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Cousin complexes and generalized fractions

Published online by Cambridge University Press:  18 May 2009

Adrian M. Riley ,
Rodney Y. Sharp  and
Hossein Zakeri
Adrian M. Riley
Affiliation:
Department of Pure Mathematics, The University of Sheffield, Hicks Building, Sheffield S3 7RH, England
Rodney Y. Sharp
Affiliation:
Department of Pure Mathematics, The University of Sheffield, Hicks Building, Sheffield S3 7RH, England
Hossein Zakeri
Affiliation:
Department of Mathematics, University for Teacher Education, 49 Mobarezan Avenue, Tehran, Iran
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Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [6, §2]: it is a complex of A-modules and A-homomorphisms

with the property that, for each n≥0,

Cohen-Macaulay rings may be characterized in terms of the Cousin complex: A is a Cohen-Macaulay ring if and only if C(A) is exact [6, (4.7)]. Also the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring: see [6, (5.4)].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 51 - 67
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Bass, H., On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28.CrossRefGoogle Scholar
2.Grothendieck, A., Local cohomology, Lecture Notes in Mathematics No. 41 (Springer, 1967).Google Scholar
3.Herzog, J. and Kunz, E., Der kanonische Modul eines Cohen-Macaulay Rings, Lecture Notes in Mathematics No. 238 (Springer, 1971).CrossRefGoogle Scholar
4.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
5.Matsumura, H., Commutative algebra (Benjamin, 1980).Google Scholar
6.Sharp, R. Y., The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356.CrossRefGoogle Scholar
7.Sharp, R. Y., Gorenstein modules, Math. Z. 115 (1970), 117–139.CrossRefGoogle Scholar
8.Sharp, R. Y., Cousin complex characterizations of two classes of commutative Noetherian rings, J. London Math. Soc. (2) 3 (1971), 621–624.CrossRefGoogle Scholar
9.Sharp, R. Y., Local cohomology and the Cousin complex for a commutative Noetherian ring, Math. Z. 153 (1977), 19–22.CrossRefGoogle Scholar
10.Sharp, R. Y., Cohen-Macaulay properties for balanced big Cohen-Macaulay modules, Math. Proc. Cambridge Philos. Soc. 90 (1981), 229–238.CrossRefGoogle Scholar
11.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
12.Sharp, R. Y., On the structure of certain exact Cousin complexes, Commutative algebra: proceedings of the Trento conference, Lecture Notes in Pure and Applied Mathematics No. 84 (Marcel Dekker, 1983), 275–290.CrossRefGoogle Scholar
13.Sharp, R. Y. and Zakeri, H., Modules of generalized fractions, Mathematika 29 (1982), 32–41.CrossRefGoogle Scholar
14.Sharp, R. Y. and Zakeri, H., Modules of generalized fractions and balanced big Cohen-Macaulay modules, Commutative algebra: Durham 1981, London Mathematical Society Lecture Notes No. 72 (Cambridge University Press, 1982), 61–82.Google Scholar
15.Zakeri, H., Modules of generalized fractions and their application in commutative algebra, Ph.D. Thesis, University of Sheffield (1982).Google Scholar