Relative and Tate homology with respect to semidualizing modules

2014 ◽  
Vol 13 (08) ◽  
pp. 1450058 ◽  
Author(s):  
Zhenxing Di ◽  
Xiaoxiang Zhang ◽  
Zhongkui Liu ◽  
Jianlong Chen
Keyword(s):  
Exact Sequence ◽  
London Math ◽  
Projective Dimension ◽  
Relative Cohomology ◽  
Relative Homology ◽  
Semidualizing Module ◽  
Coherent Ring ◽  
Semidualizing Modules ◽  
Tate Homology ◽  
Dualizing Module

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

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.

ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION

2002 ◽  
Vol 85 (2) ◽  
pp. 393-440 ◽  
Author(s):  
LUCHEZAR L. AVRAMOV ◽  
ALEX MARTSINKOVSKY
Keyword(s):  
Exact Sequence ◽  
Projective Dimension ◽  
Subject Classification ◽  
Local Rings ◽  
Tate Cohomology ◽  
Gorenstein Dimension ◽  
Relative Cohomology ◽  
Numerical Invariants ◽  
The One ◽  
Finitely Generated Modules

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

Balance for relative cohomology of complexes

2019 ◽  
Vol 19 (01) ◽  
pp. 2050005
Author(s):  
Zhenxing Di ◽  
Bo Lu ◽  
Junxiu Zhao
Keyword(s):  
Projective Dimension ◽  
Cohomology Groups ◽  
Injective Dimension ◽  
Projective Generator ◽  
Arbitrary Ring ◽  
Relative Cohomology ◽  
Injective Modules ◽  
Injective Cogenerator ◽  
Semidualizing Modules

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

scholarly journals Comparison of relative cohomology theories with respect to semidualizing modules

2009 ◽  
Vol 264 (3) ◽  
pp. 571-600 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Tirdad Sharif ◽  
Diana White
Keyword(s):  
Relative Cohomology ◽  
Semidualizing Modules

scholarly journals Presentations of rings with a chain of semidualizing modules

2017 ◽  
Vol 121 (2) ◽  
pp. 161
Author(s):  
Ensiyeh Amanzadeh ◽  
Mohammad T. Dibaei
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Semidualizing Modules ◽  
Dualizing Module ◽  
A Chain

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

Homological Dimensions with Respect to a Semidualizing Module and Tensor Products of Algebras

2015 ◽  
Vol 22 (02) ◽  
pp. 215-222
Author(s):  
Maryam Salimi ◽  
Elham Tavasoli ◽  
Siamak Yassemi
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Tensor Products ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Semidualizing Module ◽  
Homological Dimensions

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

K-theory of n-coherent rings

2021 ◽  
pp. 2350007
Author(s):  
Eugenia Ellis ◽  
Rafael Parra
Keyword(s):  
Full Subcategory ◽  
Projective Dimension ◽  
Exact Category ◽  
Finite Projective Dimension ◽  
Coherent Ring ◽  
K Theory ◽  
Coherent Rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

2017 ◽  
Vol 59 (3) ◽  
pp. 685-703 ◽  
Author(s):  
AIMIN XU
Keyword(s):  
Projective Dimension ◽  
Model Structure ◽  
Projective Modules ◽  
Cotorsion Pair ◽  
Chain Complexes ◽  
Gorenstein Projective Modules ◽  
Coherent Ring ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Modules ◽  
Model Structures

AbstractGiven a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

scholarly journals DUALIZING MODULES AND n-PERFECT RINGS

2005 ◽  
Vol 48 (1) ◽  
pp. 75-90 ◽  
Author(s):  
Edgar E. Enochs ◽  
Overtoun M. G. Jenda ◽  
J. A. López-Ramos
Keyword(s):  
Final Section ◽  
Projective Dimension ◽  
Natural Generalization ◽  
Subject Classification ◽  
Finite Projective Dimension ◽  
Flat Modules ◽  
Gorenstein Modules ◽  
Dualizing Module

AbstractIn this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.AMS 2000 Mathematics subject classification: Primary 16D20

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