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On the lengths of Koszul homology modules and generalized fractions

Published online by Cambridge University Press:  24 October 2008

N. T. Cuong  and
N. D. Minh
N. T. Cuong
Affiliation:
Hanoi Institute of Mathematics, P.O. Box 631, Bo Ho, 10.000 Hanoi, Vietnam
N. D. Minh
Affiliation:
Hanoi Institute of Mathematics, P.O. Box 631, Bo Ho, 10.000 Hanoi, Vietnam
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Extract

Throughout this paper, let A be a Noetherian local ring with maximal ideal m and M a finitely generated A-module with d = dimAM ≥ 1. Denote by N the set of all positive integers.

Let x = (x1, …, xd) be a system of parameters (s.o.p) for M and let

We consider the following two problems: (i) When is the length of Koszul homology

a polynomial in n for all k = 0, …, d and n1; …, nd sufficiently large (n ≫ 0)?

(ii) Is the length of the generalized fraction in a polynomial in n for n ≫ 0?

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 31 - 42
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Auslander, M. and Buchsbaum, D. A.. Codimension and multiplicity. Ann. Math. 68 (1958), 625–657.CrossRefGoogle Scholar
[2]Cuong, N. T.. On the length of the powers of systems of parameters in local ring. Nagoya Math. J. 120 (1990), 77–88.CrossRefGoogle Scholar
[3]Cuong, N. T.. On the dimension of the non-Cohen-Macaulay locus of local ring admitting dualizing complexes. Math. Proc. Cambridge Phil. Soc. (2) 109 (1991), 479–488.CrossRefGoogle Scholar
[4]Cuong, N. T.. On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings. Nagoya Math. J. 125 (1992), 105–114.CrossRefGoogle Scholar
[5]Cuong, N. T.. The theory of polynomial types and p-standard ideals in local rings and applications, preprint.Google Scholar
[6]Cuong, N. T., Schenzel, P. and Trung, N. V.. Verallgemeinerte Cohen-Macaulay Moduln. Math. Nach. 85 (1978), 57–75.Google Scholar
[7]Garcla Roig, J- I.. On polynomial bounds for the Koszul homology of certain multiplicity systems. J. London Math. Soc. (2) 34 (1986), 411–416.CrossRefGoogle Scholar
[8]Goto, S. and Yamagishi, K.. The theory of unconditional strong d-sequences and modules of finite local cohomology, preprint.Google Scholar
[9]Kaplansky, I.. Commutative rings (Allyn and Bacon, 1970).Google Scholar
[10]Kirby, D.. Artinian modules and Hilbert polynomials. Quart. J. Math. Oxford (2) 24 (1973), 47–57.CrossRefGoogle Scholar
[11]MacDonald, I. G.. Secondary representation of modules over a commutative ring. Sympos. Math. 11 (1973), 23–43.Google Scholar
[12]Matsumura, H.. Commutative algebra. Second Edition (Benjamin, 1980).Google Scholar
[13]Schenzel, P.. Cohomological annihilators. Math. Proc. Cambridge Phil. Soc. 91 (1982), 345–350.CrossRefGoogle Scholar
[14]Sharp, R. Y. and Hamieh, M. A.. Lengths of certain generalized fractions. J. Pure Appl. Algebra 38 (1985), 323–336.CrossRefGoogle Scholar
[15]Sharp, R. Y. and Zakeri, H.. Modules of generalized fractions. Mathematika 29 (1982), 32–41.CrossRefGoogle Scholar
[16]Sharp, R. Y. and Zakeri, H.. Local cohomology and modules of generalized fractions. Mathematika 29 (1982), 296–306.CrossRefGoogle Scholar
[17]Sharp, R. Y. and Zakeri, H.. Generalized fractions and the monomial conjecture. J. Algebra 92 (1985), 380–388.CrossRefGoogle Scholar
[18]Trung, N. V.. Toward a theory of generalized Cohen–Macaulay modules. Nagoya Math. J. 102 (1986), 1–49.CrossRefGoogle Scholar