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Infinite coverings of ideals by cosets with applications to regular sequences and balanced big Cohen–Macaulay modules

Published online by Cambridge University Press:  24 October 2008

Andrew J. Duncan
Andrew J. Duncan
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS
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Extract

This paper is devoted to the study of unions of ideals in commutative rings. The starting point is the prime avoidance lemma and an accompanying but diverse body of results on coverings of ideals by unions of ideals, which is described in Section 1. In Sections 4 and 5 these known facts, about finite and infinite unions, are combined and generalized, two distinct but overlapping cases emerging. All that is proved in Sections 4 and 5 turns on the crucial Lemma 2·1 in Section 2, which shows that a cover of an arbitrary ideal by cosets can be lifted to a cover of the entire ring. In Section 3 we introduce and define α-sieves, which provide a concise framework for the expression of applications. In Sections 6 to 9 various applications of Sections 4 and 5 are investigated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 443 - 460
Copyright
Copyright © Cambridge Philosophical Society 1990

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