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

TL;DR

This paper explicitly describes the Cartier duals of Frobenius kernels of certain finite group schemes that deform additive to multiplicative groups, generalizing prior results to more general base rings.

Contribution

It provides a detailed description of the Cartier duals for a class of finite group schemes over more general base rings, extending previous work limited to fields.

Findings

Explicit duality descriptions for Frobenius kernels

Generalization from $ ext{F}_p$-algebras to $ ext{Z}_{(p)}$-algebras

Extension of duality results to broader base rings

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.