On the Cartier Duality of Certain Finite Group Schemes of order $p^n$, II

TL;DR

This paper explicitly describes the Cartier dual of certain finite group schemes related to Frobenius kernels and Witt schemes over specific base rings, generalizing previous characteristic p results.

Contribution

It provides an explicit description of the Cartier duals of Frobenius kernels of deformation group schemes over more general base rings.

Findings

Cartier dual of $N_l$ is a Frobenius type kernel of the Witt scheme

Generalizes previous characteristic p results

Applicable over $Z_{(p)}/(p^n)$-algebras

Abstract

We explicitly describe the Cartier dual of the $l$-th Frobenius kernel $N_l$ of the deformation group scheme, which deforms the additive group scheme to the multiplicative group scheme. Then the Cartier dual of $N_l$ is given by a certain Frobenius type kernel of the Witt scheme. Here we assume that the base ring $A$ is a $Z_{(p)}/(p^n)$-algebra, where $p$ is a prime number. The obtained result generalizes a previous result by the author which assumes that $A$ is a ring of characteristic $p$.