Submersions and effective descent of étale morphisms

David Rydh

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 2, page 181-230
  • ISSN: 0037-9484

Abstract

top
Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.

How to cite

top

Rydh, David. "Submersions and effective descent of étale morphisms." Bulletin de la Société Mathématique de France 138.2 (2010): 181-230. <http://eudml.org/doc/272347>.

@article{Rydh2010,
abstract = {Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.},
author = {Rydh, David},
journal = {Bulletin de la Société Mathématique de France},
keywords = {submersive; subtrusive; universally open; descent; étale; blow-up; h-topology; algebraic spaces},
language = {eng},
number = {2},
pages = {181-230},
publisher = {Société mathématique de France},
title = {Submersions and effective descent of étale morphisms},
url = {http://eudml.org/doc/272347},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Rydh, David
TI - Submersions and effective descent of étale morphisms
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 2
SP - 181
EP - 230
AB - Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.
LA - eng
KW - submersive; subtrusive; universally open; descent; étale; blow-up; h-topology; algebraic spaces
UR - http://eudml.org/doc/272347
ER -

References

top
  1. [1] Schémas en groupes. I: Propriétés générales des schémas en groupes – Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Math., vol. 151, Springer, 1970. Zbl0207.51401
  2. [2] A. Andreotti & E. Bombieri – « Sugli omeomorfismi delle varietà algebriche », Ann. Scuola Norm. Sup Pisa23 (1969), p. 431–450. Zbl0184.24503MR266923
  3. [3] M. Artin, A. Grothendieck & J. L. Verdier (éds.) – Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Math., vol. 305, Springer, 1973. MR354654
  4. [4] M. Artin – Grothendieck topologies, Harvard University, 1962, lecture notes. Zbl0208.48701
  5. [5] —, « Algebraic approximation of structures over complete local rings », Publ. Math. I.H.É.S. 36 (1969), p. 23–58. Zbl0181.48802MR268188
  6. [6] —, « On the joins of Hensel rings », Advances in Math.7 (1971), p. 282–296. Zbl0242.13021MR289501
  7. [7] —, Théorèmes de représentabilité pour les espaces algébriques, Les Presses de l’université de Montréal, 1973. Zbl0323.14001
  8. [8] N. Bourbaki – Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No. 1308, Hermann, 1964. Zbl0205.34302MR194450
  9. [9] H. Brenner – « Grothendieck topologies and ideal closure operations », preprint arXiv:math/0612471. 
  10. [10] B. Conrad – « Deligne’s notes on Nagata compactifications », J. Ramanujan Math. Soc.22 (2007), p. 205–257. Zbl1142.14001MR2356346
  11. [11] D. E. Dobbs, M. Fontana & G. Picavet – « Generalized going-down homomorphisms of commutative rings », in Commutative ring theory and applications (Fez, 2001), Lecture Notes in Pure and Appl. Math., vol. 231, Dekker, 2003, p. 143–163. Zbl1069.13007MR2029824
  12. [12] D. Ferrand – « Conducteur, descente et pincement », Bull. Soc. Math. France131 (2003), p. 553–585. Zbl1058.14003MR2044495
  13. [13] J. Giraud – « Méthode de la descente », Bull. Soc. Math. France Mém. 2 (1964). Zbl0211.32902MR190142
  14. [14] S. Greco & C. Traverso – « On seminormal schemes », Compositio Math.40 (1980), p. 325–365. Zbl0412.14024MR571055
  15. [15] A. Grothendieck – « Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes », Publ. Math. I.H.É.S. 8 (1961). Zbl0118.36206
  16. [16] —, « Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents », Publ. Math. I.H.É.S. 11, 17 (1961, 1963). Zbl0122.16102
  17. [17] —, « Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas », Publ. Math. I.H.É.S. 20, 24, 28, 32 (1964-67). Zbl0135.39701
  18. [18] —, Éléments de géométrie algébrique. I. Le langage des schémas, second éd., Grundl. Math. Wiss., vol. 166, Springer, 1971. 
  19. [19] —(éd.) – Revêtements étales et groupe fondamental, Lecture Notes in Math., vol. 224, Springer, 1971. 
  20. [20] M. Hochster – « Prime ideal structure in commutative rings », Trans. Amer. Math. Soc.142 (1969), p. 43–60. Zbl0184.29401MR251026
  21. [21] —, « Totally integrally closed rings and extremal spaces », Pacific J. Math.32 (1970), p. 767–779. Zbl0192.38502MR257064
  22. [22] B. G. Kang & D. Y. Oh – « Lifting up an infinite chain of prime ideals to a valuation ring », Proc. Amer. Math. Soc.126 (1998), p. 645–646. Zbl0887.13004MR1415329
  23. [23] D. Knutson – Algebraic spaces, Lecture Notes in Math., vol. 203, 1971. Zbl0221.14001MR302647
  24. [24] J. Kollár – Rational curves on algebraic varieties, Ergebn. der Math. und ihrer Grenzg., vol. 32, Springer, 1996. Zbl0877.14012MR1440180
  25. [25] —, « Quotient spaces modulo algebraic groups », Ann. of Math.145 (1997), p. 33–79. Zbl0881.14017MR1432036
  26. [26] G. Laumon & L. Moret-Bailly – Champs algébriques, Ergebn. der Math. und ihrer Grenzg., vol. 39, Springer, 2000. Zbl0945.14005MR1771927
  27. [27] D. Lazard – « Autour de la platitude », Bull. Soc. Math. France97 (1969), p. 81–128. Zbl0174.33301MR254100
  28. [28] M. Manaresi – « Some properties of weakly normal varieties », Nagoya Math. J.77 (1980), p. 61–74. Zbl0403.14001MR556308
  29. [29] J. S. Milne – Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton Univ. Press, 1980. Zbl0433.14012MR559531
  30. [30] J.-P. Olivier – « Anneaux absolument plats universels et épimorphismes à buts réduit », in Séminaire d’Algèbre Commutative 1967–1968, Secrétariat mathématique, 1968, exposé no 6. Zbl0162.05701
  31. [31] —, « Le foncteur T - ∞ . Globalisation du foncteur T », in Séminaire d’Algèbre Commutative 1967–1968, Secrétariat mathématique, 1968, exposé no 9. 
  32. [32] G. Picavet – « Submersion et descente », J. Algebra103 (1986), p. 527–591. Zbl0626.14014MR864429
  33. [33] M. Raynaud – Anneaux locaux henséliens, Lecture Notes in Math., vol. 169, Springer, 1970. Zbl0203.05102MR277519
  34. [34] M. Raynaud & L. Gruson – « Critères de platitude et de projectivité. Techniques de “platification” d’un module », Invent. Math.13 (1971), p. 1–89. Zbl0227.14010MR308104
  35. [35] D. Rydh – « Existence of quotients by finite groups and coarse moduli spaces », preprint arXiv:0708.3333. 
  36. [36] M. Spivakovsky – « A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms », J. Amer. Math. Soc.12 (1999), p. 381–444. Zbl0919.13009MR1647069
  37. [37] A. Suslin & V. Voevodsky – « Singular homology of abstract algebraic varieties », Invent. Math.123 (1996), p. 61–94. Zbl0896.55002MR1376246
  38. [38] —, « Relative cycles and Chow sheaves », in Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, 2000, p. 10–86. Zbl1019.14004MR1764199
  39. [39] R. G. Swan – « On seminormality », J. Algebra67 (1980), p. 210–229. Zbl0473.13001MR595029
  40. [40] —, « Néron-Popescu desingularization », in Algebra and geometry (Taipei, 1995), Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, p. 135–192. 
  41. [41] R. W. Thomason & T. Trobaugh – « Higher algebraic K -theory of schemes and of derived categories », in The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser, 1990, p. 247–435. Zbl0731.14001MR1106918
  42. [42] C. Traverso – « Seminormality and Picard group », Ann. Scuola Norm. Sup. Pisa24 (1970), p. 585–595. Zbl0205.50501MR277542
  43. [43] V. Voevodsky – « Homology of schemes », Selecta Math. (N.S.) 2 (1996), p. 111–153. Zbl0871.14016MR1403354
  44. [44] V. Voevodsky, A. Suslin & E. M. Friedlander – Cycles, transfers, and motivic homology theories, Annals of Math. Studies, vol. 143, Princeton Univ. Press, 2000. Zbl1021.14006MR1764197
  45. [45] H. Yanagihara – « Some results on weakly normal ring extensions », J. Math. Soc. Japan35 (1983), p. 649–661. Zbl0535.13007MR714467
  46. [46] —, « On an intrinsic definition of weakly normal rings », Kobe J. Math.2 (1985), p. 89–98. Zbl0586.13006MR811809

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.