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Encyclopedia > Algebraic variety

In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...


Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory. In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

Contents

Formal definitions

Affine varieties

Let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k[x1, ..., xn] can be viewed as k-valued functions on An by evaluating f at the points in An. For each subset S of k[x1, ..., xn], define the zero-locus of S to be the set of points in An on which the functions in S vanish: In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...

Z(S) = {x in mathbb A^n mid f(x) = 0 mbox{ for all } fin S}.

A subset V of An is called an affine algebraic set if V = Z(S) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is called an affine variety. In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K, of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 and so on up to Pm(X1, ...,Xn) = 0 for some integer m. ...


Affine varieties can be given a natural topology, called the Zariski topology, by declaring all algebraic sets to be closed. In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...


Given a subset V of An, let I(V) be the ideal of all functions vanishing on V: In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

I(V) = {f in k[x_1,cdots,x_n] mid f(x) = 0 mbox{ for all } xin V}.

For any affine algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.


Projective varieties

Let Pn be a projective n-space over k. Homogeneous polynomials in k[x0, x1, ..., xn] can be viewed as k-valued functions on Pn by evaluating them on homogeneous coordinates. The homogeneity of the polynomial ensures that this construction is well-defined. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: This article does not cite its references or sources. ... In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

Z(S) = {x in mathbb P^n mid f(x) = 0 mbox{ for all } fin S}.

A subset V of Pn is called an projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety.


Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.


Given a subset V of Pn, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...


Basic results

  • An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
  • Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties (where none of the sets in the decomposition are subsets of each other).
  • Let k[V] be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k[V] over k.

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...

Discussion and generalizations

The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The current notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...


This definition works over any field K. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to problems since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. These are usually not considered varieties, and we get rid of them by requiring the schemes underlying a variety to be separated. (There is strictly speaking also a third condition, namely, that in the definition above one needs only finitely many affine patches.)


Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety simply mean that the affine charts have trivial nilradical. In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ...


A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa. In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...


These varieties have been called 'varieties in the sense of Serre', since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way. Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...


One way that leads to generalisations is to allow reducible algebraic sets (and fields K that aren't algebraically closed), so the rings R may not be integral domains. This is not a big step technically. More serious is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in co-ordinate rings aren't seen as co-ordinate functions. In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...


From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products). Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of schemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ...


There are further generalizations called stacks and algebraic spaces. In mathematics, an algebraic stack in algebraic geometry is a special case of the concept of a stack, which is useful for working on moduli questions. ... In mathematics, an algebraic space is a generalization of the affine schemes of algebraic geometry introduced by Michael Artin for use in deformation theory. ...


Isomorphism of algebraic varieties

Let V1 and V2 be algebraic varieties. We say that V1 and V2 are isomorphic, and write V1 ≅ V2, if there are regular maps φ : V1 → V2 and ψ : V2 → V1 such that the compositions ψ ° φ and φ ° ψ are the identity maps on V1 and V2 respectively. A graph isomorphism is a bijection between the vertices of two graphs and : with the property that any two vertices and from are adjacent if and only if and are adjacent in . ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...


Algebraic manifolds

Main article: Algebraic manifold

An algebraic manifold is an algebraic variety which is also a m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When k is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example. // Algebraic manifolds are an algebraic variety which are also m-dimensional manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a projective algebraic manifold is a complex manifold which is a submanifold of a complex projective space which is determined by the zeros of a set of homogeneous polynomials. ... A rendering of the Riemann Sphere. ...


See also

In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

References

  • Robin Hartshorne (1997). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9. 
  • David Cox; John Little, Don O'Shea (1997). Ideals, Varieties, and Algorithms, second edition, Springer-Verlag. ISBN 0-387-94680-2. 
  • David Eisenbud (1999). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag. ISBN 0-387-94269-6. 
  • David Dummit; Richard Foote (2003). Abstract Algebra, third edition, Wiley. ISBN 0-471-43334-9. 

This article incorporates material from Isomorphism of varieties on PlanetMath, which is licensed under the GFDL. Robin Hartshorne (born 1938) is an American mathematician. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ... Prominent people named David Cox: David Cox (Australian politician) - former member of the House of Representatives for the Division of Kingston David Cox (artist) - prominent English landscape painter Sir David Cox (statistician) - prominent English statistician David J Cox (canadian Poet, Writer) +David Cox (Murderer)-Convicted of homocide in Fairbanks, Alaska... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ... David Eisenbud (born 8 April 1947) is an American mathematician. ... Springer Science+Business Media or Springer (IPA: ) is a worldwide publishing company based in Germany which focuses on academic journals and books in the fields of science, technology, mathematics, and medicine. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
Kids.net.au - Encyclopedia Algebraic geometry - (732 words)
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry.
In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations.
Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20-th century, with the geometric applications in mind.
  More results at FactBites »

 
 

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