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Lifting Witt subgroups to characteristic zero. (English) Zbl 0919.14030
Let $$k$$ be a perfect field of characteristic $$p>0$$ with $$p$$ odd, let $$W(k)$$ denote the ring of Witt vectors over $$k$$, and let $$W_n$$ denote the group scheme of Witt vectors of finite length $$n$$. The author studies the question, when a connected subgroup scheme of $$W_n$$ lifts to $$W(k)$$ in terms of Dieudonné modules. The Dieudonné ring $$E$$ associated to $$k$$ is the non-commutative ring $$E= W(k)[F,V]$$ with relations $$FV=VF =p$$, $$Fw = w^{\sigma}F$$, $$wV = Vw^{\sigma}$$ for $$w \in W(k)$$, where $$\sigma$$ raises each component of $$w$$ to its $$p$$-th power. Under the anti-equivalence between affine commutative unipotent $$k$$-group schemes and certain modules over $$E$$ [see M. Demazure and P. Gabriel, “Groupes algébriques”, Tome I (1970; Zbl 0203.23401)] the connected subgroup schemes $$G$$ of $$W_n$$ correspond to cyclic Dieudonné modules, i.e., modules of the form $$E/I$$ for some ideal $$I \subset E$$. The author now characterizes the connected subgroup schemes $$G$$ of $$W_n$$, which lift to $$W(k)$$, in terms of the structure of the associated cyclic Dieudonné modules. The main tool is the use of finite Honda systems, and is based on work of J.-M. Fontaine [C. R. Acad. Sci., Paris, Sér. A 280, 1273-1276 (1975; Zbl 0331.14022)].

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 13K05 Witt vectors and related rings (MSC2000) 14G20 Local ground fields in algebraic geometry
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