On injective and Gorenstein injective dimensions of local cohomology modules

Zargar, Majid Rahro; Zakeri, H.

doi:10.48550/arxiv.1204.2394

Cited by 2 publications

(3 citation statements)

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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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“…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].…”

Section: Introductionmentioning

confidence: 54%

“…For example, in [45,Theorem 2.6] the author shows that if R is a Cohen-Macaulay complete local ring and H d m (R) is Gorenstein injective then R is Gorenstein. Subsequently, in [46], the authors relax the complete assumption and prove that even if H d m (R) has finite Gorenstein injective dimension, then the Cohen-Macaulay ring R is Gorenstein. On the other hand, for the injective dimension case, there exists a quasi-Gorenstein counterpart.…”

Section: Introductionmentioning

confidence: 99%