🕴🏼Elementary Algebraic Geometry Unit 4 – Coordinate Rings

What are Coordinate Rings?

• Coordinate rings are fundamental objects in algebraic geometry that associate a ring to an algebraic variety
• Defined as the ring of polynomial functions on an affine variety $V$ over a field $k$, denoted as $k[V]$
• Consist of all polynomial functions $f: V \to k$ that can be evaluated at any point of the variety
• Play a crucial role in studying the algebraic and geometric properties of varieties
• Provide a bridge between the geometric world of varieties and the algebraic world of rings
• Allow for the application of powerful algebraic techniques to study geometric problems
• Serve as a key tool in understanding the structure and behavior of algebraic varieties
• Enable the translation of geometric questions into algebraic language, making them more tractable

Key Properties of Coordinate Rings

• Coordinate rings are commutative rings, meaning that the multiplication of elements is commutative ($ab = ba$ for all $a, b \in k[V]$)
• They are integral domains, implying that the product of two non-zero elements is always non-zero
  • This property reflects the fact that algebraic varieties do not have "zero divisors"
• Coordinate rings are finitely generated $k$-algebras, as they can be generated by a finite number of polynomial functions
• The Zariski topology on an affine variety $V$ is determined by the prime ideals of its coordinate ring $k[V]$
  • Closed sets in the Zariski topology correspond to algebraic subsets of $V$
• The dimension of an affine variety is equal to the Krull dimension of its coordinate ring
• Morphisms between affine varieties can be described using homomorphisms between their coordinate rings
• The coordinate ring of a product of affine varieties is isomorphic to the tensor product of their individual coordinate rings

Constructing Coordinate Rings

• To construct the coordinate ring of an affine variety $V \subset \mathbb{A}^n$, start with the polynomial ring $k[x_1, \ldots, x_n]$
• Consider the ideal $I(V)$ consisting of all polynomials that vanish on every point of $V$
  • $I(V) = \{f \in k[x_1, \ldots, x_n] \mid f(p) = 0 \text{ for all } p \in V\}$
• The coordinate ring $k[V]$ is defined as the quotient ring $k[x_1, \ldots, x_n] / I(V)$
  • Elements of $k[V]$ are equivalence classes of polynomials, where two polynomials are equivalent if their difference lies in $I(V)$
• The quotient ring construction ensures that the polynomial functions in $k[V]$ are well-defined on the variety $V$
• The natural projection map $\pi: k[x_1, \ldots, x_n] \to k[V]$ sends a polynomial to its equivalence class in the coordinate ring
• The coordinate ring inherits the ring structure from the polynomial ring, with operations defined on equivalence classes

Relationship to Affine Varieties

• Every affine variety $V$ has an associated coordinate ring $k[V]$, which encodes its algebraic structure
• The coordinate ring $k[V]$ determines the affine variety $V$ up to isomorphism
  • Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic as $k$-algebras
• The points of an affine variety $V$ correspond to maximal ideals of its coordinate ring $k[V]$
  • This correspondence allows for the study of the geometry of $V$ through the algebraic properties of $k[V]$
• Regular functions on an affine variety $V$ are precisely the elements of its coordinate ring $k[V]$
  • A regular function is a function that can be locally expressed as a ratio of polynomials
• The dimension of an affine variety $V$ is equal to the transcendence degree of its coordinate ring $k[V]$ over the base field $k$
• Morphisms between affine varieties can be described using homomorphisms between their coordinate rings
  • A morphism $f: V \to W$ induces a homomorphism $f^*: k[W] \to k[V]$ between the coordinate rings

Important Theorems and Proofs

• Hilbert's Nullstellensatz establishes a fundamental correspondence between ideals in the coordinate ring and algebraic subsets of the affine variety
  • It states that for an algebraically closed field $k$, the maximal ideals of $k[V]$ are in one-to-one correspondence with the points of $V$
• The weak Nullstellensatz asserts that for an algebraically closed field $k$, the radical ideals of $k[V]$ are in one-to-one correspondence with the algebraic subsets of $V$
• The normalization theorem states that every reduced affine variety is birationally equivalent to a normal affine variety
  • The coordinate ring of a normal affine variety is integrally closed in its field of fractions
• The Noether normalization lemma shows that every affine variety is birationally equivalent to a hypersurface in a higher-dimensional affine space
• The Zariski's main theorem establishes that every birational morphism between affine varieties is an open immersion followed by a finite morphism
• The proof of these theorems often involves techniques from commutative algebra, such as localization, completion, and dimension theory

Applications in Algebraic Geometry

• Coordinate rings are used to study the local and global properties of algebraic varieties
• They provide a way to define and analyze regular functions, rational functions, and morphisms between varieties
• Coordinate rings are essential in the study of singularities and the resolution of singularities
  • The local ring at a point of a variety encodes information about the singularity at that point
• The prime spectrum of a coordinate ring, denoted $\text{Spec}(k[V])$, is a fundamental object in scheme theory
  • It provides a more general and intrinsic approach to studying algebraic varieties
• Coordinate rings play a role in the study of moduli spaces, which parametrize families of algebraic objects
• They are used in the construction and analysis of invariants of varieties, such as the Hilbert polynomial and the Hilbert series
• Coordinate rings are employed in the study of algebraic groups and their actions on varieties
  • The ring of invariants under a group action captures important geometric information

Common Examples and Exercises

• The coordinate ring of the affine line $\mathbb{A}^1$ over a field $k$ is isomorphic to the polynomial ring $k[x]$
• The coordinate ring of the affine plane curve $V(y - x^2) \subset \mathbb{A}^2$ is isomorphic to $k[x, y] / (y - x^2)$
• The coordinate ring of the affine space $\mathbb{A}^n$ over a field $k$ is isomorphic to the polynomial ring $k[x_1, \ldots, x_n]$
• Compute the coordinate ring of the union of two distinct points in $\mathbb{A}^1$
• Determine the coordinate ring of the intersection of two affine varieties
• Find the coordinate ring of the product of two affine varieties
• Describe the morphisms between two given affine varieties by studying homomorphisms between their coordinate rings
• Compute the dimension of an affine variety using its coordinate ring

Connections to Other Algebraic Concepts

• Coordinate rings are closely related to the concept of affine algebras in commutative algebra
  • An affine algebra over a field $k$ is a finitely generated $k$-algebra that is an integral domain
• The study of coordinate rings involves techniques from commutative algebra, such as localization, completion, and dimension theory
• Coordinate rings are used in the construction of schemes, which provide a more general framework for studying algebraic varieties
  • The prime spectrum of a coordinate ring is the underlying topological space of an affine scheme
• The theory of coordinate rings is connected to the study of modules over commutative rings
  • Modules over coordinate rings, such as the module of Kähler differentials, provide important geometric information
• Coordinate rings play a role in the study of algebraic groups and their representations
  • The coordinate ring of an algebraic group is a Hopf algebra, which encodes the group structure
• The study of coordinate rings is related to the theory of Gröbner bases and computational algebraic geometry
  • Gröbner bases provide a way to effectively compute with ideals in polynomial rings and coordinate rings