- Function field (scheme theory)
In
algebraic geometry , the function field "KX" of a scheme "X"is a generalization of the notion of a sheaf ofrational function s on a variety. In the case of varieties, such a sheaf associates to each open set "U" the ring of allrational 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".Simple cases
In the simplest cases, the definition of "KX" is straightforward. If "X" is an affine
algebraic variety , and if "U" is an open subset of "X", then "KX"("U") will be thefield 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 theconstant sheaf whose value is the fraction field of the global sections of "X".If "X" is integral but not affine, then any 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 thelocal ring of thegeneric point .General case
The trouble starts when "X" is no longer integral. Then it is possible to have
zero divisor 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 thetotal 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:
: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 localizations "SU-1"Γ("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 geometry .If "X" is a scheme 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 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.Bibliography
*Kleiman, S., "Misconceptions about "KX", "Enseign. Math." 25 (1979), 203-206, available at http://carpediem.ethz.ch:8081/swissdml.em/cntmng;jsessionid=4950B1C70AE3C05F260CDF9C8A36A85E?type=pdf&rid=ensmat-001:1979:25&did=c1:456368
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