Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic (Q432471)

Let \(p\) be a prime and \(V\) a discrete valuation ring. Let \(K\) be the field of fractions of \(V\) of characteristic \(0\) and \(k\) the residue field of \(V\) of characteristic \(p\). For a finite flat commutative group scheme \(G\) of \(p\) power order over \(V\), let \(G[p^n]\) be the schematic closure of \(G_k[p^n]\) in \(G\).NEWLINENEWLINEThe first main result of this paper isNEWLINENEWLINENEWLINETheorem 1. There exists a non-negative integer \(s\) that depends only on \(V\) such that the following property holds: For any finite flat commutative group schemes \(G\) and \(H\) over \(V\) and for any homomorphism \(f: G \to H\) whose generic fiber is an isomorphism, there exists a homomorphism \(f' : H \to G\) such that \(f' \circ f = p^s\) and \(f \circ f' = p^s\).NEWLINENEWLINENEWLINETheorem 1 is proved by \textit{M. V. Bondarko} in [Izv. Math. 70, No. 4, 661--691 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 4, 21--52 (2006; Zbl 1136.14033)] and by \textit{T. Liu} in [J. Number Theory 126, No. 2, 155--184 (2007; Zbl 1131.14051), Ann. Sci. Éc. Norm. Supér. (4) 40, No. 4, 633--674 (2007; Zbl 1163.11043)] using different methods. In this paper, the computable upper bounds of \(s\) is computed in terms of the \(e\), the ramification index of \(V\). The upper bound is \(s \leq (\mathrm{log}_pe+\mathrm{ord}_pe+2)(\mathrm{ord}_pe+2)\). It is in general weaker than the one provided by Bondarko although it will regain Bondarko's upper bound when \(p\) does not divide \(e\), but it is much stronger than the upper bounds provided in two papers of Liu.NEWLINENEWLINEThe second main result of this paper is somewhat a corollary of the first result, that is,NEWLINENEWLINE{ Theorem 2:} Let \(Y\) be a normal Noetherian integral scheme with field of functions \(L\) of characteristic zero. There exists a non-negative integer \(s_Y\) such that the following holds: Let \(\mathcal{G}\) and \(\mathcal{H}\) be truncated Barsotti-Tate groups over \(Y\) of level \(n>s_Y\) and of order a power of the prime \(p\). Let \(h:\mathcal{G}_L \to \mathcal{H}_L\) be a homomorphism. Then there exists a unique homomorphism \(g:\mathcal{G}[p^{n-s_y}] \to \mathcal{H}[p^{n-s_y}]\) that induces \(h[p^{n-s_y}]\) over \(L\).NEWLINENEWLINEThis theorem gives a sharper version of a theorem of Tate.