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.)

  • 1
    $\begingroup$ @DanielLarsson: Your reference (in Chapter I) is only of Zariski-local nature, so it doesn't catch the "global" aspect of the question posed. $\endgroup$
    – user74230
    Commented Apr 17, 2015 at 13:11
  • 12
    $\begingroup$ With all due respect to the established litterature, the Stacks project cannot be considered less respectable than anything else published. $\endgroup$
    – ACL
    Commented Apr 17, 2015 at 13:12
  • 4
    $\begingroup$ With all due respect to the Stacks Project, it is not peer-reviewed... $\endgroup$ Commented Apr 17, 2015 at 13:48
  • 5
    $\begingroup$ 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... $\endgroup$ Commented Apr 17, 2015 at 15:57
  • 3
    $\begingroup$ 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). $\endgroup$ Commented Apr 18, 2015 at 3:43

1 Answer 1


As a historical remark, I learned this lemma from Roland Huber. A possible answer is that it can be found in the adic spaces setting as Proposition 1.7.1 in Huber's book \'Etale Cohomology of Rigid analytic varieties and adic spaces (which is behind a pay wall). But I never asked Huber if he got it from an earlier paper (he may well have, the paper of Elkik on solutions... contains very similar results). I got the impression Huber thought this lemma was essentially trivial, and for trivial lemmas you can omit the reference right?

In the paper Etale Cohomology of Rigid Analytic Spaces (not behind a pay wall) you can find Huber's argument in the rigid analytic setting; see Observation 3.1.2.

When you find an even better and/or earlier reference please leave a comment on the Stacks project website so the Stacks people can add a reference there. Same for all other lemmas.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.