## 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)