Noether normalization lemma

E29377

mathematical theorem result in algebraic geometry result in commutative algebra

The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.

Observed surface forms (3)

Surface form	Occurrences
Noether's normalization lemma	1
Zariski’s lemma	1
graded Noether normalization	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in algebraic geometry ⓘ result in commutative algebra ⓘ
appearsIn	standard graduate textbooks on algebraic geometry ⓘ standard graduate textbooks on commutative algebra ⓘ
appliesTo	finitely generated k-algebras ⓘ
assumes	base field k ⓘ commutative rings with identity ⓘ
concerns	affine schemes ⓘ affine varieties ⓘ finitely generated algebras over a field ⓘ integral extensions of rings ⓘ polynomial rings ⓘ
field	algebraic geometry ⓘ commutative algebra ⓘ
framework	affine algebraic geometry ⓘ scheme theory ⓘ
hasConsequence	coordinate ring of an affine variety is module-finite over a polynomial subring ⓘ every affine variety over a field admits a finite dominant morphism to affine space of dimension equal to its Krull dimension ⓘ existence of generic linear projections with finite fibers ⓘ
hasVariant	equivariant Noether normalization ⓘ Noether normalization lemma self-linksurface differs ⓘ surface form: graded Noether normalization version for Noetherian rings ⓘ
holdsOver	any field ⓘ
implies	Krull dimension of a finitely generated k-algebra equals the transcendence degree of its field of fractions over k ⓘ any affine k-algebra is a finite module over a polynomial subring ⓘ any affine variety admits a finite surjective morphism to an affine space of the same dimension ⓘ existence of a finite injective k-algebra homomorphism from a polynomial ring to a finitely generated k-algebra ⓘ
namedAfter	Emmy Noether ⓘ
relatedTo	Hilbert basis theorem ⓘ Hilbert’s Nullstellensatz ⓘ Krull dimension ⓘ algebraic independence ⓘ finite morphisms of schemes ⓘ integral dependence ⓘ
states	every finitely generated k-algebra is integral over a polynomial k-subalgebra ⓘ if A is a finitely generated k-algebra then there exist algebraically independent elements y1,…,yd in A such that A is integral over k[y1,…,yd] ⓘ
typicalProofUses	induction on the number of generators ⓘ integral dependence and minimal polynomials ⓘ linear changes of variables ⓘ
usedFor	computing Krull dimension ⓘ dimension theory in algebraic geometry ⓘ establishing finiteness properties of morphisms of varieties ⓘ proving Hilbert’s Nullstellensatz ⓘ reducing problems on affine varieties to problems on affine space ⓘ relating affine varieties to affine space ⓘ structure theory of finitely generated algebras ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Noether normalization lemma → hasVariant → Noether normalization lemma self-linksurface differs ⓘ

this entity surface form: graded Noether normalization

Emmy → knownFor → Noether normalization lemma ⓘ

subject surface form: Emmy Noether

Emmy Noether → notableWork → Noether normalization lemma ⓘ

Hilbert basis theorem → relatedTo → Noether normalization lemma ⓘ

Hilbert’s Nullstellensatz → relatedTo → Noether normalization lemma ⓘ

this entity surface form: Zariski’s lemma

Hilbert’s Nullstellensatz → relatedTo → Noether normalization lemma ⓘ

Noether's problem → relatedTo → Noether normalization lemma ⓘ

this entity surface form: Noether's normalization lemma