scholarly journals Chains of semidualizing modules

2018 ◽  
Vol 17 (07) ◽  
pp. 1850118
Author(s):  
Ensiyeh Amanzadeh
Keyword(s):  
Local Ring ◽  
Noetherian Local Ring ◽  
Bass Numbers ◽  
Semidualizing Modules ◽  
Insight Into

Let [Formula: see text] be a commutative Noetherian local ring. We study the suitable chains of semidualizing [Formula: see text]-modules. We prove that when [Formula: see text] is Artinian, the existence of a suitable chain of semidualizing modules of length [Formula: see text] implies that the Poincar[Formula: see text] series of [Formula: see text] and the Bass series of [Formula: see text] have very specific forms. Also, in this case, we show that the Bass numbers of [Formula: see text] are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of [Formula: see text].

scholarly journals On the relative Cohen–Macaulay modules

2014 ◽  
Vol 14 (03) ◽  
pp. 1550042
Author(s):  
Majid Rahro Zargar
Keyword(s):  
Local Ring ◽  
Proper Ideal ◽  
Noetherian Local Ring ◽  
Bass Numbers

Let R be a commutative Noetherian local ring and let 𝔞 be a proper ideal of R. In this paper, as a main result, it is shown that if M is a Gorenstein R-module with c = ht M𝔞, then [Formula: see text] for all i ≠ c is completely encoded in homological properties of [Formula: see text], in particular in its Bass numbers. Notice that, this result provides a generalization of a result of Hellus and Schenzel which has been proved before, as a main result, in the case where M = R.

scholarly journals Local cohomology modules and their properties

2021 ◽  
Vol 73 (2) ◽  
pp. 268-274
Author(s):  
J. Azami ◽  
M. Hasanzad
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Cohomology Modules ◽  
The Ideal

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

Bass numbers of generalized local cohomology modules with respect to a pair of ideals

2018 ◽  
Vol 11 (02) ◽  
pp. 1850019
Author(s):  
M. Lotfi Parsa
Keyword(s):  
Local Ring ◽  
Principal Ideal ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
The Relationship ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] and [Formula: see text] are ideals of [Formula: see text], and [Formula: see text] and [Formula: see text] are [Formula: see text]-modules. We study the relationship between the Bass numbers of [Formula: see text] and [Formula: see text]. As a consequence, it follows that if one of the following holds: (a) [Formula: see text] is a principal ideal of [Formula: see text], (b) [Formula: see text], (c) [Formula: see text] (when [Formula: see text] is local and [Formula: see text] is finitely generated), (d) [Formula: see text] (when [Formula: see text] is local), (e) [Formula: see text] (when [Formula: see text] is local), then [Formula: see text] is finite for all [Formula: see text] and [Formula: see text], whenever [Formula: see text] is finitely generated and flat, [Formula: see text] is minimax, and [Formula: see text].

KUMMER COVERINGS AND SPECIALISATION

2021 ◽  
pp. 1-37
Author(s):  
Martin Olsson
Keyword(s):  
Local Ring ◽  
Fundamental Groups ◽  
Technical Result ◽  
Noetherian Local Ring ◽  
Log Schemes ◽  
Log Scheme

Abstract We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

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 Weak Formal Schemes

1972 ◽  
Vol 45 ◽  
pp. 1-38 ◽  
Author(s):  
David Meredith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Local Ring ◽  
Formal Schemes

Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m.In [5], Monsky and Washnitzer define weakly complete R-algebras with respect to m. In brief, an R-algebra A† is weakly complete if

scholarly journals POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY

2020 ◽  
Vol 63 (1) ◽  
pp. 1-5
Author(s):  
OLGUR CELIKBAS ◽  
RYO TAKAHASHI
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Positive Power ◽  
Noetherian Local Ring ◽  
Torsion Condition

AbstractWe prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

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].

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