![Cubic with double point](https://upload.wikimedia.org/wikipedia/commons/a/a6/Cubic_with_double_point.svg)
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 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 related ...
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 example ...
s of variables with coefficients in that generate a
prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine)
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
. A
Zariski open subvariety of an affine variety is called a
quasi-affine variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
.
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. Every ...
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 ration ...
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
are polynomials with coefficients in , they define an affine algebraic set
:
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 function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regula ...
s'' 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 commut ...
).
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
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
.
* In particular,
(the affine line with the origin removed) is affine.
* On the other hand,
(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
are exactly the hypersurfaces, that is the varieties defined by a single polynomial.
* 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. (Similarly, the normalization of a projective variety is a projective variety.)
Rational points
![Doubling oriented](https://upload.wikimedia.org/wikipedia/commons/4/4f/Doubling_oriented.svg)
For an affine variety
over an algebraically closed field , and a subfield of , a -''rational point'' of is a point
That is, a point of whose coordinates are elements of . The collection of -rational points of an affine variety is often denoted
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. Every ...
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 ration ...
s) are often simply called ''rational points''.
For instance, is a -rational and an -rational point of the variety
as it is in and all its coordinates are integers. The point is a real point of that is not -rational, and
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
:
where is a rational number.
The circle
is an example of an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
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
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
Singular points and tangent space
Let be an affine variety defined by the polynomials
and
be a point of .
The
Jacobian matrix of at is the matrix of the partial derivatives
:
The point is ''regular'' if the rank of equals the
codimension 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
defined by the
linear equations
:
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, an ...
.
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
and
(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
for
These basic open sets are the complements in ''k''
''n'' of the closed sets
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 ''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
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
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
: for an ideal ''J'' in
where ''k'' is an algebraically closed field,
Radical ideals (ideals which are their own radical) of ''k
' correspond to algebraic subsets of ''V''. Indeed, for radical ideals ''I'' and ''J'',
if and only if
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
). 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
if and only if
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
where
denotes the image in the quotient algebra ''R'' of the polynomial
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
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
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
morphisms 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
where
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
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.
Given an affine variety ''X'' with coordinate ring ''A'', the sheaf of ''k''-algebras
is defined by letting
be the ring of
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regula ...
s on ''U''.
Let ''D''(''f'') = for each ''f'' in ''A''. They form a base for the topology of ''X'' and so
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 mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
in the essential way, is the following:
Proof:
The inclusion ⊃ is clear. For the opposite, let ''g'' be in the left-hand side and
, 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