Purity for Barsotti-Tate groups in some mixed characteristic situations

TL;DR

This paper extends a result on the extension of Barsotti-Tate groups over certain mixed characteristic local rings, correcting previous errors and applying to regular schemes with specific local conditions.

Contribution

It generalizes Vasiu-Zink's result from dimension 2 to higher dimensions and corrects errors in existing literature on Barsotti-Tate group extensions.

Findings

Extension of Barsotti-Tate groups under specified conditions

Correction of errors in prior literature

Application to regular schemes with controlled ramification

Abstract

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