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Concept
Function field of an algebraic variety
Résumé
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in .) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.
For the Riemann sphere, which is the variety over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions).
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 as agree on the intersections of open affines. We may define the function field of V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if is an integral scheme, then for every open affine subset of the ring of sections on is an integral domain and, hence, has a field of fractions.
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