#### Vol. 8, No. 2, 2014

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
On lower ramification subgroups and canonical subgroups

### Shin Hattori

Vol. 8 (2014), No. 2, 303–330
##### 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 $\mathsc{G}$ be a finite flat commutative group scheme over ${\mathsc{O}}_{K}$ killed by some $p$-power. In this paper, we prove a description of ramification subgroups of $\mathsc{G}$ via the Breuil–Kisin classification, generalizing the author’s previous result on the case where $\mathsc{G}$ 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 $\mathsc{G}$ over ${\mathsc{O}}_{K}$ coincides with lower ramification subgroups of $\mathsc{G}$ if the Hodge height of $\mathsc{G}$ 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.

##### Keywords
finite flat group scheme, Breuil–Kisin module, canonical subgroup
##### Mathematical Subject Classification 2010
Primary: 11S23
Secondary: 14L05, 14L15
##### Milestones
Revised: 18 November 2013
Accepted: 19 November 2013
Published: 18 May 2014
##### Authors
 Shin Hattori Faculty of Mathematics Kyushu University 744 Motooka, Nishi-ku Fukuoka 819-0395 Japan