Editing Function field (scheme theory)

The '''sheaf of rational functions''' ''K<sub>X</sub>'' of a [[scheme (mathematics)|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 [[algebraic varieties]], such a sheaf associates to each open set ''U'' the [[ring (mathematics)|ring]] of all [[rational function]]s on that open set; in other words, ''K<sub>X</sub>''(''U'') is the set of fractions of [[regular function]]s on ''U''. Despite its name, ''K<sub>X</sub>'' does not always give a [[field (mathematics)|field]] for a general scheme ''X''.

== Simple cases ==

In the simplest cases, the definition of ''K<sub>X</sub>'' is straightforward. If ''X'' is an ([[Irreducible component|irreducible]]) [[affine algebraic variety]], and if ''U'' is an open subset of ''X'', then ''K<sub>X</sub>''(''U'') will be the [[fraction 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<sub>X</sub>'' will be the [[constant sheaf]] whose value is the fraction field of the global sections of ''X''.

If ''X'' is [[Glossary of scheme theory#integral|integral]] but not affine, then any non-empty affine open set will be [[Dense set|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 affine open set will be the same, so we define, for any ''U'', ''K<sub>X</sub>''(''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 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 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<sub>U</sub>'' be the set of all elements in Γ(''U'', ''O<sub>X</sub>'') that are not zero divisors in any stalk ''O<sub>X,x</sub>''. Let ''K<sub>X</sub><sup>pre</sup>'' be the presheaf whose sections on ''U'' are [[Localization of a ring|localizations]] ''S<sub>U</sub><sup>−1</sup>''Γ(''U'', ''O<sub>X</sub>'') and whose restriction maps are induced from the restriction maps of ''O<sub>X</sub>'' by the universal property of localization. Then ''K<sub>X</sub>'' is the sheaf associated to the presheaf ''K<sub>X</sub><sup>pre</sup>''.

==Further issues==

Once ''K<sub>X</sub>'' is defined, it is possible to study properties of ''X'' which depend only on ''K<sub>X</sub>''. 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]] ''K<sub>X</sub>''(''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.

== See also ==
* [[Cartier divisor]], a notion defined in terms of the function field

==Bibliography==

*Kleiman, S., "Misconceptions about ''K<sub>X</sub>''", ''Enseign. Math.'' 25 (1979), 203–206, available at https://www.e-periodica.ch/cntmng?pid=ens-001:1979:25::101

[[Category:Scheme theory]]