The sheaf of rational functions ''K
X'' of a
scheme ''X'' is the generalization to
scheme theory of the notion of
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 classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry thes ...
in classical
algebraic geometry. In the case of varieties, such a sheaf associates to each open set ''U'' the
ring of all
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s on that open set; in other words, ''K
X''(''U'') is the set of fractions of
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regul ...
s on ''U''. Despite its name, ''K
X'' does not always give a
field for a general scheme ''X''.
Simple cases
In the simplest cases, the definition of ''K
X'' is straightforward. If ''X'' is an (irreducible) affine
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 ...
, and if ''U'' is an open subset of ''X'', then ''K
X''(''U'') will 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 ring of regular functions on ''U''. Because ''X'' is affine, the ring of regular functions on ''U'' will be a localization of the global sections of ''X'', and consequently ''K
X'' will be the
constant sheaf
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific con ...
whose value is the fraction field of the global sections of ''X''.
If ''X'' is
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
but not affine, then any non-empty affine open set will be
dense in ''X''. This means there is not enough room for a regular function to do anything interesting outside of ''U'', and consequently the behavior of the rational functions on ''U'' should determine the behavior of the rational functions on ''X''. In fact, the fraction fields of the rings of regular functions on any open set will be the same, so we define, for any ''U'', ''K
X''(''U'') to be the common fraction field of any ring of regular functions on any open affine subset of ''X''. Alternatively, one can define the function field in this case to be the
local ring of the
generic point.
General case
The trouble starts when ''X'' is no longer integral. Then it is possible to have
zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the
total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
The correct solution is to proceed as follows:
:For each open set ''U'', let ''S
U'' be the set of all elements in Γ(''U'', ''O
X'') that are not zero divisors in any stalk ''O
X,x''. Let ''K
Xpre'' be the presheaf whose sections on ''U'' are
localizations ''S
U−1''Γ(''U'', ''O
X'') and whose restriction maps are induced from the restriction maps of ''O
X'' by the universal property of localization. Then ''K
X'' is the sheaf associated to the presheaf ''K
Xpre''.
Further issues
Once ''K
X'' is defined, it is possible to study properties of ''X'' which depend only on ''K
X''. This is the subject of
birational geometry.
If ''X'' is 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 ...
over a field ''k'', then over each open set ''U'' we have a field extension ''K
X''(''U'') of ''k''. The dimension of ''U'' will be equal to the
transcendence degree of this field extension. All finite transcendence degree field extensions of ''k'' correspond to the rational function field of some variety.
In the particular case of an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
''C'', that is, dimension 1, it follows that any two non-constant functions ''F'' and ''G'' on ''C'' satisfy a polynomial equation ''P''(''F'',''G'') = 0.
Bibliography
*Kleiman, S., "Misconceptions about ''K
X''", ''Enseign. Math.'' 25 (1979), 203–206, available at https://www.e-periodica.ch/cntmng?pid=ens-001:1979:25::101
Scheme theory