In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exists a non-negative integer d and algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k [y1, y2, ..., yd]. The integer d above is uniquely determined; it is the Krull dimension of the ring A. When A is an integral domain, d is also the transcendence degree of the field of fractions of A over k.
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| - Noetherscher Normalisierungssatz (de)
- Lemma di normalizzazione di Noether (it)
- Lemme de normalisation de Noether (fr)
- Noether normalization lemma (en)
- Лемма Нётер о нормализации (ru)
- 諾特正規化引理 (zh)
- Нормалізаційна лема Нетер (uk)
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| - Der noethersche Normalisierungssatz (oder auch noethersches Normalisierungslemma) (nach Emmy Noether) ist eine Strukturaussage aus dem mathematischen Teilgebiet der kommutativen Algebra. In geometrischer Sprache besagt er, dass es von einem geometrischen Objekt stets eine Abbildung in einen affinen Raum gibt, deren Fasern endlich sind. Dieser Artikel beschäftigt sich mit kommutativer Algebra. Insbesondere sind alle betrachteten Ringe kommutativ und haben ein Einselement. Für weitere Details siehe Kommutative Algebra. (de)
- En algèbre commutative, le lemme de normalisation de Noether, dû à la mathématicienne allemande Emmy Noether, donne une description des algèbres de type fini sur un corps. On fixe une algèbre commutative de type fini A sur un corps (commutatif) K. (fr)
- In matematica, il lemma di normalizzazione di Noether è un teorema dell'algebra commutativa che afferma che ogni -algebra finitamente generata (dove è un campo) è un'estensione intera di un anello di polinomi su . Prende nome da Emmy Noether, che nel 1926 lo dimostrò sotto l'ipotesi che fosse infinito. Il caso in cui è un campo finito fu dimostrato da Oscar Zariski nel 1943. (it)
- Лемма Нётер о нормализации — результат коммутативной алгебры играющий важную роль в основаниях алгебраической геометрии.Доказанa Эмми Нётер в 1926 году. Эта лемма используется в доказательстве теоремы Гильберта о нулях.Также она является важным инструментом изучения размерности Крулля. (ru)
- 在交換代數中，諾特正規化引理是一個技術性的定理，以德國數學家埃米·諾特命名。其內容如下： 設 為域， 是有限生成的 -代數，且 是整環，則存在 ，使得 在 上彼此代數獨立，且 是 的整擴張。 它的一個重要幾何結論之一是：任一射影簇均可表為仿射空間的。 (zh)
- Нормалізаційна лема Нетер — результат комутативної алгебри, що використовується при доведенні теореми Гільберта про нулі. Названа на честь Еммі Нетер. (uk)
- In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exists a non-negative integer d and algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k [y1, y2, ..., yd]. The integer d above is uniquely determined; it is the Krull dimension of the ring A. When A is an integral domain, d is also the transcendence degree of the field of fractions of A over k. (en)
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| - Der noethersche Normalisierungssatz (oder auch noethersches Normalisierungslemma) (nach Emmy Noether) ist eine Strukturaussage aus dem mathematischen Teilgebiet der kommutativen Algebra. In geometrischer Sprache besagt er, dass es von einem geometrischen Objekt stets eine Abbildung in einen affinen Raum gibt, deren Fasern endlich sind. Dieser Artikel beschäftigt sich mit kommutativer Algebra. Insbesondere sind alle betrachteten Ringe kommutativ und haben ein Einselement. Für weitere Details siehe Kommutative Algebra. (de)
- In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exists a non-negative integer d and algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k [y1, y2, ..., yd]. The integer d above is uniquely determined; it is the Krull dimension of the ring A. When A is an integral domain, d is also the transcendence degree of the field of fractions of A over k. The theorem has a geometric interpretation. Suppose A is integral. Let S be the coordinate ring of the d-dimensional affine space , and let A be the coordinate ring of some other d-dimensional affine variety X. Then the inclusion map S → A induces a surjective finite morphism of affine varieties . The conclusion is that any affine variety is a branched covering of affine space.When k is infinite, such a branched covering map can be constructed by taking a general projection from an affine space containing X to a d-dimensional subspace. More generally, in the language of schemes, the theorem can equivalently be stated as follows: every affine k-scheme (of finite type) X is finite over an affine n-dimensional space. The theorem can be refined to include a chain of ideals of R (equivalently, closed subsets of X) that are finite over the affine coordinate subspaces of the appropriate dimensions. The form of the Noether normalization lemma stated above can be used as an important step in proving Hilbert's Nullstellensatz. This gives it further geometric importance, at least formally, as the Nullstellensatz underlies the development of much of classical algebraic geometry. The theorem is also an important tool in establishing the notions of Krull dimension for k-algebras. (en)
- En algèbre commutative, le lemme de normalisation de Noether, dû à la mathématicienne allemande Emmy Noether, donne une description des algèbres de type fini sur un corps. On fixe une algèbre commutative de type fini A sur un corps (commutatif) K. (fr)
- In matematica, il lemma di normalizzazione di Noether è un teorema dell'algebra commutativa che afferma che ogni -algebra finitamente generata (dove è un campo) è un'estensione intera di un anello di polinomi su . Prende nome da Emmy Noether, che nel 1926 lo dimostrò sotto l'ipotesi che fosse infinito. Il caso in cui è un campo finito fu dimostrato da Oscar Zariski nel 1943. (it)
- Лемма Нётер о нормализации — результат коммутативной алгебры играющий важную роль в основаниях алгебраической геометрии.Доказанa Эмми Нётер в 1926 году. Эта лемма используется в доказательстве теоремы Гильберта о нулях.Также она является важным инструментом изучения размерности Крулля. (ru)
- 在交換代數中，諾特正規化引理是一個技術性的定理，以德國數學家埃米·諾特命名。其內容如下： 設 為域， 是有限生成的 -代數，且 是整環，則存在 ，使得 在 上彼此代數獨立，且 是 的整擴張。 它的一個重要幾何結論之一是：任一射影簇均可表為仿射空間的。 (zh)
- Нормалізаційна лема Нетер — результат комутативної алгебри, що використовується при доведенні теореми Гільберта про нулі. Названа на честь Еммі Нетер. (uk)
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