Hopf orders in $K[C_p^3]$ in characteristic $p$

TL;DR

This paper classifies Hopf orders in the dual of the group algebra of an elementary abelian p-group over a local field of characteristic p, and computes their duals under certain conditions.

Contribution

It provides a complete classification of Hopf orders in $(K[C_p^3])^*$ and computes dual Hopf orders in $K[C_p^3]$ under mild assumptions.

Findings

Complete classification of Hopf orders in $(K[C_p^3])^*$

Explicit computation of dual Hopf orders in $K[C_p^3]$

Results applicable to characteristic p local fields

Abstract

Let $p$ be prime, let $K$ be a non-archimedean local field of characteristic $p$ and let $C_p^3$ denote the elementary abelian group of order $p^3$. We give a complete classification of Hopf orders in the $K$-Hopf algebra $(K[C_p^3])^*$ and under a mild condition compute their dual Hopf orders in the group ring $K[C_p^3]$.