Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T06:08:09.821Z Has data issue: false hasContentIssue false
  • English
  • Français

COFINITENESS AND FINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  23 July 2009

LIZHONG CHU
LIZHONG CHU*
Affiliation:
Department of Mathematics, Suzhou University, 215006, Jiangsu, PR China (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

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 I be an ideal of a commutative Noetherian local ring R, and M and N two finitely generated modules. Let t be a positive integer. We mainly prove that (i) if HIi(M,N) is Artinian for all i<t, then HIi(M,N) is I-cofinite for all i<t and Hom(R/I,HIt(M,N)) is finitely generated; (ii) if d=pd(M)<∞ and dim N=n<∞, then HId+n(M,N) is I-cofinite. We also prove that if M is a nonzero cyclic R-module, then HIi(N) is finitely generated for all i<t if and only if HIi(M,N) is finitely generated for all i<t.

Keywords

local cohomologygeneralized local cohomologycofiniteArtinian

MSC classification

Secondary: 13D45: Local cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 244 - 250
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

Supported by the National Natural Science Foundation (No. 10771152) of China, by the Research Foundation (No. Q4107805) of Suzhou University and by the Research Foundation of Pre-research Project (No. Q3107852) of Suzhou University.

References

[1] Asadollahi, J., Khashyarmanesh, K. and Salarian, Sh., ‘On the finiteness properties of the generalized local cohomology modules’, Comm. Algebra 30 (2002), 859–867.CrossRefGoogle Scholar
[2] Bijan-Zadeh, M. H., ‘A common generalization of local cohomology theories’, Glasgow Math. J. 21 (1980), 173–181.CrossRefGoogle Scholar
[3] Brodmann, M. P. and Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[4] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Revised edn (Cambridge University Press, Cambridge, 1996).Google Scholar
[5] Chu, L. Z. and Tang, Z. M., ‘On the Artinianness of generalized local cohomology’, Comm. Algebra 35 (2007), 3821–3827.CrossRefGoogle Scholar
[6] Delfino, D. and Marley, T., ‘Cofinite modules and local cohomology’, J. Pure Appl. Algebra 121 (1997), 45–52.CrossRefGoogle Scholar
[7] Dibaei, M. T. and Yassemi, S., ‘Associated primes and cofiniteness of local cohomology modules’, Manuscripta Math. 117 (2005), 199–205.CrossRefGoogle Scholar
[8] Divaani-Aazar, K. and Sazeedeh, R., ‘Cofiniteness of generalized local cohomology modules’, Colloq. Math. 99 (2004), 283–290.CrossRefGoogle Scholar
[9] Hartshorne, R., ‘Affine duality and cofiniteness’, Invent. Math 9 (1970), 145–164.CrossRefGoogle Scholar
[10] Herzog, J., Komplexe Auflösungen und Dualität in der lokalen Algebra, Universität Regensburg, Preprint, 1974.Google Scholar
[11] Khashyarmanesh, K. and Yassi, M., ‘On the finiteness property of generalized local cohomology modules’, Algebra Colloq. 12 (2005), 293–300.CrossRefGoogle Scholar
[12] Khashyarmanesh, K., Yassi, M. and Abbasi, A., ‘Filter regular sequence and generalized local cohomology modules’, Comm. Algebra 32 (2004), 253–259.CrossRefGoogle Scholar
[13] Melkersson, L., ‘Modules cofinite with respect to an ideal’, J. Algebra 285 (2005), 649–668.CrossRefGoogle Scholar
[14] Nagel, U. and Schenzel, P., ‘Cohomological annihilators and Castelnuovo Mumford regularity’, in: Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992) (American Mathematical Society, Providence, RI, 1994), pp. 307–328.CrossRefGoogle Scholar
[15] Rotman, J., Introduction to Homological Algebra (Academic Press, New York, 1979).Google Scholar
[16] Suzuki, N., ‘On the generalized local cohomology and its duality’, J. Math. Kyoto. Univ. 18 (1978), 71–85.Google Scholar
[17] Yassemi, S., ‘Cofinite modules’, Comm. Algebra 29 (2001), 2333–2340.CrossRefGoogle Scholar