Effective models and extension of torsors over a discrete valuation ring of unequal characteristic

TL;DR

This paper studies the extension of torsors over discrete valuation rings of unequal characteristic, providing criteria and explicit calculations for when torsors extend and describing their effective models.

Contribution

It introduces invariants to determine torsor extendability over such rings and explicitly computes the effective model of the group action.

Findings

Criteria for torsor extension when G=Z/p^n Z, n<3, under certain conditions.

Explicit calculation of the effective model of G's action on the torsor.

Conditions under which the torsor structure extends from generic fiber to the whole scheme.

Abstract

Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \to X_K. If p>2, we determine, through these invariants, when Y\to X has a structure of torsor which extends that of Y_K\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y.