Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex 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.
Definition for complex manifolds
In complex geometry the objects of study are
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).
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 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.
Generalization to arbitrary scheme
In the most general setting, that of modern
Geometry of the function field
If V is a variety defined over a field K, then the function field K(V) is a finitely generated
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 algebraic plane curve defined by the equation . Its function field is the field K(x,y), generated by elements x and y that are
See also
- Function field (scheme theory): a generalization
- Algebraic function field
- Cartier divisor
References
This article needs additional citations for verification. (September 2008) |
- ISBN 0-387-95432-5.
- OCLC 13348052, section II.3 First Properties of Schemes exercise 3.6