The Stacks Project

Tag 04D1

Lemma 10.141.10. Let $R$ be a ring and let $I \subset R$ be an ideal. Let $R/I \to \overline{S}$ be an étale ring map. Then there exists an étale ring map $R \to S$ such that $\overline{S} \cong S/IS$ as $R/I$-algebras.

Proof. By Lemma 10.141.2 we can write $\overline{S} = (R/I)[x_1, \ldots, x_n]/(\overline{f}_1, \ldots, \overline{f}_n)$ as in Definition 10.135.6 with $\overline{\Delta} = \det(\frac{\partial \overline{f}_i}{\partial x_j})_{i, j = 1, \ldots, n}$ invertible in $\overline{S}$. Just take some lifts $f_i$ and set $S = R[x_1, \ldots, x_n, x_{n+1}]/(f_1, \ldots, f_n, x_{n + 1}\Delta - 1)$ where $\Delta = \det(\frac{\partial f_i}{\partial x_j})_{i, j = 1, \ldots, n}$ as in Example 10.135.8. This proves the lemma. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 37574–37580 (see updates for more information).

\begin{lemma}
\label{lemma-lift-etale}
Let $R$ be a ring and let $I \subset R$ be an ideal.
Let $R/I \to \overline{S}$ be an \'etale ring map.
Then there exists an \'etale ring map
$R \to S$ such that $\overline{S} \cong S/IS$ as $R/I$-algebras.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-etale-standard-smooth} we can write
$\overline{S} = (R/I)[x_1, \ldots, x_n]/(\overline{f}_1, \ldots, \overline{f}_n)$
as in Definition \ref{definition-standard-smooth} with
$\overline{\Delta} = \det(\frac{\partial \overline{f}_i}{\partial x_j})_{i, j = 1, \ldots, n}$
invertible in $\overline{S}$. Just take some lifts $f_i$ and set
$S = R[x_1, \ldots, x_n, x_{n+1}]/(f_1, \ldots, f_n, x_{n + 1}\Delta - 1)$
where $\Delta = \det(\frac{\partial f_i}{\partial x_j})_{i, j = 1, \ldots, n}$
as in Example \ref{example-make-standard-smooth}.
This proves the lemma.
\end{proof}

Comment #2452 by Matthieu Romagny on March 13, 2017 a 1:39 pm UTC

Probably $c$ should be $n$ in the ideal that defines $S$ and in the indices for $\Delta$.

Comment #2494 by Johan (site) on April 13, 2017 a 11:13 pm UTC

Thanks, fixed here.

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