# Universal closure of schemes à la Nagata

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.