Effective models of group schemes

TL;DR

This paper investigates models of group schemes acting on flat schemes over a discrete valuation ring, establishing conditions for the existence of faithful models and proving their representability as schemes, especially for pure schemes.

Contribution

It demonstrates that under certain conditions, models of group schemes can be faithfully extended and are representable, introducing the concept of pure schemes and their properties.

Findings

Existence of faithful models as schematic closures in automorphism sheaves.

Pure schemes are amalgamated sums of their generic fiber and finite flat subschemes.

Results extend to formal schemes.

Abstract

Let $R$ be a discrete valuation ring with fraction field $K$ and $X$ a flat $R$-scheme. Given a faithful action of a $K$-group scheme $G_K$ over the generic fibre $X_K$, we study models $G$ of $G_K$ acting on $X$. In various situations, we prove that if such a model $G$ exists, then there exists another model $G'$ that acts faithfully on $X$. This model is the schematic closure of $G$ inside the fppf sheaf $Aut_R(X)$; the major difficulty is to prove that it is representable by a scheme. For example, this holds if $X$ is locally of finite type, separated, flat and pure and $G$ is finite flat. Pure schemes (a notion recalled in the text) have many nice properties : in particular, we prove that they are the amalgamated sum of their generic fibre and the family of their finite flat closed subschemes. We also provide versions of our results in the setting of formal schemes.