Editing Function field (scheme theory) (section)

== 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]].