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Algebraic Geometry I Lectures 14 and 15 October 22, 2008

Summary: Algebraic Geometry I Lectures 14 and 15
October 22, 2008
Recall from the last lecture the following correspondences
{points on an affine variety Y } {maximal ideals of A(Y )}
SpecA A
P Z(a) maximal ideal a
Also from the last time, recall that the closed subsets defined as
V (a) = {P A|P is a prime ideal of A and a P}
gives a Zariski topology on SpecA.
{closed irreducible subsets of Y } {prime ideals of }A(Y )
So if Y is an affine variety, then Spec (A(Y )) contains more elements than
Y .
What is the "geometry" of A(Y )?
Example 0.1. Y = An, then elements of Spec (A(Y )) = k[X1, . . . , Xn]
are prime ideals in k[X1, . . . , Xn] that contains irreducible polynomials in
k[X1, . . . , Xn]. These latter elements of Spec (A(Y )) can be visualized as
Z(f) where f is irreducible.
Definition 0.2. Let A be a ring. A point P SpecA is closed if and only
if P = V (a) for some ideal a in A.
Remark 0.3.


Source: Agashe, Amod - Department of Mathematics, Florida State University


Collections: Mathematics