Models of Z/p^2 Z over a d.v.r. of unequal characteristic

TL;DR

This paper classifies and describes models of the group scheme Z/p^2 Z over a discrete valuation ring with unequal characteristic, focusing on those that match the generic fiber and relate to the Kummer sequence.

Contribution

It provides a classification and explicit description of finite flat R-group schemes of order p^2 that model Z/p^2 Z over a DVR with unequal characteristic.

Findings

Any such model fits into a short exact sequence matching the Kummer sequence.

Explicit classification of models of Z/p^2 Z over the ring R.

Connection established between models and the Kummer sequence.

Abstract

Let R be a discrete valuation ring of unequal characteristic which contains a primitive p^2-th root of unity. If K is the fraction field of R, it is well known that (Z/p^2 Z)_K is isomorphic to \mu_{p^2,K}. We prove that any finite and flat R-group scheme of order p^2 isomorphic to (Z/p^2 Z)_K on the generic fiber (i.e. a model of (Z/p^2 Z)_K), is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models.