Restatement: An affine algebraic variety, the set of common zeros of a collection of polynomials, is an irreducible closed subset of (with the induced topology). An open subset of an affine variety is called a quasi affine algebraic variety.
In this post examples affine and quasi-affine object varieties will be given after an analysis of the relationship between the Z(*) and I(*) operations are clarified.
Definition 2.1 The radical of any ideal, , is defined as
An ideal that is equal to its radical is called a radical ideal, i.e. if .
Example 2.1 The radical of the ideal , integer mutilples of , is . The radical ideal of is .
Definition 2.2 Hilbert’s Nullstellensatz. If is an algebraically closed field with an ideal in and a polynomial that vanishes at all points of , then for some integer .
Proposition 2.1 Properties of the operations Z(*) and I(*).
1) If are subsets of , then .
2) If are subsets of , then .
3) For any two subsets , of , we have .
4) For any ideal , . This is a consequence of Definition 2.2.
5) For any subset , , the closure of .
Definition 2.3 Prime ideals. An ideal of a commutative ring is prime if it has the following properties:
1) if such that , then or ,
2) is not equal to .
Definition 2.4 Maximal ideals. is a maximal ideal of a ring if there are no other ideals contained between and . Every maximal ideal is prime.
Example 2.2 In the ring of integers, , the maximal ideals are the principle ideals generated by a prime number.
Corollary 2.1 If is algebraically closed
1) There is a correspondence .
2) Every ideal not a subset of has a zero in .
Example 2.3 Let be a finite algebraic set. is generated by the function , and so is the inverse of .
Corollary 2.2 An algebraic set is an affine variety iff its ideal is a prime ideal.
Example 2.4 has a dimension of as single points are the only irreducible closed subsets of not equal to . In fact has a dimension of .
Example 2.5 If is an irreducible polynomial in , the affine variety is called a surface if and a hyper-surface of .
Any finite topological space can be decomposed uniquely into finitely many irreducible spaces.
Definition 2.5 A topological space is said to be Noetherian if it satisfies a descending chain condition for closed subsets. For any sequence of closed subsets there is an integer such that . Another way of saying this is that the closed subsets are stationary.
As is a Noetherian ring this implies that any algebraic set is a Noetherian topological space.
Definition 2.6 Inclusion-reversing is defined for two sets and with a mapping . Then is inclusion-reversing iff for every pair of sets such that .
Corollary 2.3 There is a one-to-one inclusion-reversing correspondence between algebraic sets in and radical ideals in , given by and .
To summarize the following correspondences between geometric and algebraic forms have been established:
1) points in affine space maximum ideals in a polynomial ring. (Corollary 2.1)
2) affine varieties prime ideals. (Corollary 2.2)
3) algebraic sets radical ideals. (Corollary 2.3)