6.3 Radical ideals and the Zariski topology

Radical ideals and the Zariski topology are key concepts in algebraic geometry. They provide a powerful framework for understanding the relationship between algebraic structures and geometric objects, connecting polynomial equations to their solution sets.

These ideas are crucial for grasping Hilbert's Nullstellensatz, which establishes a fundamental link between algebra and geometry. By relating radical ideals to algebraic sets, we can translate between algebraic problems and geometric ones, opening up new avenues for analysis and problem-solving.

Radical Ideals in Polynomial Rings

Definition and Characterization

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• A radical ideal is an ideal I in a ring R such that for any element x in R, if some power of x is in I, then x itself is in I
  • For example, in the ring of integers ℤ, the ideal (4) = {0, ±4, ±8, ...} is not a radical ideal because 2² = 4 ∈ (4), but 2 ∉ (4)
  • On the other hand, the ideal (2) = {0, ±2, ±4, ...} is a radical ideal in ℤ
• In a polynomial ring k[X₁, ..., Xₙ] over a field k, an ideal I is radical if and only if I is the intersection of prime ideals
  • For instance, in the ring ℚ[x], the ideal (x² - 1) is radical because it is the intersection of the prime ideals (x - 1) and (x + 1)

Radical and Nilradical

• The radical of an ideal I, denoted by √I or rad(I), is the smallest radical ideal containing I
  • It consists of all elements x in the ring such that some power of x is in I
  • For example, in the ring ℤ, the radical of the ideal (4) is (2), i.e., √(4) = (2)
• The radical of an ideal can be computed using the formula √I = {x ∈ R | xⁿ ∈ I for some n ∈ ℕ}
• The nilradical of a ring R, denoted by nil(R), is the intersection of all prime ideals in R
  • It is the set of all nilpotent elements in R (elements x such that xⁿ = 0 for some n ∈ ℕ)
  • In the ring ℤ/12ℤ, the nilradical is {0, 4, 8} because these are the elements that are annihilated by some power of themselves
• In a Noetherian ring, an ideal is radical if and only if it is the intersection of the prime ideals that contain it

Zariski Topology and Radical Ideals

Zariski Topology on Algebraic Varieties

• The Zariski topology on an affine variety V ⊆ Aⁿ over an algebraically closed field k is defined by taking the closed sets to be the algebraic subsets of V
  • Algebraic subsets are sets of the form V(I) = {x ∈ V | f(x) = 0 for all f ∈ I} for some ideal I ⊆ k[X₁, ..., Xₙ]
  • For example, in A², the algebraic subset V(xy - 1) is the hyperbola xy = 1
• The Zariski topology on a projective variety is defined similarly, with closed sets being the projective algebraic subsets
• The Zariski topology is a T₁ topology, meaning that every singleton set is closed
  • However, it is not Hausdorff, as distinct points may not have disjoint neighborhoods
  • For instance, in A¹, the points 0 and 1 do not have disjoint neighborhoods in the Zariski topology

Closure and Irreducibility

• In the Zariski topology, the closure of a set S is the smallest closed set containing S, given by V(I(S)), where I(S) is the ideal of all polynomials vanishing on S
  • For example, in A², the closure of the set {(1, 1), (2, 2)} is the line V(x - y)
• Irreducible closed sets in the Zariski topology correspond to prime ideals in the coordinate ring
  • A closed set V(I) is irreducible if and only if I is a prime ideal
  • Maximal irreducible closed sets correspond to maximal ideals
  • In A², the line V(x - y) is irreducible because the ideal (x - y) is prime

Applications of Radical Ideals and the Zariski Topology

Computations and Problem Solving

• Determine whether a given ideal in a polynomial ring is radical by checking if it is the intersection of prime ideals or using the radical ideal formula
  • For example, in ℚ[x, y], the ideal (x² - y³) is radical because it is prime, while (x² - y²) = (x - y)(x + y) is not radical
• Compute the radical of an ideal using the formula √I = {x ∈ R | xⁿ ∈ I for some n ∈ ℕ}
  • In ℚ[x], the radical of the ideal (x³ - x) is (x² - 1) because √(x³ - x) = {f ∈ ℚ[x] | f^n ∈ (x³ - x) for some n ∈ ℕ} = (x² - 1)
• Find the closure of a set in the Zariski topology by computing the ideal of polynomials vanishing on the set and then finding the corresponding algebraic subset
  • For the set {(1, 1), (-1, 1)} in A², I({(1, 1), (-1, 1)}) = (y - 1), so the closure is V(y - 1), the line y = 1

Geometry of Algebraic Varieties

• Determine the irreducible components of an algebraic set by finding the minimal prime ideals containing the ideal of the algebraic set
  • In A², the algebraic set V(xy) has two irreducible components: V(x) (the y-axis) and V(y) (the x-axis), corresponding to the minimal prime ideals (x) and (y)
• Use the correspondence between irreducible closed sets and prime ideals to study the geometry of algebraic varieties
  • For instance, the dimension of an irreducible algebraic set is equal to the Krull dimension of its coordinate ring, which is the supremum of the lengths of chains of prime ideals
• Apply the concepts of dimension, tangent spaces, and singularities in the context of the Zariski topology and radical ideals
  • The dimension of an algebraic set V(I) is the Krull dimension of the quotient ring k[X₁, ..., Xₙ]/I
  • The tangent space at a point p of an algebraic variety V is the kernel of the Jacobian matrix of the defining equations of V evaluated at p
  • A point p on an algebraic variety V is singular if the dimension of the tangent space at p is greater than the dimension of V

Hilbert's Nullstellensatz vs Radical Ideals and the Zariski Topology

Hilbert's Nullstellensatz

• Hilbert's Nullstellensatz states that if k is an algebraically closed field and I ⊆ k[X₁, ..., Xₙ] is an ideal, then I(V(I)) = √I, where I(V(I)) is the ideal of all polynomials vanishing on the algebraic set V(I)
  • In other words, for any f ∈ k[X₁, ..., Xₙ], f vanishes on V(I) if and only if some power of f is in I
  • For example, in ℂ[x, y], if I = (x² + y² - 1), then V(I) is the unit circle, and I(V(I)) = (x² + y² - 1) = √I
• The Nullstellensatz establishes a correspondence between radical ideals and algebraic subsets of affine space, which is the foundation of the Zariski topology
  • Every algebraic subset V(I) is uniquely determined by its radical ideal I(V(I)) = √I
  • Conversely, every radical ideal I is the ideal of an algebraic subset, namely V(I)

Implications and Applications

• The Nullstellensatz implies that every maximal ideal in k[X₁, ..., Xₙ] is of the form (X₁ - a₁, ..., Xₙ - aₙ) for some (a₁, ..., aₙ) ∈ kⁿ
  • This follows from the fact that maximal ideals correspond to points in the algebraic closure of k
  • Moreover, every prime ideal is the intersection of maximal ideals containing it
• The Nullstellensatz can be used to prove that the Zariski topology on an affine variety is T₁ and that irreducible closed sets correspond to prime ideals
  • If p and q are distinct points in kⁿ, then the maximal ideals (X₁ - p₁, ..., Xₙ - pₙ) and (X₁ - q₁, ..., Xₙ - qₙ) are distinct, so {p} and {q} are closed in the Zariski topology
  • If V(I) is an irreducible closed set, then I(V(I)) = √I is a prime ideal, and conversely, if I is a prime ideal, then V(I) is irreducible
• The Nullstellensatz provides a bridge between the algebraic structure of polynomial rings and the geometric structure of algebraic varieties
  • It enables the use of algebraic techniques to study geometric problems and vice versa
  • For instance, the dimension of an algebraic variety can be computed as the Krull dimension of its coordinate ring, and the singular points of a variety can be found by studying the localization of its coordinate ring at maximal ideals