\'Etale Extensions of Unipotent Torsors

TL;DR

This paper investigates extension properties of unipotent torsors over fields of positive characteristic, establishing conditions for their extension over valuation rings and algebraic curves, with applications to fundamental group schemes.

Contribution

It proves that unipotent torsors over generic points extend to finite separable extensions and ramified covers, advancing understanding of torsor extensions in positive characteristic.

Findings

Unipotent torsors over generic points extend to finite separable extensions.

Torsors over open subsets of curves extend to ramified covers étale over the subset.

Applications include isomorphisms between unipotent variants of Nori's fundamental group schemes.

Abstract

In this paper we study extension problems for torsors in positive characteristic. Let $F$ be a field of characteristic $p>0$ and $U/F$ be a unipotent algebraic group. As our first main result, we prove that every $U$-torsor defined over the generic point of a discrete valuation ring $\mathcal{O}_{K}$, containing a field $F$, extends to the normalization of $\mathcal{O}_{K}$ in some finite separable extension of its fraction field. We then globalize this result and prove that for $X/F$ a normal integral curve over an algebraically closed field $F$, every $U$-torsor over an open set $X^{\circ}\subseteq X$ extends to some ramified cover of $X$ which is \'etale over $X^{\circ}$. As an application, we are able to find isomorphisms between certain unipotent variants of Nori's fundamental group scheme for curves.