About: Affine variety

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In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn 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.

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rdfs:label	Affine variety (en) Variété algébrique affine (fr) Varietà affine (it) アフィン多様体 (ja)
rdfs:comment	En géométrie algébrique, une variété affine est un modèle local pour les variétés algébriques, c'est-à-dire que celles-ci sont obtenues par recollement de variétés affines. Grossièrement, une variété affine est un ensemble algébrique affine X avec une structure algébrique supplémentaire qui est la donnée de l'anneau des fonctions régulières sur chaque partie ouverte de X. (fr) In geometria algebrica, una varietà affine è il sottoinsieme di uno spazio affine -dimensionale su un campo algebricamente chiuso caratterizzato dall'annullarsi simultaneo di tutti i polinomi di un sottoinsieme di . Un aperto (secondo la topologia di Zariski) di una varietà affine è detto varietà quasi affine. (it) In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn 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. (en) 代数幾何学において，代数閉体 k 上のアフィン多様体（あふぃんたようたい，英: affine variety）とは，n 次元アフィン空間 kn において，k 係数の n 変数の多項式の素イデアルを生成する有限族の零点集合である．素イデアルを生成するという条件を外したときの集合は（アフィン）代数的集合と呼ばれる．アフィン多様体のザリスキ開部分多様体はと呼ばれる． X が素イデアル I によって定義されるアフィン多様体のとき，商環は X の座標環と呼ばれる．この環はちょうど X 上のすべてのがなす集合である．言い換えると，X の構造層の大域切断の空間である．はアフィン多様体のコホモロジー的特徴づけを与える．定理により代数多様体がアフィンであることとがすべての i > 0 と X 上のすべての準連接層 F に対して成り立つことは同値である（cf. カルタンの定理 B）．したがってアフィン多様体のコモロジーの研究は存在せず，直線束のコホモロジー群が中心的関心事である射影多様体とは非常に対照的である．アフィン多様体は代数多様体の局所チャートの役割を果たす，つまり，射影多様体のような一般の代数多様体はアフィン多様体を貼り合わせることで得られる．多様体に付随する線型構造も（自明に）アフィン多様体である．例えば，接空間や，のファイバーなど． (ja)
foaf:homepage	http://www.jmilne.org
name	Claim (en)
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dcterms:subject	Algebraic geometry
Wikipage page ID	320468 (xsd:integer)
Wikipage revision ID	1122223889 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ideal Principal ideal domain Product topology Quasi-coherent sheaf Saturation (commutative algebra) Basis (linear algebra) Algebraic set Algebraically closed field Unit circle Vector space Affine scheme Quotient ring General linear group Noetherian ring Normal scheme Circle Glossary of algebraic geometry Modular arithmetic Morphism Equivalence of categories Linear equation Zariski tangent space Spectrum of a ring Dual (category theory) Sheaf of modules Affine space Algebraic curve Algebraic geometry Algebraic varieties Algebraic variety Fermat's Last Theorem Field (mathematics) Base (topology) Dimension of an algebraic variety Hartogs' extension theorem Projective variety Radical of an ideal Regular function Group (mathematics) Hilbert nullstellensatz Irreducible component Jacobian matrix Associativity Algebraic geometry Polynomial Classification of finite simple groups Group of Lie type Integral closure Category (mathematics) Rational number Real number Serre's theorem on affineness Reduced ring Field of fractions Zariski topology Finite morphism Algebraic vector bundle Hilbert's nullstellensatz Affine subspace Representations on coordinate rings Springer-Verlag Cartan's theorem B Quasi-affine variety Locally ringed space
Link from a Wikipage to an external page	http://www.jmilne.org/math/CourseNotes/lec.html https://www.jmilne.org/math/CourseNotes/AG.pdf https://archive.org/details/undergraduatealg0000reid http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
sameAs	Affine variety Affine variety Affine variety Affine variety Affine variety Affine variety Affine variety
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