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

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→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 Tateʼs extension theorem for homomorphisms of p-divisible groups.