Lagrange's theorem for a family of finite flat group schemes over local Artin rings

TL;DR

This paper proves that certain finite flat group schemes over local Artin rings are killed by their order, extending Grothendieck's question to new non-commutative cases through deformation theory analysis.

Contribution

It introduces a classification for non-commutative k-group schemes over local Artin rings and confirms they are killed by their order, addressing a question in SGA 3.

Findings

Finite flat group schemes over R are killed by their order.

Classification of non-commutative k-group schemes over local Artin rings.

Positive answer to Grothendieck's question in new cases.

Abstract

Let R be a local Artin ring with residue field k of positive characteristic. We prove that every finite flat group scheme over R whose special fiber belongs to a certain explicit family of non-commutative k-group schemes is killed by its order. This is achieved via a classification result which rely on the explicit study of the infinitesimal deformation theory for such non-commutative k-group schemes. The main result answers positively in a new case a question of Grothendieck in SGA 3 on whether all finite flat group schemes are killed by their order.