Polynomial equations remember more than their zero sets at first glance: they also remember the algebra of functions that survive on those zero sets. If two polynomials agree at every point of a geometric object, the geometry has no way to distinguish them as functions on that object. The coordinate ring is the construction that makes this identification official, turning an affine algebraic set into a commutative ring whose elements are the polynomial functions living on it.

The motivating tension is already visible in the plane. Let $k$ be a field, and write $k[x,y]$ for the [polynomial ring](/page/Polynomial%20Ring) in the two variables $x$ and $y$ with coefficients in $k$. The polynomials $y$ and $x^2$ are different elements of $k[x,y]$, but on the parabola cut out by the equation $y-x^2=0$ they give the same function. Over an infinite field, this is the only polynomial relation forced by the parabola; over a finite field, extra pointwise relations can appear because a nonzero polynomial may vanish at every $k$-point. The coordinate ring records the correct equality by taking a quotient of the polynomial ring by the full vanishing ideal, not merely by the ideal generated by the equations we happened to write down.

[example: Polynomial Functions on a Parabola] Let $k$ be an infinite field and let \begin{align*} X=\{(a,b)\in k^2\mid b=a^2\}\subset k^2. \end{align*} For any point $(a,b)\in X$, the defining condition gives $b=a^2$, so evaluating the two ambient polynomials $y$ and $x^2$ at $(a,b)$ gives \begin{align*} y(a,b)=b=a^2=x^2(a,b). \end{align*} Thus $y$ and $x^2$ define the same function on $X$. For example, at a point $(a,a^2)\in X$, \begin{align*} (x^3+2y)(a,a^2)=a^3+2a^2. \end{align*} Also \begin{align*} (x^3+2x^2)(a,a^2)=a^3+2a^2. \end{align*} Therefore $x^3+2y$ and $x^3+2x^2$ have equal restrictions to $X$. Now let $f\in k[x,y]$ vanish on $X$. Since $y-x^2$ is monic as a polynomial in $y$, polynomial division in the variable $y$ gives \begin{align*} f(x,y)=q(x,y)(y-x^2)+r(x) \end{align*} for some $q\in k[x,y]$ and some $r\in k[x]$. Evaluating at $(a,a^2)\in X$ gives \begin{align*} 0=f(a,a^2)=q(a,a^2)(a^2-a^2)+r(a)=r(a). \end{align*} So $r(a)=0$ for every $a\in k$. Because $k$ is infinite, a one-variable polynomial over $k$ with infinitely many roots is the zero polynomial, hence $r=0$. Therefore \begin{align*} f(x,y)=q(x,y)(y-x^2). \end{align*} So $I(X)=(y-x^2)$, and the coordinate ring is \begin{align*} k[X]=k[x,y]/(y-x^2). \end{align*} In this quotient, the class of $y$ equals the class of $x^2$, so every polynomial function on the parabola can be represented using only the single coordinate $x$. [/example]

The point of the construction is not to replace geometry by algebra for its own sake. It is to make geometric operations visible in a ring where ordinary algebraic tools apply. The coordinate ring sits at the meeting point of [Ring](/page/Ring), ideals, and affine algebraic sets, and it is one of the basic bridges into [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry) and [Cambridge III Commutative Algebra](/page/Cambridge%20III%20Commutative%20Algebra).

## Definition

The construction should identify two ambient polynomials precisely when they give the same value at every point of the algebraic set. That means the first object to define is the quotient itself: the ring obtained by killing all polynomial functions that vanish on the set.

[definition: Coordinate Ring] Let $k$ be a field and let $X\subset k^n$ be an affine algebraic set. Let \begin{align*} I(X)=\{f\in k[x_1,\ldots,x_n]\mid f(a)=0 \text{ for every } a\in X\}. \end{align*} The coordinate ring of $X$ over $k$ is the [quotient ring](/page/Quotient%20Ring) \begin{align*} k[X]=k[x_1,\ldots,x_n]/I(X). \end{align*} [/definition]

This quotient definition is compact, but it depends on two pieces of background that deserve their own names. The next section isolates the geometric input and the vanishing ideal that makes the quotient independent of the particular equations chosen.

## Algebraic Sets and Regular Functions

Before a coordinate ring can be formed, we need to know which subsets of [affine space](/page/Affine%20Space) are legitimate geometric objects for this chapter. The guiding requirement is that the subset should be described by polynomial tests: a point belongs to the set exactly when it satisfies a chosen family of polynomial equations.

### Polynomial Equation Sets

The basic geometric object should remember only polynomial conditions, not an arbitrary list of points. This definition packages the common solution set of any chosen family of polynomial equations, so later algebraic constructions can refer to the set $X$ itself rather than to one particular presentation of its equations.

[definition: Affine Algebraic Set] Let $k$ be a field and let $n\in\mathbb N$. An affine algebraic set in $k^n$ is a subset $X\subset k^n$ for which there exists a set of polynomials $S\subset k[x_1,\ldots,x_n]$ such that \begin{align*} X=\{a\in k^n\mid f(a)=0 \text{ for every } f\in S\}. \end{align*} [/definition]

The same set $X$ can be cut out by many different collections of equations. To build the correct ring of functions on $X$, we need the collection of every polynomial that vanishes on $X$, since those are precisely the polynomials that represent the zero function on $X$.

[definition: Vanishing Ideal] Let $k$ be a field and let $X\subset k^n$. The vanishing ideal of $X$ is the subset \begin{align*} I(X)=\{f\in k[x_1,\ldots,x_n]\mid f(a)=0 \text{ for every } a\in X\}. \end{align*} [/definition]

The notation $k[X]$ emphasizes that elements of the quotient are functions on $X$, not just formal polynomial classes. The affine algebraic set definition tells us where those functions are evaluated, and the vanishing ideal tells us which ambient polynomials become zero. This raises the next question: which functions on $X$ should count as algebraic functions coming from its ambient affine space?

### Regular Functions and Coordinate Classes

Once $X$ is fixed, we want functions that are intrinsic to the algebraic geometry of $X$, not arbitrary set-theoretic functions on its points. For this introductory discussion, we first use the polynomial-restriction version: the algebraic functions on $X$ are obtained by evaluating ambient polynomials and then identifying polynomials that agree on $X$.

[definition: Regular Function on an Affine Algebraic Set] Let $k$ be a field and let $X\subset k^n$ be an affine algebraic set. A regular function on $X$ is a function $\varphi:X\to k$ for which there exists a polynomial $f\in k[x_1,\ldots,x_n]$ satisfying \begin{align*} \varphi(a)=f(a) \end{align*} for every $a\in X$. [/definition]

There is a possible mismatch between the two descriptions: an element of $k[X]$ is a polynomial class modulo $I(X)$, while a regular function is an actual function $X\to k$. For the polynomial-restriction definition just given, the comparison is direct: the class of $f$ gives the function $a\mapsto f(a)$ on $X$, and two polynomials give the same function exactly when their difference lies in $I(X)$.