Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety "V" consists of objects which are interpreted as rational functions on "V". In complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in classical algebraic geometry they are ratios of polynomials; in modern algebraic geometry they are elements of some quotient field.

More precisely, 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 is then the set of all meromorphic functions on the variety. For the purposes of comparison, it is useful to keep in mind that for the Riemann sphere, which is the variety P¹ over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions). In any case, the meromorphic functions form a field, the function field.

In classical algebraic geometry, we generalize the second point of view. Begin by noting that even 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. Again for the purposes of comparison, it is clear that we have simply 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.

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 "X" is an integral scheme, then every open affine subset "U" is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the local ring of the generic point of "X". Thus the function field of "X" is just the local ring of its generic point. This point of view is developed further in function field (scheme theory).

If "V" is a variety over a field "K", then the function field "K"("X") is a field extension of the ground field "K" over which "V" is defined; its transcendence degree is equal to the dimension of the variety. All extensions of "K" that are finitely-generated as fields arise in this way from some algebraic variety.

Properties of the variety "V" that depend only on the function field are studied in birational geometry.

Examples

The function field of a point over "K" is "K".

The function field of the affine line over "K" is isomorphic to the field "K"("t") of rational functions in one variable. This is also the function field of the projective line.

Consider the affine plane curve defined by the equation $y^2\; =\; x^5\; +\; 1$. Its function field is the field "K"("x","y"), generated by transcendental elements satisfying the algebraic relation above.

See also

* Function field (scheme theory): a generalisation.

References

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