  We show that to every p-divisible group over a p-adic ring one can associate
a display by crystalline Dieudonne 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-divisible groups and nilpotent displays over p-adic rings, and
a new proof of the equivalence due to Berthelot and Gabber between commutative
finite flat group schemes of p-power order and Dieudonne modules over perfect
rings.
