Coordinate RingEdit

A coordinate ring is a central object in algebraic geometry that encodes the regular functions on an affine algebraic set. If k is a field and X is a subset of the n-dimensional affine space defined by polynomial equations, the coordinate ring k[X] is the quotient of the polynomial ring k[x1, ..., xn] by the vanishing ideal I(X) consisting of all polynomials that vanish on every point of X. Concretely, k[X] = k[x1, ..., xn] / I(X). This construction turns geometric questions about X into algebraic questions about the ring k[X], while preserving the intuitive notion of what a regular function on X should be.

The interplay between algebra and geometry is fundamental here. Properties of the geometric object X are reflected in the algebraic structure of k[X], and vice versa. For instance, when k is algebraically closed, maximal ideals of k[X] correspond to k-rational points of X, and prime ideals correspond to irreducible subvarieties. This correspondence is a standard consequence of Hilbert’s Nullstellensatz. Morphisms between affine varieties go the other way as ring homomorphisms: a map f: X → Y induces a homomorphism k[Y] → k[X], reversing the direction in a precise way. These ideas connect the geometric intuition of shapes and equations with the algebraic machinery of rings and ideals.

In practical terms, coordinate rings are finitely generated algebras over k, and many questions in geometry become questions about generators, relations, and growth of ideals. The spectrum of k[X]—its set of prime ideals equipped with a topology and a structure sheaf—provides a bridge to the broader language of scheme. For curves, surfaces, and higher-dimensional varieties, the coordinate ring behaves as a computational and conceptual backbone: it records regular functions, supports dimension theory via Krull dimension, and interacts with foundational results such as the Noether normalization theorem and the study of function fields.

Definition and basic properties

Let X ⊂ Affine space be defined by a set of polynomials, and let I(X) be the ideal of k[x1, ..., xn] consisting of polynomials that vanish on every point of X. Then the coordinate ring is k[X] = k[x1, ..., xn] / I(X).
If X is irreducible (as a geometric object), then I(X) is a prime ideal, and k[X] is an integral domain. Conversely, k[X] being an integral domain reflects the irreducibility of X.
For a point p ∈ X with coordinates in k, the maximal ideal mx,p of k[x1, ..., xn] that vanishes at p descends to a maximal ideal of k[X], tying closed points of X to maximal ideals of the coordinate ring.

Examples

The entire affine space A^n_k has coordinate ring k[x1, ..., xn], since I(A^n_k) = {0}. This is the simplest nontrivial example that anchors the construction.
A plane curve defined by a single polynomial f(x, y) = 0 in A^2_k has coordinate ring k[x, y] / (f). For example, the circle given by x^2 + y^2 − 1 = 0 has k[X] = k[x, y] / (x^2 + y^2 − 1).
Elliptic curves arise as curves in the plane defined by a Weierstrass equation y^2 = x^3 + ax + b. Their coordinate ring is k[x, y] / (y^2 − x^3 − ax − b), illustrating how algebra encodes a rich geometric object.

Geometry and algebra in tandem

Regular functions on X are precisely the elements of k[X], and these functions are defined by polynomial expressions restricted to X.
The geometry of X, such as dimension, singularities, and irreducibility, translates into algebraic features of k[X], such as Krull dimension, the behavior of local rings, and the presence or absence of certain types of ideals.
When studying maps between varieties, the corresponding ring maps reveal how the algebraic structure of coordinate rings changes under pullbacks and pushforwards, a theme that is central to algebraic geometry.

Computation and applications

Noether normalization provides a bridge from k[X] to a polynomial ring in a smaller number of variables, showing that k[X] is finite over a polynomial subring. This has important computational consequences and underpins many algorithms in computational algebra.
Groebner bases, elimination theory, and related computational tools operate naturally in the setting of coordinate rings, enabling explicit Solving of polynomial systems and the study of dimension and equational sets.
In number theory and arithmetic geometry, coordinate rings appear in the study of curves, surfaces, and more general varieties, informing questions about rational points, function fields, and moduli spaces.

A few deeper ideas

The functorial perspective emphasizes that the data of X is captured by its coordinate ring through contravariant functors: maps of X correspond to homomorphisms of k[X] in the opposite direction.
For irreducible X, the function field k(X) is the field of fractions of k[X], tying geometric objects to their rational function behavior.
The geometric picture is enriched when passing to the language of scheme: Spec k[X] encapsulates X as an affine scheme, while more general morphisms of schemes correspond to ring homomorphisms in the opposite direction, generalizing many classical ideas from variety theory.

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