2.1 Affine space and coordinate rings

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 coordinate ring 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

• Affine n-space over a field $K$, denoted $A^n(K)$ or $K^n$, is the set of all n-tuples of elements from $K$
  • For example, $A^2(\mathbb{R})$ represents the real plane, where each point is described by a pair of real numbers $(x, y)$
  • Similarly, $A^3(\mathbb{C})$ represents 3-dimensional complex space, where each point is described by a triplet of complex numbers $(z_1, z_2, z_3)$
• The coordinate ring of affine n-space over $K$ is the polynomial ring $K[x_1, \ldots, x_n]$, where $x_1, \ldots, x_n$ are indeterminates
  • The indeterminates $x_1, \ldots, x_n$ represent the coordinates of points in the affine space
  • For instance, the coordinate ring of $A^2(\mathbb{R})$ is $\mathbb{R}[x, y]$, 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 $f(x_1, \ldots, x_n) \in K[x_1, \ldots, x_n]$ defines a function from $A^n(K)$ 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 $K[x_1, \ldots, x_n]$ 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 $R[x_1, \ldots, x_n]$ is formed by adjoining indeterminates $x_1, \ldots, x_n$ to a commutative ring $R$
  • The coefficients of the polynomials in $R[x_1, \ldots, x_n]$ come from the ring $R$
  • If $R$ is a field (such as $\mathbb{Q}$, $\mathbb{R}$, or $\mathbb{C}$), then $R[x_1, \ldots, x_n]$ is called a polynomial ring over a field
• Elements of a polynomial ring are polynomials, which are finite sums of terms of the form $a(x_1^{e_1})\ldots(x_n^{e_n})$, where $a \in R$ and $e_1, \ldots, e_n$ are non-negative integers
  • The exponents $e_1, \ldots, e_n$ 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 $\mathbb{R}[x, y]$ or $\mathbb{C}[x, y, z]$, are always integral domains

Quotient rings and ideals

• An ideal $I$ in a polynomial ring $R[x_1, \ldots, x_n]$ is a subset closed under addition and multiplication by elements of the ring
  • If $f, g \in I$, then $f + g \in I$ (closure under addition)
  • If $f \in I$ and $h \in R[x_1, \ldots, x_n]$, then $hf \in I$ (closure under multiplication by ring elements)
• The quotient ring $R[x_1, \ldots, x_n]/I$ is formed by considering the equivalence classes of polynomials modulo the ideal $I$
  • Two polynomials $f, g \in R[x_1, \ldots, x_n]$ 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 $R[x_1, \ldots, x_n]/I$ 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 $f \in K[x_1, \ldots, x_n]$ assigns a value in $K$ to each point in the affine space $A^n(K)$ by evaluating the polynomial at that point
• The zero set of a polynomial $f \in K[x_1, \ldots, x_n]$ is the set of points $(a_1, \ldots, a_n) \in A^n(K)$ such that $f(a_1, \ldots, a_n) = 0$
  • 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 $f(x, y) = x^2 + y^2 - 1$ in $\mathbb{R}[x, y]$ is the unit circle in the real plane $A^2(\mathbb{R})$
• 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 $I \subseteq K[x_1, \ldots, x_n]$ is the intersection of the zero sets of all polynomials in $I$
  • If $V(I)$ denotes the zero set of the ideal $I$, then $V(I) = \bigcap_{f \in I} V(f)$, where $V(f)$ is the zero set of the polynomial $f$
• The coordinate ring of an algebraic set $V$ is the quotient ring $K[x_1, \ldots, x_n]/I(V)$, 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 $K[x_1, \ldots, x_n]/I(V)$ 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 $I_1 \subseteq I_2 \subseteq \ldots$ eventually stabilizes (i.e., there exists an $N$ such that $I_n = I_N$ for all $n \geq N$)
  • The Noetherian property implies that every ideal in a coordinate ring has a finite set of generators
• The Krull dimension 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 $K[x_1, \ldots, x_n]$, the Krull dimension is $n$, which coincides with the dimension of the affine space $A^n(K)$

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 $K[x_1, \ldots, x_n]/I$, 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 $K[x_1, \ldots, x_n]/I$, maximal ideals are of the form $(x_1 - a_1, \ldots, x_n - a_n)/I$, where $(a_1, \ldots, a_n)$ 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 $\mathfrak{p}$ is the ring $R_{\mathfrak{p}}$ obtained by inverting all elements of $R$ not in $\mathfrak{p}$
  • For a coordinate ring $K[x_1, \ldots, x_n]/I$ and a maximal ideal $\mathfrak{m} = (x_1 - a_1, \ldots, x_n - a_n)/I$, the localization $(K[x_1, \ldots, x_n]/I)_{\mathfrak{m}}$ is the local ring at the point $(a_1, \ldots, a_n)$
  • 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