S -depth on ZD -modules and local cohomology
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Czechoslovak Mathematical Journal
10 pp
Online first
S-DEPTH ON ZD-MODULES AND LOCAL COHOMOLOGY Morteza Lotfi Parsa, Asadabad 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 CI and M be a ZD-module. As a generalization of the S-depth(I, M ) and depth(I, J, M ), the S-depth f J)}, and of (I, J) on M is defined as S-depth(I, J, M ) = inf{S-depth(a, M ) : a ∈ W(I, some properties of this concept are investigated. The relations between S-depth(I, J, M ) i i and HI,J (M ) are studied, and it is proved that S-depth(I, J, M ) = inf{i : HI,J (M ) ∈ / 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 i ordinary local cohomology modules under these conditions. Let SuppR HI,J (M ) be a finite subset of 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 m1 , m2 , . . . , mk ∈ W(I, J) such that i i i i HI,J (M ) ∼ (M ) ⊕ Hm (M ) ⊕ . . . ⊕ Hm (M ) for all i < t. = Hm 1 2 k Keywords: depth; local cohomology; Serre subcategory; ZD-module MSC 2020 : 13C15, 13C60, 13D45
1. Introduction Throughout this paper, R is a commutative Noetherian ring with nonzero identity, I and J are two ideals of R, M is an R-module and t is an integer. For notation and terminology not given in this paper, the reader is referred to [5], [6], and [13] if necessary. The local cohomology theory has been a useful and significant tool in commutative algebra and algebraic geometry. There are some extensions of this theory. BijanZadeh in [4] introduced the local cohomology modules with respect to a system of ideals. As a special case of these generalized modules, Takahashi, Yoshino, and Yoshizawa in [13] defined the local cohomology modules with respect to a pair of ideals. To be more precise, let ΓI,J (M ) = {x ∈ M : ∃ t ∈ N, I t x ⊆ Jx}. It is easy to DOI: 10.21136/CMJ.2020.0088-20
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see that ΓI,J (M ) is a submodule of M and ΓI,J (−) is a covariant R-linear functor from the category of R-modules to itself. For an integer i, the local cohomology i functor HI,J (−) with respect to (I, J) is defined to be the ith right derived functor i of ΓI,J (−). Also HI,J (M ) is called the ith local cohomology module of M with respect i to (I, J). If J = 0, then HI,J (−) coincides with the ordinary local cohomology i functor HI (−). f J) = {a 6 R : I t ⊆ J + a for some positive integer t}. One can see Let W(I, f J). Let W(I, J) = {p ∈ Spec(R) : that x ∈ ΓI,J (M ) if and only if AnnR (x) ∈ W(I, t I ⊆ J + p for some positive integer t}. It is shown in [13], Corollary 1.8, that x ∈ ΓI,J (M ) if and only if SuppR Rx ⊆ W(I, J). The notion of ZD-modules was introduced by Evans, see [9]. An R-module M is called a ZD-module (zero-divisor module) if, for any submodule N of M , the set of zero-divisors of M/N i
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