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arXiv · 1904.10659

Finite torsors on projective schemes defined over a discrete valuation ring

Phung Ho Hai·Joao Pedro dos Santos
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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.

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Phung Ho Hai, Joao Pedro dos Santos. 2019-04-24. Finite torsors on projective schemes defined over a discrete valuation ring. https://arxiv.org/abs/1904.10659

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