On the Cartier duality of certain finite group schemes of type \((p^n , \ldots , p^n )\) (Q533729)

Let \(A\) be a commutative ring of characteristic \(p\). \textit{T. Sekiguchi} and \textit{N. Suwa} [``On the unified Kummer-Artin-Schreier-Witt Theory'', Prèpublications du Laboratoire de Mathèmatiques Pures de Bordeaux, Prèpublication No. 111 (1999)] (see also [\textit{A. Mézard}, \textit{M. Romagny} and \textit{D. Tossici}, ``Sekiguchi-Suwa theory revisited'', \url{arXiv:1104.2222} (2011)]) have constructed certain group schemes \(\mathcal E_n\) over \(A\) which are \(n\)-fold iterated extensions of group schemes of the form \[ \mathcal G^{(\lambda)} = \mathrm{spec}(A[T,(1+\lambda T)^{-1}]) \] for \(\lambda \in A\). In the paper under review, the Cartier duals of finite \(p\)-subgroup schemes of \(\mathcal E_n\) arising as Frobenius kernels \[ \mathrm{ker}(\psi^{(l)}) : \mathcal E_n \to \mathcal E_n^{(p^l)} \qquad \psi^{(l)}(x_0,\dots,x_{n-1})=(x_0^{p^l},\dots,x_{n-1}^{p^l}) \] are identified, extending and completing earlier results in the case \(n=2\) of \textit{M. Amano} and the author [Tsukuba J. Math. 34, No. 1, 31--46 (2010; Zbl 1204.14020)].