Tohoku Mathematical Journal

A note on extensions of algebraic and formal groups, IV. Kummer-Artin-Schreier-Witt theory of degree $p^2$

Tsutomu Sekiguchi and Noriyuki Suwa

Full-text: Open access

Abstract

We establish a formula for homomorphisms and extensions of group schemes and formal groups, related to deformations of the multiplicative group to the additive group. As an application, we give an explicit description of the theory unifying the Kummer and Artin-Schreier-Witt theories of degree $p^2$.

Article information

Source
Tohoku Math. J. (2), Volume 53, Number 2 (2001), 203-240.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178207479

Digital Object Identifier
doi:10.2748/tmj/1178207479

Mathematical Reviews number (MathSciNet)
MR2002f:14059

Zentralblatt MATH identifier
1073.14546

Subjects
Primary: 14L05: Formal groups, $p$-divisible groups [See also 55N22]
Secondary: 13K05 14L15: Group schemes 20G10: Cohomology theory

Citation

Sekiguchi, Tsutomu; Suwa, Noriyuki. A note on extensions of algebraic and formal groups, IV. Kummer-Artin-Schreier-Witt theory of degree $p^2$. Tohoku Math. J. (2) 53 (2001), no. 2, 203--240. doi:10.2748/tmj/1178207479. https://projecteuclid.org/euclid.tmj/1178207479


Export citation

References

  • [1] N. BOURBAKI, Algebre commutative, Chapitres 8 et 9, Masson, Paris, 1983.
  • [2] M. DEMAZURE and P. GABRIEL, Groupes algebriques, Tome 1, Masson-North-Holland, Paris-Amsterdam, 1970.
  • [3] B. DWORK, On the rationalityof the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631-648
  • [4] B. GREEN and M. MATIGNON, Liftings of Galois covers of smooth curves, Compositio Math. 113 (1998), 237-272.
  • [5] A. GROTHENDIECK with J. DIEUDONNE, Elements de geometric algebrique, IV, Inst. Hautes Etudes Sci Publ. Math. No. 28, 1966, 255 pp.
  • [6] M. HAZEWINKEL, Formal groups and applications, Academic Press, New York, 1978
  • [7] L. ILLUSIE, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Ecole Norm. Sup. (4) 1 (1979), 501-661.
  • [8] M. LAZARD, Sur les groupes de Lie formels a un parametre, Bull. Soc.Math. France 83 (1955), 251-274
  • [9] T. SEKIGUCHI, On the deformations of Witt groups to tori II, J. Algebra 138 (1991), 273-297
  • [10] T. SEKIGUCHI and N. SUWA, A case of extensions of group schemes over a discrete valuation ring, Tsukub J. Math. 14 (1990), 459-^87.
  • [11] T. SEKIGUCHI and N. SUWA, Some cases of extensions of group schemes over a discrete valuation ring I, J. Fac. Sci.Univ. Tokyo Sect. IA Math. 38 (1991), 1-45.
  • [12] T. SEKIGUCHI and N. SUWA, A note on extensions of algebraic and formal groups I, Math. Z. 206 (1991), 567-575.
  • [13] T. SEKIGUCHI and N. SUWA, A note on extensions of algebraic and formal groups II, Math. Z. 217 (1994), 447-457.
  • [14] T. SEKIGUCHI and N. SUWA, On the unified Kummer-Artin-Schreier-Witt theory, Preprint series, CHU MATH No. 41, 1994.
  • [15] T. SEKIGUCHI and N. SUWA, Theories de Kummer-Artin-Schreier-Witt, C. R. Acad. Sci. Paris Ser. I Math 319(1994), 105-110.
  • [16] T. SEKIGUCHI and N. SUWA, A note on extensions of algebraic and formal groups III, Thoku Math. J. (1997), 241-257.
  • [17] T. SEKIGUCHI, F. OORT and N. SUWA, On the deformation of Artin-Schreier to Kummer, Ann. Sci. Ecol Norm. Sup.(4) 22 (1989), 345-375.

See also

  • Part III: Tsutomu Sekiguchi, Noriyuki Suwa. A note on extensions of algebraic and formal groups, III. Tohoku Math. J., Volume 49, Number 2 (1997), pp. 241-257.