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December 23, 2008 Definition 1.1. If ( a) is a homogeneous ideal of S then we define V(( a))
 

Summary: Lecture 27
Vivek Pal
December 23, 2008
1 Proj S
Definition 1.1. If ( a) is a homogeneous ideal of S then we define V(( a))
= {P Proj(S)|(a) P}
Lemma 1.2. (a) If (a) and (b) are homogeneous ideals of S, then
V( (a) (b)) = V( (a)) V( (b))
(b) If {ai}iI is a family of homogeneous ideals of S then V( ai) = V (ai)
Proof. Same as for spectra, using the fact that a homogeneous ideal P is
prime iff whenever a, s S are homogeneous elements such that ab P
then a P or b P.
Remark 1.3. Proj(S) = V (0) and = V (S)
Definition 1.4. The Zariski topology on Proj(S) is the one with closed
subsets being V(( a)) for homogeneous ideals ( a).
Example 1.5. If S = k[x1, ..., xn] with k algebraically closed, then the set
of closed points of Proj(S) (with the induced topology) is homeomorphic
to Pn
k
Example 1.6. n=2 (x - z, y - 2z) is a homogeneous prime ideal and cor-

  

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

 

Collections: Mathematics