# Annihilators and attached primes of local cohomology modules with respect to a system of ideals

Document Type : Research Paper

Author

Department of Natural Science Education, Dong Nai University, Bien Hoa city, Dong Nai province, Vietnam.

10.29252/as.2020.1959

Abstract

Let $\Phi$ be a system of ideals of a commutative Noetherian ring, we study the annihilators and attached primes of local cohomology modules with respect to a system of ideals.  We prove that if $M$ is a non-zero finitely generated $R$-module of finite dimension $d$ and $\Phi$ is a system of ideals, then$$Att(H^d_\Phi(M))=\{p\in Ass M\mid cd(\Phi,R/p)=d\}.$$ Moreover, if the cohomology dimension of $M$ with respect to $\Phi$ is $dim M-1,$ then $$Att(H^{dim M-1}_\Phi(M))=\{p\in Supp M \mid cd(\Phi,R/p)=\dim M-1\}.$$

Keywords

#### References

[1] M. Aghapournahr and L. Melkersson, Cofiniteness and coassociated primes of local cohomology modules, Math. Scand. 105 (2009), 161–170.
[2] M. Aghapournahr and K. Bahmanpour, Cofiniteness of general local cohomology modules for small dimensions, Bull. Korean Math. Soc. 53(5) (2016), 1341–1352.
[3] J. Asadollahi, K. Khashyarmanesh and Sh. Salarian, A generalization of the cofiniteness problem in local cohomology modules, J. Aust. Math. Soc. 75 (2003), 313–324.
[4] A. Atazadeh, M. Sedghi and R. Naghipour, On the annihilators and attached primes of top local cohomology modules, Arch. Math., 102(3) (2014), 225–236.
[5] A. Atazadeh, M. Sedghi and R. Naghipour, Some results on the annihilators and attached primes of local cohomology modules, Arch. Math., 109(5) (2017), 415–427.
[6] K. Bahmanpour, J. A’zami and G. Ghasemi, On the annihilators of local cohomology modules, J. Algebra 363 (2012), 8–13.
[7] K. Bahmanpour, Annihilators of local cohomology modules, Comm. Algebra 43 (2015), 2509–2515.
[8] M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherian ring, J. Lond. Math. Soc. 19(2) (1979), 402–410.
[9] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21(2) (1980), 173–181.
[10] M. H. Bijan-Zadeh, On the artinian property of certain general local cohomology modules, J. London Math. Soc. 32(2) (1985), 399–403.
[11] N. Bourbaki, Commutative Algebra, Elements of Mathematics, Herman, Paris/Addison-Wesley, Reading, MA (1972).
[12] M. T. Dibaei and S. Yassemi, Cohomological dimension of complexes, Comm. Algebra, 32(11) (2004), 4375–4386.
[13] M.T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math. 84 (2005), 292–297.
[14] K. Divaani-Aazar, R. Naghipour and M. Tousi, The Lichtenbaum-Hartshorne theorem for generalized local cohomology and connectedness, Comm. Algebra 30(8) (2002), 3687–3702.
[15] K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130(12) (2012), 3537-3544.
[16] M. Hellus, Attached primes of Matlis duals of local cohomology modules, Arch. Math. 89 (2007), 202–210.
[17] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (1991), 421–429.
[18] L.R. Lynch, Annihilators of top local cohomology, Comm. Algebra, 40 (2012), 542–551.
[19] G. Lyubeznik, Finiteness properties of local cohomology modules, (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41–55.
[20] I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973), 23–43.
[21] I. G. Macdonald and R. Y. Sharp, An elementary proof of the non-vanishing of certain local cohomology modules, Quart. J. Math. Oxford 23 (1972), 197–204.
[22] H. Matsumura, Commutative Ring Theory, Cambridge University press, cambridge, (1986).
[23] L.T. Nhan and T.N. An, On the unmixedness and universal catenaricity of local rings and local cohomology modules, J. Algebra, 321 (2009), 303–311.
[24] L.T. Nhan and T.D.M. Chau, On the top local cohomology modules, J. Algebra 349 (2012), 342–352.
[25] R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213 (2009), 582–600.
[26] H. Zoschinger, Uber koassoziierte Primideale Math. Scand. 63 (1988), 196–211.