$G$-displays of Hodge type and formal $p$-divisible groups

Patrick Daniels

Published 2020-09-18Version 1

Let $G$ be a smooth group scheme over the $p$-adic integers with reductive generic fiber, and let $\mu$ be a minuscule cocharacter for $G$ which remains minuscule in a faithful representation of $G$. We show that the category of nilpotent $(G,\mu)$-displays over $p$-nilpotent rings $R$ embeds fully faithfully into the category of formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors, and that the embedding becomes an equivalence when $R/pR$ has a $p$-basis \'etale locally. The definition of the embedding relies on the construction of a G-crystal (i.e., an exact tensor functor from representations of G to crystals in finite locally free $\mathcal{O}_{\text{Spec } R/ \mathbb{Z}_p}$-modules) associated to any adjoint nilpotent $(G,\mu)$-display, which extends the construction of the crystal associated to a nilpotent Zink display. As applications, we obtain an explicit comparison between the Rapoport-Zink functors of Hodge type defined by Kim and by B\"ueltel and Pappas, and we extend a theorem of Faltings regarding deformations of connected $p$-divisible groups with crystalline Tate tensors.