Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic

TL;DR

This paper provides a new, simpler proof of bounds on the kernel and cokernel of homomorphisms between finite flat group schemes over mixed characteristic valuation rings, improving existing bounds and extending key theorems.

Contribution

It introduces a concise method to sharpen bounds on group scheme homomorphisms and extends Tate's theorem for p-divisible groups.

Findings

Sharper bounds on kernel and cokernel of special fiber homomorphisms

New proof of extension theorem for truncated Barsotti--Tate groups

Simplified approach compared to previous results

Abstract

Let $p$ be a prime. Let $V$ be a discrete valuation ring of mixed characteristic $(0,p)$ and index of ramification $e$. Let $f: G \rightarrow H$ be a homomorphism of finite flat commutative group schemes of $p$ power order over $V$ whose generic fiber is an isomorphism. We provide a new proof of a result of Bondarko and Liu that bounds the kernel and the cokernel of the special fiber of $f$ in terms of $e$. For $e < p-1$ this reproves a result of Raynaud. Our bounds are sharper that the ones of Liu, are almost as sharp as the ones of Bondarko, and involve a very simple and short method. As an application we obtain a new proof of an extension theorem for homomorphisms of truncated Barsotti--Tate groups which strengthens Tate's extension theorem for homomorphisms of $p$-divisible groups.