The Stacks Project has the following really nice Lemma concerning étale maps of rings:

Let $A\rightarrow B$ be a finitely presented, tale morphism of rings. Then there exists a presentation $$ B\cong \frac{A[x_1,\ldots,x_n]}{(f_1,\ldots,f_n)} $$ such that the matrix $(\partial f_i/\partial x_j)$ is invertible in $B$. In other words, any étale map is globally 'standard smooth'.

Now, I would like to use this result in a paper I'm writing, and while the proof given seems correct to me, it would obviously be better to have a slightly more 'respectable' reference for this fact.

Question: Does anyone know of a good (i.e. peer-reviewed) reference for this Lemma? Is it in EGA somewhere? (I looked, but I couldn't find it anywhere, only the weaker claim that every étale map is locally standard étale.)

The Stacks Project has the following really nice Lemma concerning étale maps of rings:

Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ B\cong \frac{A[x_1,\ldots,x_n]}{(f_1,\ldots,f_n)} $$ such that the matrix $(\partial f_i/\partial x_j)$ is invertible in $B$. In other words, any étale map is globally 'standard smooth'.

Now, I would like to use this result in a paper I'm writing, and while the proof given seems correct to me, it would obviously be better to have a slightly more 'respectable' reference for this fact.

Question: Does anyone know of a good (i.e. peer-reviewed) reference for this Lemma? Is it in EGA somewhere? (I looked, but I couldn't find it anywhere, only the weaker claim that every étale map is locally standard étale.)