Volume 16, Issue 1 (4-2021)                   IJMSI 2021, 16(1): 169-180 | Back to browse issues page

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Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local cohomology modules $H^{i}_{R_{+}}(M,N)$. Finally, the tame
loci $T^{i}(M,N)$ of $(M,N)$ will be considered and some sufficient conditions are proposed for the openness of these sets in the Zariski topology.

Type of Study: Research paper | Subject: Special

1. M. P. Brodmann, Asymptotic behaviour of cohomology:tameness, support and associated primes, in commutative algebra and
2. algebraic geometry, S. Ghorpade, H. Srinivasan and J. Verma, eds.,contemp. Math 390 (2005) 31-61. [DOI:10.1090/conm/390/07292]
3. M. P. Brodmann, A cohomological stability result for projective schemes over surfaces, J. Reine angew. Math, 606,(2007) 179-92. [DOI:10.1515/CRELLE.2007.040]
4. M. P. Brodmann, S. Fumasoli and C. S. Lim, Low-codimensional associated primes of graded components of local cohomology modules, J. Alg. 275 (2004) 867-882. [DOI:10.1016/j.jalgebra.2003.12.003]
5. M. P. Brodmann and M. Jahangiri, Tame loci of certain local cohomology modules, J. Commut. Alg. 4(1),(2012) 79-100. [DOI:10.1216/JCA-2012-4-1-79]
6. M. P. Brodmann and R. Y. Sharp, Local cohomology -An Algebraic introduction with geometric applications, (Cambridge Studies in Advanced Mathematics 60, Cambridge University Press (1998). [DOI:10.1017/CBO9780511629204]
7. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in Advanced Mathematics 39, Revised edition, Cambridge University Press (1998). [DOI:10.1017/CBO9780511608681]
8. S. D. Cutkosky and J. Herzog, Failure of tameness for local cohomology, J. Pure Appl. Alg. 211 (2007) 428-432. [DOI:10.1016/j.jpaa.2007.01.017]
9. F. Dehghani-Zadeh and H. Zakeri, Some Results on Graded Generalized Local Cohomology Modules, J. Math. Ext 5(1),(2010) ,9-73.
10. K. Divaani-Azar and A. Hajikarimi, Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals,Canad. Math. Bull (2011) 1-7. [DOI:10.4153/CMB-2011-046-8]
11. J. Herzog, Komplexe, Aufl"{o}sungen und Dualit"{a}t in der Lokalen Algebra, Habilitationsschrift, Universit"{a}t Regensburg,1974.
12. M. Jahangiri, N. Shirmohammadi and sh. Tahamtan,Tameness and Artinianness of graded generalized local cohomology modules, Alg. Colloq. 22(1) (2015) 131-146. [DOI:10.1142/S1005386715000127]
13. K. Khashyarmanesh, Associated primes of graded components of generalized local cohomology modules, Comm. Alg. 33,(2005),3081-3090. [DOI:10.1081/AGB-200066119]
14. D. Kirby, Artinian modules and Hilbert polynomials,Q. J. Math 24(2) (1973) 17-57.C. S. Lim, Graded local cohomology modules and their associated primes: the Cohen-Macauly case, J. Pure Appl. Alg.185(2003) 225-238. [DOI:10.1016/S0022-4049(03)00091-4]
15. H. Matsumura, Commutative Ring Theory, Cambridge, UK:Cambridge University Press (1986).
16. L. Melkersson,Properties of cofinite modules and applications to local cohomology , Math. Proc. Camb. Phil. Soc. 125, (1999), 417-423. [DOI:10.1017/S0305004198003041]
17. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, Orlando (1979).
18. C. Rotthaus and L. M. Sega, Some properties of graded local cohomology modules, J. Algebra, 283, (2005), 232- 247. [DOI:10.1016/j.jalgebra.2004.07.034]
19. N. Suzuki, On the generalized local cohomology and its duality J. Math. Kyoto. Univ 18 (1978) 71-85. [DOI:10.1215/kjm/1250522630]
20. S. Yassemi, Generalized section functors, J. Pure. Appl. Alg. 95 (1994) 103-119. [DOI:10.1016/0022-4049(94)90121-X]
21. N. Zamani, On graded generalize local cohomology, Arch.Math. 86 (2006) 321-330. [DOI:10.1007/s00013-005-1524-6]