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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."

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    $\begingroup$ What about finite étale coverings? $\endgroup$
    – abx
    Commented Dec 13, 2013 at 20:48
  • $\begingroup$ yes, it the second case f is finite etale. I guess not much can be said if I relax noetherian condition. $\endgroup$ Commented Dec 14, 2013 at 14:50

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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.

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  • $\begingroup$ yes, you are right on both counts. the premise of the question is more or less wrong with regards to formally étale morphisms. and, for étale morphisms, the best one can say is finite étale covers. but of course one can study finite pro-étale covers or finite formally étale covers. $\endgroup$ Commented Dec 21, 2013 at 19:26

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