Models of torsors over affine spaces

Abstract

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

13 pages. arXiv admin note: substantial text overlap with arXiv:1210.1522