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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 Tates extension theorem for homomorphisms of $p$-divisible groups.
Let $cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear f
Let $(R, mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/mathfrak{m} = mathbb{F}_{q}$. Given a monic polynomial $P(t) in R[t]$ whose reduction modulo $mathfrak{m}$ gives an irreducible polynomial $bar{P}(t) in mat
We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular com
Let $p$ be a prime. Let $(R,ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $Spec Rsetminus{ideal{m}}$ extends to an abelian schem
Let $Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $lambda$ of the field of algebraic numbers which is prime to p, consider the $lambda$-adic pro-semisimple completion of $Pi$ as an object of the groupoi