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).
An affine space of positive dimension is not complete
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0Does someo$n$e help me? – 2011-02-23
3 Answers
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.)
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0@charm: Dear Charm, Firstly, do you know the statement of Noether normalization? Regards, – 2011-02-24
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.
Matt E, all I know about Noether normalization is the existence of an algebraically independent elements of a finitely generated commutative algebra.
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0P.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