Affine variety (original) (raw)
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.
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| dbo:abstract | 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. 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 k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the zero-locus is considered (that is, the points of the affine variety are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two. (en) 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) 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． アフィン多様体は，圏同値の違いを除いて，アフィンスキームすなわち環のスペクトルの特別な場合である．複素幾何学において，アフィン多様体はシュタイン多様体の類似である． (ja) 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) |
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| 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) |
| rdfs:label | Affine variety (en) Variété algébrique affine (fr) Varietà affine (it) アフィン多様体 (ja) |
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