About: Morphism of algebraic varieties

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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 regular function.A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well.

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rdfs:label	Morphismus (Varietät) (de) 多様体の射 (ja) Morphism of algebraic varieties (en) Морфізм алгебричних многовидів (uk)
rdfs:comment	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 regular function.A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well. (en) Ein Morphismus von Varietäten ist in der algebraischen Geometrie eine Abbildung von Varietäten mit bestimmten Regularitätseigenschaften. Ein Morphismus affiner Varietäten ist eine polynomiale Abbildung. Morphismen affiner Varietäten entsprechen eindeutig Homomorphismen ihrer Koordinatenringe. Die Definition kann auf quasiaffine, projektive und quasiprojektive Varietäten verallgemeinert werden, indem man Morphismen mit Hilfe regulärer Funktionen lokal definiert. Morphismen abstrakter Varietäten sind lokale Garbenmorphismen. (de) 代数幾何学においてアフィン多様体の間の写像が正則写像（せいそくしゃぞう、英: regular map）であるとは、それが多項式によって与えられることを言う。陽に書けば、X, Y がそれぞれアフィン多様体 An, Am の（あるいは代数的集合）であるとき、X から Y への正則写像 f は、各 fi が座標環 k[x1, …, xn]/I（I は X を定義するイデアル）に属するものとして、なる形に書ける。ゆえに像 f(X) は Y に含まれる（つまり、Y の定義方程式を満たす）。より一般に、抽象代数多様体間の写像 ƒ: X → Y が一点 x において正則 (regular at a point x)とは、x の近傍 U と f(x) の近傍 V が存在して、制限写像 ƒ: U → V が U と V との上の写像として正則となることを言う。さらに ƒ が X の任意の点において正則であるとき、ƒ は正則 (regular) であるという。上の有理函数が正則であるための必要十分条件は、それが極を持たぬことである。これはハルトークスの拡張定理の類似である。正則写像は定義によりアフィン多様体の圏における射である。特にアフィン多様体の間の正則写像は、その座標環の間の環準同型に反変的に一対一対応する。 f が代数多様体の間の射ならば、f の像はその閉包の稠密開集合を含む（を参照）。 (ja) В алгебричній геометрії, морфізмом між алгебричними многовидами називається відображення між многовидами, що локально є многочленом. Морфізми також називають регулярними відображеннями. Морфізм із алгебричного многовида в афінну пряму називається регулярною функцією. (uk)
rdfs:seeAlso	Morphism of schemes Degree of a continuous mapping Zariski's connectedness theorem
dcterms:subject	Algebraic varieties Types of functions Functions and mappings
Wikipage page ID	24457573 (xsd:integer)
Wikipage revision ID	1088910717 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ideal Morphism of schemes Normal variety Algebra homomorphism Algebraic function Algebraically closed field Algebraic varieties Types of functions Holomorphic function Resolution of singularities Integral domain Integrally closed domain Affine scheme Ringed space Complex analysis Continuous map Generic point Quasi-projective variety Generic freeness Constructible set (topology) Contraction morphism Liouville's theorem (complex analysis) Smooth function Complete variety Complex algebraic variety Function field of an algebraic variety Ideal (ring theory) Leray spectral sequence Maximal ideal Local diffeomorphism Sheaf of modules Riemann–Hurwitz formula Affine varieties Affine variety Algebraic curve Algebraic geometry Algebraic varieties Functions and mappings Flat morphism Abstract variety Noether's normalization lemma Birational Isomorphism Hartogs' extension theorem Projective line Projective variety Rational function Removable singularity Springer Science+Business Media Coordinate ring Étale morphism Image (mathematics) Finite morphism Residue field Smooth morphism Zariski's main theorem Affine line Universally catenary ring Upper-semicontinuous Frobenius morphism

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