FINITE TORSORS ON PROJECTIVE SCHEMES DEFINED OVER A DISCRETE VALUATION RING

Overview & Synopsis

Given a Henselian and Japanese discrete valuation ring A and a flat and projective A-scheme X, we follow the approach of [BdS11] 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 → 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 → X has a reduction of structure group to a finite one.