On the structure of affine flat group schemes over discrete valuation rings, II

TL;DR

This paper investigates the structure of affine flat group schemes over discrete valuation rings, focusing on Neron blowups and their role in Tannakian categories of D-modules, culminating in a criterion for differential Galois groups.

Contribution

It introduces the concept of prudence in affine group schemes and applies it to analyze the structure of differential Galois groups over complete DVRs.

Findings

Neron blowups of formal subgroups are typical in affine group schemes of infinite type.

Prudence is a Tannakian property that helps verify if the ring of functions is free.

A general structure theorem for differential Galois groups over complete DVRs is established.

Abstract

In the 1st part of this work [DHdS18], 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.