Noether normalization lemma

The following proof is due to Nagata, following Mumford's red book. A more geometric proof is given on page 127 of the red book.

The ring A in the lemma is generated as a k-algebra by some elements, ${\displaystyle y_{1},...,y_{m}}$. We shall induct on m. If ${\displaystyle m=0}$, then the assertion is trivial. Assume now ${\displaystyle m>0}$. It is enough to show that there is a subring S of A that is generated by ${\displaystyle m-1}$ elements, such that A is finite over S. Indeed, by the inductive hypothesis, we can find algebraically independent elements ${\displaystyle x_{1},...,x_{d}}$ of S such that S is finite over ${\displaystyle k[x_{1},...,x_{d}]}$.

Since otherwise there would be nothing to prove, we can also assume that there is a nonzero polynomial f in m variables over k such that

${\displaystyle f(y_{1},\ldots ,y_{m})=0}$.

Given an integer r which is determined later, set

${\displaystyle z_{i}=y_{i}-y_{1}^{r^{i-1}},\quad 2\leq i\leq m.}$

Then the preceding reads:

${\displaystyle f(y_{1},z_{2}+y_{1}^{r},z_{3}+y_{1}^{r^{2}},\ldots ,z_{m}+y_{1}^{r^{m-1}})=0}$.

Now, if ${\displaystyle ay_{1}^{\alpha _{1}}\prod _{2}^{m}(z_{i}+y_{1}^{r^{i-1}})^{\alpha _{i}}}$ is a monomial appearing in the left-hand side of the above equation, with coefficient ${\displaystyle a\in k}$, the highest term in ${\displaystyle y_{1}}$ after expanding the product looks like

${\displaystyle ay_{1}^{\alpha _{1}+r\alpha _{2}+\cdots +\alpha _{m}r^{m-1}}.}$

Whenever the above exponent agrees with the highest ${\displaystyle y_{1}}$ exponent produced by some other monomial, it is possible that the highest term in ${\displaystyle y_{1}}$ of ${\displaystyle f(y_{1},z_{2}+y_{1}^{r},z_{3}+y_{1}^{r^{2}},...,z_{m}+y_{1}^{r^{m-1}})}$ will not be of the above form, because it may be affected by cancellation. However, if r is larger than any exponent appearing in f, then each ${\displaystyle \alpha _{1}+r\alpha _{2}+\cdots +\alpha _{m}r^{m-1}}$ encodes a unique base r number, so this does not occur. Thus ${\displaystyle y_{1}}$ is integral over ${\displaystyle S=k[z_{2},...,z_{m}]}$. Since ${\displaystyle y_{i}=z_{i}+y_{1}^{r^{i-1}}}$ are also integral over that ring, A is integral over S. It follows A is finite over S, and since S is generated by m-1 elements, by the inductive hypothesis we are done.

If A is an integral domain, then d is the transcendence degree of its field of fractions. Indeed, A and ${\displaystyle S=k[y_{1},...,y_{d}]}$ 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 the polynomial ring S is d. (This is also a consequence of dimension theory.) We induct on d, with the case ${\displaystyle d=0}$ being trivial. Since ${\displaystyle 0\subsetneq (y_{1})\subsetneq (y_{1},y_{2})\subsetneq \cdots \subsetneq (y_{1},\dots ,y_{d})}$ is a chain of prime ideals, the dimension is at least d. To get the reverse estimate, let ${\displaystyle 0\subsetneq {\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{m}}$ be a chain of prime ideals. Let ${\displaystyle 0\neq u\in {\mathfrak {p}}_{1}}$. We apply the noether normalization and get ${\displaystyle T=k[u,z_{2},\dots ,z_{d}]}$ (in the normalization process, we're free to choose the first variable) such that S is integral over T. By the inductive hypothesis, ${\displaystyle T/(u)}$ has dimension d - 1. By incomparability, ${\displaystyle {\mathfrak {p}}_{i}\cap T}$ is a chain of length ${\displaystyle m}$ and then, in ${\displaystyle T/({\mathfrak {p}}_{1}\cap T)}$, it becomes a chain of length ${\displaystyle m-1}$. Since ${\displaystyle \operatorname {dim} T/({\mathfrak {p}}_{1}\cap T)\leq \operatorname {dim} T/(u)}$, we have ${\displaystyle m-1\leq d-1}$. Hence, ${\displaystyle \dim S\leq d}$.

The following refinement appears in Eisenbud's book, which builds on Nagata's idea:[2]

Geometrically speaking, the last part of the theorem says that for ${\displaystyle X=\operatorname {Spec} A\subset \mathbf {A} ^{m}}$ any general linear projection ${\displaystyle \mathbf {A} ^{m}\to \mathbf {A} ^{d}}$ induces a finite morphism ${\displaystyle X\to \mathbf {A} ^{d}}$ (cf. the lede); besides Eisenbud, see also .

A typical nontrivial application of the normalization lemma is the generic freeness theorem: Let ${\displaystyle A,B}$ be rings such that ${\displaystyle A}$ is a Noetherian integral domain and suppose there is a ring homomorphism ${\displaystyle A\to B}$ that exhibits ${\displaystyle B}$ as a finitely generated algebra over ${\displaystyle A}$. Then there is some ${\displaystyle 0\neq g\in A}$ such that ${\displaystyle B[g^{-1}]}$ is a free ${\displaystyle A[g^{-1}]}$-module.

To prove this, let ${\displaystyle F}$ be the fraction field of ${\displaystyle A}$. We argue by induction on the Krull dimension of ${\displaystyle F\otimes _{A}B}$. The base case is when the Krull dimension is ${\displaystyle -\infty }$; i.e., ${\displaystyle F\otimes _{A}B=0}$; that is, when there is some ${\displaystyle 0\neq g\in A}$ such that ${\displaystyle gB=0}$ , so that ${\displaystyle B[g^{-1}]}$ is free as an ${\displaystyle A[g^{-1}]}$-module. For the inductive step, note that ${\displaystyle F\otimes _{A}B}$ is a finitely generated ${\displaystyle F}$-algebra. Hence by the Noether normalization lemma, ${\displaystyle F\otimes _{A}B}$ contains algebraically independent elements ${\displaystyle x_{1},\dots ,x_{d}}$ such that ${\displaystyle F\otimes _{A}B}$ is finite over the polynomial ring ${\displaystyle F[x_{1},\dots ,x_{d}]}$. Multiplying each ${\displaystyle x_{i}}$ by elements of ${\displaystyle A}$, we can assume ${\displaystyle x_{i}}$ are in ${\displaystyle B}$. We now consider:

${\displaystyle A':=A[x_{1},\dots ,x_{d}]\to B.}$

Now ${\displaystyle B}$ may not be finite over ${\displaystyle A'}$, but it will become finite after inverting a single element as follows. If ${\displaystyle b}$ is an element of ${\displaystyle B}$, then, as an element of ${\displaystyle F\otimes _{A}B}$, it is integral over ${\displaystyle F[x_{1},\dots ,x_{d}]}$; i.e., ${\displaystyle b^{n}+a_{1}b^{n-1}+\dots +a_{n}=0}$ for some ${\displaystyle a_{i}}$ in ${\displaystyle F[x_{1},\dots ,x_{d}]}$. Thus, some ${\displaystyle 0\neq g\in A}$ kills all the denominators of the coefficients of ${\displaystyle a_{i}}$ and so ${\displaystyle b}$ is integral over ${\displaystyle A'[g^{-1}]}$. Choosing some finitely many generators of ${\displaystyle B}$ as an ${\displaystyle A'}$-algebra and applying this observation to each generator, we find some ${\displaystyle 0\neq g\in A}$ such that ${\displaystyle B[g^{-1}]}$ is integral (thus finite) over ${\displaystyle A'[g^{-1}]}$. Replace ${\displaystyle B,A}$ by ${\displaystyle B[g^{-1}],A[g^{-1}]}$ and then we can assume ${\displaystyle B}$ is finite over ${\displaystyle A':=A[x_{1},\dots ,x_{d}]}$. To finish, consider a finite filtration ${\displaystyle B=B_{0}\supset B_{1}\supset B_{2}\supset \cdots \supset B_{r}}$ by ${\displaystyle A'}$-submodules such that ${\displaystyle B_{i}/B_{i+1}\simeq A'/{\mathfrak {p}}_{i}}$ for prime ideals ${\displaystyle {\mathfrak {p}}_{i}}$ (such a filtration exists by the theory of associated primes). For each i, if ${\displaystyle {\mathfrak {p}}_{i}\neq 0}$, by inductive hypothesis, we can choose some ${\displaystyle g_{i}\neq 0}$ in ${\displaystyle A}$ such that ${\displaystyle A'/{\mathfrak {p}}_{i}[g_{i}^{-1}]}$ is free as an ${\displaystyle A[g_{i}^{-1}]}$-module, while ${\displaystyle A'}$ is a polynomial ring and thus free. Hence, with ${\displaystyle g=g_{0}\cdots g_{r}}$, ${\displaystyle B[g^{-1}]}$ is a free module over ${\displaystyle A[g^{-1}]}$. ${\displaystyle \square }$

• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001
• "Noether theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]. NB the lemma is in the updating comments.
• Noether, Emmy (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen: 28–35, archived from the original on March 8, 2013

• Robertz, D.: Noether normalization guided by monomial cone decompositions. J. Symbolic Comput. 44(10), 1359–1373 (2009)