Nagata's compactification theorem

In algebraic geometry, Nagata's compactification theorem, introduced by Nagata (1962, 1963), implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper mapping. Deligne showed, in unpublished notes expounded by Conrad, that the condition that S is Noetherian can be replaced by the condition that S is quasi-compact and quasi-separated.

Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow. Lütkebohmert (1993) gave a scheme-theoretic proof of Nagata's theorem.

Nagata's theorem is used to define the analogue in algebraic geometry of cohomology with compact support, or more generally higher direct image functors with proper support.