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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On symmetric orders and separable algebras

Author: T. V. Fossum
Journal: Trans. Amer. Math. Soc. 180 (1973), 301-314
MSC: Primary 16A16
MathSciNet review: 0318203
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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.

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Article copyright: © Copyright 1973 American Mathematical Society