In
algebraic geometry, the function field of an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''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
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
these are
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s and their higher-dimensional analogues; in
modern algebraic geometry they are elements of some quotient ring's
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
.
Definition for complex manifolds
In complex algebraic geometry the objects of study are complex
analytic varieties
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible
and (or) reduced or complex analytic space is a generali ...
, 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).
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
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
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
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
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
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. Furthermore, it can be verified that these are all the same, and are all equal to the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
of the
generic point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
In classical algebraic g ...
of
. Thus the function field of
is just the local ring of its generic point. This point of view is developed further in
function field (scheme theory) The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open ...
. See .
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
field extension of the ground field ''K''; its
transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
is equal to the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the variety. All extensions of ''K'' that are finitely-generated as fields over ''K'' arise in this way from some algebraic variety. These field extensions are also known as
algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
s over ''K''.
Properties of the variety ''V'' that depend only on the function field are studied in
birational geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
.
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
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
.
Consider the affine plane curve defined by the equation
. Its function field is the field ''K''(''x'',''y''), generated by elements ''x'' and ''y'' that are
transcendental over ''K'' and satisfy the algebraic relation
.
See also
*
Function field (scheme theory) The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open ...
: a generalization
*
Algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
*
Cartier divisor
References
*
* {{Citation , last1=Hartshorne , first1=Robin , author1-link=Robin Hartshorne , title=
Algebraic Geometry , publisher=
Springer-Verlag , location=Berlin, New York , isbn=978-0-387-90244-9 , oclc=13348052 , mr=0463157 , year=1977, section II.3 First Properties of Schemes exercise 3.6
Algebraic varieties