FINITENESS PROPERTIES GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO AN IDEAL CONTAINING THE IRRELEVANT IDEAL

Title & Authors
FINITENESS PROPERTIES GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO AN IDEAL CONTAINING THE IRRELEVANT IDEAL

Abstract
The membership of the generalized local cohomology modules $\small{H_a^i}$(M,N) of two R-modules M and N with respect to an ideal a in certain Serre subcategories of the category of modules is studied from below ($\small{i}$<$\small{t}$). Furthermore, the behaviour of the $\small{n}$th graded component $\small{H_a^i(M,N)_n}$ of the generalized local cohomology modules with respect to an ideal containing the irrelevant ideal as $\small{n{\rightarrow}-{\infty}}$ is investigated by using the above result, in certain graded situations.
Keywords
generalized local cohomology;finiteness;Serre subcategories;
Language
English
Cited by
References
1.
M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), no. 2, 173-181.

2.
M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Univ. Press 1998.

3.
W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge Univ. Press 1993.

4.
F. Dehghani-Zadeh, Cofinite modules and asymptotic behaviour of generalized local cohomology, Proceedings of the 5-th Asian Mathematical Conference, Malaysia 2009, 592- 602.

5.
F. Dehghani-Zadeh, On the finiteness properties of generalized local cohomology modules, Int. Electron. J. Algebra 10 (2011), 113-122.

6.
F. Dehghani-Zadeh and H. Zakeri, Some results on graded generalized local cohomology modules, J. Math. Ext. 5 (2010), no. 1, 59-73.

7.
J. Herzog, Komplexe, Auflosungen und Dualitat in der Lokalen Algebra, Habilitationsschrift, Universitat Regensburg, 1974.

8.
M. Jahangiri, N. Shirmohammadi, and Sh. Tahamtan, Tameness and Artinianness of graded generalized local cohomology modules, preprint.

9.
M. Jahangiri and H. Zakeri, Local cohomology modules with respect to an ideal containing the irrelevant ideal, J. Pure Appl. Algebra 213 (2009), no. 4, 573-581.

10.
T. Marley and J. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), no. 1, 180-193.

11.
H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press 1986.

12.
L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417-423.

13.
L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668.

14.
J. Rotman, An Introduction to Homological Algebra, Academic Press, Orlando, 1979.