Tokyo Journal of Mathematics

On the Extensions of Group Schemes Deforming $\mathbf{G}_a$ to $\mathbf{G}_m$

Takashi KONDO

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For given group schemes $\mathcal{G}^{(\lambda_i)}\ (i=1,2,\ldots)$ deforming the additive group scheme $\mathbf{G}_a$ to the multiplicative group scheme $\mathbf{G}_m$, T. Sekiguchi and N. Suwa constructed extensions: $$0\rightarrow\mathcal{G}^{(\lambda_2)}\rightarrow\mathcal{E}^{(\lambda_1,\lambda_2)}\rightarrow\mathcal{G}^{(\lambda_1)}\rightarrow0,\ \ldots,\ 0\rightarrow\mathcal{G}^{(\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})}\rightarrow0,\ldots$$ inductively, by calculating the group of extensions $\mathrm{Ext}^1(\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})},\mathcal{G}^{(\lambda_n)})$. Here we treat the group $\mathrm{Ext}^1(\mathcal{G}^{(\lambda_0)},\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)})$ of extensions in the case of $n=2,\ 3$. The case of $n=2$ was studied by D. Horikawa.

Article information

Tokyo J. Math., Volume 33, Number 2 (2010), 283-309.

First available in Project Euclid: 31 January 2011

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Zentralblatt MATH identifier

Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)


KONDO, Takashi. On the Extensions of Group Schemes Deforming $\mathbf{G}_a$ to $\mathbf{G}_m$. Tokyo J. Math. 33 (2010), no. 2, 283--309. doi:10.3836/tjm/1296483471.

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  • M. Demazure and P. Gabriel, Groups algébriques, Tome 1, Masson-North-Holland, Paris-Amsterdam, 1970.
  • D. Horikawa, On the extensions of some group schemes, Master thesis (in Japanese), Chuo University, 2002.
  • T. Sekiguchi, F. Oort and N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Scient. Éc. Norm. Sup., $4^e$ série 22 (1989), 345–375.
  • T. Sekiguchi and N. Suwa, On the unified Kummer-Artin-Schreier-Witt theory, Prépublication No. 11, Université de Bordeaux (1999).
  • T. Sekiguchi and N. Suwa, A note on extensions of algebraic and formal groups, IV, Tohoku Math. J. 53(2001), 203-240.
  • T. Sekiguchi and N. Suwa, A note on extensions of algebraic and formal groups, V, Japan. J. Math. Vol. 29, No. 2 (2003), 221–284.
  • J. P. Serre, Algebraic Groups and Class Fields, GTM., 117, Springer-Verlag, 1988.
  • W. C. Waterhouse and B. Weisfeiler, One-dimensional Affine group schemes, Journal of Algebra. 66 (1980).