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arXiv · 2409.13599

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

Otto Overkamp
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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.

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Otto Overkamp. 2024-09-20. On Bosch-L\"utkebohmert-Raynaud's Conjecture I. https://arxiv.org/abs/2409.13599

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