Extensions of group schemes of $\mu$-type by a constant group scheme

TL;DR

This paper investigates conditions under which certain group scheme extensions split over rings of $S$-integers in number fields, focusing on extensions of $oldsymbol{\mu_p}$ by $oldsymbol{\Z/p extbf{ extbackslash Z}}$.

Contribution

It provides criteria for splitting of extensions of $oldsymbol{\mu_p}$ by $oldsymbol{\Z/p extbf{ extbackslash Z}}$ over $O_S$, advancing understanding of group scheme structures over number rings.

Findings

Criteria for splitting over $O_S$ established

Extensions split under specific prime and ring conditions

Enhanced understanding of group scheme extensions in number theory

Abstract

For a number field $K$, a finite set of primes $S$ not containing a fixed prime $p$, we explain when extensions of group schemes of $\mu_p$ by $\Z/p\Z$ split over the ring of $S$-integers $O_S$ of $K$.