Smoothness of the truncated display functor

TL;DR

This paper establishes a smooth functor from truncated Barsotti-Tate groups to truncated displays using crystalline Dieudonne theory, providing new proofs of key equivalences in p-adic geometry and group schemes.

Contribution

It introduces a smooth display functor for truncated p-divisible groups and offers new proofs of fundamental equivalences in the theory of p-adic groups and Dieudonne modules.

Findings

The functor from truncated Barsotti-Tate groups to displays is smooth.

New proof of equivalence between infinitesimal p-divisible groups and nilpotent displays.

New proof of equivalence between finite flat p-group schemes and Dieudonne modules.

Abstract

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.