Finiteness of fppf cohomology

TL;DR

This paper proves the finiteness of the first fppf cohomology group for certain group schemes over Henselian DVRs with finite residue fields and applies this to integral models of Shimura varieties.

Contribution

It establishes a finiteness result for fppf cohomology groups of flat group schemes over Henselian DVRs and applies it to PEL type Shimura varieties.

Findings

$H^1_{fppf}(Spec R,G)$ is finite for the given group schemes.

Application to finiteness of integral models of Shimura varieties.

Provides tools for studying arithmetic properties of algebraic groups.

Abstract

Let $R$ be a Henselian DVR with finite residue field. Let $G$ be a finite type, flat $R$-group scheme (not necessarily commutative) with smooth generic fiber. We show that $H^1_{\mathrm{fppf}}(\mathrm{Spec} R,G)$ is finite. We then give an application of the global analogue of this finiteness result to PEL type integral models of Shimura varieties.