Noether Normalization

Question

I teach a course of undergraduate algebraic geometry. I noticed that students have difficulty grasping the proof of Noether normalizsation Lemma (given in Reid's undergraduate algebraic geometry). Also, the proof given in Mumford red book is also non-elementary. Does anyone know of some easier proof of the result? Thank you

Answer 1

Here is the proof from my class notes last time I taught the course. I don't remember where I borrowed it from. Hope it helps. It is tedious but elementary in the sense of Martin's comment. It is the special case where the base field is infinite. It might become clearer if you give the various steps as exercises to your class.

Noether normalization lemma: Given any infinite field $k$, and any finitely generated $k$-algebra $B$, then we can decompose the extension $k$ in $B$, into $k$ in $A$ in $B$ where $A$ is isomorphic to a polynomial algebra over $k$, and $B$ is a module-finite extension of $A$.

Proof: Proof by induction on the number of $k$-algebra generators of $B$ over $k$. First do the case of one generator. If $B = k[c]$ and $c$ is transcendental over $k$ we are done. If $c$ is algebraic over $k$ then $B\cong k[T]/(f(T))$ where $f$ is the monic irreducible polynomial of least degree $d$ satisfied by $c$ over $k$. Then $B = k[c]$ is generated over $k$ by the elements $1,c,…,c_{d-1}$, and in particular $B$ is a finite vector space over $k$. So take $A = k$, and we are done.

Now assume $B = k[c_1,...,c_n]$ where $n\geq 2$. Consider the canonical surjective $k$-algebra map $k[T_1,...,T_n]\to B$ taking $T_i\mapsto c_i$. If this map is also injective we are done, since then we may take $A = B$.

If this map is not injective then some non-constant polynomial $f$ maps to zero. If $f$, considered as a polynomial in $T_n$, is monic of degree $d\geq1$, then $k[c_1,\dots,c_n]$ is generated as a module over $k[c_1,\dots,c_{n-1}]$ by the finite set of monomials of degree less than $d$ in $c_n$. In this case we are done, since by the inductive hypothesis $k[c_1,…,c_{n-1}]$ is module finite over some subalgebra isomorphic to a polynomial algebra over $k$. Then by transitivity of finiteness, $B = k[c_1,...,c_n]$ is also finite over that same polynomial subalgebra of $B$.

We claim we can “change coordinates” linearly in $k[T_1,\dots,T_n]$, until the polynomial $f$ becomes monic in $T_n$. That will mean there is another $k$-algebra surjection $$k[T_1,\dots,T_n]\to B=k[e_1,\dots,e_n],$$taking $T_i\mapsto e_i$ such that some element of the kernel is monic in $T_n$. Then we are done by the same inductive argument. So all we have to do is the change of variables argument, sometimes known as the Weierstrass preparation lemma.

Let $f(T_1,\dots,T_n)$ be any non constant polynomial, and consider the linear change of coordinates $$T_i\mapsto(T_i+a_iT_n), 1≤i≤n-1,\quad \text{and}\quad T_n\mapsto T_n,$$ where the $a_i$’s are elements of $k$. Write $f$ as $f^d + $ lower degree terms, where $d =$ degree $f$, and $f^d$ is homogeneous of degree $d$.

Under this transformation, $f\mapsto f(T_1+a_1T_n,\dots,T_{n-1}+a_{n-1}T_n, T_n)$. We claim it is possible to choose the coefficients $a_i$ so that this new polynomial has non zero coefficient of $T_n^d$. Just expand, and get $$f(T_1+a_1T_n,\dots,T_{n-1}+a_{n-1}T_n, T_n)=f^d(T_1+a_1T_n,\dots,T_{n-1}+a_{n-1}T_n, T_n) + \text{lower degree terms}.$$

Then expand the top degree term and get
$$f^d(T_1+a_1T_n,\dots,T_{n-1}+a_{n-1}T_n, T_n) = f^d(a_1,\dots,a_{n-1}, 1)T_n^d + \text{lower degree terms in }T_n.$$
Adding gives $$f(T_1+a_1T_n,\dots,T_{n-1}+a_{n-1}T_n, T_n) = f^d(a_1,\dots,a_{n-1}, 1)T_n^d + \text{lower degree terms in }T_n.$$
Thus we only have to choose the $a_i$’s so that $f^d(a_1,\dots,a_{n-1}, 1)\neq0$. Since $f^d$ is a non-zero homogeneous polynomial of degree $d\geq 1$, $f^d(a_1,\dots,a_{n-1}, 1)$ is a non-zero polynomial of degree less than or equal to $d$ in the $a_i$’s. By induction on $d$, since our field $k$ is infinite, it does not vanish for all choices of the $a_i$’s. QED.