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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0After proving what's asked for in the hint, do you see how to proceed? (Consider the images of X under all of the coordinate projections. What happens if X is compact?) – 2011-02-23
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0In fact, I have no idea how to prove the image of X under the projection map onto the 1st coordinate is an open subset in Zariski topology. – 2011-02-23
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0Does someone 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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0Thanks Matt E & Plop. Matt E, could you show me how to use the Noether normalization to solve my problem? – 2011-02-24
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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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1Dear Charm, First a practical comment: if you leave your comment as an answer, I don't get notified; if you leave it as a comment under my answer, as you did before, I get notified, which makes it easier to reply. – 2011-02-24
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0Okay, now for some mathematics! Noether normalization is a much stronger statement than what you wrote. The wikipedia entry http://en.wikipedia.org/wiki/Noether_normalization_lemmagives a good description which you might want to read; in particular, it gives the geometric interpretation, which is most relevant here. You could also look at this MO answer: http://mathoverflow.net/questions/42275/choosing-the-algebraic-independent-elements-in-noethers-normalization-lemma/42363#42363 which sketches a geometric proof of Noether normalization. ... – 2011-02-24
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0My suggestion is that you spend some time understanding Noether normalization and its geometric meaning first (and spend some time thinking about the general process of projection, perhaps by looking for and solving some exercises in Hartshorne Chapter I that are related to projections) before trying to solve your original question. This will have the added advantage that Chevalley's theorem discussed above (among other things) will make more sense, and the solution of your original question, will become reasonably obvious (following either the original hint or my Noether normalization hint). – 2011-02-24
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0Sorry, there was a typo in the above wikipedia link; it should be http://en.wikipedia.org/wiki/Noether_normalization_lemma – 2011-02-24
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0Matt E, could you give me a proof using the original hint (i.e. projection to the first coordinate is either open or a point)? – 2011-03-01
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0Dear Charm, Such a proof is already in the second para. of Plop's answer. Regards, – 2011-03-03
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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