Construction in Algebraic Geometry
In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. We have defined the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
Read more about this topic: Function Field Of An Algebraic Variety
Famous quotes containing the words construction in, construction, algebraic and/or geometry:
“There’s no art
To find the mind’s construction in the face.”
—William Shakespeare (1564–1616)
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)