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

Abstract

In this paper we show that the finite subgroup scheme Spec $A[X, Y]/(X^{p^l}, Y^{p^l})$ of $\mathscr{E}^{\lambda, \mu, D} \in {\rm Ext}^1(\mathscr{G}^{(\lambda)}, \mathscr{G}^{(\mu)})$ is a Cartier dual of a certain finite subgroup scheme of the fiber product $W_{l,A} \times_{{\rm Spec} A} W_{l,A}$ of Witt vectors of length $l$ in positive characteristic $p$. After this, we treat the kernel of the type $F^2 + [a]F + [b]: W_{l,A} \to W_{l,A}$, where $F$ is the Frobenius endomorphism and $[a]$ is the Teichmüller lifting of $a \in A$, respectively.