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 $R = \mathcal{O}_K[\![s_1, \ldots, s_d]\!]$ and $R = \mathcal{O}_K\langle t_1^{\pm 1}, \ldots, t_d^{\pm 1}\rangle$. 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 $e \geq p$. 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 $e \geq p$, we prove that such a locus is not $p$-adically closed.

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在相对情况下扩展 p 可分群和 Barsotti-Tate 变形环

令 $k$ 是特征 $p > 2$ 的完美域，令 $K$ 是 $W(k)[\frac{1}{p}]$ 的分枝度 $e$ 的有限完全分枝扩展. 我们考虑满足某些条件的 $W(k)$ 上的未分支基环 $R_0$，并让 $R = R_0\otimes_{W(k)}\mathcal{O}_K$。这种 $R$ 的例子包括 $R = \mathcal{O}_K[\![s_1, \ldots, s_d]\!]$ 和 $R = \mathcal{O}_K\langle t_1^{\pm 1} , \ldots, t_d^{\pm 1}\rangle$。我们表明，当 $e < p-1$ 时，雷诺定理关于扩展 $p$-可分群的推广适用于基环 $R$，而当 $R = \mathcal{O}_K[\ ![s]\!]$ 与 $e \geq p$。作为一个应用，我们证明如果 $R$ 有 Krull 维 $2$ 并且 $e < p-1$，那么 $\mathrm{Gal}(\overline{R}[\frac{1}{p}]/R[\frac{1}{p}])$ 的 Barsotti-Tate 表示的轨迹切出一个封闭的子方案的通用变形方案。如果 $R = \mathcal{O}_K[\![s]\!]$ 和 $e \geq p$，我们证明这样的轨迹不是 $p$-adically 封闭的。