On the group of extensions Ext\(^{1}(\mathcal{G}^{(\lambda _{0})}, \mathcal{E}^{(\lambda _{1}, \ldots , \lambda _n )})\) over a discrete valuation ring (Q533738)

Let \(A\) be a discrete valuation ring with maximal ideal \(\mathfrak{m}\). For \(\lambda\in\mathfrak{m},\) \(\lambda\neq0\), we let \(\mathcal{G}^{\left( \lambda\right) }\) be the \(A\)-group scheme \(\mathcal{G}^{\left( \lambda\right) }\mathcal{=}\)Spec\(\left( A\left[ X,\left( 1+\lambda X\right) ^{-1}\right] \right) ,\) where the multiplication on \(\mathcal{G} ^{\left( \lambda\right) }\) is induced from the comultiplication on \(A\left[ X,\left( 1+\lambda X\right) ^{-1}\right] \) given by \(X\mapsto X\otimes1+1\otimes X+\lambda X\otimes X.\) Such group schemes are generically isomorphic to the multiplicative group scheme \(\mathbb{G}_{m},\) isomorphic to the additive group scheme \(\mathbb{G}_{a}\) over \(A/\mathfrak{m}\), and constitute the group schemes of dimension one. For \(\lambda_{1},\lambda_{2} \in\mathfrak{m}\setminus\left\{ 0\right\} \) one can compute extensions \(\mathcal{E}^{\left( \lambda_{1},\lambda_{2}\right) }\) of \(\mathcal{G} ^{\left( \lambda_{1}\right) }\) by \(\mathcal{G}^{\left( \lambda_{2}\right) };\) one can then take \(\lambda_{3}\in\mathfrak{m}\setminus\left\{ 0\right\} \) and compute the extensions of \(\mathcal{E}^{\left( \lambda_{1},\lambda _{2}\right) }\) by \(\mathcal{G}^{\left( \lambda_{3}\right) }.\) Inductively, extensions of the form \(\mathcal{E}^{\left( \lambda_{1},\dots\lambda _{n}\right) }\) can be constructed. In the work under review, the author constructs the group Ext\(^{1}\left( \mathcal{G}^{\left( \lambda_{0}\right) },\mathcal{E}^{\left( \lambda _{1},\dots\lambda_{n}\right) }\right) .\) The main result is an adaptation of the results in [\textit{D. Horikawa}, On the extensions of some group schemes. Master thesis, Chuo University (2002)] and [\textit{T. Kondo}, ``On the extensions of group schemes deforming \(\mathbb{G}_{a}\) to \(\mathbb{G}_{m}\)'', Tokyo J. Math. 33, No. 2, 283--309 (2010; Zbl 1213.14080)] in the cases \(n=2,3\). He obtains a rather explicit description of the extensions in terms of the Artin-Hasse exponential series, and the proof of his construction constitutes half of this paper. It is also natural to be interested in the group Ext\(^{1}\left( \mathcal{E}^{\left( \lambda_{1},\dots\lambda_{n-1}\right) },\mathcal{G} ^{\left( \lambda_{n}\right) }\right) .\) However, this group is already known, having appeared in [\textit{T. Sekiguchi} and \textit{N. Suwa}, ``On the unified Kummer-Artin-Schreier-Witt theory'', Prépublication No. 11, Université de Bordeaux (1999)]. The results of [loc. cit.] are used in this new construction.