Models of torsors over affine spaces

TL;DR

This paper proves that pointed torsors over affine spaces over a field can be extended over a discrete valuation ring after an automorphism, with an effective proof approach and discussions on special cases.

Contribution

It introduces a method to extend pointed torsors over affine spaces from a field to a ring, possibly after automorphism, advancing understanding of torsor behavior over base changes.

Findings

Extension of torsors is possible after automorphism.

Effective proof method provided.

Discussion of special cases like $ ext{alpha}_{p,R}$.

Abstract

Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type group scheme can be extended to a torsor over $\mathbb{A}^{n}_{R}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}^{n}_{K}$. The proof is effective. Other cases, including $X=\alpha_{p,R}$, will also be discussed.