On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

In the first part of this work [12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of D-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of infinite type, Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of D-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

Automated summary

This work studies affine group schemes over a discrete valuation ring using Neron blowups, focusing on a certain class of infinite-type affine group schemes known as Neron blowups of formal subgroups. The study shows how these group schemes naturally appear in Tannakian categories of D-modules, and introduces a Tannakian property named prudence for affine group schemes, which aids in verifying the underlying ring of functions. This property is then successfully applied to derive a general result on the structure of differential Galois groups over complete DVRs.