2.4 Coordinate rings of affine varieties

Coordinate rings are the mathematical backbone of affine varieties. They're like a secret code that unlocks the mysteries of these geometric objects. By studying these rings, we can uncover important properties and relationships within affine varieties.

Think of coordinate rings as a bridge between algebra and geometry. They allow us to translate geometric problems into algebraic ones, making it easier to analyze and solve complex issues in affine varieties. It's a powerful tool that connects different areas of math.

Coordinate Rings of Affine Varieties

Definition and Properties

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• The coordinate ring of an affine variety V, denoted by A(V), is the ring of regular functions on V
  • Regular functions on an affine variety V are functions that can be expressed as polynomials in the coordinate variables (x₁, ..., xₙ)
  • The coordinate ring A(V) is the quotient ring k[x₁, ..., xₙ]/I(V), where I(V) is the ideal of polynomials vanishing on V
    • The elements of A(V) are equivalence classes of polynomials, where two polynomials are equivalent if their difference lies in I(V)
• The coordinate ring A(V) is an integral domain if and only if V is an irreducible variety
  • An irreducible variety cannot be expressed as the union of two proper subvarieties
• The dimension of the coordinate ring A(V) is equal to the dimension of the affine variety V
  • The dimension of an affine variety is the maximum length of a chain of irreducible subvarieties

Examples

• Consider the affine variety V = V(y - x²) ⊆ A². The coordinate ring A(V) is isomorphic to k[x], the polynomial ring in one variable
• For the affine space Aⁿ, the coordinate ring A(Aⁿ) is isomorphic to the polynomial ring k[x₁, ..., xₙ]

Coordinate Rings vs Polynomial Rings

Relationship

• The coordinate ring A(V) is a quotient ring of the polynomial ring k[x₁, ..., xₙ]
  • The quotient map k[x₁, ..., xₙ] → A(V) sends a polynomial to its equivalence class in the coordinate ring
• The ideal I(V) used to define the coordinate ring consists of all polynomials that vanish on the affine variety V
  • A polynomial f vanishes on V if f(p) = 0 for all points p ∈ V
• The coordinate ring A(V) inherits the grading from the polynomial ring k[x₁, ..., xₙ], making it a graded ring
  • The grading is defined by the degree of polynomials: A(V) = ⨁ₙ₌₀ A(V)ₙ, where A(V)ₙ consists of equivalence classes of polynomials of degree n

Nullstellensatz

• The Nullstellensatz establishes a correspondence between radical ideals in k[x₁, ..., xₙ] and affine varieties in kⁿ
  • Every radical ideal I ⊆ k[x₁, ..., xₙ] is the ideal of an affine variety V(I)
  • Every affine variety V ⊆ kⁿ is the zero set of a radical ideal I(V) ⊆ k[x₁, ..., xₙ]

Computing Coordinate Rings

Determining the Ideal

• To compute the coordinate ring A(V), first determine the ideal I(V) of polynomials vanishing on V
  • Express the affine variety V as the zero set of a collection of polynomials f₁, ..., fₘ in k[x₁, ..., xₙ]
  • The ideal I(V) is the radical of the ideal generated by the polynomials f₁, ..., fₘ
    • The radical of an ideal I, denoted by √I, is the set of all polynomials f such that fⁿ ∈ I for some integer n ≥ 1

Computing the Quotient Ring

• Compute the quotient ring k[x₁, ..., xₙ]/I(V) to obtain the coordinate ring A(V)
  • The elements of the quotient ring are equivalence classes of polynomials, where two polynomials are equivalent if their difference lies in I(V)
• Use Gröbner basis techniques to simplify the computation of the quotient ring and its elements
  • A Gröbner basis is a special generating set of an ideal that allows for efficient computation in the quotient ring
  • Gröbner bases can be computed using algorithms like Buchberger's algorithm or the F4/F5 algorithms

Examples

• For the affine variety V = V(y² - x³ - x) ⊆ A², the ideal I(V) is generated by the polynomial y² - x³ - x
  • The coordinate ring A(V) is isomorphic to k[x, y]/(y² - x³ - x)
• Consider the affine variety V = V(x² + y² - 1) ⊆ A². The ideal I(V) is generated by the polynomial x² + y² - 1
  • The coordinate ring A(V) is isomorphic to k[x, y]/(x² + y² - 1)

Coordinate Rings for Studying Affine Varieties

Correspondence with Points and Subvarieties

• The Hilbert Nullstellensatz relates the maximal ideals of A(V) to the points of the affine variety V
  • Every maximal ideal of A(V) corresponds to a point of V, and every point of V corresponds to a maximal ideal of A(V)
• The prime ideals of A(V) correspond to the irreducible subvarieties of V
  • An ideal I ⊆ A(V) is prime if and only if V(I) is an irreducible subvariety of V
  • The height of a prime ideal in A(V) equals the codimension of the corresponding irreducible subvariety

Regular Functions and Local Properties

• The units in the coordinate ring A(V) are precisely the non-vanishing regular functions on V
  • A regular function f ∈ A(V) is a unit if and only if f(p) ≠ 0 for all points p ∈ V
• The coordinate ring A(V) can be used to study the local properties of V, such as the tangent space and the local ring at a point
  • The tangent space at a point p ∈ V is the dual of the maximal ideal corresponding to p in A(V)
  • The local ring at a point p ∈ V is the localization of A(V) at the maximal ideal corresponding to p

Morphisms and Coordinate Rings

• Morphisms between affine varieties can be studied using homomorphisms between their coordinate rings
  • A morphism φ: V → W between affine varieties induces a homomorphism φ*: A(W) → A(V) between their coordinate rings
  • The properties of the morphism φ (injectivity, surjectivity, isomorphism) are reflected in the properties of the induced homomorphism φ*

Examples

• For the affine variety V = V(y - x²) ⊆ A², the point (a, a²) ∈ V corresponds to the maximal ideal (x - a, y - a²) ⊆ A(V)
• Consider the affine varieties V = V(y² - x³ - x) and W = V(y - x²). The morphism φ: V → W given by (x, y) ↦ (x, y²) induces a homomorphism φ*: A(W) → A(V) that sends x to x and y to y²