Noether normalization lemma

Noether normalization lemma. Let $k$ be a field, and let $A$ be a finitely generated $k$ -algebra. Then there exist algebraically independent elements

y_1,\dots,y_d \in A

such that $A$ is integral over the polynomial subalgebra $k[y_1,\dots,y_d]$ . Equivalently, $A$ is a finitely generated module over $k[y_1,\dots,y_d]$ .

Equivalent characterizations

Equivalently, there is an injective $k$ -algebra homomorphism

k[t_1,\dots,t_d]\hookrightarrow A

over whose image $A$ is module-finite.

Remarks

The integer $d$ necessarily equals the Krull dimension of $A$ .

Examples

Polynomial rings normalize themselves. If $A=k[x_1,\dots,x_n]$ , take $y_i=x_i$ and $d=n$ . Then $A=k[y_1,\dots,y_n]$ , so $A$ is integral over the chosen polynomial subalgebra in the strongest possible way (equality).

A plane curve coordinate ring. Let $A = k[x,y]/(y^2-x^3-x).$ Let $\bar x,\bar y$ be the residue classes of $x,y$ in $A$ . Then $\bar y$ satisfies a monic polynomial over $k[\bar x]$ : $\bar y^2 - \bar x^3 - \bar x = 0.$ Hence $\bar y$ is integral over $k[\bar x]$ , and $A$ is integral over the polynomial subalgebra $k[\bar x]\cong k[t]$ (so here $d=1$ ).

A reducible example: $k[x,y]/(xy)$ . Let $A = k[x,y]/(xy).$ Set $u=\bar x+\bar y\in A$ (bars denote residue classes). Then $\bar x$ satisfies the monic equation $T^2-uT=0$ in $A[T]$ (since $\bar x^2-u\bar x=\bar x(\bar x-(\bar x+\bar y))=-\bar x\bar y=0$ ), so $\bar x$ is integral over $k[u]$ . Similarly, $\bar y$ is integral over $k[u]$ . Therefore $A$ is integral over the polynomial subalgebra $k[u]\cong k[t]$ .