Function field (scheme theory)
The sheaf of rational functions KX of a scheme X is thekshdfjhsdbjhzbkjbnasVas;hdgvklbds ;avdsg;jhvbasljdhvas Vas;vdv asvh asv hav agv agv agv oaihdsvohaOHIWG IGH TG
DFHG DKG ABGD SO ABG g fro nhg dgh dgbhd ghaction 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 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-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 geometry.
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.
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