2.4 Coordinate rings of affine varieties
5 min read•july 30, 2024
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 of an V, denoted by A(V), is the 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 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 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
- 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 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 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 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
Key Terms to Review (15)
Affine Variety: An affine variety is a subset of affine space defined as the common zero set of a collection of polynomials. These varieties are fundamental objects in algebraic geometry, connecting geometric concepts with algebraic expressions through their coordinate rings and properties.
Coordinate Ring: The coordinate ring of an affine variety is a way to represent the algebraic structure of the variety through polynomials. Specifically, it consists of all polynomial functions defined on the affine space corresponding to the variety, allowing for a bridge between geometric objects and algebraic expressions. This concept is crucial in understanding properties such as dimension, ideals, and relationships between different types of varieties.
Embedding of Varieties: An embedding of varieties is a way of placing one algebraic variety into another in such a manner that the first variety retains its structure within the second. This is done by associating points in the first variety to points in the second variety via a morphism that is both injective and respects the algebraic structure, often allowing for a clearer understanding of geometric properties and relationships. This concept is crucial when examining how different varieties relate to one another and interact through their coordinate rings.
Graded Ring: A graded ring is a ring that can be decomposed into a direct sum of abelian groups, where each group corresponds to a non-negative integer index called the grade. This structure allows for the organization of elements by their degrees, which is essential in algebraic geometry for understanding polynomial functions and their behavior under various operations.
Hilbert's Nullstellensatz: Hilbert's Nullstellensatz is a fundamental theorem in algebraic geometry that establishes a connection between ideals in polynomial rings and the geometric properties of algebraic varieties. It essentially states that there is a correspondence between the radical of an ideal and the points of the affine variety it defines, linking algebraic expressions to their geometric counterparts.
Homomorphism of Rings: A homomorphism of rings is a function between two rings that preserves the ring operations, specifically addition and multiplication. This means that if you have two rings, say A and B, a homomorphism will map elements from A to B in such a way that the sum and product of any two elements in A maps to the sum and product of their images in B, respectively. This concept is crucial for understanding how different algebraic structures can relate to each other, especially when dealing with coordinate rings of affine varieties.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors and a multiplicative identity. This means that if the product of two non-zero elements is zero, then at least one of those elements must be zero. Integral domains are important because they allow for cancellation in equations, which is crucial for solving polynomial equations and understanding algebraic structures.
Isomorphism of Varieties: An isomorphism of varieties is a bijective morphism between two algebraic varieties that has a morphism in both directions, making them essentially the same in structure and properties. This concept highlights the idea that two varieties can be considered the same if they can be transformed into one another via continuous and smooth mappings, retaining the algebraic structure defined by their coordinate rings. Isomorphic varieties share the same geometric and algebraic characteristics, such as dimension and singularities, making them interchangeable in many contexts.
Localization of a Ring: Localization of a ring is the process of creating a new ring from an existing ring by inverting a subset of its elements, typically aimed at focusing on properties that are true locally at specific elements. This technique allows mathematicians to study properties of the ring more conveniently by 'zooming in' on particular elements or behaviors, which is particularly useful in the context of coordinate rings associated with affine varieties.
Morphism: A morphism is a structure-preserving map between two mathematical objects, typically within the context of algebraic geometry where it connects varieties. This concept allows for the exploration of relationships between different varieties, examining how their structures relate and translate through functions and coordinates.
Prime ideal: A prime ideal is a special kind of ideal in a ring that has the property that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal. This concept is crucial because it helps in understanding the structure of polynomial rings and their ideals, as well as how these ideals relate to varieties in algebraic geometry. Prime ideals serve as building blocks for the definition of irreducibility in varieties and play a significant role in determining the properties of coordinate rings.
Quotient Ring: A quotient ring is a mathematical structure formed by partitioning a ring into equivalence classes using an ideal, effectively allowing us to simplify problems in ring theory. This concept connects the properties of polynomials, ideals, and coordinate rings, enabling the study of algebraic structures in a more manageable way. By treating elements in a ring as equivalent if they differ by an element of the ideal, we can focus on the essential features of the algebraic system without getting bogged down by extraneous details.
Radical Ideal: A radical ideal is an ideal in a ring such that if a power of an element is in the ideal, then the element itself is also in the ideal. This concept connects deeply with the structure of coordinate rings, where radical ideals help describe the properties of affine varieties and their points. Radical ideals play a crucial role in the Zariski topology, as they relate to the closure of sets and help understand the relationship between algebraic sets and their corresponding coordinate rings.
Ring of regular functions: A ring of regular functions consists of the set of polynomial functions defined on an affine variety, which behaves nicely under addition and multiplication. This concept connects algebra and geometry, as these rings help describe the structure of varieties by capturing the algebraic properties of functions that are well-behaved, or 'regular', on those varieties. These rings play a critical role in understanding morphisms and mappings between different geometric spaces.
Zariski's Lemma: Zariski's Lemma is a fundamental result in algebraic geometry that states that if a point is contained in an affine variety, then the maximal ideal of the coordinate ring corresponding to that point can be represented by a set of generators consisting of polynomials vanishing at that point. This lemma connects deeply with various aspects of algebraic varieties, such as coordinate rings, projective varieties, isomorphisms, and primary decomposition, by providing a clear relationship between geometric properties and algebraic structures.