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A case of extensions of group schemes over a discrete valuation ring. (English) Zbl 0755.14014
Let \(A\) be a discrete valuation field of characteristic 0 with residue field of prime characteristic \(p\). Let \(G\) be a group scheme over \(\text{Spec}(A)\) with generic fiber isomorphic to the multiplicative group and special fiber isomorphic to the additive group. It is known [cf. W. C. Waterhouse and B. Weisfeiler, J. Algebra 66, 550- 568 (1980; Zbl 0452.14013); see also F. Oort and the authors, Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 3, 345-375 (1989; Zbl 0714.14024)] that \(G\) is isomorphic to the group scheme \(G^{(\lambda)}=\text{Spec} A[x,1/(\lambda x+1)]\) with the group law \({(x,y)\mapsto\lambda xy+x+y}\), where \(\lambda\) is a non-zero element of the maximal ideal of \(A\).
The paper under review contains an explicit description of extensions of \(G^{(\lambda)}\) by \(G^{(\mu)}\).

14L15 Group schemes
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
14B12 Local deformation theory, Artin approximation, etc.
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