On the local-global principles for the $CD_{ < n}$ of local cohomology modules

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran.

2 Department of Basic Sciences, Arak University of Technology, P. O. Box 38135-1177, Arak, Iran.



The concept of Faltings' local-global principle for $CD_{ < n}$ of local cohomology modules over a Noetherian ring $R$ is introduced, and it is shown that this principle holds at levels 1, 2 over local rings. We also establish the same principle at all levels over an arbitrary Noetherian local ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in [9].


[1] D. Asadollahi and R. Naghipour, Faltings' local-global principle for the finiteness of local cohomology modules, Comm. Algebra, 43 (2015) 953-958.
[2] A. Abbasi and H. Roshan Shekalgourabi, Serre subcategory properties of generalized local cohomology modules, Korean Annals of Math., 28 No. 1 (2011) 25-37.
[3] M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly laskerian local cohomology modules. Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 105 No. 4 (2014) 347-356.
[4] M. Asgharzadeh and M. Tousi, A unified approach to local cohomology modules using Serre classes, Canada. Math. Bull., 53 (2010) 577-586.
[5] K. Bahmanpour, On the category of weakly Laskerian cofinite modules, Math. Scand., 115 No. 1 (2014) 62-68.
[6] K. Bahmanpour, Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra, 321 (2009) 1997-2011.
[7] K. Bahmanpour, R. Naghipour and M. Sedghi, Minimaxness and cofiniteness properties of local cohomology modules. Comm. Algebra, 41 (2013) 2799-2814.
[8] M. P. Brodmann and F. A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128 (2000) 2851-2853.
[9] M. P. Brodmann, Ch. Rotthaus and R. Y. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure and Appl. Algebra, 153 (2000) 197-227.
[10] M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.
[11] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules. Proc. Amer. Math. Soc., 133 No. 3 (2005) 655-660.
[12] G. Faltings, Der endlichkeitssatz in der lokalen kohomologie, Math. Ann., 255 (1981) 45-56.
[13] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005) 649-668.
[14] A. A. Mehrvarz, R. Naghipour, and M. Sedghi, Faltings' local-global principle for the finiteness of local cohomology modules over noetherian rings, Comm. Algebra, 43 (2015) 4860-4872.
[15] R. Naghipour, R. Maddahalli, and K. Ahmadi Amoli, Faltings' local-global principle for the in dimension < n of local cohomology modules, Comm. Algebra, 46 No. 8 (2018) 3496-3509
[16] P. H. Quy, On the _niteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc., 138 (2010) 1965-1968.
[17] K. N. Raghavan, Local-global principle for annihilation of local cohomology, Contemporary Math., 159 (1994) 329-331.
[18] T. Yoshizawa, Subcategories of extension modules by Serre subcategories, Proc. Amer. Math. Soc., 140 (2012) 2293-2305.
[19] H. Zöschinger, Koatomare moduln, Math. Z., 170 (1980) 221-232.
[20] H. Zöschinger, Minimax moduln, J. Algebra, 102 (1986) 1-32.