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

TL;DR

This paper explores how properties of Breuil-Kisin modules are inherited by p-divisible groups under a faithful algebra action, constructing a new category of modules and analyzing their structural properties.

Contribution

It introduces a new category of Breuil-Kisin modules over a specific ring and investigates their freeness and projectiveness when associated with p-divisible groups.

Findings

Constructed a new category of Breuil-Kisin modules over  ring.

Analyzed the freeness and projectiveness properties of these modules.

Established conditions under which these modules inherit structural properties from p-divisible groups.

Abstract

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 construct a category of new Breuil-Kisin module $\mathfrak{M}$ defined over the ring $\mathfrak{S}:=W(\kappa)[\![u]\!]$ and study the freeness and projectiveness properties of such a module.