Commutative algebra | Algebraic varieties | Lemmas in algebra | Algebraic geometry
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. (Wikipedia).
Brill-Noether part 4: Noether's Theorem
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refs: arabello, cornalba, griffiths, harris
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