Group schemes of square free order

TL;DR

This paper studies the structure of finite flat group schemes of square-free order over schemes, showing they can be decomposed into extensions involving prime order components, with splitting after base change.

Contribution

It provides a structural decomposition theorem for square-free order group schemes, detailing how they can be expressed as extensions of etale and prime order group schemes.

Findings

Any such group scheme can be written as an extension of an etale group scheme by a sum of prime order schemes.

The extension splits after a finite etale surjective base change.

The structure theorem applies to all finite flat finitely presented group schemes of square-free order.

Abstract

Every finite flat finitely presented group scheme G of square free order over a scheme S can be written as an extension of a finite etale S-group scheme G" by a commutative finite flat finitely presented S-group scheme G' that is a direct sum of group schemes of prime order. Such an extension splits over a finite etale surjective base change.