Schemes as closed subschemes in étale schemes

Question

The Nagata compactification theorem asserts that every scheme, separated and of finite type over a Noetherian base scheme $S$, admits an open immersion into a proper $S$-scheme.

It would be useful for me if (what I think of as somehow the dual holds, that is if) every scheme, unramified and of finite presentation over a reasonable base $S$, admits a closed immersion into an étale $S$-scheme.

Is such a statement known to be true, false, or open?

Well, it is trivially true when $S$ is a field, I wonder if things change when $S$ is taken to be an arbitrary Noetherian scheme.

Does the statement hold without assuming unramification, if we only ask for a closed immersion into a smooth $S$-scheme?

PS Apologies if this is widely known for algebraic geometers, I am not one, yet.

Answer 1

Here is what is classically known:

Theorem 1 [EGAIV$_4$, Cor. 18.4.7]. Let $X$ and $Y$ be schemes, and let $f\colon X \to Y$ be a morphism locally of finite type. Then, for every point $x\in X$, the following are equivalent:

1. $f$ is formally unramified at $x$;
2. There exists an open neighborhood $U$ of $x$ such that there is a diagram $$\require{AMScd} \begin{CD} U @>j>> X'\\ @ViVV @VVhV\\ X @>f>> Y \end{CD}$$ where $j$ is a closed embedding and $h$ is étale.

This says that what you ask for is locally true.

Recently, on the other hand, Rydh showed the following, which gives a (sort of) "global" version of the property above:

Theorem 2 [Rydh, Thm. 1.2]. Let $X$ and $Y$ be schemes, and let $f\colon X \to Y$ be a formally unramified morphism that is locally of finite type. Then, there exists a canonical algebraic space $E_{X/Y}$ such that there exists a commutative diagram $$\begin{CD} X @>i>> E_{X/Y}\\ @| @VVeV\\ X @>f>> Y \end{CD}$$ where $i$ is a closed immersion and $e$ is étale.

Moreover, Rydh shows that $E_{X/Y}$ is not necessarily a scheme [Rydh, Ex. 2.5], but will be a scheme in the case that $f$ is a local immersion [Rydh, Rem. 1.3]. I don't know if there could exist a scheme that could be chosen instead to give the factorization in Theorem 2.