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Tag 00U9

Chapter 10: Commutative Algebra > Section 10.141: Étale ring maps

Lemma 10.141.2. Any étale ring map is standard smooth. More precisely, if $R \to S$ is étale, then there exists a presentation $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$ such that the image of $\det(\partial f_j/\partial x_i)$ is invertible in $S$.

Proof. Let $R \to S$ be étale. Choose a presentation $S = R[x_1, \ldots, x_n]/I$. As $R \to S$ is étale we know that $$ \text{d} : I/I^2 \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} S\text{d}x_i $$ is an isomorphism, in particular $I/I^2$ is a free $S$-module. Thus by Lemma 10.134.6 we may assume (after possibly changing the presentation), that $I = (f_1, \ldots, f_c)$ such that the classes $f_i \bmod I^2$ form a basis of $I/I^2$. It follows immediately from the fact that the displayed map above is an isomorphism that $c = n$ and that $\det(\partial f_j/\partial x_i)$ is invertible in $S$. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 37279–37285 (see updates for more information).

    \begin{lemma}
    \label{lemma-etale-standard-smooth}
    Any \'etale ring map is standard smooth. More precisely, if
    $R \to S$ is \'etale, then there exists a presentation
    $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$ such that
    the image of $\det(\partial f_j/\partial x_i)$ is invertible in $S$.
    \end{lemma}
    
    \begin{proof}
    Let $R \to S$ be \'etale. Choose a presentation $S = R[x_1, \ldots, x_n]/I$.
    As $R \to S$ is \'etale we know that
    $$
    \text{d} :
    I/I^2
    \longrightarrow
    \bigoplus\nolimits_{i = 1, \ldots, n} S\text{d}x_i
    $$
    is an isomorphism, in particular $I/I^2$ is a free $S$-module.
    Thus by Lemma \ref{lemma-huber} we may assume (after possibly changing
    the presentation), that $I = (f_1, \ldots, f_c)$ such that the classes
    $f_i \bmod I^2$ form a basis of $I/I^2$. It follows immediately from
    the fact that the displayed map above is an isomorphism that $c = n$ and
    that $\det(\partial f_j/\partial x_i)$ is invertible in $S$.
    \end{proof}

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