The construction of a module of finite projective dimension from a finitely generated module of finite injective dimension

1975 ◽  
Vol 50 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Rodney Y. Sharp
Keyword(s):  
Projective Dimension ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Finitely Generated Module ◽  
Finite Projective Dimension

scholarly journals EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION

2012 ◽  
Vol 55 (1) ◽  
pp. 85-96
Author(s):  
SEAN SATHER-WAGSTAFF
Keyword(s):  
Projective Dimension ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Locally Finite ◽  
Semidualizing Module ◽  
Finite Projective Dimension ◽  
Finitely Generated Modules ◽  
The Way

AbstractThis paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

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 Cohomological dimensions with respect to sum and intersection of ideals

2017 ◽  
Vol 102 (116) ◽  
pp. 115-120
Author(s):  
Alireza Vahidi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Equivalent Condition ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Finite Projective Dimension

Let R be a commutative Noetherian ring with non-zero identity, a and b proper ideals of R,M a finitely generated R-module with finite projective dimension, and X a finitely generated R-module. We study the cohomological dimensions of M and X with respect to a + b and a ? b. We show that the inequality cda+b(M,X) ? cda(M,X) + cdb(X) holds true and we find an equivalent condition for it to be equality.

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 Homological aspects of semidualizing modules

2010 ◽  
Vol 106 (1) ◽  
pp. 5 ◽  
Author(s):  
Ryo Takahashi ◽  
Diana White
Keyword(s):  
Projective Dimension ◽  
Dual Theory ◽  
Injective Dimension ◽  
Relative Cohomology ◽  
Semidualizing Module ◽  
Finite Projective Dimension ◽  
Semidualizing Modules ◽  
The Relationship ◽  
Cohomology Modules

We investigate the notion of the $C$-projective dimension of a module, where $C$ is a semidualizing module. When $C=R$, this recovers the standard projective dimension. We show that three natural definitions of finite $C$-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite $C$-projective dimension. In parallel, we develop the dual theory for injective dimension and $C$-injective 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.

On a Class of Automaton Algebras

2016 ◽  
Vol 60 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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