Affine variety

A cubic plane curve given by $y^2 = x^2(x+1)$

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space $k^n$ of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety.

If X is an affine variety defined by a prime ideal I, then the quotient ring

$k[x_1, \ldots, x_n]/I$

is called the coordinate ring of X. This ring is precisely the set of all regular functions on X; in other words, it is the space of global sections of the structure sheaf of X. A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if

$H^i(X, F) = 0$

for any $i > 0$ and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.

An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles.

An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring. In complex geometry, an affine variety is an analog of a Stein manifold.

Introduction

The most concrete point of view to describe an affine algebraic variety is that it is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if $f_1, \ldots, f_k$ are polynomials with coefficients in k, they define an affine variety (or affine algebraic set)

$V(f_1,\ldots, f_k) = \left\{(a_1,\ldots,a_n)\in k^n \;|\;f_1(a_1,\ldots, a_n)=\ldots=f_k(a_1,\ldots, a_n)=0\right\}.$

By Hilbert's Nullstellensatz, the points of the variety are in one to one correspondence with the maximal ideals of its coordinate ring, the k-algebra $R=k[x_1, \ldots, x_n]/\langle f_1, \ldots, f_k\rangle,$ through the map $(a_1,\ldots, a_n) \mapsto \langle \overline{x_1-a_1}, \ldots, \overline{x_n-a_n}\rangle,$ where $\overline{x_i-a_i}$ denotes the image in the quotient algebra R of the polynomial $x_i-a_i.$ In scheme theory, this correspondence has been extended to prime ideals to define the affine scheme $\operatorname{Spec}(R),$ which may be identified to the variety, through an equivalence of categories.

The elements of the coordinate ring R are also called the regular functions or the polynomial functions on the variety. They form the ring of the regular functions on the variety, or, simply, the ring of the variety. In fact an element $\overline{f}\in R$ is the image of a polynomial $f\in k[x_1,\ldots, x_n],$ which defines a function from kⁿ into k; The restriction of f to the variety does not depend on the choice of $f$ among the polynomials mapped on $\overline{f}$ by the quotient.

The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety).

Structure sheaf

Equipped with the structure sheaf described below, an affine variety is a locally ringed space.

Given an affine variety X with coordinate ring A, we define the sheaf of k-algebras $\mathcal{O}_X$ by letting $\mathcal{O}_X(U) = \Gamma(U, \mathcal{O}_X)$ be the ring of regular functions on U.

We let D(f) = { x | f(x) ≠ 0 } for each f in A. They form a base for the topology of X and so $\mathcal{O}_X$ is determined by its values on the open sets D(f). (See also: sheaf of modules#Sheaf associated to a module.)

The key fact, which relies on Hilbert nullstellensatz in the essential way, is the following:

Claim — $\Gamma(D(f), \mathcal{O}_X) = A[f^{-1}]$ for any f in A.

Proof:^[1] The inclusion ⊃ is clear. For the opposite, let g be in the left-hand side and $J = \{ h \in A | hg \in A \}$, which is an ideal. If x is in D(f), then, since g is regular near x, there is some open affine neighborhood D(h) of x such that $g \in k[D(h)] = A[h^{-1}]$; that is, h^m g is in A and thus x is not in V(J). In other words, $V(J) \subset \{ x | f(x) = 0 \}$ and thus the Hilbert nullstellensatz implies f is in the radical of J; i.e., $f^n g \in A$. $\square$

The claim, first of all, implies that X is a "locally ringed" space since

$\mathcal{O}_{X, x} = \varinjlim_{f(x) \ne 0} A[f^{-1}] = A_{\mathfrak{m}_x}$

where $\mathfrak{m}_x = \{ f \in A | f(x) = 0 \}$. Secondly, the claim implies that $\mathcal{O}_X$ is a sheaf; indeed, it says if a function is regular (pointwise) on D(f), then it must be in the coordinate ring of D(f); that is, "regular-ness" can be patched together.

Hence, $(X, \mathcal{O}_X)$ is a locally ringed space.

Examples

• The complement of a hypersurface of an affine variety X; i.e., X - { f = 0 } for some regular function f on X is affine; its coordinate ring is the localization $k[X][f^{-1}]$. In particular, A¹ - 0 (the affine line with the origin removed) is affine.
• Every closed subvariety of the affine space $\mathbf{A}^n$ of codimension one is defined by a prime ideal of the polynomial ring of height one, which is principal; thus, they are hypersurfaces (i.e., defined by a single polynomial.)
• C² - 0 is an open subset of the affine variety that is not affine; cf. Hartogs' extension theorem
• The normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (It turns out the normalization of a projective variety is a projective variety.)

Rational points

Tangent space

Tangent spaces may be defined just as in calculus. Let $X = \operatorname{spec} A, A = k[x_1, \dots, x_n]/(f_1, \dots, f_r)$ be the affine variety. Then the affine subvariety of $k^n$ defined by the linear equations

$\sum_{i=1}^n {\partial f_j \over \partial {x_i} }(a_1, \dots, a_n) (x_i - a_i) = 0, \quad j = 1, \dots, r$

is called the tangent space at $x = (a_1, \dots, a_n).$^[2] (A more intrinsic definition is given by Zariski tangent space.) If the tangent space at x and the variety X have the same dimension, the point x is said to be smooth; otherwise, singular.

The important difference from calculus is that the inverse function theorem fails. To alleviate this problem, one has to consider the étale topology instead of the Zariski topology. (cf. Milne, Étale)

Notes

1. Mumford, Ch. I, § 4. Proposition 1.
2. Milne AG, Ch. 5

See also

• Representations on coordinate rings

References

The original article was written as a partial human translation of the corresponding French article.

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Milne, Algebraic geometry
• Milne, Lectures on Étale cohomology
• Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 354063293X.