# Noether Normalization

The [Noether normalization lemma](https://www.wikiwand.com/en/Noether_normalization_lemma "Noether normalization lemma") says, in geometric terms, that every affine scheme _X_ of finite type over a field _k_ has a finite surjective morphism to affine space **A**_n_ over _k_, where _n_ is the dimension of _X_. Likewise, every [projective scheme](https://www.wikiwand.com/en/Projective_scheme "Projective scheme") _X_ over a field has a finite surjective morphism to [projective space](https://www.wikiwand.com/en/Projective_space "Projective space") **P**_n_, where _n_ is the dimension of _X_.
![Pasted image 20220114184700.png](attachments/Pasted%20image%2020220114184700.png)
![Pasted image 20220114184736.png](attachments/Pasted%20image%2020220114184736.png)

Proposition $4.5$ (Noether normalization). Any affine variety $X \subset \mathbb{A}^{n}$ of dimension $d$ admits a finite morphism $X \rightarrow \mathbb{A}^{d}$, which is the restriction to $X$ of a linear $\operatorname{map} \mathbb{A}^{n} \rightarrow \mathbb{A}^{d}$

Theorem $8.4$ (Noether normalization). For a projective variety $X \subset \mathbb{P}^{n}$ of dimension $d$ the following hold.
(a) There is a finite morphism $X \rightarrow \mathbb{P}^{d}$, which is the restriction to $X$ of a linear projection $\mathbb{P}^{n} \rightarrow \mathbb{P}^{d}$ with center at a linear subspace $\mathbb{P}^{k} \subset \mathbb{P}^{n}$ disjoint with $X$, where $k+d=n-1$.
(b) There exists a (unique) normal projective variety $X_{\text {norm }}$ and a finite birational morphism $\nu: X_{\text {norm }} \rightarrow X$ such that any morphism $f: X \rightarrow Y$, where $Y$ is normal, admits a unique lift to a morphism $\bar{f}: X_{\text {norm }} \rightarrow Y$ closing the commutative diagram (1).