Affine and Quasi-affine Varieties – 2

Restatement: An affine algebraic variety, the set of common zeros of a collection of polynomials, is an irreducible closed subset of {A_k}^n (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, \tilde {a}, is defined as

\sqrt{\tilde {a}}:=\lbrace f\in A, such \; that \; f^r\in {\tilde {a}}\; for \; some \; r>0 \rbrace.

An ideal that is equal to its radical is called a radical ideal, i.e. if I=\sqrt I.

Example 2.1 The radical of the ideal 4\mathbb{Z}, integer mutilples of 4, is 2\mathbb{Z}. The radical ideal of 5\mathbb{Z} is 5\mathbb{Z}.

Definition 2.2 Hilbert’s Nullstellensatz. If k is an algebraically closed field with \tilde {a} an ideal in A=k[x_1,\dots ,x_n] and f\in A a polynomial that vanishes at all points of Z(\tilde {a}), then f^r\in \tilde {a} for some integer r>0.

Proposition 2.1 Properties of the operations Z(*) and I(*).

1) If T_1\subseteq T_2 are subsets of A, then Z(T_1) \supseteq Z(T_2).

2) If Y_1\subseteq Y_2 are subsets of {A_k}^n, then I(Y_1) \supseteq I(Y_2).

3) For any two subsets Y_1, Y_2 of {A_k}^n, we have I(Y_1\cup Y_2)=I(Y_1)\cap I(Y_2).

4) For any ideal \tilde {a}\subseteq A, I(Z(\tilde {a}))=\sqrt{\tilde {a}}. This is a consequence of Definition 2.2.

5) For any subset Y\subseteq {A_k}^n, Z(I(Y))=\bar{Y}, the closure of Y.

Definition 2.3 Prime ideals. An ideal I of a commutative ring R is prime if it has the following properties:

1) if a,b\in R such that ab\in I, then a\in I or b\in I,

2) I is not equal to R.

Definition 2.4 Maximal ideals. I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Every maximal ideal is prime.

Example 2.2 In the ring of integers, \mathbb{Z}, the maximal ideals are the principle ideals generated by a prime number.

Corollary 2.1 If k is algebraically closed

1) There is a 1:1 correspondence \{points\; in \; {A_k}^n\}\Leftrightarrow \{ maximal \; ideals \; of \; k[x_1, \dots, x_n]\}.

2) Every ideal I not a subset of k[x_1,\dots ,x_n] has a zero in {A_k}^n.

Example 2.3 Let X=\{a_1, \dots , a_n \} \subset A_k^1 be a finite algebraic set. I(X) is generated by the function (x-a_1)\dots (x-a_r), and Z(I(X))=X so Z(*) is the inverse of I(*).

Corollary 2.2 An algebraic set X\subset {A_k}^n is an affine variety iff its ideal I(X)=k[x_1, \dots, x_n] is a prime ideal.

Example 2.4 {A_k}^1 has a dimension of 1 as single points are the only irreducible closed subsets of {A_k}^1 not equal to {A_k}^1. In fact {A_k}^n has a dimension of n.

Example 2.5 If f is an irreducible polynomial in A=k[x_1, \dots , x_n], the affine variety Y=Z(f) is called a surface if n=3 and a hyper-surface of n>3.

Any finite topological space can be decomposed uniquely into finitely many irreducible spaces.

Definition 2.5 A topological space X is said to be Noetherian if it satisfies a descending chain condition for closed subsets. For any sequence Y_1\supseteq Y_2\supseteq \dots of closed subsets there is an integer r such that Y_r=Y_{r+1}=\dots. Another way of saying this is that the closed subsets are stationary.

As k[x_1,\dots ,x_n] 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 A and B with a mapping \phi: A \rightarrow S. Then \phi is inclusion-reversing iff for every pair of sets a_1, a_2 \in A such that a_1 \subseteq a_2 : \phi(a_2) \subseteq \phi (a_1).

Corollary 2.3 There is a one-to-one inclusion-reversing correspondence between algebraic sets in {A_k}^n and radical ideals in A, given by Y \mapsto I(Y) and \tilde {a} \mapsto Z(\tilde {a}).

To summarize the following correspondences between geometric and algebraic forms have been established:

1) points in affine space \Leftrightarrow maximum ideals in a polynomial ring. (Corollary 2.1)

2) affine varieties \Leftrightarrow prime ideals. (Corollary 2.2)

3) algebraic sets \Leftrightarrow radical ideals. (Corollary 2.3)

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