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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 $$G^{(\mu)}$$.

##### MSC:
 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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