Note 1. Varieties (1)

Note 1. Varieties (1)

Part 1: Definitions

Definition 1.1. (Variety)

1. Let $k$ be an algebraically closed field, then an affine $n-$space is $\mathbb{A}_k^n = k^n$.
2. For an ideal $I \triangleleft R = k[x_i \mid 1 \leqslant i \leqslant n]$, an affine algebraic set is defined by: $$V(I) = \{a \in \mathbb{A}_k^n \mid f(a) = 0 \text{ for all } f \in I\}$$

Proposition 1.2.

• Affine algebraic set satisfy the following:

1. $V(R) = \varnothing$, $V((0)) = \mathbb{A}_k^n$.
2. $V(I) \cup V(J) = V(IJ)$
3. For aribitrary index set $A$, we have $\displaystyle \bigcap_{\alpha \in A}V(I_\alpha) = V \left(\sum_{\alpha \in A}I_{\alpha}\right)$
4. These fact shows that by defining $\{V(I) \mid I \triangleleft R\}$ be exactly the closed set of $\mathbb{A}_k^n$, we may induce a topological space, namely, Zariski topology.

Definition 1.2.

1. A toplogical space $X$ is said to be irreducible if for any proper pair of closed set $C_1$, $C_2$, $X \neq C_1 \cup C_2$. ($X$ is not a union of proper closed set)
2. An affine algebraic set $V(I)$ is an affine variety if it is irreducible in Zariski topology.
3. The \textbf{ideal of $S \subset \mathbb{A}_k^n$} is defined by $$I(S) = \{f \in R \mid f(x) = 0 \text{ for all }x \in S\}$$

Proposition 1.3.

1. Suppose $I_1 \subset I_2 \subset R$, then $V(I_1) \supset V(I_2)$.
2. Suppose $S_1 \subset S_2 \subset \mathbb{A}_k^n$, then $I(S_1) \supset I(S_2)$.
3. Suppose $S_1, S_2 \subset \mathbb{A}_k^n$, then $I(S_1 \cup S_2) = I(S_1) \cap I(S_2)$.
4. Suppose $S \subset \mathbb{A}_k^n$, then $V(I(S)) = \overline{S}$.

Proof. 

For 4., obviously $S \subset V(I(S))$, and hence $\overline{S} \subset \overline{V(I(S))} = V(I(S))$. Now suppose $C = V(J)$ is any closed set containing $S$, which is $S \subset V(J)$ and giving $I(S) \supset I(V(J))$. We thus have $J \subset I(S)$ and $V(J) \supset V(I(S))$, so $V(I(S)) = \overline{S}$.

$\square$