Function field (scheme theory)
Encyclopedia
In algebraic geometry
, the function field KX of a scheme
X
is a generalization of the notion of a sheaf
of rational function
s on a variety. In the case of varieties, such a sheaf associates to each open set U the ring
of all rational function
s on that open set; in other words, KX(U) is the set of fractions of regular functions on U. Despite its name, KX does not always give a field
for a general scheme X.
, and if U is an open subset of X, then KX(U) will be the field of fractions
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 KX will be the constant sheaf
whose value is the fraction field of the global sections of X.
If X is integral 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, KX(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
.
s 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, not only can this fail to give a sheaf, in general it does not even give a presheaf! The well-known article of Kleiman, listed in the bibliography, gives such an example.
The correct solution is to proceed as follows:
.
If X is an algebraic variety
over a field k, then over each open set U we have a field extension KX(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
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the function field KX of a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
X
is a generalization of the notion of a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of rational function
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...
s on a variety. In the case of varieties, such a sheaf associates to each open set U the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of all rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s on that open set; in other words, KX(U) is the set of fractions of regular functions on U. Despite its name, KX does not always give a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
for a general scheme X.
Simple cases
In the simplest cases, the definition of KX is straightforward. If X is an affine algebraic varietyAlgebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
, and if U is an open subset of X, then KX(U) will be the field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
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 KX will be the constant sheaf
Constant sheaf
In mathematics, the constant sheaf on a topological space X associated to a set A is a sheaf of sets on X whose stalks are all equal to A. It is denoted by or AX. The constant presheaf with value A is the presheaf that assigns to each open subset of X the value A, and all of whose restriction maps...
whose value is the fraction field of the global sections of X.
If X is integral 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, KX(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
Local ring
In abstract algebra, more particularly in 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 number fields examined at a particular place, or...
of the generic point
Generic point
In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
.
General case
The trouble starts when X is no longer integral. Then it is possible to have zero divisorZero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s 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
Total quotient ring
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of a domain to commutative rings that may have zero divisors. The construction embeds the ring in a larger ring, giving every non-zerodivisor of the...
, that is, to invert every element that is not a zero divisor. Unfortunately, not only can this fail to give a sheaf, in general it does not even give a presheaf! 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 SU be the set of all elements in Γ(U, OX) that are not zero divisors in any stalk OX,x. Let KXpre be the presheaf whose sections on U are localizationsLocalization of a ringIn abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
SU-1Γ(U, OX) and whose restriction maps are induced from the restriction maps of OX by the universal property of localization. Then KX is the sheaf associated to the presheaf KXpre.
Further issues
Once KX is defined, it is possible to study properties of X which depend only on KX. This is the subject of birational geometryBirational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
.
If X is an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over a field k, then over each open set U we have a field extension KX(U) of k. The dimension of U will be equal to the transcendence degree
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...
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
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
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.