Generalization To Arbitrary Scheme
In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if X is an integral scheme, then every open affine subset U is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the local ring of the generic point of X. Thus the function field of X is just the local ring of its generic point. This point of view is developed further in function field (scheme theory).
Read more about this topic: Function Field Of An Algebraic Variety
Famous quotes containing the words arbitrary and/or scheme:
“Me, what’s that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)
“We doubt not the destiny of our country—that she is to accomplish great things for human nature, and be the mother of a nobler race than the world has yet known. But she has been so false to the scheme made out at her nativity, that it is now hard to say which way that destiny points.”
—Margaret Fuller (1810–1850)