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