On the Lau group scheme

TL;DR

This paper explicitly describes the group scheme banding the gerbe relating truncated Barsotti-Tate groups and displays, generalizing Lau's construction and proposing conjectures for broader contexts.

Contribution

It provides an explicit description of the banding group scheme and extends Lau's results to a more general setting involving (G,μ)-pairs.

Findings

Explicit description of the banding group scheme.

Generalization of the stack of truncated Barsotti-Tate groups.

Conjectural descriptions for broader (G,μ) contexts.

Abstract

In a 2013 article, Eike Lau constructed a canonical morphism from the stack of $n$-truncated Barsotti-Tate groups over $F_p$ to the stack of $n$-truncated displays. He also proved that this morphism is a gerbe banded by a commutative group scheme. In this paper we describe the group scheme explicitly. The stack of $n$-truncated Barsotti-Tate groups over $F_p$ has a generalization related to any pair $(G,\mu)$, where $G$ is a smooth group scheme over $Z/p^n$ and $\mu$ is a 1-bounded cocharacter of $G$. The same is true for the stack of $n$-truncated displays. We conjecture that in this more general situation the first stack is a gerbe over the second one banded by a commutative group scheme, and we give a conjectural description of this group scheme. We also give a conjectural description of the stack of $n$-truncated Barsotti-Tate groups over the formal spectrum of $Z_p$ and of its $(G,\mu)$-generalization.