Smoothness of the truncated display functor

Abstract

We show that to every p p -divisible group over a p p -adic ring one can associate a display by crystalline Dieudonné theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p p -divisible groups and nilpotent displays over p p -adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p p -power order and Dieudonné modules over perfect rings.