On basic and Bass quaternion orders

Abstract

A quaternion order O \mathcal {O} over a Dedekind domain R R is Bass if every R R -superorder is Gorenstein, and O \mathcal {O} is basic if it contains an integrally closed quadratic R 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.