Multidimensional theorem of Shafarevich and Serre

G. T. Konovalov

Matematičeskie zametki, Tome 13 (1973) no. 4, pp. 573-576.

Résumé

Let $H$ be the group obtained by taking the product of $n$ copies of the maximal ideal of the ring of integers $\mathfrak o$ of a local field of characteristic 0 with an algebraically closed residue field $k$ of characteristic $p>0$, and let the composition law be defined as for an n-parametric commutative formal group over 0. Let the kernel of multiplication by $p$ in $H$ be finite. A filtration $p^mH$ ($m\ge0$ is an integer) in $H$ is introduced whose properties allow us to obtain an exact sequence of proalgebraic groups $0\to Z_p^r\to W^s\to H\to0$, where $Z_p$ and $W$ are the additive groups of $p$-adic integers and Witt vectors of infinite length over $k$, respectively; $r\ge0$ and $s>0$ are integers.