Nagata's compactification

Question

Let $f:X \to Y$ be a separated and finite type morphism of schemes. We assume that there is an open immersion $i:U \to Y$ and a proper morphism $f_U:X \to U$ such that $f$ and $i \circ f_U $ coincide. Then can we choose a compactification $\overline{f}:\overline{X} \to Y$ satisfying $X = \overline{X} \times_{Y} U$?

Answer 1

Yes, this is true over a noetherian base. See

B. Conrad, Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc.22(2007), no.3, 205–257. [erratum, same journal 24 (2009), no. 4, 427–428.]

and

Deligne, Le théorème de plongement de Nagata. Kyoto J. Math.50(2010), no.4, 661–670.