A question about Klaus Hulek algebraic geometry (regarding Noether normalization)

Question

This is the proof of Noether normalization on p.30 of Klaus Hulek's elementary algebraic geometry.

And on the next page, the book says that "Analyzing the above proof, we see that y1, .., ym can be taken to be any 'general' choice of linear forms in a1, .., an"

But how is the fact that y1, .. , ym can be linear forms implied in the proof? I

Answer 1

First, let us clarify what a "linear form in the $a_i$" is. We say that $y\in A$ is a linear combination of the $a_i$ if there are certain $\gamma_1,\ldots,\gamma_n\in k$ such that $y=\gamma_1a_1+\cdots+\gamma_na_n$.

Now, you can prove by induction that the $y_i$ have this property. The induction start was done by setting $y_i=a_i$, which clearly satisfies the above condition. In the induction step, you get $y_1,\ldots,y_m$ which are (by induction hypothesis) linear forms in the $a'_i$. This means that there are certain $\delta_{i,1},\ldots,\delta_{i,n-1}\in k$ with $y_i=\delta_{i,1}a'_1+\cdots+\delta_{i,n-1}a'_{n-1}$. Since $a'_j=a_j-\alpha_ja_n$, we set

• $\gamma_{i,j}:=\delta_{i,j}$ for $1\le j\le n-1$ and
• $\gamma_{i,n}:=-\sum_{j=1}^{n-1} \delta_{ij}\alpha_j$.

We get
$$y_i = \sum_{j=1}^{n-1} \delta_{i,j} a_j - \delta_{i,j}\alpha_j a_n = \sum_{j=1}^n \gamma_{i,j} a_j.$$ Therefore, the $y_i$ are linear combinations in the $a_i$.

Does this answer your question or are you curious about why the $\gamma_{ij}$ can be chosen generically?