Affine Algebraic Geometry
The following are notes on affine algebraic geometry from a graduate seminar I gave in 2024, covering the Zariski topology, affine varieties, and morphisms.
General Information: $S$ is a set of polynomials, $V(S)$ is the set of all points in a space such that all the polynomials in $S$ evaluate to $0$ at those points.
\[ \begin{align*} S &\subseteq \mathbb{C}[x_1,\dots, x_n] \\ V(S) &:= \{z \in \mathbb{C}^n : f(z) = 0 \quad \forall f \in S\} \end{align*} \]Example:
\[ \begin{align*} S &= \{x_1, x_2\} \subseteq \mathbb{C}[x_1,x_2] \quad : \quad x_1(z_1,z_2) = z_1 \quad x_2(z_1,z_2) = z_2 \\ V(S) &= \{z \in \mathbb{C}^2 \quad :\quad x_1(z) = 0 \text{ and } x_2(z) = 0\} \\ &= \{0,0\} \end{align*} \]Example 2:
\[ \begin{align*} S &= \{x_1, x_2\} \subseteq \mathbb{C}[x_1,x_2] \quad : \quad x_1(z_1,z_2) = z_1 \quad x_2(z_1,z_2) = z_2 \\ V(S) &= \{z \in \mathbb{C}^2 \quad :\quad x_1(z)x_2(z) = 0\} \\ &= \text{the coordinate axes} \end{align*} \]If all of the above is the same but instead of $x_1()$ and $x_2()$ we have just $(x_1x_2)(z) = 0 \Rightarrow z_1 z_2 = 0$
The following collection of closed sets defines the Zariski topology on $\mathbb{C}^n$:
\[ \begin{align*} \{V(S) : S \subseteq \mathbb{C}[x_1, \dots , x_n]\} \end{align*} \]Where if $S \subseteq \mathbb{C}[x_1, \dots , x_n]$ then $V(S)$ inherits the Zariski subspace topology.
Nice functions: (on $V(S)$)
\[ \begin{align*} \mathbb{C}[V(S)] &= \{f\big|_{V(S)} : f \in \mathbb{C}[x_1, \dots , x_n]\} \\ &\underset{\text{subalg}}{\subseteq} C^0(V(S), \mathbb{C}) \end{align*} \]Where $V(S)$ is equipped with the Zariski subspace topology and $\mathbb{C}$ is equipped with the Zariski topology.
Affine Varieties: Intuition: An affine algebraic variety is a pair $(V(S), \mathbb{C}[V(S)])$ (the vanishing locus of a set of function and the nice function on the vanishing locus) up to isomorphism.
Definition: An affine algebraic variety is a pair $(X, \mathbb{C}[X])$ of a topological space $X$ and a subalgebra $\mathbb{C}[X] \subseteq C^0 (X, \mathbb{C})$ such that $\exists n \in \mathbb{Z}_{\geq 0}$, a subset $S \subseteq \mathbb{C}[x_1, \dots x_n]$ and a homomorphism $f: X \mapsto V(S)$ such that $f^*(\mathbb{C}[V(S)]) = \mathbb{C}[X]$
Examples of Affine Varieties:
- $S \subseteq \mathbb{C}[x_1, \dots, x_n] \Rightarrow (V(S), \mathbb{C}[V(S)])$ is an algebraic variety. Trivially, if $S={0}$, then $V(S) = \mathbb{C}^n$, $\mathbb{C}[V(S)] = \mathbb{C}[x_1, \dots, x_n]$ \\ $\Rightarrow (\mathbb{C}^n, \mathbb{C}[x_1,\dots, x_n])$ is an affine algebraic variety.
- Let $V$ be a f.d. v.s. with $n = \dim V$ and $\text{Sym}(V^*)$ the symmetric algebra of $V^*$ ... There is more, but I don't feel confident enough in all the concepts in this example to properly explain it.
Morphisms of Affine Varieties:
Examples: Suppose that $\varphi : \mathbb{C}^m \to \mathbb{C}^n$ is a map with $\varphi = (\varphi_1,\dots, \varphi_n)$ where $\varphi_i : \mathbb{C}^m \to \mathbb{C}$. For example:
\[ \begin{align*} \varphi : \mathbb{C}^3 &\to \mathbb{C}^2 \\ \varphi(z_1,z_2,z_3) &= (z_1^3z_2^2z_3^5 - 18z_1z_2^4+\pi, -8z_1z_2^3) = (\varphi_1,\varphi_2) \end{align*} \]is a morphism.
\[ \begin{align*} \varphi : \mathbb{C}^3 &\to \mathbb{C}^2 \\ \varphi(z_1,z_2,z_3) &= (e^z_1\log(z_2)z_3, |z_1|) \end{align*} \]is not a morphism.
Equivalence with the Reduced, Finitely Generated Algebras: Define the category AVar as follows: $\ob_{\avar}$ are affine algebraic varieties, $\mor_{\avar}$ are morphisms between algebraic varieties.
Lemma: let $X,Y,Z$ be affine algebraic varieties, $f : X \to Y$, $g: Y \to Z$ morphisms. Then $g \circ f : X \to Z$ is a morphism of affine algebraic varieties.
Recall: Let $X,Y$ be topological spaces with $f: X \mapsto Y$ a continuous map. Consider the algebras: $C^0(X, \mathbb{C})$ and $C^0(Y,\mathbb{C})$ where $\mathbb{C}$ has the Zariski topology. We have an algebra homomorphism:
\[ \begin{align*} f^*: C^0(Y,\mathbb{C}) &\mapsto C^0(X, \mathbb{C}) \\ \varphi &\mapsto \varphi \circ f \end{align*} \]Now define another category $\alg$ as follows: $\ob_{\alg}$ are reduced, finitely generated, commutative algebras (over $\mathbb{C}$). That is, we have an algebra $A$ over $\mathbb{C}$ such that:
- if $a \in A$ and $a^n = 0$ for some $n \in \mathbb{N}$, then $a=0$ (Reduced).
- $ab = ba \quad \forall a,b \in A$ (Commutative).
- $\exists n \in \mathbb{Z}_{\geq 0}$ and a surjective algebra morphism \[ \begin{align*} \mathbb{C}[x_1,\dots,x_n] \overset{\varphi}{\to} A \end{align*} \] \[ \begin{align*} \forall a \in A \Rightarrow \exists f \in \mathbb{C}[x_1,\dots,x_n] \text{ s.t. } \varphi(f) = a \end{align*} \] And, \[ \begin{align*} f = \sum a_{i_1,\dots, i_n}x_1^{i_1}\dots x_n^{i_n} \Rightarrow a = \varphi(f) = \sum a_{i_1,\dots, i_n}\varphi(x_1^{i_1})\dots \varphi(x_n^{i_n}) \end{align*} \] (Finitely Generated)
$\mor_{\alg}$ = algebra morphisms.
Lemma: let $A,B,C$ be algebras, with $f: A \to B$ and $g: B \to C$ algebra morphisms. Then $g \circ f: A \to C$ is an algebra morphism.
Define a functor $\mathcal{F}: \avar \to \alg^{\opp}$ where the opposite (op) category has the same objects, but with morphisms in the opposite direction. And:
- Objects: \[ \begin{align*} \mathcal{F}(X) = \mathbb{C}[X] \text{ where } X \text{ is an affine variety} \end{align*} \]
- Morphisms: Let $X,Y$ be affine algebraic varieties with $f \in \homm_{\avar}(X,Y)$. Then $\mathcal{F}(f) \in \homm_{\alg^{\opp}}(\mathbb{C}[X],\mathbb{C}[Y]) = \homm_{\alg}(\mathbb{C}[Y],\mathbb{C}[X])$ with: \[ \begin{align*} f^* : \mathbb{C}[Y] &\to \mathbb{C}[X] \\ \varphi &\mapsto \varphi \circ f \end{align*} \]