Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-25T21:15:19.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexes of Cousin type and modules of generalized fractions

Published online by Cambridge University Press:  22 January 2016

Sang-Cho Chung
Sang-Cho Chung*
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Nagoya, 464-01, Japan
*
Department of Mathematics, Chungnam National University, Taejon 305-764, Korea
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a commutative (Noetherian) ring, M an R-module and let ℱ = (Fi)i≥0 be a filtration of Spec(R) which admits M.

A complex of R-modules is said to be of Cousin type if it satisfies the four conditions of ([GO], 3.2) which are reproduced below (Definition (1.5)). In ([RSZ], 3.4), Riley, Sharp and Zakeri proved that the complex, which is constructed from a chain of special triangular subsets defined in terms of ℱ (Example (1.3)(3)), is of Cousin type for M with respect to ℱ (Corollary (3.5)(2)). Gibson and O’carroll ([GO], 3.6) showed that the complex, which is obtained by means of a chain = (Ui)i≥1 of saturated triangular subsets and the filtration = (Gi)i≥0 induced by and M, is of Cousin type for M with respect to (Corollary (3.5)(3)).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 136 , December 1994 , pp. 17 - 34
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[B] Bourbaki, N., “Commutative algebra,” Addison-Wesley publishing company, 1972.Google Scholar
[C] Chung, S. C., Associated prime ideals and isomorphisms of modules of generalized fractions, to appear in Math. J. Toyama Univ., 17 (1994).Google Scholar
[GO] Gibson, G. J. and O’carroll, L., Direct limit systems, generalized fractions and complexes of Cousin type, J. Pure Appl. Algebra, 54 (1988), 249–259.Google Scholar
[HS] Hamieh, M. A. and Sharp, R. Y., Krull dimension and generalized fractions, Proc. Edinburgh Math. Soc., 28 (1985), 349–353.Google Scholar
[M] Matsumuta, H., “Commutative ring theory,” Cambridge University Press, 1986.Google Scholar
[O] O’carroll, L., On the generalized fractions of Sharp and Zakeri, J. London Math. Soc., (2) 28 (1983), 417–427.CrossRefGoogle Scholar
[RSZ] Riley, A. M., Sharp, R. Y. and Zakeri, H., Cousin complexes and generalized fractions, Glasgow Math. J., 26 (1985), 51–67.Google Scholar
[SI] Sharp, R. Y., The Cousin complex for a module over a commutative Noetherian ring, Math. Z., 112 (1969), 340–356.Google Scholar
[S2] Sharp, R. Y., Gorenstein modules, Math. Z., 115 (1970), 117–139.Google Scholar
[S3] Sharp, R. Y., On the structure of certain exact Cousin complexes, LN in Pure Appl. Math., 84 (1981), 275–290.Google Scholar
[S4] 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
[ST] Sharp, R. Y. and Tang, Z., On the structure of Cousin complexes, J. Math. Kyoto Univ., 33-1 (1993), 285–297.Google Scholar
[SY] Sharp, R. Y. and Yassi, M., Generalized fractions and Hughes’ grade-theoretic analogue of the Cousin complex, Glasgow Math. J., 32 (1990), 173–188.Google Scholar
[SZ1] Sharp, R. Y. and Zakeri, H., Modules of generalized fractions, Mathematika, 29 (1982), 32–41.Google Scholar
[SZ2] 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, 72 (Cambridge University Press, 1982), 61–82.Google Scholar
[SZ3] Sharp, R. Y. and Zakeri, H., Local cohomology and modules of generalized fractions, Mathematika, 29 (1982), 296–306.Google Scholar
[SZ4] Sharp, R. Y. and Zakeri, H., Generalized fractions, Buchsbaum modules and generalized Cohen-Macaulay modules, Math. Proc. Camb. Phil. Soc., 98 (1985), 429–436.Google Scholar
[SV] Stückrad, J. and Vogel, W., “Buchsbaum rings and applications,” Springer-Verlag, 1986.Google Scholar