On the freeness and projectiveness of Breuil-Kisin module

Abstract: In the work we have considered Breuil-Kisin module over the ring of witt vectors $W(\kappa)$ over the residue field $\kappa$ of characteristic $p$ and a finite flat $\mathbb{Z}p$-algebra $R$. Then considered Breuil-Kisin modules $M$ over the ring $W(\kappa)$ and taking the action of $R$ on $W(\kappa)$, we get again a Breuil-Kisin module $M$ over the ring $R \otimes{\mathbb{Z}_p} W(\kappa)$. We have studied freeness and projectiveness of this module.