In the last days I came to consider the following question which I'd be happy to see answered by the affirmative:



> 
> if $f:X\to S$ is a morphism of schemes which is formally étale, quasicompact,
>  universally bijective, universally schematically dominant, with $S$ noetherian,
>  is $f$ an isomorphism?
> 
> 
> 


Comments :


1. you may assume $X$ and $S$ affine if you like,
2. I can show that it is enough to show that $f$ is a monomorphism, i.e. the diagonal morphism $\Delta:X\to X\times\_SX$ is an isomorphism. Under the assumptions, $\Delta$ is a closed bijective immersion defined by an ideal $I$ such that $I^2=I$. If $I$ were nilpotent this would imply $I=0$, but in general...


**EDIT.** In fact my comment no 2 assumes that $f$ is quasicompact (an assumption which was absent from the first version of the question). Here is how it goes. If $f$ is quasicompact and universally schematically dominant, it follows from results of Olivier and Mesablishvili (see Mesablishvili, *More on descent theory for schemes*, Georgian Math. J. 2004; treated also in the Stacks Project [here](http://stacks.math.columbia.edu/tag/08WE "here")) that $f$ is an effective epimorphism. It is a categorical fact that an effective epi which is a mono is iso. This explains comment no 2. Since I am pleased to assume that $f$ is quasicompact and $S$ is noetherian, I modified the question accordingly.



Is this formally étale morphism of schemes an isomorphism?
[I originally gave what I thought to be a counterexample in the affine case, but I realized it violates universal schematic dominance, so below I give a non-qc counterexample that was originally a comment.]


Let $A = \mathbf{F}\_2^I$ be a product of copies of $\mathbf{F}\_2$ indexed by an infinite set $I$, and let $S = {\rm{Spec}}(A)$. Observe that every local ring on $S$ is $\mathbf{F}\_2$ since every element of $A$ is idempotent. Let $X$ be the disjoint union of the evident collection of clopen points (indexed by $I$) and the reduced structure $Y$ on the closed complement of their union. Let $f:X \rightarrow S$ be the natural map. This is a universal bijection (built from the stratification by $Y$ and $S-Y$) and an isomorphism on local rings, so faithfully flat. In particular, it is universally schematically dominant. It is not even qc (since $I$ is infinite), so not an isomorphism. The interesting thing is that it is formally etale. This amounts to showing $f|\_Y:Y \rightarrow S$ is formally etale.


More generally, if $A$ is any ring and $J$ is an ideal such that $J^2 = J$ (such as $A$ as above and $J$ the ideal of $Y$ in $S$) then $A \rightarrow A/J$ is formally etale (that being a non-flat map when $J \ne 0$ and ${\rm{Spec}}(A/J)$ is not open in Spec($A$), such as happens above, so perhaps slightly surprising at first sight). This is the standard counterexample to EGA 0$\_{\rm{IV}}$ 19.10.3(i) and to EGA IV$\_4$ 18.4.6(i) (whose "proof" ends by invoking EGA 0$\_{\rm{IV}}$, 19.10.3(i)).


