Finite torsors on projective schemes defined over a discrete valuation ring

TL;DR

This paper develops a Tannakian framework for finite torsors on projective schemes over a discrete valuation ring, establishing a classification analogous to Nori's fundamental group and comparing with recent theories.

Contribution

It introduces a Tannakian category of coherent modules on such schemes and proves the pro-finiteness of the associated affine group, extending Nori's theory.

Findings

The affine group classifying finite torsors is pro-finite and Mittag-Leffler.

Any quasi-finite torsor admits a reduction to a finite group scheme.

The developed theory aligns with and extends recent approaches by Mehta-Subramanian and Antei-Emsalem-Gasbarri.

Abstract

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Q\to X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Q\to X$ has a reduction of structure group to a finite one.