# The Stacks Project

## Tag: 00SV

This tag has label algebra-lemma-relative-global-complete-intersection-conormal and it points to

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Lemma 9.128.13. Let $R$ be a ring. Let $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ be a relative global complete intersection. For every prime $\mathfrak q$ of $S$, let $\mathfrak q'$ denote the corresponding prime of $R[x_1, \ldots, x_n]$. Then
1. $f_1, \ldots, f_c$ is a regular sequence in the local ring $R[x_1, \ldots, x_n]_{\mathfrak q'}$,
2. each of the rings $R[x_1, \ldots, x_n]_{\mathfrak q'}/(f_1, \ldots, f_i)$ is flat over $R$, and
3. the $S$-module $(f_1, \ldots, f_c)/(f_1, \ldots, f_c)^2$ is free with basis given by the elements $f_i \bmod (f_1, \ldots, f_c)^2$.

Proof. First, by Lemma 9.68.2, part (3) follows from part (1). Parts (1) and (2) immediately reduce to the Noetherian case by Lemma 9.128.12 (some minor details omitted). Assume $R$ is Noetherian. By Lemma 9.127.4 for example we see that $f_1, \ldots, f_c$ form a regular sequence in the local ring $R[x_1, \ldots, x_n]_{\mathfrak q'} \otimes_R \kappa(\mathfrak p)$. Moreover, the local ring $R[x_1, \ldots, x_n]_{\mathfrak q'}$ is flat over $R_{\mathfrak p}$. Since $R$, and hence $R[x_1, \ldots, x_n]_{\mathfrak q'}$ is Noetherian we may apply Lemma 9.94.3 to conclude. $\square$

\begin{lemma}
\label{lemma-relative-global-complete-intersection-conormal}
Let $R$ be a ring. Let $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$
be a relative global complete intersection. For every prime
$\mathfrak q$ of $S$, let $\mathfrak q'$ denote the corresponding
prime of $R[x_1, \ldots, x_n]$. Then
\begin{enumerate}
\item $f_1, \ldots, f_c$ is a regular sequence in the local ring
$R[x_1, \ldots, x_n]_{\mathfrak q'}$,
\item each of the rings
$R[x_1, \ldots, x_n]_{\mathfrak q'}/(f_1, \ldots, f_i)$ is flat over $R$, and
\item the $S$-module $(f_1, \ldots, f_c)/(f_1, \ldots, f_c)^2$
is free with basis given by the elements $f_i \bmod (f_1, \ldots, f_c)^2$.
\end{enumerate}
\end{lemma}

\begin{proof}
First, by Lemma \ref{lemma-regular-quasi-regular}, part (3) follows
from part (1). Parts (1) and (2) immediately reduce to the Noetherian case
by Lemma \ref{lemma-relative-global-complete-intersection-Noetherian}
(some minor details omitted). Assume $R$ is Noetherian.
By Lemma \ref{lemma-lci} for example we see
that $f_1, \ldots, f_c$ form a regular sequence in the local ring
$R[x_1, \ldots, x_n]_{\mathfrak q'} \otimes_R \kappa(\mathfrak p)$.
Moreover, the local ring $R[x_1, \ldots, x_n]_{\mathfrak q'}$
is flat over $R_{\mathfrak p}$. Since $R$, and hence
$R[x_1, \ldots, x_n]_{\mathfrak q'}$ is Noetherian we
may apply Lemma \ref{lemma-grothendieck-regular-sequence}
to conclude.
\end{proof}


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