Noether normalization lemma

In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926.^[1] A simple version states that for any field k, and any finitely generated commutative k-algebra A, there exists a nonnegative integer d and algebraically independent elements y₁, y₂, ..., y_d in A such that A is a finitely generated module over the polynomial ring S:=k[y₁, y₂, ..., y_d].

The integer d 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 ${\mathbb A}_{k}^{d}$, and A as 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 $X\to {\mathbb A}_{k}^{d}$. 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.^[2]

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.

Proof

The following proof is due to Nagata and is taken from Mumford's red book. A proof in the geometric flavor is also given in the page 127 of the red book and this mathoverflow thread.

The ring A in the lemma is generated as a k-algebra by elements, say, $y_{1},...,y_{m}$ such that $y_{1},...,y_{d}$ (some d) are algebraically independent over k and the rest are algebraic over $k[y_{1},...,y_{d}]$.^[3] We shall induct on m. If $m=d$, then the assertion is trivial. Assume now $m>d$. It is enough to show that there is a subring S of A that is generated by $m-1$ elements and is such that A is finite over S, for, by the inductive hypothesis, we can find algebraically independent elements $x_{1},...,x_{d}$ of S such that S is finite over $k[x_{1},...,x_{d}]$. Since $m>d$, 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

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

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, the highest term in $y_{1}$ of $ay_{1}^{{\alpha _{1}}}\prod _{2}^{m}(z_{i}+y_{1}^{{r^{{i-1}}}})^{{\alpha _{i}}},a\in k,$ looks

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

Thus, if r is larger than any exponent $\alpha _{i}$ appearing in f, then the highest term of $f(y_{1},z_{2}+y_{1}^{r},z_{3}+y_{1}^{{r^{2}}},...,z_{m}+y_{1}^{{r^{{m-1}}}})$ in $y_{1}$ also has the form as above. In other words, $y_{1}$ is integral over $S=k[z_{2},...,z_{m}]$. Since $y_{i}=z_{i}+y_{1}^{{r^{{i-1}}}}$ are also integral over that ring, A is integral over S. It follows A is finite over S.

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

Refinement

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 $I_{1}\subset \dots \subset I_{m}$ be a chain of ideals such that ${\displaystyle \operatorname {dim} (A/I_{i})=d_{i}>d_{i+1}.}$ Then there exists algebraically independent elements y₁, ..., y_d in A such that

1. A is a finitely generated module over the polynomial subring S = k[y₁, ..., y_d].
2. $I_{i}\cap S=(y_{{d_{i}+1}},\dots ,y_{d})$.
3. If the $I_{i}$'s are homogeneous, then y_i's may be taken to be homogeneous.

Moreover, if k is an infinite field, then any sufficiently general choice of y_I's has Property 1 above ("sufficiently general" is made precise in the proof).

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

Corollary — Let A be an integral domain that is a finitely generated algebra over a field. If ${\mathfrak {p}}$ is a prime ideal of A, then

$\dim A=\operatorname {height}{\mathfrak {p}}+\dim A/{\mathfrak {p}}$.

In particular, the Krull dimension of the localization of A at any maximal ideal is dim A.

Corollary — Let $A\subset B$ be integral domains that are finitely generated algebras over a field. Then

$\dim B=\dim A+\operatorname {tr.deg}_{{Q(A)}}Q(B)$

(the special case of Nagata's altitude formula).

Notes

1. Noether 1926
2. Eisenbud 1995, Theorem 13.3
3. cf. Exercise 16 in Ch. 5 of Atiyah-MacDonald, Introduction to commutative algebra.

References

• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001
• Hazewinkel, Michiel, ed. (2001), "Noether theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4. 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