Noether normalization lemma

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In mathematics the Noether normalization lemma is a technical result of commutative algebra, introduced in (Noether 1926). In a simple version, it states that for any field K, and any commutative K-algebra A that is finitely generated over K and an integral domain, there are algebraically independent elements

y₁, y₂, ..., y_d,

in A, such that A is an integral extension of

K[y₁, y₂, ..., y_d] = B.

In geometric terms, B is the coordinate ring of affine space of dimension d, and A is the coordinate ring of some algebraic variety which must be of the same dimension (i.e. d is the transcendence degree of the field of fractions of A, and so is determined by A). The inclusion map of B into A gives as affine schemes a finite morphism

φ: Spec(A) → Spec(B).

The conclusion is that any variety is a branched covering of affine space.

[edit] References

• Noether, Emmy (1926), “Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p”, Nachr. Ges. Wiss. Göttingen: 28-35, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971>