scholarly journals Acyclic complexes of finitely generated free modules over local rings

2009 ◽  
Vol 105 (1) ◽  
pp. 85 ◽  
Author(s):  
Meri T. Hughes ◽  
David A. Jorgensen ◽  
Liana M. Sega
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Reflexive Module ◽  
Commutative Local Ring ◽  
Standard Construction ◽  
Acyclic Complexes ◽  
Free Modules

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.

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].

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

Bounds on Betti Numbers

1982 ◽  
Vol 34 (3) ◽  
pp. 589-592
Author(s):  
Mark Ramras
Keyword(s):  
Natural Number ◽  
Local Ring ◽  
Residue Class ◽  
Asymptotic Properties ◽  
Betti Numbers ◽  
Commutative Local Ring ◽  
Artin Ring ◽  
Residue Class Field ◽  
Image Position ◽  
Free Modules

The Betti numbers βn(k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn(k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn(k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn(k) ≧ 2n.In this paper, we examine the sequence {βn(M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds:123where l(X) is the length of X.

𝐿-dimension for modules over a local ring

2021 ◽  
pp. 83-91
Author(s):  
Courtney Gibbons ◽  
David Jorgensen ◽  
Janet Striuli
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Residue Field ◽  
Free Resolution ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Content Type ◽  
Commutative Local Ring ◽  
Finitely Generated Modules

We introduce a new homological dimension for finitely generated modules over a commutative local ring R R , which is based on a complex derived from a free resolution L L of the residue field of R R , and called L L -dimension. We prove several properties of L L -dimension, give some applications, and compare L L -dimension to complete intersection dimension.

Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

2014 ◽  
Vol 21 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Le Thanh Nhan ◽  
Tran Do Minh Chau
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Noetherian Dimension ◽  
Base Ring ◽  
Finitely Generated Modules

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

scholarly journals ON A NEW INVARIANT OF FINITELY GENERATED MODULES OVER LOCAL RINGS

2010 ◽  
Vol 09 (06) ◽  
pp. 959-976 ◽  
Author(s):  
NGUYEN TU CUONG ◽  
DOAN TRUNG CUONG ◽  
HOANG LE TRUONG
Keyword(s):  
Local Ring ◽  
Negative Integer ◽  
Local Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Type ◽  
Systems Of Parameters ◽  
Finitely Generated Modules

Let M be a finitely generated module on a local ring R and [Formula: see text] a filtration of submodules of M such that do < d1 < ⋯ < dt = d, where di = dim Mi. This paper is concerned with a non-negative integer [Formula: see text] which is defined as the least degree of all polynomials in n1, …, nd bounding above the function [Formula: see text] We prove that [Formula: see text] is independent of the choice of good systems of parameters [Formula: see text]. When [Formula: see text] is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen–Macaulay locus of M to compute [Formula: see text], and also to study the behavior of it under local flat homomorphisms.

scholarly journals Some Remarks on Non-Commutative Extensions of Local Rings

1959 ◽  
Vol 14 ◽  
pp. 45-51
Author(s):  
Edward H. Batho
Keyword(s):  
Local Ring ◽  
Ideal Theory ◽  
Dimension Theory ◽  
Local Rings ◽  
Important Class ◽  
Matrix Rings ◽  
Commutative Local Ring

In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.

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 Note on Non-Commutative Semi-Local Rings

1960 ◽  
Vol 17 ◽  
pp. 161-166 ◽  
Author(s):  
Yukitoshi Hinohara
Keyword(s):  
Local Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Noetherian Rings ◽  
Local Rings ◽  
Finitely Generated

Our aim in this note is to generalize some topological results of commutative noetherian rings to non-commutative rings. As a supplemental remark of [2] we prove in § 1 that any right ideal of a complete right semi-local ring is closed, and thatfor any finitely generated right module M over a complete right semi-local ring Λ where J is the Jacobson radical of Λ.

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