Nagata compactification theorem is the following fundamental result:

Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ and a quasi-compact $S$-open immersion $X\rightarrow \overline{X}$.

B. Conrad has written a detailed proof with some exposition of the importance of the result too.

In some context, I need to work with schemes that are not locally of finite type. It would be great if there is a result along the lines of

Let $S$ be a qcqs scheme. Let $X$ be a separated quasi-compact $S$-scheme. Then there exists a separated universally closed $S$-scheme $\overline{X}$ and a quasi-compact $S$-open immersion $X\rightarrow \overline{X}$. (Desideratum, not a theorem).

For me specifically it is OK to assume that $S$ is the Spec of a perfect field and that $X$ is Noetherian regular. I guess, however, most general result (with a reasonably short formulation) is aesthetically speaking more appropriate.

It may be germane to mention that universally closed morphisms are quasi-compact.


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