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Deligne’s notes on Nagata compactifications. (English) Zbl 1142.14001

J. Ramanujan Math. Soc. 22, No. 3, 205-257 (2007); erratum ibid. 24, No. 4, 427-428 (2009).
In modern abstract algebraic geometry, a fundamental theorem of M. Nagata can be stated as follows: If \(S\) is a noetherian scheme and \(f: X\to S\) is a separated morphism of finite type in the category of schemes, then there exists a proper \(S\)-scheme \(\overline X\) and an open immersion \(j: X\hookrightarrow X\) over \(S\).
This version is known as “Nagata’s compactification theorem”, and its genesis is truly remarkable. In fact, M. Nagata proved the original version of this important result in [J. Math. Kyoto Univ. 2, 1–10 (1962; Zbl 0109.39503)], together with a certain generalization of it shortly thereafter [cf.: M. Nagata, J. Math. Kyoto Univ. 3, 89–102 (1963; Zbl 0223.14011)]. However, Nagata’s proof is given in terms of pre-Grothendieck algebro-geometric terminology and rather difficult to understand in the framework of contemporary algebraic geometry. His arguments use the Zariski-Riemann space attached to an algebraic function field, on the one hand, and a complicated induction process with respect to the rank of certain valuations on the other. Due to these particular circumstances concerning common mathematical communication, there remained some uncertainty among modern algebraic geometers about the validity of Nagata’s compactification theorem for a scheme \(X\) over a general noetherian base scheme \(S\).
Although a short proof of Nagata’s theorem in the noetherian case was delivered by W. Lütkebohmert [Manuscr. Math. 80, No.1, 95–111 (1993; Zbl 0822.14010)] in the early 1990s, the precise understanding of Nagata’s methods of proof continued to stay the same challenge as before.
Now, after 45 years, the work under review provides the first published account of Nagata’s approach in the modern scheme-theoretic context, and that in its full generality and with its far-reaching applications. Actually, P. Deligne had worked out such a scheme-theoretic version of Nagata’s proof some years ago, but only in the form of personal notes of sketchy character. In view of the crucial importance of Nagata’s theorem, and with P. Deligne’s permission to disseminate an elaborated exposition of his unpublished notes on Nagata’s proof, the author of the current paper decided to write out complete proofs of Deligne’s generalizing assertations to the benefit of the general public. The outcome of this highly rewarding undertaking is a detailed proof of the Nagata compactification theorem in its most general setting (Theorem 4.1.):
Let \(f: X\to S\) be a separated morphism of finite type between schemes, where \(S\) is quasi-compact and quasi-separated. Then there exists an open immersion \(j:X\hookrightarrow\overline X\) of \(S\)-schemes such that \(\overline X\) is proper over \(S\).
This result generalizes the “classical” case where \(S\) is assumed to be a noetherian scheme in a significant way. Though its proof requires a rather complicated series of partial steps, including a general variant of Nagata’s so-called quasi-dominations as well as refinements concerning classical blow-up techniques, Chow’s lemma, and proper birational morphisms, the exposition lucidly reveals the great power of Nagata’s methods and their translation by P. Deligne and the author into the modern framework. For the convenience of the reader, the author has separated the (easier) case of a noetherian base scheme from the most general case, for the treatment of which he uses a new trick based upon some more recent results by R. Thomason and T. Trobaugh of general character [Higher algebraic K-theory of schemes and of derived categories, in: The Grothendieck Festschrift, Vol. III, 247–430, Progress in Mathematics, Birkhäuser Verlag (1990; Zbl 0731.14001)]. As the author points out, in a recent paper of M. Temkin [Relative Riemann-Zariski spaces, Preprint, arXiv:0804.2843 (2007)] on applying valuation-theoretic methods to problems in algebraic geometry, Nagata’s original viewpoint is also resurrected in a modern form, including another new proof of Nagata’s compactification theorem.

MSC:

14A15 Schemes and morphisms
14E99 Birational geometry
14E25 Embeddings in algebraic geometry
14E05 Rational and birational maps
13A18 Valuations and their generalizations for commutative rings
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