Sekiguchi-Suwa theory revisited

Ariane Mézard[1]; Matthieu Romagny[2]; Dajano Tossici[3]

  • [1] Institut de Mathématiques de Jussieu Université Pierre et Marie Curie 4 place Jussieu 75252 Paris Cedex 05, France
  • [2] Institut de Recherche Mathématique de Rennes Université de Rennes 1 Campus de Beaulieu (Bât. 22) 35042 Rennes Cedex, France
  • [3] Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 163-200
  • ISSN: 1246-7405

Abstract

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We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity.

How to cite

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Mézard, Ariane, Romagny, Matthieu, and Tossici, Dajano. "Sekiguchi-Suwa theory revisited." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 163-200. <http://eudml.org/doc/275720>.

@article{Mézard2014,
abstract = {We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity.},
affiliation = {Institut de Mathématiques de Jussieu Université Pierre et Marie Curie 4 place Jussieu 75252 Paris Cedex 05, France; Institut de Recherche Mathématique de Rennes Université de Rennes 1 Campus de Beaulieu (Bât. 22) 35042 Rennes Cedex, France; Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy},
author = {Mézard, Ariane, Romagny, Matthieu, Tossici, Dajano},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {smooth group schema; Artin-Schreier-Witt isogeny; Witt vector; Artin-Hasse exponential},
language = {eng},
month = {4},
number = {1},
pages = {163-200},
publisher = {Société Arithmétique de Bordeaux},
title = {Sekiguchi-Suwa theory revisited},
url = {http://eudml.org/doc/275720},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Mézard, Ariane
AU - Romagny, Matthieu
AU - Tossici, Dajano
TI - Sekiguchi-Suwa theory revisited
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 163
EP - 200
AB - We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity.
LA - eng
KW - smooth group schema; Artin-Schreier-Witt isogeny; Witt vector; Artin-Hasse exponential
UR - http://eudml.org/doc/275720
ER -

References

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  2. N. Bourbaki, Algèbre commutative, Chapitre 9. Anneaux locaux noethériens complets, Masson (1983). Zbl0579.13001MR722608
  3. P. Cartier, Groupes formels associés aux anneaux de Witt généralisés, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A49–A52. Zbl0168.27501MR218361
  4. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. IHES 20 (1964). Zbl0136.15901
  5. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math., Springer-Verlag (1995). Zbl0819.13001
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  7. M. Artin, A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, tome 1, Springer Lecture Notes in Mathematics 269 (1972). Zbl0234.00007
  8. T. Sekiguchi, On the deformations of Witt groups to tori. II, J. Algebra 138, no. 2 (1991), 273–297. Zbl0735.14033
  9. T. Sekiguchi, F. Oort, N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4) 22 no. 3 (1989), 345–375. Zbl0714.14024MR1011987
  10. T. Sekiguchi and N. Suwa, A note on extensions of algebraic and formal groups. IV. Kummer-Artin-Schreier-Witt theory of degree p 2 , Tohoku Math. J. (2) 53 no. 2 (2001), 203–240. Zbl1073.14546MR1829979
  11. T. Sekiguchi and N. Suwa, A note on extensions of algebraic and formal groups. V, Japan. J. Math. 29, no. 2 (2003), 221–284. Zbl1075.14045
  12. T. Sekiguchi and N. Suwa, On the unified Kummer-Artin-Schreier-Witt Theory, no. 111 in the preprint series of the Laboratoire de Mathématiques Pures de Bordeaux (1999). 
  13. T. Sekiguchi and N. Suwa, Some cases of extensions of group sche,es over a discrete valuation ring I, J. Fac. Sci. Univ. Tokyo, Sect IA, Math, 38 (1991), 1–45. Zbl0793.14035MR1104364
  14. D. Tossici, Models of μ p 2 , K over a discrete valuation ring, with an appendix by X. Caruso. J. Algebra 323 no. 7 (2010), 1908–1957. Zbl1193.14059MR2594655
  15. W. Waterhouse, B. Weisfeiler, One-dimensional affine group schemes, J. Algebra 66 no. 2 (1980), 550–568. Zbl0452.14013MR593611
  16. T. Yasuda, Non-adic formal schemes, Int. Math. Res. Not. IMRN no. 13 (2009), 2417–2475. Zbl1245.14004MR2520785

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