Editing Function field (scheme theory) (section)

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''.