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Nagata compactification for algebraic spaces

Part of: Algebraic geometry: Foundations Birational geometry

Published online by Cambridge University Press:  13 July 2012

Brian Conrad ,
Max Lieblich  and
Martin Olsson
Brian Conrad
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA ([email protected])
Max Lieblich
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA ([email protected])
Martin Olsson
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA ([email protected])
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Abstract

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

Keywords

algebraic spacesNagata compactificationNoetherian approximation

MSC classification

Secondary: 14A15: Schemes and morphisms 14E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 4 , October 2012 , pp. 747 - 814
Copyright
Copyright © Cambridge University Press 2012

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References

1.Artin, M., Algebraization of formal moduli, II, Existence of modifications, Annals Math. 91(1) (1970), 88–135.CrossRefGoogle Scholar
2.Conrad, B., Deligne's notes on Nagata compactifications, J. Ramanujan Math. Soc. 22(3) (2007), 205–257.Google Scholar
3.Conrad, B., Keel–Mori theorem via stacks, preprint (available at http://math.stanford.edu/~conrad).Google Scholar
4.Conrad, B. and de Jong, A. J., Approximation of versal deformations, J. Alg. 255(2) (2002), 489–515.CrossRefGoogle Scholar
5.Deligne, P., Le théorème de plongement de Nagata, personal notes (unpublished).Google Scholar
6.Demazure, M. and Grothendieck, A., Schémas en groupes, Volumes I–III, Lecture Notes in Mathematics, Volumes 151–153 (Springer, 1970).Google Scholar
7.Fujiwara, K. and Kato, F., Foundations of rigid geometry, preprint (2011).Google Scholar
8.Grothendieck, A., Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Volume 224 (Springer, 1971).Google Scholar
9.Grothendieck, A., Élements de géométrie algébrique, Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32 (1960–7).CrossRefGoogle Scholar
10.Gruson, L. and Raynaud, M., Critères de platitude et de projectivité, Invent. Math. 13 (1971), 1–89.Google Scholar
11.Illusie, L., Complexe cotangent et déformations, Volume I, Lecture Notes in Mathematics, Volume 239 (Springer, 1971).Google Scholar
12.Keel, S. and Mori, S., Quotients by groupoids, Annals Math. 145 (1997), 193–213.CrossRefGoogle Scholar
13.Knutson, D., Algebraic spaces, Lecture Notes in Mathematics, Volume 203 (Springer, 1971).Google Scholar
14.Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 39 (Springer, 2000).Google Scholar
15.Lazlo, Y. and Olsson, M., The six operations for sheaves on Artin stacks, I, Publ. Math. IHES 107 (2008), 109–168.CrossRefGoogle Scholar
16.Lazlo, Y. and Olsson, M., The six operations for sheaves on Artin stacks, II, Publ. Math. IHES 107 (2008), 169–210.CrossRefGoogle Scholar
17.Lütkebohmert, W., On compactification of schemes, Manuscr. Math. 80 (1993), 95–111.CrossRefGoogle Scholar
18.Nagata, M., Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2(1) (1962), 1–10.Google Scholar
19.Nagata, M., A generalization of the imbedding problem, J. Math. Kyoto Univ. 3(1) (1963), 89–102.Google Scholar
20.Raoult, J.-C., Compactification des espaces algébriques normaux, C. R. Acad. Sci. Paris Sér. A–B 273 (1971), A766–A767.Google Scholar
21.Raoult, J.-C., Compactification des espaces algébriques, C. R. Acad. Sci. Paris Sér. A–B 278 (1974), 867–869.Google Scholar
22.Raoult, J.-C., Le théorème de compactification de Nagata pour les espaces algébriques, Thesis (1974).Google Scholar
23.Rydh, D., Existence and properties of geometric quotients, preprint (available at www.math.kth.se/~dary).Google Scholar
24.Rydh, D., Noetherian approximation of algebraic spaces and stacks, preprint (arXiv: 0904.0227; 2009).Google Scholar
25.Thomason, R. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Volume III, Progress in Mathematics, pp. 247–435 (Birkhäuser, 1990).CrossRefGoogle Scholar
26.Varshavsky, Y., Lefschetz–Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Analysis 17 (2007), 271–319.CrossRefGoogle Scholar
27.Vojta, P., Nagata's embedding theorem, preprint (arXiv:0706.1907; 2007).Google Scholar