Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On symmetric orders and separable algebras
HTML articles powered by AMS MathViewer

by T. V. Fossum PDF
Trans. Amer. Math. Soc. 180 (1973), 301-314 Request permission

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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A16
  • Retrieve articles in all journals with MSC: 16A16
Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 301-314
  • MSC: Primary 16A16
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0318203-1
  • MathSciNet review: 0318203