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.

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].

Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

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.

A Complex of Modules and Its Applications to Local Cohomology and Extension Functors

2015 ◽  
Vol 117 (1) ◽  
pp. 150 ◽  
Author(s):  
Kamal Bahmanpour
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Residue Field ◽  
Regular Sequence ◽  
Injective Hull ◽  
Finitely Generated ◽  
Complete Local Ring ◽  
Dual Functor ◽  
Matlis Dual Functor

Let $(R,m)$ be a commutative Noetherian complete local ring and let $M$ be a non-zero Cohen-Macaulay $R$-module of dimension $n$. It is shown that, if $\operatorname{projdim}_R(M)<\infty$, then $\operatorname{injdim}_R(D(H^n_{\mathfrak{m}}(M)))<\infty$, and if $\operatorname{injdim}_R(M)<\infty$, then $\operatorname{projdim}_R(D(H^n_{\mathfrak{m}}(M)))<\infty$, where $D(-):= \operatorname{Hom}_{R}(-,E)$ denotes the Matlis dual functor and $E := E_R(R/\mathfrak{m})$ is the injective hull of the residue field $R/\mathfrak{m}$. Also, it is shown that if $(R,\mathfrak{m})$ is a Noetherian complete local ring, $M$ is a non-zero finitely generated $R$-module and $x_1,\ldots,x_k$, $(k\geq 1)$, is an $M$-regular sequence, then \[ D(H^k_{(x_1,\ldots,x_k)}(D(H^k_{(x_1,\ldots,x_k)}(M))))\simeq M. \] In particular, $\operatorname{Ann} H^k_{(x_1,\ldots,x_k)}(M)=\operatorname{Ann} M$. Moreover, it is shown that if $R$ is a Noetherian ring, $M$ is a finitely generated $R$-module and $x_1,\ldots,x_k$ is an $M$-regular sequence, then \[ \operatorname{Ext}^{k+1}_R(R/(x_1,\ldots,x_k),M)=0. \]

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 Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals

2019 ◽  
Vol 125 (2) ◽  
pp. 163-178
Author(s):  
Lars Winther Christensen ◽  
Oana Veliche ◽  
Jerzy Weyman
Keyword(s):  
Recent Work ◽  
Local Ring ◽  
Complete Intersection ◽  
Dynkin Diagram ◽  
Free Resolution ◽  
Free Resolutions ◽  
Regular Local Ring ◽  
Minimal Free Resolution ◽  
Grade 3

Recent work on generic free resolutions of length $3$ attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade $3$ perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.

FAILURE OF CANCELLATION FOR QUARTIC AND HIGHER-DEGREE ORDERS

2002 ◽  
Vol 01 (04) ◽  
pp. 469-481 ◽  
Author(s):  
RYAN KARR
Keyword(s):  
Mild Condition ◽  
Residue Field ◽  
Unit Group ◽  
Field Extension ◽  
Integral Closure ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Quotient Field ◽  
Almost All ◽  
Finitely Generated Modules

Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let [Formula: see text] denote the integral closure of D in K. Let [Formula: see text] where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form [Formula: see text], where p ∈ D is prime.

Grothendieck groups for categories of complexes

2008 ◽  
Vol 3 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Hans-Bjørn Foxby ◽  
Esben Bistrup Halvorsen
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Full Subcategory ◽  
Grothendieck Group ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Projective Modules ◽  
Finitely Generated ◽  
Grothendieck Groups ◽  
Finitely Generated Modules

AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

Sign in / Sign up

Export Citation Format

Share Document