A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally closed?


If $f$ is a morphism between locally noetherian schemes $X$ and $Y$ which is formally etale and universally closed, then it will be quasi-compact, whence etale and moreover quasi-finite. In this case, I am tempted to conjecture that, if $f$ is surjective, then


$$X \cong \bigsqcup\_{i=1}^{n} Y$$
 in this case. This is probably overly optimistic, but I have something like this in mind when I use the word "characterize."



formally étale morphisms which are also universally closed
The statement "A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion." is not correct. An example is to consider the map from the zero dimensional local ring k[x\_1, x\_2, ...]/(x\_ix\_j) to its completion. It is also not true that a universally closed formally \'etale morphism of Noetherian schemes is \'etale. An example is to consider the map from a field into its separable algebraic closure.


