Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-16T12:33:34.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Certain countably generated big Cohen-Macaulay modules are balanced

Published online by Cambridge University Press:  24 October 2008

R. Y. Sharp
R. Y. Sharp
Affiliation:
Department of Pure Mathematics, The University, Sheffield S3 7RH
Get access Rights & Permissions [Opens in a new window]

Extract

Throughout this note, A will denote a (commutative, Noetherian) local ring (with identity) having maximal ideal m and dimension d. Let x1, …, xd be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay A-module with respect to x1, …, xd, if x1, …, xd is an M-sequence. In the last ten or fifteen years there has been substantial interest in such modules, initially stemming from M. Hochster's discoveries that, if A contains a field as a subring, and x1, …,xd is any s.o.p. for A, then there exists a big Cohen-Macaulay A-module with respect to x1, …,xd, and that the existence of such modules has important consequences for the local homological conjectures in commutative algebra: see [6].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 73 - 77
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bartijn, J. and Strooker, J. R.. Modifications monomiales. In Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, Paris, 1982, Lecture Notes in Math. vol. 1029 (Springer-Verlag, 1983), pp. 192–217.CrossRefGoogle Scholar
[2]Bourbaki, N.. Algèbre homologique. Chapitre 10 of Algèbre (Masson, 1980).Google Scholar
[3]Foxby, H.-B.. On the μ in a minimal injective resolution II. Math. Scand. 41 (1977), 19–44.CrossRefGoogle Scholar
[4]Griffith, P.. A representation theorem for complete local rings. J. Pure Appl. Algebra 7 (1976), 303–315.CrossRefGoogle Scholar
[5]Griffith, P.. Maximal Cohen-Macaulay modules and representation theory. J. Pure Appl. Algebra 13 (1978), 321–334.CrossRefGoogle Scholar
[6]Hochster, M.. Topics in the Homological Theory of Modules over Commutative Rings. C.B.M.S. Regional Conference Series in Mathematics no. 24 (American Mathematical Society, 1975).CrossRefGoogle Scholar
[7]Hochster, M.. Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors. In Proceedings of the Conference on Commutative Algebra, Queen's University, Kingston, Ontario,1975 (Queen's University papers on Pure and Applied Mathematics no. 42, 1975), pp. 106–195.Google Scholar
[8]Matsumura, H.. Commutative Algebra (Benjamin/Cummings, 1980).Google Scholar
[9]Carroll, L. O'. Balanced big Cohen-Macaulay modules and ring extensions. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), 171–172.CrossRefGoogle Scholar
[10]Ogoma, T.. Fibre products of Noetherian rings and their applications. Math. Proc. Cambridge Philos. Soc. 97 (1985), 231–241.CrossRefGoogle Scholar
[11]Riley, A. M.. Balanced big Cohen-Macaulay modules and free extensions of local rings. Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 41–45.CrossRefGoogle Scholar
[12]Sharp, R. Y.. Cohen-Macaulay properties for balanced big Cohen-Macaulay modules. Math. Proc. Cambridge Philos. Soc. 90 (1981), 229–238.CrossRefGoogle Scholar
[13]Sharp, R. Y.. A Cousin complex characterization of balanced big Cohen-Macaulay modules. Quart. J. Math. Oxford Ser. (2) 33 (1982), 471–485.CrossRefGoogle Scholar
[14]Sharp, R. Y. and Vámos, P.. Baire's category theorem and prime avoidance in complete local rings. Arch. Math. (Basel) 44 (1985), 243–248.CrossRefGoogle Scholar
[15]Takeuchi, Y.. On localizations of a balanced big Cohen-Macaulay module. Kobe J. Math. 1 (1984), 43–46.Google Scholar
[16]Zarzuela, S.. Balanced big Cohen-Macaulay modules and flat extensions of rings. Math. Proc. Cambridge Philos. Soc. 102 (1987), 203–209.CrossRefGoogle Scholar