On the Cartier duality of certain finite group schemes of order $p^n$, III

Abstract

Let $\mathcal{G}^{(\lambda)}$ be a group scheme which deforms $\mathbb{G}_a$ to $\mathbb{G}_m$. We explicitly describe the Cartier dual of the $l$-th Frobenius type kernel $N_l$ of the group scheme $\mathcal{E}^{(\lambda,\mu;D)}$ which is an extension of $\mathcal{G}^{(\lambda)}$ by $\mathcal{G}^{(\mu)}$. Here we assume that the base ring $A$ is a $\mathbb{Z}_{(p)}$-algebra containing some nilpotent elements. The obtained result generalizes a previous result by N. Aki and M. Amano (Tsukuba J.Math. $\textbf{34}$ (2010)) which assumes that $A$ is an $\mathbb{F}_p$-algebra.