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 the field 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 the constant sheaf whose value is the fraction field of the global sections of X.
If X is integral but not affine, then any non-empty 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 the local ring of the generic point.
Read more about this topic: Function Field (scheme Theory)
Famous quotes containing the words simple and/or cases:
“The simple opposition between the people and big business has disappeared because the people themselves have become so deeply involved in big business.”
—Walter Lippmann (1889–1974)
“Colonel, never go out to meet trouble. If you will just sit still, nine cases out of ten someone will intercept it before it reaches you.”
—Calvin Coolidge (1872–1933)