Orders in Separable Algebras

Abstract

The module ${P^ \ast }/m{P^ \ast }$, where $P$ is an order in a separable algebra over the quotient field of an integrally closed, quasi-local domain, is studied. It is shown that if the domain is complete, ${P^ \ast }/m{P^ \ast }$ contains one element from each isomorphism class of irreducible $P$ modules. Also, in general, if the global dimension of $P$ is finite, then it equals the homological dimension of ${P^ \ast }/m{P^ \ast }$.