Breuil's classification of $p$-divisible groups over regular local rings of arbitrary dimension

Abstract

Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $$ W(k) [[t_1, \dots, t_r, u]]/(u^e + pb_{e-1} u^{e-1} + \dots + pb_1 u + pb_0), $$ where $b_0, \dots, b_{e-1} \in W(k) [[t, \dots, t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.