Models of the group schemes of roots of unity

TL;DR

This paper constructs and analyzes finite flat models of p^n-th roots of unity over certain valuation rings, comparing Kummer group schemes with Breuil-Kisin modules, and conjectures their equivalence.

Contribution

It introduces the concept of Kummer group schemes as finite flat models of roots of unity and compares them with Breuil-Kisin modules, providing explicit computations for small n.

Findings

Construction of Kummer group schemes from Sekiguchi and Suwa's work.

Comparison between Kummer group schemes and Breuil-Kisin modules.

Evidence supporting the conjecture that all models are Kummer group schemes.

Abstract

Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of \mu_{p^n,K}, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n < 4. This leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer group schemes.