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.