scholarly journals Dual of the Auslander-Bridger formula and GF-perfectness

2007 ◽  
Vol 101 (1) ◽  
pp. 5 ◽  
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Injective Dimension ◽  
Gorenstein Dimension ◽  
Finite Module ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let $(R,\mathfrak m)$ be a local ring. We say that an $R$-module $M$ with $\dim_R M=n$ is a Grothendieck module if the $n$-th local cohomology module of $M$ with respect to $\mathfrak m$, $\mathrm{H}_{\mathfrak m}^n (M)$, is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 95-101
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Injective Resolution ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Gorenstein Injective ◽  
Cousin Complexes ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

Gorenstein homological dimensions and some duality results

2017 ◽  
Vol 10 (03) ◽  
pp. 1750048
Author(s):  
Fatemeh Mohammadi Aghjeh Mashhad
Keyword(s):  
Local Ring ◽  
Duality Theorem ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Duality Results ◽  
Dualizing Complex ◽  
Matlis Duality ◽  
Homological Dimensions ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Let [Formula: see text] be a local ring and [Formula: see text] denote the Matlis duality functor. Assume that [Formula: see text] possesses a normalized dualizing complex [Formula: see text] and [Formula: see text] and [Formula: see text] are two homologically bounded complexes of [Formula: see text]-modules with finitely generated homology modules. We will show that if G-dimension of [Formula: see text] and injective dimension of [Formula: see text] are finite, then [Formula: see text] Also, we prove that if Gorenstein injective dimension of [Formula: see text] and projective dimension of [Formula: see text] are finite, then [Formula: see text] These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

scholarly journals Faltings’ Finiteness Dimension of Local Cohomology Modules Over Local Cohen–Macaulay Rings

2017 ◽  
Vol 60 (2) ◽  
pp. 225-234
Author(s):  
Kamal Bahmanpour ◽  
Reza Naghipour
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Local Cohomology ◽  
Nilpotent Ideal ◽  
Local Cohomology Module ◽  
Injective Dimension ◽  
Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet (R, m) denote a local Cohen–Macaulay ring and I a non-nilpotent ideal of R. The purpose of this article is to investigate Faltings’ finiteness dimension fI(R) and the equidimensionalness of certain homomorphic images of R. As a consequence we deduce that fI(R) = max{1, ht I}, and if mAssR(R/I) is contained in AssR(R), then the ring is equidimensional of dimension dim R−1. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module , in the case where (R,m) is a complete equidimensional local ring.

scholarly journals Krull dimension and generalized fractions

1985 ◽  
Vol 28 (3) ◽  
pp. 349-353 ◽  
Author(s):  
M. A. Hamieh ◽  
R. Y. Sharp
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Image Position ◽  
System Of Parameters

Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

scholarly journals Top Local Cohomology Modules with Specified Attached Primes

2008 ◽  
Vol 15 (02) ◽  
pp. 341-344 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Raheleh Jafari
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Proper Subset ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of Assh R(M) can be expressed as the set of attached primes of the top local cohomology module [Formula: see text] for some ideal 𝔞 of R. Moreover, if 𝔞 is an ideal of R such that the set of attached primes of [Formula: see text] is a non-empty proper subset of Assh R(M), then [Formula: see text] for some ideal 𝔟 of R with dim R(R/𝔟) = 1.

GORENSTEIN INJECTIVE MODULES AND A GENERALIZATION OF ISCHEBECK FORMULA

2013 ◽  
Vol 12 (04) ◽  
pp. 1250197
Author(s):  
REZA SAZEEDEH
Keyword(s):  
Local Ring ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Noetherian Local Ring ◽  
Injective Modules ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Let (R,[Formula: see text]) be a commutative Noetherian local ring and let M and N be nonzero finitely generated R-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper, we prove a generalization of Ischebeck formula, that is [Formula: see text].

A Bass Formula for Gorenstein Injective Dimension

2007 ◽  
Vol 35 (6) ◽  
pp. 1882-1889 ◽  
Author(s):  
Leila Khatami ◽  
Siamak Yassemi
Keyword(s):  
Injective Dimension ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

scholarly journals On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

2015 ◽  
Vol 22 (spec01) ◽  
pp. 935-946 ◽  
Author(s):  
Majid Rahro Zargar ◽  
Hossein Zakeri
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Dualizing Complex ◽  
Local Cohomology Modules ◽  
Gorenstein Injective ◽  
Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht M𝔞. We prove that injdim M < ∞ if and only if [Formula: see text] and that [Formula: see text]. We also prove that if R has a dualizing complex and Gid RM < ∞, then [Formula: see text]. Moreover if R and M are Cohen-Macaulay, then Gid RM < ∞ whenever [Formula: see text]. Next, for a finitely generated R-module M of dimension d, it is proved that if [Formula: see text] is Cohen-Macaulay and [Formula: see text], then [Formula: see text]. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

2019 ◽  
Vol 30 (02) ◽  
pp. 379-396
Author(s):  
V. H. Jorge Pérez ◽  
T. H. Freitas
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Artinian Modules ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Artinian Module ◽  
Samuel Multiplicity

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

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