Noether normalization lemma

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, ..., y_d 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, ..., y_d]. 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 ${\displaystyle \mathbb {A} _{k}^{d}}$. Then the inclusion map ${\displaystyle S\hookrightarrow A}$ induces a surjective finite morphism of affine varieties ${\displaystyle X\to \mathbb {A} _{k}^{d}}$: that is, 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: 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.