What does Noether's Normalization lemma even mean?

Question

Noether's Normalization lemma states that

For any field $k$, and any finitely generated commutative $k$-algebra $A$, there exists a non-negative integer $d$ and algebraically independent elements $y_1,y_2,\dots,y_d\in A$ such that $A$ is a finitely generated module over the polynomial ring $k[y_1,y_2,\dots,y_d]$.

I'm trying to understand the statement of this lemma. A finitely generated commutative $k$-algebra would be $k[x_1,x_2,\dots,x_n]$. Clearly, $x_1,x_2,\dots,x_n$ are algebraically independent. Aren't we essentially saying that $k[x_1,x_2,\dots,x_n]$ is a finitely generated module over $k[x_1,x_2,\dots,x_n]$? What is there to prove in that?

Answer 1

It is important to know that finitely generated $k$-algebras are not just of the form $k[x_1,...,x_n]$. They can also be quotients of these rings by ideals. For instance $A=\mathbb{C}[x,y]/(x^2-y^3)$. This is a $\mathbb{C}$-algebra, but will not be isomorphic to any polynomial ring over $\mathbb{C}$. One way to see this is that its localization at the maximal ideal $(x,y)$ will not be integrally closed in the fraction field of $A$. The element $x/y$, for instance, will be integral over $A_{(x,y)}$ but not actually in $A$.

However, if we localize any polynomial ring $P$ over $\mathbb{C}$ at a maximal ideal $m$, we find that $P_m$ is integrally closed in the fraction field of $P$.

This may not be the best way to see that $A$ is not a polynomial ring, but it is very geometric. The ring $A$ is the coordinate ring of the plane curve given by $x^2 = y^3$. This curve has a cusp at the origin and is quite singular. This geometric fact is somehow wrapped up in the algebra of the local ring $A_{(x,y)}$. Any polynomial ring is the coordinate ring of a certain affine space. Affine spaces are nice and smooth, and this fact is again reflected somehow in the algebra of its localizations at maximal ideals.

What normalization tells us is that finitely $k$-algebras are reasonable objects. They are integral extensions (very nice extensions) of polynomial rings (very nice rings).

One geometric consequence (at least over an algebraically closed field) that normalization gives us is that any algebraic set is a cover of an affine space. Normalization gives us some inclusion of a polynomial ring $P=k[y_1,...,y_d]$ into the ring $A$. If we think of this geometrically, $A$ is the coordinate ring of some variety, and $P=k[y_1,...,y_d]$ is the coordinate ring of some $d$-dimensional linear subspace. The geometric picture is the reversal of the algebraic picture, in that an inclusion of $P$ into $A$ means the variety associated with $A$ projects (surjectively) onto the linear space associated with $P$. An important consequence of this is the Nullstellensatz.

Answer 2

Even though the question already got answered, let me share my way of understanding this lemma since it might be helpful for other people.

I will illustrate it with an example. Suppose you have the $\mathbb{Q}$-algebra $\mathbb{Q}[\sqrt{2}, \pi]$, which is finitely generated (by $\sqrt{2}$ and $\pi$). You can think of it as the quotient $\mathbb{Q}(X, Y)/(X^2-2)$, but I will write it the other way since it becomes more intuitive.

Noether's normalization lemma says that you can write it as a finitely generated $\mathbb{Q}[z_1, \dots, z_d]$-module, with $z_1, \dots, z_d$ algebraically independent. In our specific example, we can write $\mathbb{Q}[\sqrt{2},\pi]$ as a finitely generated $\mathbb{Q}[\pi]$-module in the following way: $$ \mathbb{Q}[\sqrt{2}, \pi] = \mathbb{Q}[\pi] \oplus \mathbb{Q}[\pi] \sqrt{2}.$$

What this is actually doing is separating the transcendental part of $\mathbb{Q}[\sqrt{2}, \pi]$ and inserting it into the coefficients. Then, since the original algebra will be algebraic over $\mathbb{Q}[\pi]$, it will be finitely generated (as a module over these coefficients).

In general, if you have a finitely generated $\mathbb{K}$-algebra, which we can write as $\mathbb{K}[a_1, \dots, a_n]$, we can insert its transcendental part into the coefficients and hence obtain the coefficient ring $\mathbb{K}[z_1, \dots, z_d]$. Everything that's left will be algebraic, so $\mathbb{K}[a_1, \dots, a_n]$ will be a finitely generated $\mathbb{K}[z_1, \dots, z_d]$-module.