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Tag 04D1

Chapter 10: Commutative Algebra > Section 10.138: Étale ring maps

Lemma 10.138.11. 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 10.138.2 we can write $\overline{S} = (R/I)[x_1, \ldots, x_n]/(\overline{f}_1, \ldots, \overline{f}_n)$ as in Definition 10.132.6 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 $S = R[x_1, \ldots, x_n, x_{n+1}]/(f_1, \ldots, f_c, x_{n + 1}\Delta - 1)$ where $\Delta = \det(\frac{\partial f_i}{\partial x_j})_{i, j = 1, \ldots, c}$ as in Example 10.132.8. 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 35255–35261 (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
    $S = R[x_1, \ldots, x_n, x_{n+1}]/(f_1, \ldots, f_c, x_{n + 1}\Delta - 1)$
    where $\Delta = \det(\frac{\partial f_i}{\partial x_j})_{i, j = 1, \ldots, c}$
    as in Example \ref{example-make-standard-smooth}.
    This proves the lemma.
    \end{proof}

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