On Bosch-L\"utkebohmert-Raynaud's Conjecture I

TL;DR

This paper investigates Bosch-Lütkebohmert-Raynaud's conjecture on the existence of Néron lft-models for algebraic groups over certain schemes, providing new counterexamples and a proof in the perfect residue field case.

Contribution

It introduces a new construction of counterexamples and offers an elementary proof of the conjecture for perfect residue fields, utilizing the concept of weakly permawound unipotent groups.

Findings

Counterexamples to the conjecture are more numerous and conceptually explained.

The conjecture is proven for perfect residue fields using new methods.

A new framework for understanding Néron lft-models is developed.

Abstract

Let $G$ be a smooth algebraic group over the field of rational functions of an excellent Dedekind scheme $S$ of equal characteristic $p>0.$ A N\'eron lft-model of $G$ is a smooth separated model $\mathscr{G} \to S$ of $G$ satisfying a universal property. Predicting whether a given $G$ admits such a model is a very delicate (and, in general, open) question if $S$ has infinitely many closed points, which is the subject of Conjecture I due to Bosch-L\"utkebohmert-Raynaud. This conjecture was recently proven by T. Suzuki and the author if the residue fields of $S$ at closed points are perfect, but refuted in general. The aim of the present paper is two-fold: firstly, we give a new construction of counterexamples which is more general and provides a conceptual explanation for the only counterexamples known previously, as well as providing many new counterexamples. Secondly, we shall give a new and elementary proof of Conjecture I in the case of perfect residue fields. Both parts make use of the concept of weakly permawound unipotent groups recently introduced by Rosengarten.