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

Abstract

We explicitly describe the Cartier dual of the $l$-th Frobenius kernel $N_1$ of the group scheme $\mathscr{G}^\lambda$, which deforms $\mathbf{G}_a$ to $\mathbf{G}_m$. Then the Cartier dual of $N_1$ is given by a certain Frobenius type kernel of the Witt scheme. Here we assume that the base ring $A$ is a $\mathbf{Z}_{(p)} / (p^n)$-algebra, where $p$ is a prime number. The obtained result generalizes a previous result by the author [1] which assumes that $A$ is an $\mathbf{F}_p$--algebra.