Affine spaces and coordinate rings are the building blocks of algebraic geometry. They provide a way to study geometric objects using algebraic tools, bridging the gap between algebra and geometry.
In this chapter, we explore how affine n-space represents points as n-tuples, while its consists of polynomial functions on that space. This connection allows us to analyze geometric properties through algebraic techniques.
Affine n-space and its coordinate ring
Definition and notation
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Affine n-space over a K, denoted An(K) or Kn, is the set of all n-tuples of elements from K
For example, A2(R) represents the real , where each is described by a pair of real numbers (x,y)
Similarly, A3(C) represents 3-dimensional complex space, where each point is described by a triplet of complex numbers (z1,z2,z3)
The coordinate of affine n-space over K is the polynomial ring K[x1,…,xn], where x1,…,xn are indeterminates
The indeterminates x1,…,xn represent the coordinates of points in the
For instance, the coordinate ring of A2(R) is R[x,y], the ring of polynomials in two variables with real coefficients
Properties and significance
The elements of the coordinate ring are polynomial functions on the affine space
Each polynomial f(x1,…,xn)∈K[x1,…,xn] defines a function from An(K) to K by evaluating the polynomial at points in the affine space
The coordinate ring is a commutative ring with identity
The ring operations (addition, subtraction, and multiplication) of polynomials in K[x1,…,xn] are performed component-wise, inheriting the commutativity and identity properties from the field K
The affine space and its coordinate ring provide a foundation for studying algebraic geometry
Algebraic geometry explores the connections between geometric objects (such as algebraic sets and varieties) and their algebraic counterparts (such as ideals in coordinate rings)
The interplay between affine spaces and their coordinate rings allows for the application of algebraic techniques to geometric problems, and vice versa
Polynomial rings and quotients
Constructing polynomial rings
A polynomial ring R[x1,…,xn] is formed by adjoining indeterminates x1,…,xn to a commutative ring R
The coefficients of the polynomials in R[x1,…,xn] come from the ring R
If R is a field (such as Q, R, or C), then R[x1,…,xn] is called a polynomial ring over a field
Elements of a polynomial ring are polynomials, which are finite sums of terms of the form a(x1e1)…(xnen), where a∈R and e1,…,en are non-negative integers
The exponents e1,…,en determine the degree of each term, and the overall degree of the polynomial is the maximum degree among its terms
Polynomial rings are integral domains if the coefficient ring R is an integral domain
An integral domain is a commutative ring with no zero divisors (i.e., if ab=0, then either a=0 or b=0)
Polynomial rings over fields, such as R[x,y] or C[x,y,z], are always integral domains
Quotient rings and ideals
An I in a polynomial ring R[x1,…,xn] is a subset closed under addition and multiplication by elements of the ring
If f,g∈I, then f+g∈I (closure under addition)
If f∈I and h∈R[x1,…,xn], then hf∈I (closure under multiplication by ring elements)
The quotient ring R[x1,…,xn]/I is formed by considering the equivalence classes of polynomials modulo the ideal I
Two polynomials f,g∈R[x1,…,xn] are equivalent modulo I if their difference f−g belongs to the ideal I
The equivalence class of a polynomial f is denoted by f+I or [f]
The quotient ring R[x1,…,xn]/I is a ring with zero element I and operations induced by the polynomial ring
Addition: (f+I)+(g+I)=(f+g)+I
Multiplication: (f+I)(g+I)=(fg)+I
Quotient rings of polynomial rings are used to study algebraic sets and varieties
Algebraic sets are defined as the zero sets of collections of polynomials
Varieties are irreducible algebraic sets, which can be studied using the coordinate rings of their affine open subsets
Geometric meaning of polynomial functions
Zero sets and algebraic sets
Polynomial functions on affine spaces are elements of the coordinate ring
A polynomial function f∈K[x1,…,xn] assigns a value in K to each point in the affine space An(K) by evaluating the polynomial at that point
The zero set of a polynomial f∈K[x1,…,xn] is the set of points (a1,…,an)∈An(K) such that f(a1,…,an)=0
The zero set of a polynomial is the collection of points in the affine space where the polynomial vanishes
For example, the zero set of the polynomial f(x,y)=x2+y2−1 in R[x,y] is the unit circle in the real plane A2(R)
The zero set of a polynomial is an algebraic set in the affine space
An algebraic set is the zero set of a collection of polynomials
Algebraic sets are the basic closed sets in the Zariski topology on affine spaces
Ideals and coordinate rings of algebraic sets
The zero set of an ideal I⊆K[x1,…,xn] is the intersection of the zero sets of all polynomials in I
If V(I) denotes the zero set of the ideal I, then V(I)=⋂f∈IV(f), where V(f) is the zero set of the polynomial f
The coordinate ring of an algebraic set V is the quotient ring K[x1,…,xn]/I(V), where I(V) is the ideal of all polynomials vanishing on V
The ideal I(V) consists of all polynomials that evaluate to zero at every point in the algebraic set V
The coordinate ring K[x1,…,xn]/I(V) captures the algebraic properties of the algebraic set V
The elements of the coordinate ring can be viewed as polynomial functions on V, as they are equivalence classes of polynomials modulo the ideal of functions vanishing on V
Properties of coordinate rings
Noetherian property and Krull dimension
Coordinate rings are Noetherian rings, meaning that every ideal is finitely generated
A ring is Noetherian if it satisfies the ascending chain condition: every ascending chain of ideals I1⊆I2⊆… eventually stabilizes (i.e., there exists an N such that In=IN for all n≥N)
The Noetherian property implies that every ideal in a coordinate ring has a finite set of generators
The Krull of a coordinate ring is equal to the dimension of the corresponding affine space
The Krull dimension of a ring is the supremum of the lengths of all chains of prime ideals in the ring
For a coordinate ring K[x1,…,xn], the Krull dimension is n, which coincides with the dimension of the affine space An(K)
Integral domain and units
Coordinate rings are integral domains, as they are quotient rings of polynomial rings over fields
Polynomial rings over fields are integral domains, and quotients of integral domains by prime ideals are again integral domains
The coordinate ring of an algebraic set V is an integral domain because the ideal I(V) of polynomials vanishing on V is a prime ideal
The units in a coordinate ring are precisely the nonzero constant polynomials
A unit in a ring is an element that has a multiplicative inverse
In a coordinate ring K[x1,…,xn]/I, the units are the equivalence classes of nonzero constant polynomials (i.e., polynomials of degree 0)
The units in a coordinate ring form a subgroup of the multiplicative group of the ring
Maximal ideals and local rings
Maximal ideals in a coordinate ring correspond to points in the affine space
A maximal ideal is an ideal that is not contained in any other proper ideal
In a coordinate ring K[x1,…,xn]/I, maximal ideals are of the form (x1−a1,…,xn−an)/I, where (a1,…,an) is a point in the algebraic set defined by the ideal I
The localization of a coordinate ring at a maximal ideal yields the local ring at the corresponding point
The localization of a ring R at a prime ideal p is the ring Rp obtained by inverting all elements of R not in p
For a coordinate ring K[x1,…,xn]/I and a maximal ideal m=(x1−a1,…,xn−an)/I, the localization (K[x1,…,xn]/I)m is the local ring at the point (a1,…,an)
Local rings capture the local behavior of algebraic sets near a specific point
Coordinate rings provide an algebraic approach to studying geometric properties of affine spaces and algebraic sets
The algebraic properties of coordinate rings, such as their prime ideals, Krull dimension, and local rings, correspond to geometric features of affine spaces and algebraic sets
By studying coordinate rings and their associated algebraic objects, one can gain insights into the geometry of algebraic sets and their singularities
Key Terms to Review (20)
Affine coordinate system: An affine coordinate system is a mathematical framework used to define points in an affine space through coordinates that are related by linear transformations. This system allows for the representation of geometric properties such as lines, planes, and transformations without the need for a fixed origin. Affine coordinate systems play a key role in understanding both the geometric structures and the associated coordinate rings that describe algebraic varieties.
Affine space: An affine space is a geometric structure that generalizes the concept of Euclidean space by allowing for points to be defined without a fixed origin. In this structure, points can be added together and scaled by real numbers, but there is no inherent notion of distance or angles. This framework is crucial for understanding various mathematical concepts, such as coordinate rings and algorithms that operate on polynomials in a more abstract setting.
Affine variety: An affine variety is a subset of affine space that is defined as the common zero set of a collection of polynomials. It represents the solution set to polynomial equations, allowing for the study of geometric properties using algebraic techniques, and serves as a fundamental building block in algebraic geometry.
Bezout's Theorem: Bezout's Theorem states that if two projective plane curves intersect, the number of intersection points, counted with multiplicity, is equal to the product of their degrees. This theorem is crucial in understanding the relationships between algebraic varieties and their properties in affine space and coordinate rings, as it establishes a foundational connection between algebraic equations and geometric intersections.
Coordinate ring: A coordinate ring is a mathematical structure that represents the algebraic functions on an algebraic set, allowing for a bridge between geometry and algebra. It is formed from polynomial functions defined on affine space, where the points of the affine space correspond to maximal ideals in the coordinate ring. This connection enables a geometric interpretation of algebraic sets, establishing a foundation for further study in algebraic geometry.
David Hilbert: David Hilbert was a prominent German mathematician in the late 19th and early 20th centuries, renowned for his foundational contributions to various areas of mathematics, including algebra, number theory, and geometry. His work laid the groundwork for modern computational algebraic geometry, influencing methods for solving polynomial systems and establishing key principles such as the Hilbert's Nullstellensatz.
Dimension: Dimension is a fundamental concept that describes the degree of freedom or the number of coordinates needed to specify a point in a space. In algebraic geometry, it relates to the complexity and structure of varieties, where the dimension can provide insights into their properties and relationships with other geometric objects.
Embedding: Embedding refers to a mathematical concept where one space is contained within another space, often in a way that preserves the structure of the original space. In algebraic geometry, this term is crucial as it helps in understanding how varieties can be represented and related through morphisms. When working with both projective varieties and affine spaces, embeddings play a key role in transforming geometric objects into a higher-dimensional setting or into a different type of variety while maintaining their essential properties.
Field: A field is a set equipped with two operations, typically called addition and multiplication, satisfying certain properties that allow for the manipulation of elements in a way that generalizes the arithmetic of rational and real numbers. Fields play a crucial role in algebraic structures, providing the foundational building blocks for various mathematical systems, including vector spaces and polynomial rings, which are essential for understanding relationships between algebra and geometry as well as coordinate systems in affine spaces.
Hilbert's Nullstellensatz: Hilbert's Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep connection between ideals in polynomial rings and algebraic sets. It provides a way to understand the relationship between solutions of polynomial equations and the corresponding algebraic varieties, thus linking algebraic concepts with geometric intuition.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. In the context of affine spaces and coordinate rings, homomorphisms allow us to relate different coordinate rings and facilitate understanding the geometric properties of the spaces they represent.
Ideal: An ideal is a special subset of a ring that allows for the creation of a new ring structure, facilitating algebraic operations and enabling the manipulation of polynomial equations. Ideals are fundamental in algebraic geometry as they connect algebraic properties with geometric shapes, helping to define solutions to polynomial equations and establish relationships between algebra and geometry.
Isomorphism: Isomorphism is a mathematical concept that describes a structural similarity between two algebraic objects, indicating that they can be considered equivalent in some sense. This idea connects various areas, such as relating birational equivalence and understanding the properties of projective varieties, showing that different varieties can have the same essential structure. It emphasizes how algebraic structures, such as ideals and varieties, correspond to each other and reflects the deep connections between algebra and geometry.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has had a profound impact on various mathematical fields, particularly in the development of sheaf theory and the study of projective varieties, linking many concepts together that are crucial for understanding modern algebraic geometry.
Line: A line is a straight, one-dimensional figure that extends infinitely in both directions and is defined by two points in a plane. In the context of affine space, lines can be described using equations in coordinate rings, where they serve as fundamental geometric objects that represent linear relationships between variables. Lines help establish the connection between points, and their interactions can lead to various geometric properties and algebraic structures.
Module: A module is a mathematical structure that generalizes the concept of vector spaces by allowing scalars to come from a ring instead of a field. This flexibility means that modules can be studied over different types of rings, providing a richer framework for understanding linear algebraic concepts in more complex settings. Modules play a critical role in connecting algebra and geometry, particularly through their relationship with affine spaces and coordinate rings.
Morphism: A morphism is a structure-preserving map between two algebraic objects, such as varieties or algebraic sets, that allows us to understand their relationship in a geometric and algebraic context. Morphisms play a crucial role in linking different varieties and understanding their properties, enabling us to study their intersections, projections, and other geometric features.
Plane: A plane is a flat, two-dimensional surface that extends infinitely in all directions and is defined by three non-collinear points in a three-dimensional space. In the context of affine geometry, a plane can be seen as an affine space where each point can be represented using coordinates in a given system, allowing for the study of geometric properties and relationships between points, lines, and shapes.
Point: A point is a fundamental concept in geometry and algebraic geometry representing a specific location in space that has no dimensions, only coordinates. In affine space, points are essential as they form the foundation for defining geometric objects, establishing relationships, and analyzing their properties using coordinate systems. The concept of a point is also deeply connected to coordinate rings, which allow for the algebraic representation of these locations and the manipulation of their associated data.
Ring: A ring is a mathematical structure consisting of a set equipped with two operations, typically called addition and multiplication, that generalizes the arithmetic properties of integers. Rings must satisfy certain properties, such as being closed under addition and multiplication, having an additive identity (zero), and every element having an additive inverse. This concept is pivotal in establishing connections between algebra and geometry, as it provides the framework for coordinate rings in affine spaces, where geometric objects are studied through algebraic equations.