Extending torsors under quasi-finite flat group schemes

TL;DR

This paper investigates extending torsors from the generic fiber to the entire model over a discrete valuation ring, providing simpler descriptions in semistable cases and solutions without finite flat models under certain conditions.

Contribution

It offers a simplified approach for semistable curves and extends torsor existence results to cases lacking finite flat models when R is Henselian and Japanese.

Findings

Simplified description for semistable curves.

Extension of torsors without finite flat models.

Applicable under Henselian and Japanese conditions.

Abstract

Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$. In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the curves in the case when the structural $K$-group scheme of the torsor admits a finite flat model over $R$. In this paper, we first give a simpler description of the problem in the case where the curve is semistable. Secondly, if $R$ is assumed to be Henselian and Japanese, we solve the problem of extending torsors even if the structural group does not admit a finite flat $R$-model.