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.

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 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.

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 An Euler characteristic for modules of finite G-dimension

2008 ◽  
Vol 102 (2) ◽  
pp. 206 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Diana White
Keyword(s):  
Euler Characteristic ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Finite Projective Dimension ◽  
Extremal Behavior ◽  
Classical Invariant ◽  
Category Of Modules ◽  
Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111 ◽  
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp&gt; 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

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