On bass orders in division algebras
Abstract
A result by F.-T. Tu dating from 2011 states that the orders in a local matrix algebra of dimension four that can be written as a finite intersection of maximal orders are those having an Eichler Gorenstein closure. In a more recent paper, we gave a similar characterization for orders with a Bass Gorenstein closure. More precisely, we prove that these orders are the ones that can be written as a ghost intersection, i.e., the set of points defined over the base field of an intersection of maximal orders that is defined over a field extension. Although we proved this result only for the local four-dimensional matrix algebra, the statement makes sense also for a local quaternion division algebra, since the latter turns into a matrix algebra when we extend the base field to an even degree extension. In this work, we prove an analog of our previous result in the latter setting.
Related
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- A stacky resolution of singularities for orders by Azumaya algebras
- Simple Torsion-Free Linear Groups from Global Division Algebras
BibTeX
@misc{arenascarmona2026bass,
title = {{On bass orders in division algebras}},
author = {Luis Arenas-Carmona},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16002},
eprint = {2610.16002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16002},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On bass orders in division algebras
AU - Luis Arenas-Carmona
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16002
UR - https://arxiv.org/abs/2610.16002
AB - A result by F.-T. Tu dating from 2011 states that the orders in a local matrix algebra of dimension four that can be written as a finite intersection of maximal orders are those having an Eichler Gorenstein closure. In a more recent paper, we gave a similar characterization for orders with a Bass Gorenstein closure. More precisely, we prove that these orders are the ones that can be written as a ghost intersection, i.e., the set of points defined over the base field of an intersection of maximal orders that is defined over a field extension. Although we proved this result only for the local four-dimensional matrix algebra, the statement makes sense also for a local quaternion division algebra, since the latter turns into a matrix algebra when we extend the base field to an even degree extension. In this work, we prove an analog of our previous result in the latter setting.
ER -