affine variety

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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, an affine variety, or affine algebraic variety, over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
is the zero-locus in the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
of some finite family of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
s of variables with coefficients in that generate a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
. 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 do not require a prime ideal, and call ''irreducible'' an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said ''defined over'' , and the points of the variety that belong to are said ''-rational'' or ''rational over'' . In the common case where is the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, a -rational point is called a ''real point''. When the field is not specified, a ''rational point'' is a point that is rational over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s. For example,
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
asserts that the affine algebraic variety (it is a curve) defined by has no rational points for any integer greater than two.

# Introduction

An affine algebraic set is the set of solutions in an algebraically closed field of a system of polynomial equations with coefficients in . More precisely, if $f_1, \ldots, f_m$ are polynomials with coefficients in , they define an affine algebraic set :$V\left(f_1,\ldots, f_m\right) = \left\.$ 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 is an affine algebraic set, and is the ideal of all polynomials that are zero on , then the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
is called the 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 In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutat ...
).

# Examples

* The complement of a hypersurface in an affine variety (that is for some polynomial ) is affine. Its defining equations are obtained by saturating by the defining ideal of . The coordinate ring is thus the localization . * In particular, $\mathbb C - 0$ (the affine line with the origin removed) is affine. * On the other hand, $\mathbb C^2 - 0$ (the affine plane with the origin removed) is not an affine variety; cf. Hartogs' extension theorem. * The subvarieties of codimension one in the affine space $k^n$ are exactly the hypersurfaces, that is the varieties defined by a single polynomial. * The
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
of an irreducible affine variety is affine; the coordinate ring of the normalization is the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' i ...
of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)

# Rational points

For an affine variety $V\subseteq K^n$ over an algebraically closed field , and a subfield of , a -''rational point'' of is a point $p\in V\cap k^n.$ That is, a point of whose coordinates are elements of . The collection of -rational points of an affine variety is often denoted $V\left(k\right).$ Often, if the base field is the complex numbers , points which are -rational (where is the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s) are called ''real points'' of the variety, and -rational points ( the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s) are often simply called ''rational points''. For instance, is a -rational and an -rational point of the variety $V = V\left(x^2+y^2-1\right)\subseteq\mathbf^2,$ as it is in and all its coordinates are integers. The point is a real point of that is not -rational, and $\left(i,\sqrt\right)$ is a point of that is not -rational. This variety is called a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, because the set of its -rational points is the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. It has infinitely many -rational points that are the points :$\left\left(\frac,\frac\right\right)$ where is a rational number. The circle $V\left(x^2+y^2-3\right)\subseteq\mathbf^2$ is an example of an algebraic curve of degree two that has no -rational point. This can be deduced from the fact that, modulo , the sum of two squares cannot be . It can be proved that an algebraic curve of degree two with a -rational point has infinitely many other -rational points; each such point is the second intersection point of the curve and a line with a rational slope passing through the rational point. The complex variety $V\left(x^2+y^2+1\right)\subseteq\mathbf^2$ has no -rational points, but has many complex points. If is an affine variety in defined over the complex numbers , the -rational points of can be drawn on a piece of paper or by graphing software. The figure on the right shows the -rational points of $V\left(y^2-x^3+x^2+16x\right)\subseteq\mathbf^2.$

# Singular points and tangent space

Let be an affine variety defined by the polynomials and $a=\left(a_1, \dots,a_n\right)$ be a point of . The
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables a ...
of at is the matrix of the partial derivatives :$\frac \left(a_1, \dots, a_n\right).$ The point is ''regular'' if the rank of equals the
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals t ...
of , and ''singular'' otherwise. If is regular, the tangent space to at is the
affine subspace In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of $k^n$ defined by the
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s :$\sum_^n \frac \left(a_1, \dots, a_n\right) \left(x_i - a_i\right) = 0, \quad j = 1, \dots, r.$ If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point. A more intrinsic definition, which does not use coordinates is given by
Zariski tangent space In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, a ...
.

# The Zariski topology

The affine algebraic sets of ''k''''n'' form the closed sets of a topology on ''k''''n'', called the Zariski topology. This follows from the fact that $V\left(1\right)=\emptyset,$ $V\left(S\right)\cup V\left(T\right)=V\left(ST\right),$ and $V\left(S\right)\cap V\left(T\right)=V\left(S+T\right)$ (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 = \$ for These basic open sets are the complements in ''k''''n'' of the closed sets $V\left(f\right)=D_f=\,$ zero loci of a single polynomial. If ''k'' is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
(for instance, if ''k'' is a field or a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
), 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 ''k''''n'' the Zariski topology on ''V'' is simply the subspace topology inherited from the Zariski topology on ''k''''n''.

# 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 ', the coordinate ring of an affine variety ''V''. Let ''I(V)'' be the set of all polynomials in which vanish on ''V'', and let $\sqrt$ 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 where ''k'' is an algebraically closed field, $I\left(V\left(J\right)\right)=\sqrt.$ Radical ideals (ideals which are their own radical) of ''k ' correspond to algebraic subsets of ''V''. Indeed, for radical ideals ''I'' and ''J'', $I\subseteq J$ if and only if $V\left(J\right)\subseteq V\left(I\right).$ 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 $V\left(I\right)=V\left(J\right)\cup V\left(K\right)$). 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 ' correspond to points of ''V''. If ''I'' and ''J'' are radical ideals, then $V\left(J\right)\subseteq V\left(I\right)$ 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 this correspondence becomes explicit through the map $\left(a_1,\ldots, a_n\right) \mapsto \langle \overline, \ldots, \overline\rangle,$ where $\overline$ 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:

# Products of affine varieties

A product of affine varieties can be defined using the isomorphism then embedding the product in this new affine space. Let and have coordinate rings and respectively, so that their product has coordinate ring . Let be an algebraic subset of and an algebraic subset of Then each is a polynomial in , and each is in . The product of and is defined as the algebraic set in The product is irreducible if each , is irreducible. It is important to note that the Zariski topology on is not the topological product of the Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets and Hence, polynomials that are in but cannot be obtained as a product of a polynomial in with a polynomial in will define algebraic sets that are in the Zariski topology on but not in the product topology.

# Morphisms of affine varieties

A morphism, or regular map, of affine varieties is a function between affine varieties which is polynomial in each coordinate: more precisely, for affine varieties and , a morphism from to is a map of the form where for each These are the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of affine varieties. There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field and homomorphisms of coordinate rings of affine varieties over going in the opposite direction. Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over and their coordinate rings, the category of affine varieties over is dual to the category of coordinate rings of affine varieties over The category of coordinate rings of affine varieties over is precisely the category of finitely-generated, nilpotent-free algebras over More precisely, for each morphism of affine varieties, there is a homomorphism between the coordinate rings (going in the opposite direction), and for each such homomorphism, there is a morphism of the varieties associated to the coordinate rings. This can be shown explicitly: let and be affine varieties with coordinate rings and respectively. Let be a morphism. Indeed, a homomorphism between polynomial rings factors uniquely through the ring and a homomorphism is determined uniquely by the images of Hence, each homomorphism corresponds uniquely to a choice of image for each . Then given any morphism from to a homomorphism can be constructed which sends to $\overline,$ where $\overline$ is the equivalence class of in Similarly, for each homomorphism of the coordinate rings, a morphism of the affine varieties can be constructed in the opposite direction. Mirroring the paragraph above, a homomorphism sends to a polynomial $f_i\left(X_1,\dots,X_n\right)$ in . This corresponds to the morphism of varieties defined by

# Structure sheaf

Equipped with the structure sheaf described below, an affine variety is a
locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf o ...
. Given an affine variety ''X'' with coordinate ring ''A'', the sheaf of ''k''-algebras $\mathcal_X$ is defined by letting $\mathcal_X\left(U\right) = \Gamma\left(U, \mathcal_X\right)$ be the ring of regular functions on ''U''. Let ''D''(''f'') = for each ''f'' in ''A''. They form a base for the topology of ''X'' and so $\mathcal_X$ 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: Proof: The inclusion ⊃ is clear. For the opposite, let ''g'' be in the left-hand side and $J = \$, 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

# 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 $H^i\left(X, F\right) = 0$ for any $i > 0$ and any
quasi-coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
''F'' on ''X''. (cf.
Cartan's theorem B In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development o ...
.) 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.

# Affine algebraic groups

An affine variety over an algebraically closed field is called an affine algebraic group if it has: * A ''multiplication'' , which is a regular morphism that follows the
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
axiom—that is, such that for all points , and in * An ''identity element'' such that for every in * An ''inverse morphism'', a regular bijection such that for every in Together, these define a group structure on the variety. The above morphisms are often written using ordinary group notation: can be written as , or ; the inverse can be written as or Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as: , and . The most prominent example of an affine algebraic group is the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of degree This is the group of linear transformations of the
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
if a basis of is fixed, this is equivalent to the group of invertible matrices with entries in It can be shown that any affine algebraic group is isomorphic to a subgroup of . For this reason, affine algebraic groups are often called linear algebraic groups. Affine algebraic groups play an important role in the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else ...
, as the groups of Lie type are all sets of -rational points of an affine algebraic group, where is a finite field.

# 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s. * An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as
projective varieties In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
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 a special case of an
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
, a locally-ringed space which is isomorphic to the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
of a commutative ring (up to an equivalence of categories). Each affine variety has an affine scheme associated to it: if is an affine variety in with coordinate ring then the scheme corresponding to is the set of prime ideals of The affine scheme has "classical points" which correspond with points of the variety (and hence maximal ideals of the coordinate ring of the variety), and also a point for each closed subvariety of the variety (these points correspond to prime, non-maximal ideals of the coordinate ring). 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

# See also

*
Algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
*
Affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
* Representations on coordinate rings

# References

The original article was written as a partial human translation of the corresponding French article. * * * * Milne,
Lectures on Étale cohomology
' * *{{cite book , last=Reid , first=Miles , authorlink=Miles Reid , title=Undergraduate Algebraic Geometry , date=1988 , publisher=Cambridge University Press , isbn=0-521-35662-8 , url=https://archive.org/details/undergraduatealg0000reid , url-access=registration Algebraic geometry