# On basic and Bass quaternion orders

## TL;DR

This paper proves that in quaternion orders over Dedekind domains, being Bass and being basic are equivalent properties, with basicness being a local property, unifying local and global perspectives.

## Contribution

It establishes the equivalence of Bass and basic conditions for quaternion orders, highlighting the local nature of basicness in these structures.

## Key findings

- Bass and basic properties are equivalent in quaternion orders.
- Basicness is a local property of quaternion orders.
- The equivalence holds in both local and global settings.

## Abstract

A quaternion order O over a Dedekind domain R is Bass if every R-superorder is Gorenstein, and O is basic if it contains an integrally closed quadratic R-order. In this article, we show that these conditions are equivalent in local and global settings: a quaternion order is Bass if and only if it is basic. In particular, we show that the property of being basic is a local property of a quaternion order.