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- Noether_normalization_lemma abstract "In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. A simple version states that for any field k, and any finitely generated commutative k-algebra A, there exists a nonnegative 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 is the Krull dimension of A (since A and S have the same dimension.) When A is an integral domain, d is the transcendence degree of the field of fractions of A over k.The theorem has geometric interpretation. Suppose A is integral. Let S be the coordinate ring of d-dimensional affine space , and A as 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 prime ideals of R (equivalently, irreducible 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.".
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- Noether_normalization_lemma wikiPageExternalLink 42363.
- Noether_normalization_lemma wikiPageID "3003553".
- Noether_normalization_lemma wikiPageRevisionID "597949663".
- Noether_normalization_lemma hasPhotoCollection Noether_normalization_lemma.
- Noether_normalization_lemma id "n/n066790".
- Noether_normalization_lemma title "Noether theorem".
- Noether_normalization_lemma subject Category:Algebraic_geometry.
- Noether_normalization_lemma subject Category:Algebraic_varieties.
- Noether_normalization_lemma subject Category:Commutative_algebra.
- Noether_normalization_lemma subject Category:Lemmas.
- Noether_normalization_lemma type Abstraction100002137.
- Noether_normalization_lemma type AlgebraicVarieties.
- Noether_normalization_lemma type Assortment108398773.
- Noether_normalization_lemma type Collection107951464.
- Noether_normalization_lemma type Communication100033020.
- Noether_normalization_lemma type Group100031264.
- Noether_normalization_lemma type Lemma106751833.
- Noether_normalization_lemma type Lemmas.
- Noether_normalization_lemma type Message106598915.
- Noether_normalization_lemma type Proposition106750804.
- Noether_normalization_lemma type Statement106722453.
- Noether_normalization_lemma comment "In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926.".
- Noether_normalization_lemma label "Lemma di normalizzazione di Noether".
- Noether_normalization_lemma label "Lemme de normalisation de Noether".
- Noether_normalization_lemma label "Noether normalization lemma".
- Noether_normalization_lemma label "Noetherscher Normalisierungssatz".
- Noether_normalization_lemma label "諾特正規化引理".
- Noether_normalization_lemma sameAs Noetherscher_Normalisierungssatz.
- Noether_normalization_lemma sameAs Lemme_de_normalisation_de_Noether.
- Noether_normalization_lemma sameAs Lemma_di_normalizzazione_di_Noether.
- Noether_normalization_lemma sameAs m.08jz00.
- Noether_normalization_lemma sameAs Q1287074.
- Noether_normalization_lemma sameAs Q1287074.
- Noether_normalization_lemma sameAs Noether_normalization_lemma.
- Noether_normalization_lemma wasDerivedFrom Noether_normalization_lemma?oldid=597949663.
- Noether_normalization_lemma isPrimaryTopicOf Noether_normalization_lemma.