On lower ramification subgroups and canonical subgroups

Abstract

Let $p$ be a rational prime, $k$ be a perfect field of characteristic $p$ and $K$ be a finite totally ramified extension of the fraction field of the Witt ring of $k$. Let $\mathcal{G}$ be a finite flat commutative group scheme over ${\mathcal{O}}_K$ killed by some $p$-power. In this paper, we prove a description of ramification subgroups of $\mathcal{G}$ via the Breuil–Kisin classification, generalizing the author’s previous result on the case where $\mathcal{G}$ is killed by $p\ge 3$. As an application, we also prove that the higher canonical subgroup of a level $n$ truncated Barsotti–Tate group $\mathcal{G}$ over ${\mathcal{O}}_K$ coincides with lower ramification subgroups of $\mathcal{G}$ if the Hodge height of $\mathcal{G}$ is less than $\left(p-1\right)∕p^n$, and the existence of a family of higher canonical subgroups improving a previous result of the author.