Extending $p$-divisible groups and Barsotti-Tate deformation ring in the relative case

Abstract

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions, and let $R=R_0{\otimes }_{W(k)}{\mathcal{O}}_K$. Examples of such $R$ include $R={\mathcal{O}}_K[[s_1,\dots ,s_d]]$ and $R={\mathcal{O}}_K⟨t_1^{\pm 1},\dots ,t_d^{\pm 1}⟩$. We show that the generalization of Raynaud's theorem on extending $p$-divisible groups holds over the base ring $R$ when $e<p-1$, whereas it does not hold when $R={\mathcal{O}}_K[[s]]$ with $e\ge p$. As an application, we prove that if $R$ has Krull dimension $2$ and $e<p-1$, then the locus of Barsotti-Tate representations of $\mathrm{Gal}(\overline{R}[\frac{1}{p}]/R[\frac{1}{p}])$ cuts out a closed subscheme of the universal deformation scheme. If $R={\mathcal{O}}_K[[s]]$ with $e\ge p$, we prove that such a locus is not $p$-adically closed.

18 pages; minor corrections and more details added

Cited by in corpus (2)

• Relative crystalline representations and $p$-divisible groups in the small ramification case
• Relative Fontaine-Messing theory over power series rings