# Reference for a lemma on étale maps

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.)

• @DanielLarsson: Your reference (in Chapter I) is only of Zariski-local nature, so it doesn't catch the "global" aspect of the question posed. – user74230 Apr 17 '15 at 13:11
• With all due respect to the established litterature, the Stacks project cannot be considered less respectable than anything else published. – ACL Apr 17 '15 at 13:12
• With all due respect to the Stacks Project, it is not peer-reviewed... – Matthieu Romagny Apr 17 '15 at 13:48
• There was no editor overseeing a peer review process, however peers certainly do read it and I would hope point out any errors... Perhaps even more frequently than referee's point out errors in published works... – Karl Schwede Apr 17 '15 at 15:57
• I know for a fact that journals accept Stacks Projects tags as references. Especially for such standard (pun sadly intended) facts (this does not count as repetition -- according to Just a Minute rules -- the former was "fact", which is singular). – bananastack Apr 18 '15 at 3:43