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Let $M$ be an affine algebraic variety and consider the ring of polynomial functions on $M$, $\mathcal{O}(M):=\{f: M\to k : f\text{ a polynomial}\}$. If $k$ is algebraically closed we can recover our manifold by mapping maximal ideals in $\mathcal{O}(M)$ back to points on $M$, since maximal ideals will be of the form $(x-a)$.

This seems to me analogous to the notion of a dual space (the set of functionals on some vector space), where under good conditions (reflexivity) we can recover $M$ by considering the dual of the dual. Is anything like that going on here? Is the analogy superficial or is something deeper going on?

EDITED: I had originally revealed a deep confusion by calling the affine algebraic variety a manifold. I misunderstood my professor's comments at the end of class.

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What is a polynomial on a manifold? –  t.b. Apr 6 '12 at 21:57
    
I believe the correct definition should those functions from $M$ to $k$ that are polynomials in local coordinates, but someone who has a better background in algebraic geometry may find that this is not the best definition. –  Helmut Apr 6 '12 at 22:03
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I suspect you're having some kind of (smooth, projective?) algebraic varieties in mind and there this and this might help. If you're really talking about manifolds then I'm not aware of any reasonable notion of polynomials (note that in your proposed definition this would have to entail that coordinate changes are given by polynomials if you want them to be independent of a particular chart). –  t.b. Apr 6 '12 at 22:08
    
Thank you t.b. I was indeed thinking about an affine algebraic variety. Sorry for the confusion, I'm obviously very new to this. –  Helmut Apr 7 '12 at 20:16

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