Relative ampleness in rigid geometry

Brian Conrad[1]

  • [1] University of Michigan Department of Mathematics Ann Arbor, MI 48109 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 1049-1126
  • ISSN: 0373-0956

Abstract

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We develop a rigid-analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for morphisms and proper geometric objects. The basic definition is fibral, but pointwise arguments from the algebraic and complex-analytic cases do not apply, so we use cohomological properties of formal schemes over completions of local rings on rigid spaces. An analytic notion of quasi-coherence is introduced so that we can recover a proper object from sections of an ample bundle via suitable Proj construction. The locus of relative ampleness in the base is studied, as is the behavior of relative ampleness with respect to analytification and arbitrary extension of the base field. In particular, we obtain a quick new proof of the relative GAGA theorem over affinoids.

How to cite

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Conrad, Brian. "Relative ampleness in rigid geometry." Annales de l’institut Fourier 56.4 (2006): 1049-1126. <http://eudml.org/doc/10167>.

@article{Conrad2006,
abstract = {We develop a rigid-analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for morphisms and proper geometric objects. The basic definition is fibral, but pointwise arguments from the algebraic and complex-analytic cases do not apply, so we use cohomological properties of formal schemes over completions of local rings on rigid spaces. An analytic notion of quasi-coherence is introduced so that we can recover a proper object from sections of an ample bundle via suitable Proj construction. The locus of relative ampleness in the base is studied, as is the behavior of relative ampleness with respect to analytification and arbitrary extension of the base field. In particular, we obtain a quick new proof of the relative GAGA theorem over affinoids.},
affiliation = {University of Michigan Department of Mathematics Ann Arbor, MI 48109 (USA)},
author = {Conrad, Brian},
journal = {Annales de l’institut Fourier},
keywords = {Ampleness; rigid geometry; descent; ampleness},
language = {eng},
number = {4},
pages = {1049-1126},
publisher = {Association des Annales de l’institut Fourier},
title = {Relative ampleness in rigid geometry},
url = {http://eudml.org/doc/10167},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Conrad, Brian
TI - Relative ampleness in rigid geometry
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 1049
EP - 1126
AB - We develop a rigid-analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for morphisms and proper geometric objects. The basic definition is fibral, but pointwise arguments from the algebraic and complex-analytic cases do not apply, so we use cohomological properties of formal schemes over completions of local rings on rigid spaces. An analytic notion of quasi-coherence is introduced so that we can recover a proper object from sections of an ample bundle via suitable Proj construction. The locus of relative ampleness in the base is studied, as is the behavior of relative ampleness with respect to analytification and arbitrary extension of the base field. In particular, we obtain a quick new proof of the relative GAGA theorem over affinoids.
LA - eng
KW - Ampleness; rigid geometry; descent; ampleness
UR - http://eudml.org/doc/10167
ER -

References

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  1. M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira) (1969), 21-71, Univ. Tokyo Press, Tokyo Zbl0205.50402MR260746
  2. V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33 (1990), Amer. Math. Soc. Zbl0715.14013MR1070709
  3. V. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHÉS 78 (1993), 7-161 Zbl0804.32019MR1259429
  4. E. Bierstone, P. Milman, Semianalytic sets and subanalytic sets, Publ. Math. IHÉS 67 (1988), 5-42 Zbl0674.32002MR972342
  5. S. Bosch, U. Görtz, Coherent modules and their descent on relative rigid spaces, J. reine angew. Math. 495 (1998), 119-134 Zbl0884.14009MR1603849
  6. S. Bosch, U. Günzter, R. Remmert, Non-Archimedean analysis, (1984), Springer-Verlag Zbl0539.14017MR746961
  7. S. Bosch, W. Lütkebohmert, Formal and rigid geometry I, Math. Annalen 295 (1993), 291-317 Zbl0808.14017MR1202394
  8. S. Bosch, W. Lütkebohmert, Formal and rigid geometry II, Math. Annalen 296 (1993), 403-429 Zbl0808.14018MR1225983
  9. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, (1990), Springer-Verlag Zbl0705.14001MR1045822
  10. S. Bosch, W. Lütkebohmert, M. Raynaud, Formal and rigid geometry III. The relative maximum principle, Math. Annalen 302 (1995), 1-29 Zbl0839.14013MR1329445
  11. B. Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier Grenoble 49 (1999), 473-541 Zbl0928.32011MR1697371
  12. B. Conrad, Higher-level canonical subgroups in abelian varieties, (2005) 
  13. B. Conrad, Modular curves and rigid-analytic spaces, (2005) Zbl1156.14312MR2217566
  14. B. Conrad, Rigid spaces and algebraic spaces, (2005) 
  15. J. Dieudonné, A. Grothendieck, Éléments de géométrie algébrique, Publ. Math. IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967) Zbl0203.23301
  16. J. Fresnel, M. Matignon, Sur les espaces analytiques quasi-compact de dimenion 1 sur un corps valué complet ultramétrique, Ann. Mat. Pura Appl. 145 (1986), 159-210 Zbl0623.32020MR886711
  17. J. Fresnel, M. van der Put, Rigid analytic geometry and its applications, (2004), Birkhäuser, Boston Zbl1096.14014MR2014891
  18. J. Frisch, Points de platitude d’un morphisme d’espaces analytiques complexes, Inv. Math. 4 (1967), 118-138 Zbl0167.06803MR222336
  19. R. Godement, Théorie des faiseaux, (1973), Hermann, Paris MR345092
  20. H. Grauert, R. Remmert, Bilder und Urbilder analytishcer Garben, Annals of Math. 68 (1958), 393-443 Zbl0089.06003MR102612
  21. H. Grauert, R. Remmert, Theory of Stein spaces, 239 (1979), Springer-Verlag Zbl0433.32007MR580152
  22. H. Grauert, R. Remmert, Coherent analytic sheaves, 265 (1984), Springer-Verlag Zbl0537.32001MR755331
  23. A. Grothendieck, Techniques de construction en géométrie analytique IX: quelques problèmes de modules, exposé 16, Séminaire H. Cartan (1960/61), 1-20 Zbl0142.33504
  24. A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV: les schémas de Hilbert, exposé 221, Séminaire Bourbaki 1960/61 (1961), Secrétariat mathématique, 11 rue Pierre Curie, Paris Zbl0236.14003MR1611207
  25. A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique V. Les schémas de Picard: théorèmes d’existence, exposé 232, Séminaire Bourbaki 1961/62 (1962), Secrétariat mathématique, Paris Zbl0238.14014MR146040
  26. A. Grothendieck, Revêtements étales et groupe fondamental, Lecture Note in Math. 224 (1971), Springer-Verlag, New York MR354651
  27. A. Grothendieck et al., Théorie des intersections et théorème de Riemann-Roch, Lecture Note in Math. 225 (1971), Springer-Verlag, New York Zbl0229.14008MR354655
  28. W. Gubler, Local heights on subvarieties over non-Archimedean fields, J. reine angew. Math. 498 (1998), 61-113 Zbl0906.14013MR1629925
  29. M. Hakim, Topos annelés et schémas relatifs, 64 (1972), Springer-Verlag Zbl0246.14004MR364245
  30. C. Houzel, Géométrie analytique locale: I, II, exposés 18–19, Séminaire H. Cartan (1960/61) Zbl0121.15906
  31. R. Kiehl, Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie, Inv. Math. 2 (1967), 191-214 Zbl0202.20101MR210948
  32. M. Kisin, Local constancy in p -adic families of Galois representations, Math. Z. 230 (1999), 569-593 Zbl0932.32028MR1680032
  33. U. Köpf, Über eigentliche familien algebraischer varietäten über affinoiden räumen, Schriftenreihe Math. Inst. Münster, 2. Serie. Heft 7 (1974) Zbl0275.14006MR422671
  34. W. Lütkebohmert, Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie, Math. Z. 139 (1974), 69-84 Zbl0283.32022MR352527
  35. W. Lütkebohmert, Formal-algebraic and rigid-analytic geometry, Math. Annalen 286 (1990), 341-371 Zbl0716.32022MR1032938
  36. W. Lütkebohmert, On compactification of schemes, Manus. Math. 80 (1993), 95-111 Zbl0822.14010MR1226600
  37. H. Matsumura, Commutative ring theory, (1986), Cambridge Univ. Press Zbl0603.13001MR879273
  38. S. Mochizuki, Foundations of p -adic Teichmüller theory, AMS/IP Studies in Adv. Math. 11 (1999), Intl. Press, Cambridge Zbl0969.14013MR1700772
  39. D. Mumford, Abelian varieties, (1970), Oxford University Press, Bombay Zbl0223.14022MR282985
  40. H. Schoutens, Blowing up in rigid analytic geometry, Bull. Belg. Math. Soc. 2 (1995), 399-417 Zbl0854.32020MR1355829
  41. Y-T. Siu, Noetherianness of rings of holomorphic functions on Stein compact subsets, Proc. Amer. Math. Soc. 21 (1969), 483-489 Zbl0175.37402MR247135
  42. M. Temkin, On local properties of non-Archimedean analytic spaces, Math. Annalen 318 (2000), 585-607 Zbl0972.32019MR1800770

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