Purity for Barsotti–Tate groups in some mixed characteristic situations

Let $p$ be a prime. $R$ regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ Cohen the same residue field $k$ as and $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo does not belong ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ $k[[x_1,\ldots,x_d]]$. We extend result Vasiu-Zink (for $d=2$) show each Barsotti-Tate group over $\text{Frac}(R)$ which extends every $\text{Spec}(R)$ $1$, uniquely $R$. This corrects in many cases several errors literature. As an application, we get if $Y$ integral scheme characteristic formal power series some complete discrete valuation absolute ramification index $e\le p-1$, then generic point $Y$.