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Let $k$ be a perfect field of characteristic $p \geq 3$. We classify$p$-divisible groups over regular local rings of the form$W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where$b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element.This classification was in the case $r = 0$ conjectured by Breuil and proved byKisin.

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