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

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.

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.

DOI : https://doi.org/10.5802/jtnb.713
Classification:  13B05,  14L15,  12G05
@article{JTNB_2010__22_1_219_0,
     author = {Tsuno, Yuji},
     title = {Degeneration of the Kummer sequence in characteristic $p>0$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     pages = {219-257},
     doi = {10.5802/jtnb.713},
     mrnumber = {2675882},
     zbl = {1237.14055},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2010__22_1_219_0}
}
Tsuno, Yuji. Degeneration of the Kummer sequence in characteristic $p>0$. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 219-257. doi : 10.5802/jtnb.713. http://www.numdam.org/item/JTNB_2010__22_1_219_0/

[1] M. Demazure and P. Gabriel, Groupes algébriques, Tome I. Masson & Cie, Editeur, Paris; North-Holland Publishing, Amsterdam, 1970. | MR 302656 | Zbl 0203.23401

[2] A. Grothendieck, Le groupe de Brauer. Dix exposés sur la cohomologie des schémas, 46–188. North-Holland, 1968. | MR 244269

[3] B. Mazur, L. Roberts, Local Euler Characteristics. Invent. math. 9 (1970), 201–234. | MR 258844 | Zbl 0191.19202

[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 2345371 | Zbl 1155.14025

[5] T. Sekiguchi and N. Suwa, Théorie de Kummer-Artin-Schreier et applications. J. Théor. Nombres Bordeaux 7 (1995), 177–189. | Numdam | MR 1413576 | Zbl 0920.14023

[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 1011987 | Zbl 0714.14024

[7] J. P. Serre, Groupes algébriques et corps de classes. Hermann, Paris, 1959. | MR 103191 | Zbl 0097.35604

[8] R. P. Stanley, Enumerative Combinatorics, vol. 1. Cambridge Stud. Adv. Math. vol. 49, Cambridge University Press, Cambridge, 1997. | MR 1442260 | Zbl 0889.05001

[9] N. Suwa, Twisted Kummer and Kummer-Artin-Schreier theories. Tôhoku Math. J. 60 (2008), 183–218. | MR 2428860 | Zbl 1145.13005

[10] N. Suwa, Around Kummer theories. RIMS Kôkyûroku Bessatsu B12 (2009), 115–148. | Zbl pre05635228

[11] J. Tate and F. Oort, Group scheme of prime order. Ann. Sci. Éc. Norm. Sup. (4) 3 (1970), 1–21. | Numdam | MR 265368 | Zbl 0195.50801

[12] W. C. Waterhouse, Introduction to affine group schemes. Springer, 1979. | MR 547117 | Zbl 0442.14017

[13] W. C. Waterhouse, A unified Kummer-Artin-Schreier sequence. Math. Ann. 277 (1987), 447–451. | MR 891585 | Zbl 0608.12026

[14] W. C. Waterhouse and B. Weisfeiler, One-dimensional affine group schemes. J. Algebra 66 (1980), 550–568. | MR 593611 | Zbl 0452.14013