In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926.
[1] It states that for any field k, and any finitely generated commutative k-algebra A, there exist elements y1, y2, ..., yd in A that are algebraically independent over k and such that A is a finitely generated module over the polynomial ring S = k [y1, y2, ..., yd].
The integer d is equal to the Krull dimension of the ring A; and if 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 the coordinate ring of an affine variety X, and consider S as the coordinate ring of a d-dimensional affine space
induces a surjective finite morphism of affine varieties
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: 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 corresponding dimensions.
[2] The Noether normalization lemma can be used as an important step in proving Hilbert's Nullstellensatz, one of the most fundamental results of classical algebraic geometry.
The normalization theorem is also an important tool in establishing the notions of Krull dimension for k-algebras.
If A is an integral domain, then d is also the transcendence degree of the field of fractions of A over k. The following proof is due to Nagata and appears in Mumford's red book.
A more geometric proof is given on page 176 of the red book.
elements, such that A is finite over S. Indeed, by the inductive hypothesis, we can find, for some integer d,
are not algebraically independent, so that there is a nonzero polynomial f in m variables over k such that Given an integer r which is determined later, set and, for simplification of notation, write
is a monomial appearing in the left-hand side of the above equation, with coefficient
after expanding the product looks like Whenever the above exponent agrees with the highest
exponent produced by some other monomial, it is possible that the highest term in
encodes a unique base r number, so this does not occur.
, A is in fact finite over S. This completes the proof of the claim, so we are done with the first part.
Moreover, if A is an integral domain, then d is the transcendence degree of its field of fractions.
have the same transcendence degree (i.e., the degree of the field of fractions) since the field of fractions of A is algebraic over that of S (as A is integral over S) and S has transcendence degree d. Thus, it remains to show the Krull dimension of S is d. (This is also a consequence of dimension theory.)
is a chain of prime ideals, the dimension is at least d. To get the reverse estimate, let
(in the normalization process, we're free to choose the first variable) such that S is integral over T. By the inductive hypothesis,
The following refinement appears in Eisenbud's book, which builds on Nagata's idea:[2] Theorem — Let A be a finitely generated algebra over a field k, and
Then there exists algebraically independent elements y1, ..., yd in A such that Moreover, if k is an infinite field, then any sufficiently general choice of yI's has Property 1 above ("sufficiently general" is made precise in the proof).
Corollary — Let A be an integral domain that is a finitely generated algebra over a field.
be integral domains that are finitely generated algebras over a field.
Then (the special case of Nagata's altitude formula).
A typical nontrivial application of the normalization lemma is the generic freeness theorem: Let
is a Noetherian integral domain and suppose there is a ring homomorphism