# stacks/stacks-project

Improve a lemma

Thanks to Rankeya Datta
https://stacks.math.columbia.edu/tag/07NA#comment-4481
 @@ -27063,8 +27063,8 @@ \section{Noether normalization} \begin{lemma} \label{lemma-Noether-normalization-over-a-domain} Let $R \to S$ be an injective finite type map of domains. Then there exists an integer $d$ and a factorization Let $R \to S$ be an injective finite type ring map. Assume $R$ is a domain. Then there exists an integer $d$ and a factorization $$R \to R[y_1, \ldots, y_d] \to S' \to S$$ @@ -27083,15 +27083,20 @@ \section{Noether normalization} Lemma \ref{lemma-finite-is-integral}) we can find monic $P_i \in K[y_1, \ldots, y_d][T]$ such that $P_i(x_i) = 0$ in $S_K$. Let $f \in R$ be a nonzero element such that $fP_i \in R[y_1, \ldots, y_d][T]$ for all $i$. Set $x_i' = fx_i$ and let $S' \subset S$ be the subalgebra generated by $y_1, \ldots, y_d$ and $x'_1, \ldots, x'_n$. Note that $x'_i$ is integral over $fP_i \in R[y_1, \ldots, y_d][T]$ for all $i$. Then $fP_i(x_i)$ maps to zero in $S_K$. Hence after replacing $f$ by another nonzero element of $R$ we may also assume $fP_i(x_i)$ is zero in $S$. Set $x_i' = fx_i$ and let $S' \subset S$ be the $R$-subalgebra generated by $y_1, \ldots, y_d$ and $x'_1, \ldots, x'_n$. Note that $x'_i$ is integral over $R[y_1, \ldots, y_d]$ as we have $Q_i(x_i') = 0$ where $Q_i = f^{\deg_T(P_i)}P_i(T/f)$ which is a monic polynomial in $T$ with coefficients in $R[y_1, \ldots, y_d]$ by our choice of $f$. Hence $R[y_1, \ldots, y_n] \subset S'$ is finite by Lemma \ref{lemma-characterize-finite-in-terms-of-integral}. By construction $S'_f \cong S_f$ and we win. Since $S' \subset S$ we have $S'_f \subset S_f$ (localization is exact). On the other hand, the elements $x_i = x'_i/f$ in $S'_f$ generate $S_f$ over $R_f$ and hence $S'_f \to S_f$ is surjective. Whence $S'_f \cong S_f$ and we win. \end{proof}