Maximal models of torsors over a local field

TL;DR

This paper explores the concept of maximal models of torsors over a local field, providing new results on their maximality, compatibility, and existence for certain group schemes, advancing the understanding of ramification in torsor theory.

Contribution

It offers a detailed investigation into maximal models of torsors, including proofs of their maximality, compatibility, and existence for semi-direct product group schemes, building on Raynaud's and Lewin-Ménégaux's work.

Findings

Proved maximality of the models.

Established compatibility along inductions.

Proved existence for semi-direct product group schemes.

Abstract

Let $R$ be a discrete valuation ring, and $K$ its fraction field. In 1967, Raynaud initiated the notion of maximal $R$-model for torsors over $K$, and it was further developed by Lewin-M\'en\'egaux. In this paper, motivated by a conjectural ramification theory for infinitesimal torsors, we investigate this notion of maximal model in greater detail. We prove the maximality, the compatibility along inductions, and an existence result for group schemes of semi-direct products.