Affine variety

(Redirected from Affine algebraic set) A cubic plane curve given by $y^{2}=x^{2}(x+1)$ In algebraic geometry, an affine variety, or affine algebraic variety, over a algebraically closed field k is the zero-locus in 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.

Some texts, such as Miles Reid's Undergraduate Algebraic Geometry, do not require the field to be algebraically closed. Other texts do not require affine varieties to be irreducible; this article refers to such objects as affine algebraic sets.

Introduction

An affine algebraic set is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if $f_{1},\ldots ,f_{m}$  are polynomials with coefficients in k, they define an affine algebraic set

$V(f_{1},\ldots ,f_{m})=\left\{(a_{1},\ldots ,a_{n})\in k^{n}\;|\;f_{1}(a_{1},\ldots ,a_{n})=\ldots =f_{m}(a_{1},\ldots ,a_{n})=0\right\}.$

An affine (algebraic) variety is an affine algebraic set which is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible.

If X is an affine algebraic set defined by an ideal I, then the quotient ring $R=k[x_{1},\ldots ,x_{n}]/I$  is called the coordinate ring of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain. 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 regular functions on the variety, or, simply, the ring of the variety; in other words (see #Structure sheaf), it is the space of global sections of the structure sheaf of X.

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).

The Zariski topology

The affine algebraic sets of kn form the closed sets of a topology on kn, called the Zariski topology. This follows from the fact that $V(0)=k[x_{1},\ldots ,x_{n}],$  $V(1)=\emptyset ,$  $V(S)\cup V(T)=V(ST),$  and $V(S)\cap V(T)=V(S+T)$  (in fact, a countable intersection of affine algebraic sets is an affine algebraic set).

The Zariski topology can also be described by way of basic open sets, where Zariski-open sets are countable unions of sets of the form $U_{f}=\{p\in k^{n}:f(p)\neq 0\}$  for $f\in k[x_{1},\ldots ,x_{n}].$  These basic open sets are the complements in kn of the closed sets $V(f)=D_{f}=\{p\in k^{n}:f(p)=0\},$  zero loci of a single polynomial. If k is Noetherian (for instance, if k is a field or a principal ideal domain), then every ideal of k is finitely-generated, so every open set is a finite union of basic open sets.

If V is an affine subvariety of kn the Zariski topology on V is simply the subspace topology inherited from the Zariski topology on kn.

Geometry-algebra correspondence

The geometric structure of an affine variety is linked in a deep way to the algebraic structure of its coordinate ring. Let I and J be ideals of k[V], the coordinate ring of an affine variety V. Let I(V) be the set of all polynomials in $k[x_{1},\ldots ,x_{n}],$  which vanish on V, and let ${\sqrt {I}}$  denote the radical of the ideal I, the set of polynomials f for which some power of f is in I. The reason that the base field is required to be algebraically closed is that affine varieties automatically satisfy Hilbert's nullstellensatz: for an ideal J in $k[x_{1},\ldots ,x_{n}],$  where k is an algebraically closed field, $I(V(J))={\sqrt {J}}.$

Radical ideals (ideals which are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, $I\subseteq J$  if and only if $V(J)\subseteq V(I).$  Hence V(I)=V(J) if and only if I=J. Furthermore, the function taking an affine algebraic set W and returning I(W), the set of all functions which also vanish on all points of W, is the inverse of the function assigning an algebraic set to a radical ideal, by the nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.

Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set V(I) can be written as the union of two other algebraic sets if and only if I=JK for proper ideals J and K not equal to I (in which case $V(I)=V(J)\cup V(K)$ ). This is the case if and only if I is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain.

Maximal ideals of k[V] correspond to points of V. If I and J are radical ideals, then $V(J)\subseteq V(I)$  if and only if $I\subseteq J.$  As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those which contain no proper algebraic subsets), which are points in V. If V is an affine variety with coordinate ring $R=k[x_{1},\ldots ,x_{n}]/\langle f_{1},\ldots ,f_{m}\rangle ,$  this correspondence becomes explicit 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 $x_{i}-a_{i}.$  An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field.

The following table summarises this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring:

Type of algebraic set Type of ideal
affine subvariety prime ideal
point maximal ideal

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: 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, hm 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

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

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.

Serre's theorem on affineness

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.

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.)

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 $x=(a_{1},\dots ,a_{n}).$  (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)[clarification needed]

Generalizations

• If an author requires the base field of an affine variety to be algebraically closed (as this article does), then irreducible affine algebraic sets over non-algebraically closed fields are a generalization of affine varieties. This generalization notably includes affine varieties over the real numbers.
• 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, a locally-ringed space which is isomorphic to the spectrum of a commutative ring. Each variety has an affine scheme associated to it. The classical points of the variety correspond to maximal ideals of the variety, and the affine scheme also has points corresponding to closed subvarieties of the variety. If V(I) is an affine variety in kn with coordinate ring $R=k[x_{1},\ldots ,x_{n}]/I,$  then the scheme corresponding to V(I) is $\operatorname {Spec} (R),$  the set of prime ideals of R. This creates a more well-defined notion of the "generic point" of an affine variety, by assigning to each closed subvariety an open point which is dense in the subvariety. More generally, an affine scheme is an affine variety if it is reduced, irreducible, and of finite type over an algebraically closed field k.