On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

Zargar, Majid Rahro; Zakeri, H.

doi:10.1142/s1005386715000784

Cited by 14 publications

(8 citation statements)

References 11 publications

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“…An ideal I ⊂ R is called a cohomologically complete intersection whenever H i I (R) = 0 for all i = c for some c (see [6] for the definition and a characterization). If I is a cohomologically complete interssection, then in the paper of Zargar and Zakeri (see [15]) the ring R is called Cohen-Macaulay with respect to I.…”

Section: On a Duality For Cohomologically Complete Intersectionsmentioning

confidence: 99%

Notes on local cohomology and duality

Hellus

Schenzel

2014

Journal of Algebra

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Abstract. We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of R/p where p is a one dimensional prime ideal in a local complete Gorenstein domain (R, m). This is related to results of Enochs and Xu (see [4] and [3]). We prove a certain 'dual' version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2). There is a generalization of local duality to cohomologically complete intersection ideals I in the sense that for I = m we get back the classical Local Duality Theorem. We determine the exact class of modules to which a characterization of cohomologically complete intersection from [6] generalizes naturally (see Theorem 4.4).In this paper we prove a Matlis dual version of Hartshorne-Lichtenbaum Vanishing Theorem and generalize the Local Duality Theorem. The latter generalization is done for ideals which are cohomologically complete intersections, a notion which was introduced and studied in [6]. The generalization is such that local duality becomes the special case when the ideal I is the maximal ideal m of the given local ring (R, m). We often use formal local cohomology, a notion which was introduced and studied by the second author in [13]. Formal local cohomology is related to Matlis duals of local cohomology modules (see [5, Sect. 7.1 and 7.2] and Corollary 3.4).We start in Section 1 with the study of the Matlis duals of local cohomology modules H n−1 I (R), where n = dim R. The latter is also the formal local cohomology module lim ← − H 1 m (R/I α ) provided R is a Gorenstein ring. We describe this module as the cokernel of a certain canonical map. As a consequence we derive a formula (see Theorem 1.5) for the Matlis dual of E R

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Section: On a Duality For Cohomologically Complete Intersectionsmentioning

confidence: 99%

Notes on local cohomology and duality

Hellus

Schenzel

2014

Journal of Algebra

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“…That is, R is a quasi-Gorenstein ring if and only if Id R H d This encouraged us to investigate the Gorenstein injective version of the Aoyama's theorem which is, indeed, a theorem whenever R is Cohen-Macaulay. We proved the following fact which also recovers the Cohen-Macaulay case of[46].…”

mentioning

confidence: 53%

A study of quasi-Gorenstein rings

Tavanfar

Tousi

2018

Journal of Pure and Applied Algebra

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“…Then, as a corollary, we prove that if M is a relative Cohen-Macaulay R-module with respect to a, where is defined as in [20], and…”

Section: Introductionmentioning

confidence: 91%

“…The following theorem, which is an immediate consequence of Theorem 2.2, is a generalization of [20,Theorem 2.5].…”

Section: Right Derived Section Functor Injective Dimension and (Goren...mentioning

confidence: 99%

Interplay between homological dimensions of a complex and its right derived section

Jalali

2014

Preprint

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Let (R, m) be a commutative Noetherian local ring, a be a proper ideal of R and M be an R-complex in D(R). We prove that if, then idRRΓa(M ) = idRM (respectively, fdRRΓa(M ) = fdRM ). Next, it is proved that the right derived section functor of a complex M ∈ D❁(R) (R is not necessarily local) can be computed via a genuine left-bounded complex G ≃ M of Gorenstein injective modules. We show that if R has a dualizing complex and M is an R-complex in D f (R), then GfdRRΓa(M ) = GfdRM and GidRRΓa(M ) = GidRM . Also, we show that if M is a relative Cohen-Macaulay R-module with respect to a (respectively, Cohen-Macaulay R-module of dimension n), then GfdRH ht M a a (M ) = GfdRM + n (respectively, GidRH n m (M ) = GidRM − n). The above results generalize some known results and provide characterizations of Gorenstein rings.

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On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

Cited by 14 publications

References 11 publications

Notes on local cohomology and duality

Notes on local cohomology and duality

A study of quasi-Gorenstein rings

Interplay between homological dimensions of a complex and its right derived section

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