On Bosch-Lütkebohmert-Raynaud's Conjecture I
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913510430081024 |
|---|---|
| author | Overkamp, Otto |
| author_facet | Overkamp, Otto |
| contents | 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 Néron 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-Lütkebohmert-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Bosch-Lütkebohmert-Raynaud's Conjecture I Overkamp, Otto Algebraic Geometry Number Theory 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 Néron 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-Lütkebohmert-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. |
| title | On Bosch-Lütkebohmert-Raynaud's Conjecture I |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2409.13599 |