The Stacks project

Lemma 10.143.2. 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 10.136.6 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 $S$. $\square$

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