scholarly journals A note on grade

1975 ◽  
Vol 59 ◽  
pp. 149-152 ◽  
Author(s):  
P. Jothilingam
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Finite Projective Dimension

All rings that occur in this note will be assumed to be commutative with unity and all modules will be finitely generated and unitary.The grade of a module M over a noetherian local ring R is defined to be the length of a maximal R-sequence contained in the annihilator of M. If M has finite projective dimension it is well-known that grade M ≤ proj. dim M. We can say more when R is a regular local ring.

Co-Cohen-Macaulay Modules and Generalized Local Cohomology

2011 ◽  
Vol 18 (spec01) ◽  
pp. 807-813
Author(s):  
Amir Mafi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finite Dimension ◽  
Cohomological Dimension ◽  
Projective Dimension ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Finite Projective Dimension

Let (R,𝔪) be a Noetherian local ring, 𝔞 a proper ideal of R, and M, N two finitely generated R-modules of finite projective dimension m and of finite dimension n, respectively. It is shown that if n ≤ 2, then the generalized local cohomology module [Formula: see text] is a co-Cohen-Macaulay module. Additionally, we show that [Formula: see text] for all i > m+s and [Formula: see text], where s is the cohomological dimension of N with respect to 𝔞.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF THE LINEARITY DEFECT

2017 ◽  
Vol 230 ◽  
pp. 35-47 ◽  
Author(s):  
HOP D. NGUYEN ◽  
THANH VU
Keyword(s):  
Asymptotic Behavior ◽  
Local Ring ◽  
General Result ◽  
Free Resolution ◽  
Graded Algebra ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Minimal Free Resolution ◽  
Noetherian Local Ring ◽  
Graded Module

This work concerns the linearity defect of a module $M$ over a Noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted $\text{ld}_{R}M$. Roughly speaking, $\text{ld}_{R}M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. It is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_{R}(I^{n}M))_{n}$ and $(\text{ld}_{R}(M/I^{n}M))_{n}$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_{R}C_{n})_{n}$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$.

scholarly journals Local rings with quasi-decomposable maximal ideal

2018 ◽  
Vol 168 (2) ◽  
pp. 305-322 ◽  
Author(s):  
SAEED NASSEH ◽  
RYO TAKAHASHI
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Maximal Ideal ◽  
Projective Dimension ◽  
Complete Classification ◽  
Local Rings ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Direct Sum

AbstractLet (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

scholarly journals Special homological dimensions and intersection theorem

2005 ◽  
Vol 96 (2) ◽  
pp. 161
Author(s):  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Gorenstein Dimension ◽  
Homological Dimensions

Let $(R,m)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen-Macaulay ring if there exists a Cohen-Macaulay finite (i.e. finitely generated) $R$-module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi-projective dimension.

scholarly journals CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY

2021 ◽  
pp. 1-18
Author(s):  
BENJAMIN BRIGGS ◽  
ELOÍSA GRIFO ◽  
JOSH POLLITZ
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Residue Field ◽  
Projective Dimension ◽  
Complete Intersections ◽  
Finitely Generated ◽  
Finite Projective Dimension ◽  
The World ◽  
Small Test ◽  
Test Module

Abstract A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.

scholarly journals Maximal depth property of finitely generated modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850202 ◽  
Author(s):  
Ahad Rahimi
Keyword(s):  
Local Ring ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Maximal Depth ◽  
Finitely Generated Modules ◽  
Associated Prime

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. We say [Formula: see text] has maximal depth if there is an associated prime [Formula: see text] of [Formula: see text] such that depth [Formula: see text]. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of [Formula: see text] are considered for [Formula: see text].

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Toward a theory of generalized Cohen-Macaulay modules

1986 ◽  
Vol 102 ◽  
pp. 1-49 ◽  
Author(s):  
Ngô Viêt Trung
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Annihilator of local cohomology of homogeneous parts of a graded module

2020 ◽  
pp. 2150092
Author(s):  
Tran Do Minh Chau ◽  
Nguyen Thi Kieu Nga ◽  
Le Thanh Nhan
Keyword(s):  
Asymptotic Stability ◽  
Local Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

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