Homological aspects of semidualizing 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.

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Relative and Tate homology with respect to semidualizing modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500583 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450058 ◽

Cited By ~ 3

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].

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Balance for relative cohomology of complexes

Journal of Algebra and Its Applications ◽

10.1142/s021949882050005x ◽

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.

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EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION

Glasgow Mathematical Journal ◽

10.1017/s0017089512000353 ◽

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.

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Coreflexive Modules and Semidualizing Modules with Finite Projective Dimension

Taiwanese Journal of Mathematics ◽

10.11650/tjm/8009 ◽

2017 ◽

Vol 21 (6) ◽

pp. 1283-1324

Author(s):

Xi Tang ◽

Zhaoyong Huang

Keyword(s):

Projective Dimension ◽

Finite Projective Dimension ◽

Semidualizing Modules

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

Algebra Colloquium ◽

10.1142/s100538671500019x ◽

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.

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Gorenstein cohomology of N-complexes

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501741 ◽

2019 ◽

Vol 19 (09) ◽

pp. 2050174

Author(s):

Bo Lu ◽

Zhenxing Di

Keyword(s):

Projective Dimension ◽

Complex Case ◽

Binomial Coefficients ◽

Cohomology Groups ◽

Injective Dimension ◽

Relative Cohomology ◽

Gorenstein Injective Dimension ◽

Gorenstein Injective ◽

Gaussian Binomial Coefficients

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

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