2
$\begingroup$

If $f:X_0\rightarrow X$ is a closed immersion of locally noetherian schemes such that the topological spaces of $X_0$ and $X$ are identical (or, more generally, if $f$ is a universal homeomorphism), then it is known (see e.g. SGA1, Thm I.8.2, or, more generally, SGA1, Thm. IX.4.10) that pullback along $f$ induces an equivalence between the categories of etale $X$-schemes and etale $X_0$-schemes.

The main property of étale morphisms is the "infinitesimal lifting criterion", after which the definition of formally étale morphisms is modeled.

Is a "topological invariance result" as above also true for the categories of formally étale $X_0$- and $X$-schemes?

$\endgroup$
5
  • $\begingroup$ I thought etale and formally etale are the same? At least that's what remark I.3.22 in Milne's book seems to say $\endgroup$
    – Moshe
    May 4, 2012 at 17:09
  • $\begingroup$ Not precisely: etale=formally etale + locally of finite presentation. $\endgroup$
    – Lars
    May 4, 2012 at 18:15
  • $\begingroup$ It would be great if you add what you've tried so far. At first sight, this seems to be just the definition of formally etale. If not, please explain. $\endgroup$ May 5, 2012 at 12:57
  • $\begingroup$ Martin, that's what I thought at first. I agree that the fully faithfulness follows more or less directly from the definition of a formally etale morphism. The essential surjectivity (for $X_0\rightarrow X$ a nilp. thickening) is usually proven using the reduction to "standard etale" morphisms, which to my knowledge does not work for formally etale morphisms. The more general case ($X_0\rightarrow X$ a universal homeomorphism) is an application of faithfully flat descent (that's why it is in Exp. IX of SGA1). $\endgroup$
    – Lars
    May 5, 2012 at 15:29
  • $\begingroup$ So if you have a nice "formal" argument for essential surjectivity, I would be very interested. $\endgroup$
    – Lars
    May 5, 2012 at 15:31

0

Your Answer

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

Browse other questions tagged or ask your own question.