4.2 Prime and maximal ideals in coordinate rings

Prime and maximal ideals in coordinate rings are crucial for understanding the structure of affine varieties. They bridge the gap between algebra and geometry, allowing us to translate geometric properties into algebraic language and vice versa.

These ideals help us identify important subsets of varieties, like irreducible components and individual points. By studying them, we gain insights into the dimension, irreducibility, and local properties of varieties, which are essential for deeper algebraic geometry concepts.

Prime and maximal ideals in coordinate rings

Defining prime and maximal ideals

• A proper ideal $P$ in a ring $R$ is prime if for any $a,b \in R$ such that $ab \in P$, either $a \in P$ or $b \in P$
  • Example: In the ring $\mathbb{Z}$, the ideal $(2)$ is prime since if $ab \in (2)$, then either $a$ or $b$ must be even
• A proper ideal $M$ in a ring $R$ is maximal if there is no proper ideal $I$ such that $M \subsetneq I \subsetneq R$
  • Example: In the ring $\mathbb{Z}$, the ideal $(5)$ is maximal since there is no proper ideal strictly between $(5)$ and $\mathbb{Z}$

Correspondence with subvarieties and points

• In the coordinate ring $k[V]$ of an affine variety $V$, prime ideals correspond to irreducible subvarieties of $V$
  • Example: In the coordinate ring $\mathbb{C}[x,y]/(y^2-x^3-x)$ of the cubic curve $V(y^2-x^3-x)$, the prime ideal $(x,y)$ corresponds to the irreducible subvariety consisting of the origin
• Maximal ideals in $k[V]$ correspond to points of the affine variety $V$
  • Example: In the coordinate ring $\mathbb{R}[x,y]/(x^2+y^2-1)$ of the unit circle, the maximal ideal $(x-1,y)$ corresponds to the point $(1,0)$ on the circle
• The maximal ideals in the coordinate ring $k[x_1, \ldots, x_n]$ are of the form $(x_1 - a_1, \ldots, x_n - a_n)$ for some $a_1, \ldots, a_n \in k$
  • Example: In $\mathbb{C}[x,y]$, the maximal ideal $(x-2,y+3)$ corresponds to the point $(2,-3)$ in $\mathbb{C}^2$

Spectrum of a coordinate ring

Definition and notation

• The spectrum $\text{Spec}(R)$ of a ring $R$ is the set of all prime ideals of $R$
• The spectrum of a coordinate ring $k[V]$ is denoted by $\text{Spec}(k[V])$ and consists of all prime ideals in $k[V]$
  • Example: For the coordinate ring $\mathbb{R}[x,y]/(x^2+y^2-1)$ of the unit circle, $\text{Spec}(\mathbb{R}[x,y]/(x^2+y^2-1))$ contains prime ideals like $(0)$ and $(x-1,y)$

Zariski topology

• The spectrum $\text{Spec}(k[V])$ can be given the Zariski topology, where closed sets are defined by ideals in $k[V]$
  • The Zariski topology on $\text{Spec}(k[V])$ is defined by taking the closed sets to be $V(I) = \{P \in \text{Spec}(k[V]) : I \subseteq P\}$ for ideals $I$ in $k[V]$
  • The Zariski topology makes $\text{Spec}(k[V])$ into a topological space
• There is a one-to-one correspondence between irreducible closed subsets of $\text{Spec}(k[V])$ and prime ideals in $k[V]$
  • Example: In $\text{Spec}(\mathbb{C}[x,y])$, the irreducible closed subset $V(x-1,y)$ corresponds to the prime ideal $(x-1,y)$
• The maximal ideals in $\text{Spec}(k[V])$ are closed points in the Zariski topology
  • Example: In $\text{Spec}(\mathbb{R}[x])$, the maximal ideal $(x-2)$ is a closed point

Ideals and subvarieties

Correspondence between ideals and subvarieties

• Every ideal $I$ in the coordinate ring $k[V]$ defines a subvariety $V(I) = \{x \in V : f(x) = 0 \text{ for all } f \in I\}$ of the affine variety $V$
  • Example: In $\mathbb{C}[x,y]$, the ideal $I=(x^2+y^2-1)$ defines the unit circle $V(I)=\{(x,y) \in \mathbb{C}^2 : x^2+y^2=1\}$
• Conversely, every subvariety $W$ of $V$ defines an ideal $I(W) = \{f \in k[V] : f(x) = 0 \text{ for all } x \in W\}$ in $k[V]$
  • Example: The parabola $W=\{(x,y) \in \mathbb{R}^2 : y=x^2\}$ defines the ideal $I(W)=(y-x^2)$ in $\mathbb{R}[x,y]$
• The correspondence between ideals and subvarieties reverses inclusions: if $I \subseteq J$ are ideals in $k[V]$, then $V(J) \subseteq V(I)$
  • Example: In $\mathbb{C}[x,y]$, $(x) \subseteq (x,y)$, so $V(x,y)=\{(0,0)\} \subseteq V(x)=\{(0,y) : y \in \mathbb{C}\}$

Irreducibility and dimension

• The subvariety $V(I)$ is irreducible if and only if the ideal $I$ is prime
  • Example: The ideal $(x^2-y^2)=(x-y)(x+y)$ in $\mathbb{C}[x,y]$ is not prime, so $V(x^2-y^2)$ is reducible
• The dimension of a subvariety $V(P)$ defined by a prime ideal $P$ is equal to the Krull dimension of the coordinate ring $k[V]/P$
  • Example: The dimension of the parabola $V(y-x^2)$ in $\mathbb{C}^2$ is equal to the Krull dimension of $\mathbb{C}[x,y]/(y-x^2)$, which is $1$

Properties of prime and maximal ideals

Nullstellensatz and irreducible components

• The Nullstellensatz establishes a correspondence between maximal ideals in $k[V]$ and points of the affine variety $V$
  • The Weak Nullstellensatz states that if $k$ is algebraically closed, then every maximal ideal in $k[x_1, \ldots, x_n]$ is of the form $(x_1 - a_1, \ldots, x_n - a_n)$ for some $a_1, \ldots, a_n \in k$
  • The Strong Nullstellensatz states that if $k$ is algebraically closed and $I$ is an ideal in $k[x_1, \ldots, x_n]$, then $I(V(I)) = \sqrt{I}$
• The prime ideals in $k[V]$ determine the irreducible components of the affine variety $V$
  • Example: The irreducible components of $V(xy)$ in $\mathbb{C}^2$ are $V(x)$ and $V(y)$, corresponding to the prime ideals $(x)$ and $(y)$ in $\mathbb{C}[x,y]$

Height, dimension, and localization

• The height of a prime ideal $P$ in $k[V]$ is equal to the codimension of the subvariety $V(P)$ in $V$
  • Example: In $\mathbb{C}[x,y,z]$, the prime ideal $(x,y)$ has height $2$ since $V(x,y)$ has codimension $2$ in $\mathbb{C}^3$
• The Krull dimension of the coordinate ring $k[V]$ is equal to the dimension of the affine variety $V$
  • Example: The Krull dimension of $\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ is $2$, equal to the dimension of the unit sphere in $\mathbb{R}^3$
• The localization of $k[V]$ at a prime ideal $P$ corresponds to the local ring of the subvariety $V(P)$ at the generic point
  • Example: The localization of $\mathbb{C}[x,y]$ at the prime ideal $(x-1,y)$ is isomorphic to the local ring of the variety $V(x-1,y)$ at the point $(1,0)$