Abstract
Let \((R, {\mathfrak{m}})\) be a Noetherian local ring, I an ideal of R and M, N two finitely generated R-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that \(H^j_I(M, N) = 0\) for all j > dim(R), provided M is of finite projective dimension. Next, we study and give characterizations for the least and the last integer r such that Supp\((H^r_I(M, N))\) is infinite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bijan-Zadeh M.H. (1980). A common generalization of local cohomology theories. Glasgow Math. J. 21: 173–181
Brodmann M.P. and Sharp R.Y. (1998). Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge University Press, Cambridge
Bruns W. and Herzog J. (1998). Cohen-Macaulay Rings. Cambridge University Press, Cambridge
Cuong N.T. and Hoang N.V. (2005). Some finite properties of generalized local cohomology modules. East-West J. Math. (2) 7: 107–115
Eisenbud D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin
Herzog, J.: Komplexe, Auflösungen und dualität in der localen Algebra, Habilitationsschrift. Universität Regensburg (1970)
Herzog J. and Zamani N. (2003). Duality and vanishing of generalized local cohomology. Arch. Math. J. (5) 81: 512–519
Huneke, C.: Problems on local cohomology. In: Free Resolution in Commutative Algebraic Geometry, vol. 2, pp. 93–108. Bartlett, Boston, MA (1992)
Huneke C. and Sharp R.Y. (1993). Bass numbers of local cohomology modules. Trans. Amer. Math. Soc. 339: 765–779
Kaplansky, I.: Commutative ring. University of Chicago Press (revised edition) (1974)
Katzman M. (2002). An example of an infinite set of associated primes of local cohomology module. J. Alg. 252: 161–166
Khashyarmanesh K. and Salarian Sh. (1999). On the associated primes of local cohomology modules. Comm. Alg. 27: 6191–6198
Marley Th. (2001). Associated primes of local cohomology module over rings of small dimension. Manuscripta Math. (4) 104: 519–525
Nhan L.T. (2005). On generalized regular sequences and the finiteness for associated primes of local cohomology modules. Comm. Alg. 33: 793–806
Rotman J. (1979). Introduction to Homological Algebra. Academic Press, New York
Suzuki N. (1978). On the generalized local cohomology and its duality. J. Math. Kyoto Univ. 18: 71–78
Vasconcelos W. (1974). Divisor Theory in Module Categories. North-Holand, Amsterdam
Yassemi S. (1994). Generalized section functors. J. Pure Appl. Alg. 95: 103–119
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported in part by the National Basis Research Programme in Natural Science of Vietnam.
Rights and permissions
About this article
Cite this article
Cuong, N.T., Van Hoang, N. On the vanishing and the finiteness of supports of generalized local cohomology modules. manuscripta math. 126, 59–72 (2008). https://doi.org/10.1007/s00229-007-0162-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-007-0162-7