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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.

Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow. Deligne showed, in unpublished notes expounded by Conrad, that Nagata's proof can be translated into scheme theory and that the condition that S is Noetherian can be replaced by the much weaker condition that S is quasi-compact and quasi-separated. Lütkebohmert (1993) gave another scheme-theoretic proof of Nagata's theorem.

An important application of Nagata's theorem is in defining the analogue in algebraic geometry of cohomology with compact support, or more generally higher direct image functors with proper support.

References

Conrad, B, Deligne's notes on Nagata's compactifications (PDF)
Lütkebohmert, Werner (1993), "On compactification of schemes", Manuscripta Mathematica, 80 (1): 95–111, doi:10.1007/BF03026540, ISSN 0025-2611
Nagata, Masayoshi (1962), "Imbedding of an abstract variety in a complete variety", Journal of Mathematics of Kyoto University, 2 (1): 1–10, doi:10.1215/kjm/1250524969, ISSN 0023-608X, MR 0142549
Nagata, Masayoshi (1963), "A generalization of the imbedding problem of an abstract variety in a complete variety", Journal of Mathematics of Kyoto University, 3 (1): 89–102, doi:10.1215/kjm/1250524859, ISSN 0023-608X, MR 0158892

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