Degeneration of the Kummer sequence in characteristic p>0
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 219-257.

Nous étudions une déformation de la suite de Kummer à la suite radicielle sur une 𝔽 p -algèbre, qui est duale en un sens pour la déformation de la suite d’Artin-Schreier à la suite radicielle, étudiée par Saidi. Nous examinons aussi quelques relations entre nos suites et l’immersion d’un schéma en groupes commutatifs, fini et plat dans un schéma en groupes commutatifs, lisse, affine et connexe, construite par Grothendieck.

We study a deformation of the Kummer sequence to the radicial sequence over an 𝔽 p -algebra, which is somewhat dual for the deformation of the Artin-Schreier sequence to the radicial sequence, studied by Saidi. We also discuss some relations between our sequences and the embedding of a finite flat commutative group scheme into a connected smooth affine commutative group schemes, constructed by Grothendieck.

DOI : 10.5802/jtnb.713
Classification : 13B05, 14L15, 12G05
Tsuno, Yuji 1

1 Department of Mathematics Chuo University 1-13-27 Kasuga Bunkyo-ku, Tokyo 112-8551, JAPAN
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Tsuno, Yuji. Degeneration of the Kummer sequence in characteristic $p>0$. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 219-257. doi : 10.5802/jtnb.713. http://www.numdam.org/articles/10.5802/jtnb.713/

[1] M. Demazure and P. Gabriel, Groupes algébriques, Tome I. Masson & Cie, Editeur, Paris; North-Holland Publishing, Amsterdam, 1970. | MR | Zbl

[2] A. Grothendieck, Le groupe de Brauer. Dix exposés sur la cohomologie des schémas, 46–188. North-Holland, 1968. | MR

[3] B. Mazur, L. Roberts, Local Euler Characteristics. Invent. math. 9 (1970), 201–234. | MR | Zbl

[4] M. Saidi, On the degeneration of étale /p and /p 2 -torsors in equal characteristic p>0. Hiroshima. Math. J. 37 (2007), 315–341. | MR | Zbl

[5] T. Sekiguchi and N. Suwa, Théorie de Kummer-Artin-Schreier et applications. J. Théor. Nombres Bordeaux 7 (1995), 177–189. | Numdam | MR | Zbl

[6] T. Sekiguchi, F. Oort and N. Suwa, On the deformation of Artin-Schreier to Kummer. Ann. Sci. École Norm. Sup. (4) 22 (1989), 345–375. | Numdam | MR | Zbl

[7] J. P. Serre, Groupes algébriques et corps de classes. Hermann, Paris, 1959. | MR | Zbl

[8] R. P. Stanley, Enumerative Combinatorics, vol. 1. Cambridge Stud. Adv. Math. vol. 49, Cambridge University Press, Cambridge, 1997. | MR | Zbl

[9] N. Suwa, Twisted Kummer and Kummer-Artin-Schreier theories. Tôhoku Math. J. 60 (2008), 183–218. | MR | Zbl

[10] N. Suwa, Around Kummer theories. RIMS Kôkyûroku Bessatsu B12 (2009), 115–148.

[11] J. Tate and F. Oort, Group scheme of prime order. Ann. Sci. Éc. Norm. Sup. (4) 3 (1970), 1–21. | Numdam | MR | Zbl

[12] W. C. Waterhouse, Introduction to affine group schemes. Springer, 1979. | MR | Zbl

[13] W. C. Waterhouse, A unified Kummer-Artin-Schreier sequence. Math. Ann. 277 (1987), 447–451. | MR | Zbl

[14] W. C. Waterhouse and B. Weisfeiler, One-dimensional affine group schemes. J. Algebra 66 (1980), 550–568. | MR | Zbl

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