On invariants dual to the Bass numbers

Enochs, Edgar E.; Xu, Jinzhong

doi:10.1090/s0002-9939-97-03662-9

Cited by 19 publications

(14 citation statements)

References 12 publications

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“…As a consequence we derive a formula (see Theorem 1.5) for the Matlis dual of E R (R/p), where p ∈ Spec R is a 1-dimensional prime ideal. In some sense this is related to results by Enochs and Xu (see [4] and [3]).…”

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confidence: 81%

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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confidence: 81%

Notes on local cohomology and duality

Hellus

Schenzel

2014

Journal of Algebra

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“…We now establish some additional results on cotorsion modules which are needed in sections 3 and 4. We thank Edgar Enochs for showing us a proof of part (b) of the following lemma, which is implicit in [5]: Proof. To prove (a), let A be a flat S-module.…”

Section: Flat Covers and Cotorsion Modulesmentioning

confidence: 99%

“…However, flat resolutions with this kind of minimality condition do exist for a large class of modules (i.e., cotorsion modules), which we show is sufficient to prove Theorem 1.1. The theory of flat covers, cotorsion modules, and minimal flat resolutions, as developed by Enochs and Xu in [4] and [5], are essential ingredients in all our arguments. In Section 2, we summarize some basic properties of these notions as well as prove some auxiliary results which are used in later sections.…”

Section: Introductionmentioning

confidence: 98%

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The acyclicity of the Frobenius functor for modules of finite flat dimension

Marley

Webb

2016

Journal of Pure and Applied Algebra

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Abstract. Let R be a commutative Noetherian local ring of prime characteristic p and f : R−→R the Frobenius ring homomorphism. For e ≥ 1 let R (e) denote the ring R viewed as an R-module via f e . Results of Peskine, Szpiro, and Herzog state that for finitely generated modules M , M has finite projective dimension if and only if Tor R i (R (e) , M ) = 0 for all i > 0 and all (equivalently, infinitely many) e ≥ 1. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.

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“…In this paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu [22, Theorem 3.2], we show that C is dualizing if and only if for an R-module M , the necessary and sufficient condition for M to be C-injective is that π i (p, M ) = 0 for all p ∈ Spec (R) and all i = ht (p), where π i is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu [8].…”

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confidence: 98%

Dual of bass numbers and dualizing modules

Rahmani

Taherizadeh

2016

Communications in Algebra

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Abstract. Let R be a Noetherian ring and let C be a semidualizing R-module. In this paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu [22, Theorem 3.2], we show that C is dualizing if and only if for an R-module M , the necessary and sufficient condition for M to be C-injective is that π i (p, M ) = 0 for all p ∈ Spec (R) and all i = ht (p), where π i is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu [8].

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On invariants dual to the Bass numbers

Cited by 19 publications

References 12 publications

Notes on local cohomology and duality

Notes on local cohomology and duality

The acyclicity of the Frobenius functor for modules of finite flat dimension

Dual of bass numbers and dualizing modules

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