A generalization of the dualizing complex structure and its applications
type: Item
instance of: scholarly article
title: A generalization of the dualizing complex structure and its applications
zbMATH Open document ID: 0982.13010
DOI: 10.1016/S0022-4049(99)00160-7
published in: Journal of Pure and Applied Algebra
publication date: 2002-03-18T00:00:00Z
review text: Let \((A, {\mathcal M})\) be a local commutative noetherian ring with has a fundamental dualizing complex \(I^\bullet\), let \(M\) be a finitely generated \(A\)-module satisfying \((S_2)\) and the property that the minimal elements of \(\text{Supp}_A(M)\) coincide with the set of prime ideals \(p\in \text{Supp}_A(M)\) with \(\dim(A/p)= \dim M\). The authors show that \[ 0\to H^{d-s}( \text{Hom}_A(M,I^\bullet))\to \text{Hom}_A(M,I^{d-s})\to \text{Hom}_A(M,I^{d-s+1})\to \dots \] is the Cousin complex of \(H^{d-s}(\text{Hom}_A(M,I^\bullet))\) with respect to the dimension filtration of \(\text{Supp}_A(M)\) where \(d=\dim A\) and \(s= \dim_A(M)\). As an application it is shown that all cohomology modules of the Cousin complex of modules satisfying the preceding conditions are finitely generated. The relation between the Serre condition of the canonical module and the local cohomology of the ring is also investigated. In the last section, it is shown that if \(A\) is an \((S_2)\)-local ring, with Cohen-Macaulay formal fibres, and admits a canonical module, then it has a dualizing complex.
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 1562999
zbMATH Keywords: dualizing complex
zbMATH Keywords: Cousin complex
zbMATH Keywords: generalized Cohen-Macaulay ring
zbMATH Keywords: canonical module
zbMATH Keywords: local cohomology
author: Mohammad T. Dibaei
author: Massoud Tousi
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
cites work: Some basic results on canonical modules
cites work: On the endomorphism ring of the canonical module
cites work: On Grothendieck's local duality theorem
cites work: The structure of dualizing complex for a ring which is \((S_2)\)
cites work: Dualizing Complex and the Canonical Element Conjecture II
cites work: FUNDAMENTAL DUALIZING COMPLEXES FOR COMMUTATIVE NOETHERIAN RINGS
cites work: On Macaulayfication of certain quasi-projective schemes
cites work: Q3739243
cites work: Existence of dualizing complexes
cites work: Cousin complexes and flat ring extensions
cites work: The Cousin complex for a module over a commutative Noetherian ring
cites work: Dualizing complexes for commutative Noetherian rings
cites work: A commutative Noetherian ring which possesses a dualizing complex is acceptable
cites work: Q3853753
cites work: A COUSIN COMPLEX CHARACTERIZATION OF BALANCED BIG COHEN-MACAULAY MODULES
cites work: Cousin Complexes and Generalized Hughes Complexes
Wikidata QID: Q126809409
description: scientific article; zbMATH DE number 1562999