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I'm trying to get a feel for basic algebraic geometry. One of the first things I read about is the localization of a polynomial ring at a point, yielding a ring of rational functions defined locally at this point.

In complex geometry, replacing the ring of polynomials by the ring of holomorphic functions and localizing to get meromorphic functions makes sense because holomorphic functions are often too scarce to give geometrical information. Even here though, it's not clear to me why meromorphic functions are actually enough to do wonders.

  1. What's the justification for doing this with polynomials, i.e why should rational functions yield great results and why won't polynomials suffice?
  2. Why do meromorphic functions give so much geometrical information about Riemann surfaces?
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    $\begingroup$ Localizing at a point does not give a field in general, unless the point is the generic point of an irreducible component. Localizing at a point gives a local ring. $\endgroup$ – Bruno Joyal Oct 20 '15 at 21:03
  • $\begingroup$ @BrunoJoyal ah, my mistake. Edited to fix. $\endgroup$ – Arrow Oct 20 '15 at 21:22

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