Torsion graded pieces of Nyggard filtration for crystalline representation

Abstract: Let $K$ be a unramified $p$-adic field with the absolute Galois group $G_K$ and $T$ a crystalline $\mathbb Z_p$-representation of $G_K$. We study the graded pieces of integral filtration on $D_{\rm dR}(T)$ given by Nyggard filtration of the attached Breuil-Kisin module of $T$. We show that the $i$-graded piece has nontrivial $p$-torsion only if $ i = r_j +m p$ for a Hodge-Tate weight $ r_j$ of $T$ and $m$ a positive integer.