so the image has to be a finite set. Projecting on each coordinate, you get that the variety is finite.

0

Matt E, all I know about Noether normalization is the existence of an algebraically independent elements of a finitely generated commutative algebra.

  • 0
    P.S. If you write @Matt E, I get notified of your comment. If you just write my name, as you did, I will only find your comment if I happen to read over this page again (which is what happened this time).2011-03-03