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 and individual points. By studying them, we gain insights into the , irreducibility, and of varieties, which are essential for deeper algebraic geometry concepts.
Prime and maximal ideals in coordinate rings
Defining prime and maximal ideals
Top images from around the web for Defining prime and maximal ideals
ag.algebraic geometry - Maximum area of intersection between annulus and circle? - MathOverflow View original
Is this image relevant?
ring theory - Complement of maximal multiplicative set is a prime ideal - Mathematics Stack Exchange View original
Is this image relevant?
algebraic geometry - determine the position of axis in 3d space - Mathematics Stack Exchange View original
Is this image relevant?
ag.algebraic geometry - Maximum area of intersection between annulus and circle? - MathOverflow View original
Is this image relevant?
ring theory - Complement of maximal multiplicative set is a prime ideal - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Defining prime and maximal ideals
ag.algebraic geometry - Maximum area of intersection between annulus and circle? - MathOverflow View original
Is this image relevant?
ring theory - Complement of maximal multiplicative set is a prime ideal - Mathematics Stack Exchange View original
Is this image relevant?
algebraic geometry - determine the position of axis in 3d space - Mathematics Stack Exchange View original
Is this image relevant?
ag.algebraic geometry - Maximum area of intersection between annulus and circle? - MathOverflow View original
Is this image relevant?
ring theory - Complement of maximal multiplicative set is a prime ideal - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
A proper ideal P in a ring R is prime if for any a,b∈R such that ab∈P, either a∈P or b∈P
Example: In the ring Z, the ideal (2) is prime since if ab∈(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⊊I⊊R
Example: In the ring Z, the ideal (5) is maximal since there is no proper ideal strictly between (5) and Z
Correspondence with subvarieties and points
In the k[V] of an V, correspond to of V
Example: In the coordinate ring C[x,y]/(y2−x3−x) of the cubic curve V(y2−x3−x), the (x,y) corresponds to the irreducible consisting of the origin
Maximal ideals in k[V] correspond to points of the affine variety V
Example: In the coordinate ring R[x,y]/(x2+y2−1) of the unit circle, the (x−1,y) corresponds to the point (1,0) on the circle
The maximal ideals in the coordinate ring k[x1,…,xn] are of the form (x1−a1,…,xn−an) for some a1,…,an∈k
Example: In C[x,y], the maximal ideal (x−2,y+3) corresponds to the point (2,−3) in C2
Spectrum of a coordinate ring
Definition and notation
The [Spec](https://www.fiveableKeyTerm: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 Spec(k[V]) and consists of all prime ideals in k[V]
Example: For the coordinate ring R[x,y]/(x2+y2−1) of the unit circle, Spec(R[x,y]/(x2+y2−1)) contains prime ideals like (0) and (x−1,y)
Zariski topology
The spectrum Spec(k[V]) can be given the , where are defined by ideals in k[V]
The Zariski topology on Spec(k[V]) is defined by taking the closed sets to be V(I)={P∈Spec(k[V]):I⊆P} for ideals I in k[V]
The Zariski topology makes Spec(k[V]) into a topological space
There is a one-to-one correspondence between irreducible closed subsets of Spec(k[V]) and prime ideals in k[V]
Example: In Spec(C[x,y]), the irreducible closed subset V(x−1,y) corresponds to the prime ideal (x−1,y)
The maximal ideals in Spec(k[V]) are closed points in the Zariski topology
Example: In Spec(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∈V:f(x)=0 for all f∈I} of the affine variety V
Example: In C[x,y], the ideal I=(x2+y2−1) defines the unit circle V(I)={(x,y)∈C2:x2+y2=1}
Conversely, every subvariety W of V defines an ideal I(W)={f∈k[V]:f(x)=0 for all x∈W} in k[V]
Example: The parabola W={(x,y)∈R2:y=x2} defines the ideal I(W)=(y−x2) in R[x,y]
The correspondence between ideals and subvarieties reverses inclusions: if I⊆J are ideals in k[V], then V(J)⊆V(I)
Example: In C[x,y], (x)⊆(x,y), so V(x,y)={(0,0)}⊆V(x)={(0,y):y∈C}
Irreducibility and dimension
The subvariety V(I) is irreducible if and only if the ideal I is prime
Example: The ideal (x2−y2)=(x−y)(x+y) in C[x,y] is not prime, so V(x2−y2) 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−x2) in C2 is equal to the Krull dimension of C[x,y]/(y−x2), 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 states that if k is algebraically closed, then every maximal ideal in k[x1,…,xn] is of the form (x1−a1,…,xn−an) for some a1,…,an∈k
The states that if k is algebraically closed and I is an ideal in k[x1,…,xn], then I(V(I))=I
The prime ideals in k[V] determine the irreducible components of the affine variety V
Example: The irreducible components of V(xy) in C2 are V(x) and V(y), corresponding to the prime ideals (x) and (y) in 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 C[x,y,z], the prime ideal (x,y) has height 2 since V(x,y) has codimension 2 in C3
The Krull dimension of the coordinate ring k[V] is equal to the dimension of the affine variety V
Example: The Krull dimension of R[x,y,z]/(x2+y2+z2−1) is 2, equal to the dimension of the unit sphere in R3
The localization of k[V] at a prime ideal P corresponds to the of the subvariety V(P) at the generic point
Example: The localization of 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)