Properties of Breuil-Kisin modules inherited by $p$-divisible groups

Absos Ali Shaikh, Mabud Ali Sarkar

Published 2021-03-19Version 1

In this paper, by assuming a faithful action of a finite flat $\mathbb{Z}_p$-algebra $\mathscr{R}$ on a $p$-divisible group $\mathcal{G}$ defined over the ring of $p$-adic integers $\mathscr{O}_K$, we have constructed a new Breuil-Kisin module $\mathfrak{M}$ defined over the ring $\mathfrak{S}:=W(\kappa)[[u]]$ and studied freeness and projectiveness properties of such a module. Finally we have lifted the Breuil-Kisin module to \'etale level following the $\mathscr{R}$-action and showed that if the $p$-adic Tate module of a $p$-divisible group $\mathcal{G}$ over $\mathscr{O}_K$ is free then the corresponding lifted Breuil-Kisin module is free.