Grothendieck Groups of Bass Orders

Abstract

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [ 1 ]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [ 3 ] and Roïter [ 6 ] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [ 2; 8 ]). We shall use the results of Drozd, Kiricenko, and Roïter [ 3 ] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.