Models of mu_{p^2,K} over a discrete valuation ring

TL;DR

This paper classifies and explicitly describes models of the group scheme _{p^2,K} over a discrete valuation ring, extending the understanding of finite flat group schemes in algebraic geometry.

Contribution

It proves that any finite flat R-group scheme matching _{p^2,K} on the generic fiber fits into a short exact sequence akin to the Kummer sequence, and provides explicit classifications.

Findings

Classified models of _{p^2,K} over discrete valuation rings.

Established the structure of finite flat group schemes isomorphic to _{p^2,K} on the generic fiber.

Connected the models to the Kummer sequence in a geometric setting.

Abstract

Let R be a discrete valuation ring with residue field of characteristic p>0. Let K be its fraction field. We prove that any finite and flat R-group scheme, isomorphic to \mu_{p^2,K} on the generic fiber, is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models. In the appendix X. Caruso shows how to classify models of \mu_{p^2,K}, in the case of unequal characteristic, using the Breuil-Kisin theory.