Czechoslovak Mathematical Journal, first online, pp. 1-10

$S$-depth on $ZD$-modules and local cohomology

Morteza Lotfi Parsa

Received February 27, 2020.   Published online October 29, 2020.

Abstract:  Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-${\rm depth}(I, M)$ and ${\rm depth}(I, J, M)$, the $S$-${\rm depth}$ of $(I, J)$ on $M$ is defined as $S$-${\rm depth}(I, J, M)=\inf\{S$-${\rm depth}(\frak{a}, M) \colon\frak{a}\in\widetilde{\rm W}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-${\rm depth}(I, J, M)$ and $H^i_{I,J}(M)$ are studied, and it is proved that $S$-${\rm depth}(I, J, M)=\inf\{i \colon H^i_{I,J}(M)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let ${\rm Supp}_R H^i_{I,J}(M)$ be a finite subset of ${\rm Max}(R)$ for all $i<t$, where $M$ is an arbitrary $R$-module and $t$ is an integer. It is shown that there are distinct maximal ideals $\frak m_1, \frak m_2,\ldots,\frak m_k\in{\rm W}(I, J)$ such that $H^i_{I,J}(M)\cong H^i_{\frak m_1}(M)\oplus H^i_{\frak m_2}(M)\oplus\cdots \oplus H^i_{\frak m_k}(M)$ for all $i<t$.
Keywords:  depth; local cohomology; Serre subcategory; $ZD$-module
Classification MSC:  13C15, 13C60, 13D45
DOI:  10.21136/CMJ.2020.0088-20

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[1] M. Aghapournahr, K. Ahmadi-Amoli, M. Y. Sadeghi: The concept of $(I,J)$-Cohen-Macaulay modules. J. Algebr. Syst. 3 (2015), 1-10. DOI 10.22044/JAS.2015.482 | MR 3534204
[2] M. Aghapournahr, L. Melkersson: Local cohomology and Serre subcategories. J. Algebra 320 (2008), 1275-1287. DOI 10.1016/j.jalgebra.2008.04.002 | MR 2427643 | Zbl 1153.13014
[3] M. Asadollahi, K. Khashyarmanesh, S. Salarian: A generalization of the cofiniteness problem in local cohomology modules. J. Aust. Math. Soc. 75 (2003), 313-324. DOI 10.1017/s1446788700008132 | MR 2015320 | Zbl 1096.13522
[4] M. H. Bijan-Zadeh: Torsion theories and local cohomology over commutative Noetherian rings. J. London Math. Soc., II. Ser. 19 (1979), 402-410. DOI 10.1112/jlms/s2-19.3.402 | MR 0540052 | Zbl 0404.13010
[5] M. P. Brodmann, R. Y. Sharp: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511629204 | MR 1613627 | Zbl 0903.13006
[6] W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608681 | MR 1251956 | Zbl 0909.13005
[7] L. Chu, Q. Wang: Some results on local cohomology modules defined by a pair of ideals. J. Math. Kyoto Univ. 49 (2009), 193-200. DOI 10.1215/kjm/1248983036 | MR 2531134 | Zbl 1174.13024
[8] K. Divaani-Aazar, M. A. Esmkhani: Artinianness of local cohomology modules of ZD-modules. Commun. Algebra 33 (2005), 2857-2863. DOI 10.1081/agb-200063983 | MR 2159511 | Zbl 1090.13012
[9] E. G. Evans, Jr.: Zero divisors in Noetherian-like rings. Trans. Am. Math. Soc. 155 (1971), 505-512. DOI 10.1090/s0002-9947-1971-0272773-9 | MR 0272773 | Zbl 0216.32603
[10] M. Lotfi Parsa: Depth of an ideal on ZD-modules. Publ. Inst. Math., Nouv. Sér. 106(120) (2019), 29-37. DOI 10.2298/pim1920029l | MR 4040296
[11] M. Lotfi Parsa, S. Payrovi: Lower bounds for local cohomology modules with respect to a pair of ideals. Algebra Colloq. 23 (2016), 329-334. DOI 10.1142/s1005386716000341 | MR 3475055 | Zbl 1344.13012
[12] S. Payrovi, M. Lotfi Parsa: Finiteness of local cohomology modules defined by a pair of ideals. Commun. Algebra 41 (2013), 627-637. DOI 10.1080/00927872.2011.631206 | MR 3011786 | Zbl 1263.13016
[13] R. Takahashi, Y. Yoshino, T. Yoshizawa: Local cohomology based on a nonclosed support defined by a pair of ideals. J. Pure Appl. Algebra 213 (2009), 582-600. DOI 10.1016/j.jpaa.2008.09.008 | MR 2483839 | Zbl 1160.13013

Affiliations:  \!\! Morteza Lotfi Parsa, Sayyed Jamaleddin Asadabadi University, Asadabad, 6541861841, Iran, e-mail:,

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