Construction in Algebraic Geometry
In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. We have defined the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
Read more about this topic: Function Field Of An Algebraic Variety
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