Calling a variety would appear to conflict with the preexisting notion of an affine (http://planetmath.org/AffineVariety) or projective variety. However, it can be shown that if is algebraically closed, then there is an equivalence of categories between affine abstract varieties over and affine varieties over , and another between projective abstract varieties over and projective varieties over .
This equivalence of categories identifies an abstract variety with the set of its -points; this can be thought of as simply ignoring all the generic points. In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in defined by the ideal
is identified with
A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes. To see this, suppose we have a projective variety in given by the homogeneous ideal . If we delete the hyperplane , then we obtain an affine variety: let ; then the affine variety is the set of common zeros of
In this way, we can get overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.
|Date of creation||2013-03-22 14:16:43|
|Last modified on||2013-03-22 14:16:43|
|Last modified by||mps (409)|