4.1 Coordinate rings of affine and projective varieties

Coordinate rings are the algebraic backbone of varieties, connecting geometry to algebra. They encode crucial information about a variety's structure, dimension, and properties, allowing us to study geometric objects using powerful algebraic techniques.

For affine varieties, the coordinate ring consists of polynomial functions. For projective varieties, we use homogeneous polynomials in the graded coordinate ring. These rings provide a bridge between the concrete geometry of varieties and abstract algebra.

Coordinate rings of varieties

Definition and structure

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• The coordinate ring of an affine variety $V$, denoted $A(V)$, is the ring of polynomial functions on $V$, the set of all polynomial functions from $V$ to the base field $k$
• $A(V)$ is a finitely generated $k$-algebra, generated by the coordinate functions on $V$ (e.g., $x$, $y$, and $z$ for a variety in $\mathbb{A}^3$)
• The coordinate ring $A(V)$ is isomorphic to the quotient ring $k[x_1, \ldots, x_n]/I(V)$, where $I(V)$ is the ideal of polynomials vanishing on $V$
  • This isomorphism allows for the study of the algebraic properties of $A(V)$ using the tools of commutative algebra
  • Example: For the variety $V = V(xy - 1) \subset \mathbb{A}^2$, $A(V) \cong k[x, y]/(xy - 1)$

Properties and correspondences

• The Nullstellensatz establishes a correspondence between affine varieties and radical ideals in $k[x_1, \ldots, x_n]$
  • Every radical ideal $I$ corresponds to a unique affine variety $V(I)$
  • Every affine variety $V$ corresponds to a unique radical ideal $I(V)$
• The dimension of an affine variety $V$ is equal to the Krull dimension of its coordinate ring $A(V)$
  • The Krull dimension is the supremum of the lengths of chains of prime ideals in $A(V)$
  • Example: The variety $V = V(x^2 + y^2 - 1) \subset \mathbb{A}^2$ has dimension 1, as $A(V) \cong k[x, y]/(x^2 + y^2 - 1)$ has Krull dimension 1
• The coordinate ring $A(V)$ is an integral domain if and only if $V$ is an irreducible variety
  • An irreducible variety cannot be written as the union of two proper subvarieties
  • Example: The variety $V = V(xy) \subset \mathbb{A}^2$ is reducible, as $V = V(x) \cup V(y)$, and $A(V) \cong k[x, y]/(xy)$ is not an integral domain
• The function field of an irreducible affine variety $V$ is the field of fractions of $A(V)$
  • The function field consists of rational functions on $V$, i.e., quotients of polynomials in $A(V)$
  • Example: For the irreducible variety $V = V(y - x^2) \subset \mathbb{A}^2$, the function field is $k(V) = \operatorname{Frac}(k[x, y]/(y - x^2))$

Homogeneous coordinate rings

Definition and grading

• The homogeneous coordinate ring of a projective variety $V$, denoted $S(V)$, is the graded ring of homogeneous polynomial functions on $V$
• $S(V)$ is a graded $k$-algebra, with the grading given by the degree of the homogeneous polynomials
  • A polynomial $f \in k[x_0, \ldots, x_n]$ is homogeneous of degree $d$ if $f(\lambda x_0, \ldots, \lambda x_n) = \lambda^d f(x_0, \ldots, x_n)$ for all $\lambda \in k$
  • Example: The polynomial $x_0^2 + x_1x_2$ is homogeneous of degree 2
• The homogeneous coordinate ring $S(V)$ is isomorphic to the quotient ring $k[x_0, \ldots, x_n]/I(V)$, where $I(V)$ is the homogeneous ideal of polynomials vanishing on $V$
  • A homogeneous ideal is an ideal generated by homogeneous polynomials
  • Example: For the projective variety $V = V(x_0x_2 - x_1^2) \subset \mathbb{P}^2$, $S(V) \cong k[x_0, x_1, x_2]/(x_0x_2 - x_1^2)$

Hilbert function and polynomial

• The Hilbert function of $S(V)$ encodes information about the dimensions of the graded components of $S(V)$
  • The Hilbert function $h_V: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ is defined by $h_V(d) = \dim_k S(V)_d$, where $S(V)_d$ is the $d$-th graded component of $S(V)$
  • Example: For the projective variety $V = V(x_0x_2 - x_1^2) \subset \mathbb{P}^2$, the Hilbert function is $h_V(d) = d + 1$ for all $d \geq 0$
• The Hilbert polynomial of $S(V)$ is a polynomial that agrees with the Hilbert function for sufficiently large degrees and provides information about the dimension and degree of $V$
  • The Hilbert polynomial $P_V(t)$ is defined as the unique polynomial such that $P_V(d) = h_V(d)$ for all $d \gg 0$
  • The degree of $P_V(t)$ is equal to the dimension of $V$, and the leading coefficient of $P_V(t)$ is related to the degree of $V$
  • Example: For the projective variety $V = V(x_0x_2 - x_1^2) \subset \mathbb{P}^2$, the Hilbert polynomial is $P_V(t) = t + 1$, indicating that $V$ has dimension 1 and degree 1

Affine vs projective rings

Embedding affine varieties into projective space

• Any affine variety $V$ can be embedded into a projective space as a quasi-projective variety, denoted $\bar{V}$
  • The embedding is given by the map $\varphi: V \to \mathbb{P}^n$, $(a_1, \ldots, a_n) \mapsto (1 : a_1 : \ldots : a_n)$
  • The image of $\varphi$ is an open subset of the projective closure of $V$, which is the smallest projective variety containing $\varphi(V)$
  • Example: The affine variety $V = V(y - x^2) \subset \mathbb{A}^2$ can be embedded into $\mathbb{P}^2$ as $\bar{V} = V(x_1x_2 - x_0x_1) \subset \mathbb{P}^2$

Homogenization and dehomogenization

• The homogeneous coordinate ring $S(\bar{V})$ of the projective closure $\bar{V}$ is related to the affine coordinate ring $A(V)$ by the process of homogenization and dehomogenization of polynomials
• The affine coordinate ring $A(V)$ can be recovered from $S(\bar{V})$ by dehomogenizing with respect to a non-vanishing homogeneous coordinate
  • Dehomogenization of a homogeneous polynomial $f(x_0, \ldots, x_n)$ with respect to $x_0$ is the polynomial $f(1, x_1, \ldots, x_n)$
  • Example: Dehomogenizing the polynomial $x_0x_2 - x_1^2$ with respect to $x_0$ yields the polynomial $x_2 - x_1^2$
• The projective coordinate ring $S(\bar{V})$ can be obtained from $A(V)$ by homogenizing the polynomials in $A(V)$ with respect to a new variable
  • Homogenization of a polynomial $f(x_1, \ldots, x_n)$ of degree $d$ with respect to $x_0$ is the polynomial $x_0^d f(x_1/x_0, \ldots, x_n/x_0)$
  • Example: Homogenizing the polynomial $y - x^2$ with respect to $x_0$ yields the polynomial $x_0y - x^2$

Applications of coordinate rings

Determining geometric properties

• Use the coordinate ring to determine the dimension, irreducibility, and singularities of a variety
  • The dimension of a variety is equal to the Krull dimension of its coordinate ring
  • A variety is irreducible if and only if its coordinate ring is an integral domain
  • Singularities of a variety correspond to prime ideals in its coordinate ring that are not maximal
  • Example: The variety $V = V(y^2 - x^3) \subset \mathbb{A}^2$ has a singularity at the origin, as the maximal ideal $(x, y)$ in $A(V) \cong k[x, y]/(y^2 - x^3)$ is not a regular local ring

Computing Hilbert functions and polynomials

• Compute the Hilbert function and Hilbert polynomial of a projective variety to study its geometric properties
  • The Hilbert function provides information about the dimensions of the graded components of the homogeneous coordinate ring
  • The Hilbert polynomial encodes the dimension and degree of the projective variety
  • Example: For the projective variety $V = V(x_0^2x_2 - x_1^3) \subset \mathbb{P}^2$, the Hilbert polynomial is $P_V(t) = 3t + 1$, indicating that $V$ has dimension 1 and degree 3

Correspondence between ideals and varieties

• Utilize the correspondence between ideals and varieties to solve problems related to the structure of coordinate rings
  • The Nullstellensatz establishes a bijective correspondence between radical ideals and affine varieties
  • The projective Nullstellensatz establishes a bijective correspondence between homogeneous radical ideals and projective varieties
  • Example: To find the ideal of polynomials vanishing on the affine variety $V = V(x^2 + y^2 - 1) \subset \mathbb{A}^2$, compute the radical of the ideal $(x^2 + y^2 - 1)$

Applying the Nullstellensatz

• Apply the Nullstellensatz to establish the relationship between the ideal-theoretic and geometric properties of varieties
  • The Nullstellensatz states that for any ideal $I \subset k[x_1, \ldots, x_n]$, $I(V(I)) = \sqrt{I}$
  • The projective Nullstellensatz states that for any homogeneous ideal $I \subset k[x_0, \ldots, x_n]$, $I(V(I)) = \sqrt{I}$
  • Example: To show that the affine varieties $V(x^2 - y)$ and $V(x - y^2)$ in $\mathbb{A}^2$ are not isomorphic, prove that their coordinate rings are not isomorphic using the Nullstellensatz

Graded rings and projective varieties

• Use the properties of graded rings to study the homogeneous coordinate ring of a projective variety
  • The homogeneous coordinate ring of a projective variety is a graded ring, with the grading given by the degree of the homogeneous polynomials
  • Graded modules over the homogeneous coordinate ring correspond to coherent sheaves on the projective variety
  • Example: The twisted cubic curve $V = V(x_1x_3 - x_2^2, x_0x_3 - x_1x_2, x_0x_2 - x_1^2) \subset \mathbb{P}^3$ has homogeneous coordinate ring $S(V) \cong k[x_0, x_1, x_2, x_3]/(x_1x_3 - x_2^2, x_0x_3 - x_1x_2, x_0x_2 - x_1^2)$, which is a graded ring

Switching between affine and projective settings

• Employ the techniques of homogenization and dehomogenization to switch between affine and projective settings
  • Homogenization allows for the study of affine varieties using tools from projective geometry
  • Dehomogenization allows for the study of projective varieties using tools from affine geometry
  • Example: To find the singular points of the affine variety $V = V(y^2 - x^3 - x^2) \subset \mathbb{A}^2$, homogenize the defining equation to obtain the projective variety $\bar{V} = V(x_0x_2^2 - x_1^3 - x_0x_1^2) \subset \mathbb{P}^2$, find the singular points of $\bar{V}$, and then dehomogenize to obtain the singular points of $V$

Function fields of irreducible varieties

• Solve problems involving the function field of an irreducible variety using the properties of the coordinate ring
  • The function field of an irreducible affine variety is the field of fractions of its coordinate ring
  • The function field of an irreducible projective variety is the field of fractions of the degree 0 part of its homogeneous coordinate ring
  • Example: To find the genus of the smooth projective curve $V = V(x_0x_2 - x_1^2) \subset \mathbb{P}^2$, compute the dimension of the space of global sections of the canonical sheaf on $V$ using the properties of the function field $k(V) = \operatorname{Frac}(k[x_0, x_1, x_2]/(x_0x_2 - x_1^2))$