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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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References

REFERENCES

[1]Atiyah, M. F. and MacDonald, I. G.. Introduction to Commutative Algebra (Addison-Wesley. 1969).Google Scholar
[2]Auslander, M. and Buchsbaum, D. A.. Codimension and multiplicity. Ann. of Math. 68 (1958), 625–657.CrossRefGoogle Scholar
[3]Bartijn, J.. Flatness, completion, regular sequences; un ménage à trois. Ph.D. thesis, University of Utrecht (1985).Google Scholar
[4]Burch, L.. Codimension and analytic spread. Proc. Cambridge Philos. Soc. 72 (1972), 369–373.CrossRefGoogle Scholar
[5]Davis, E. D.. Ideals of the principle class, R-sequences and a certain monoidal transformation. Pacific J. Math. 20 (1967), 197–205.CrossRefGoogle Scholar
[6]Duncan, A.. Two topics in commutative ring theory. Ph.D. thesis, University of Edinburgh (1988).Google Scholar
[7]Foxby, H-B.. On the μi in a minimal injective resolution II. Math. Scand. 41 (1977), 19–44.CrossRefGoogle Scholar
[8]Gilmer, R.. Contracted ideals in Krull domains. Duke Math. J. 37 (1970). 769–774.CrossRefGoogle Scholar
[9]Griffith, P.. A representation theorem for complete local rings. J. Pure Appl. Algebra 13 (1976). 303–315.CrossRefGoogle Scholar
[10]Griffith, P.. Maximal Cohen–Macaulay modules and representation theory. J. Pure Appl. Algebra 13 (1978), 321–334.CrossRefGoogle Scholar
[11]Hochster, M.. Big Cohen–Macaulay modules and algebras and embeddability in rings of Wit vectors. In Proceedings of the conference on Commutative Algebra, Queen's University Kingston, Ontario, 1975 (Queen's University papers on Pure and Applied Mathematic no. 42, 1975), pp. 106–195.Google Scholar
[12]Kaplansky, I.. Commutative Rings (Allyn and Bacon, Inc., 1970).Google Scholar
[13]Matsumura, H.. Commutative Ring Theory. Cambridge Studies in Advanced Math. no. 5(Cambridge University Press, 1986).Google Scholar
[14]McAdam, S.. Finite coverings by ideals. In Ring Theory, Proceedings of the Oklahonu Conference, 1973 (Marcel Dekker Inc., 1974), pp. 163–171.Google Scholar
[15]McCoy, N. H.. A note on finite unions of ideals and subgroups. Proc. Amer. Math. Soc. 8 (1957) 633–637.CrossRefGoogle Scholar
[16]Sharp, R. Y.. Cohen–Macaulay properties for balanced big Cohen–Macaulay modules. Math Proc. Cambridge Philos. Soc. 90 (1981), 229–238.CrossRefGoogle Scholar
[17]Sharp, R. Y.. Certain countably generated big Cohen–Macaulay modules are balanced. Math Proc. Cambridge Philos. Soc. 105 (1989), 73–77.CrossRefGoogle Scholar
[18]Sharp, R. Y. and Vámos, P.. Baire's category theorem and prime avoidance in complete loca rings. Arch. Math. (Basel) 44 (1985), 243–248.CrossRefGoogle Scholar