On symmetric orders and separable algebras

Abstract:

Let $K$ be an algebraic number field, and let $\Lambda$ be an $R$-order in a separable $K$-algebra $A$, where $R$ is a Dedekind domain with quotient field $K$; let $\Delta$ denote the center of $\Lambda$. A left $\Lambda$-lattice is a finitely generated left $\Lambda$-module which is torsion free as an $R$-module. For left $\Lambda$-modules $M$ and $N, \operatorname {Ext} _\Lambda ^1(M,N)$ is a module over $\Delta$. In this paper we examine ideals of $\Delta$ which are the annihilators of $\operatorname {Ext} _\Lambda ^1(M,\_)$ for certain classes of left $\Lambda$-lattices $M$ related to the central idempotents of $A$, and we compute these ideals explicitly if $\Lambda$ is a symmetric $R$-algebra. For a group algebra, these ideals determine the defect of a block. We then compare these annihilator ideals with another set of ideals of $\Delta$ which are closely related to the homological different of $\Lambda$, and which in a sense measure deviation from separability. Finally we show that, for $\Lambda$ to be separable over $R$, it is necessary and sufficient that $\Lambda$ is a symmetric $R$-algebra, $\Delta$ is separable over $R$, and the center of each localization of $\Lambda$ at the maximal ideals of $R$ maps onto the center of its residue class algebra.