Lifting Witt subgroups to characteristic zero
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 \textit{M. Demazure} and \textit{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 \textit{J.-M. Fontaine} [C. R. Acad. Sci., Paris, Sér. A 280, 1273-1276 (1975; Zbl 0331.14022)].
scientific article; zbMATH DE number 1215962