Semidualizing Module Research Articles

Strongly Ding projective modules with respect to a semidualizing module

Strongly Ding projective modules with respect to a semidualizing module

Tate Homology of Modules Based on Tate $$\mathcal {F}_C$$ F C -Resolutions

Let R be a commutative Noetherian ring and C a semidualizing module. Based on a kind of Tate \(\mathcal {F}_C\)-resolutions of modules constructed by Hu et al., in this paper, we introduce and investigate Tate homology for R-modules. For modules admitting a Tate \(\mathcal {F}_C\)-resolution, we first connect the Tate homology and corresponding relative homologies via a long exact sequence. Then we show that the vanishing of such Tate homology characterizes the finiteness of \(\mathcal {F}_C\)-projective dimension of modules. Finally, if in addition R is a Cohen–Macaulay ring with a dualizing module, we establish a balance result for such Tate homology.

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.

Gorenstein homological dimensions with respect to a semidualizing module

In this paper, let R be a commutative ring and C a semidualizing module. We investigate the (weak) C-Gorenstein global dimension of R and we get a simple formula to compute the C-Gorenstein global dimension. Moreover, we compare it with the classical (weak) global dimension of R and get the relations between them. At last, we compare the weak C-Gorenstein global dimension with the C-Gorenstein global dimension and we get that they are equal when R is Noetherian.

Coreflexive Modules and Semidualizing Modules with Finite Projective Dimension

Let $R$ and $S$ be rings and $_S\omega_R$ a semidualizing bimodule. For a subclass $\mathcal{T}$ of the class of $\omega$-coreflexive modules and $n \geq 1$, we introduce and study modules of $\omega$-$\mathcal{T}$-class $n$. By using the properties of such modules, we get some equivalent characterizations for $\omega_S$ having finite projective dimension. In particular, we prove that the projective dimension of $\omega_S$ is at most $n$ if and only if any module of $\omega$-$\mathcal{T}$-class $n$ is $\omega$-coreflexive. Moreover, we get some equivalent characterizations for $\omega_S$ having finite projective dimension at most two or one in terms of the properties of (adjoint) $\omega$-coreflexive and $\omega$-cotorsionless modules. Finally, we give some partial answers to the Wakamatsu tilting conjecture.

Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.

Remarks on torsionfreeness and its applications

ABSTRACTIn this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (Sn). As its applications, we give a characterization of Cohen-Macaulay rings R such that R𝔭 is Gorenstein for all prime ideals 𝔭 of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.

DC-projective dimensions, Foxby equivalence and SDC-projective modules

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study [Formula: see text]-projective dimensions and [Formula: see text]-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to [Formula: see text]-projective modules and [Formula: see text]-injective modules. Thirdly, we characterize Ding projective modules in [Formula: see text] and Ding injective modules in [Formula: see text]. At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.

Gorenstein dimensions over some rings of the form R ⊕ C

Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].

$\V\W$-Gorenstein categories

Let A be an abelian category, and V , W two additive full subcategories of A. We introduce and study the VW -Gorenstein subcategory of A , which unies many known notions, such as the Gorenstein category and the category consisting of GC -projective (injective) modules, although they were dened in a different way. We also prove that the Bass class with respect to a semidualizing module is one kind of VW -Gorenstein category. The connections between VW -Gorenstein categories and Gorenstein categories are discussed. Some applications are given.

Stability of Gorenstein flat categories with respect to a semidualizing module

We first introduce in the paper the $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: \[\xymatrix @C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C \ar @\lt 0.5ex>[r]^{C\otimes _R-} \amp \,\,\,\,\mathcal {G}(\mathcal {W}_F) \ar @\lt 0.5ex>[l]^{\textrm {Hom}_R(C,-)}} \] where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C$ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules, respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence $\mathbb {G}_\bullet =\cdots \rightarrow G_1\rightarrow G_0\rightarrow G^0\rightarrow G^1\rightarrow \cdots $ in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong $ $\im \,(G_0\rightarrow G^0) $ and $\mathbb {G}_\bullet $ is $\mbox {Hom}_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes _R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when $R$ is a commutative $GF$-closed ring.

Ding projective modules with respect to a semidualizing bimodule

In this paper, we introduce and discuss the notion of DC- projective modules over commutative rings, where C is a semidualizing module. This extends Gillespie and Ding, Mao's notion of Ding projective modules. The properties of DC-projective dimensions are also given.

Auslander Class, G C , and C–projective Modules Modulo Exact Zero-divisors

For a semidualizing module C over a ring R, we study the following classes modulo exact zero divisors: G C –projectives, 𝒢 C ; the Auslander class 𝒜 C ; the Bass class ℬ C ; 𝒫 C –projective; ℱ C –projective; and ℐ C –injective dimensions.

Tensor and Torsion Products of Relative Injective Modules with Respect to a Semidualizing Module

Let R be a commutative Cohen–Macaulay ring, and let C be a semidualizing module of R. In this paper, we show that C is generically dualizing if and only if the tensor products of injective and C-injective R-modules are injective. This leads to a characterization of dualizing modules as well as generalizes a result of Enochs and Jenda.

Gorenstein Flat Complexes with Respect to a Semidualizing Module

In this paper, we introduce and study G C-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that G C-flat complexes are actually the complexes of G C-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a [Formula: see text]-cover, where [Formula: see text] is the class of G C-flat complexes. We also give a characterization of complexes of modules in [Formula: see text] that are defined by Sather-Wagstaff, Sharif and White.

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

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.

Duality Pairs Induced by Gorenstein Projective Modules with Respect to Semidualizing Modules

Let C be a semidualizing module over a commutative Noetherian ring R. We investigate duality pairs induced by C-Gorenstein projective modules. It is proven that R is Artinian if and only if \((\mathcal {GPC},\mathcal {GIC})\) is a duality pair if and only if \((\mathcal {GIC},\mathcal {GPC})\) is a duality pair and \(M^{+}\in {\mathcal {GIC}}\) whenever \(M\in {\mathcal {GPC}}\), where \(\mathcal {GPC}\) (\(\mathcal {GIC}\)) is the class of \(\mathcal {GC}\)-Gorenstein projective (\(\mathcal {GC}\)-Gorenstein injective) R-modules. In particular, we give a necessary and sufficient condition for a commutative Artinian ring to be virtually Gorenstein. Moreover, we get that R is Artinian if and only if the class \(\mathcal {GP}\) of Gorentein projective R-modules is preenveloping. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.

Reflexive modules with finite Gorenstein dimension with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. It is shown that a finitely generated R-module M with finite G C -dimension is C-reflexive if and only if $M_{\mathfrak {p}}$ is $C_{\mathfrak {p}}$ -reflexive for $\mathfrak {p} \in \text {Spec}\,(R) $ with $\text {depth}\,(R_{\mathfrak {p}}) \leq 1$ , and $G_{C_{\mathfrak {p}}}-\dim _{R_{\mathfrak {p}}} (M_{\mathfrak {p}}) \leq \text {depth}\,(R_{\mathfrak {p}})-2 $ for $\mathfrak {p} \in \text {Spec}\, (R) $ with $\text {depth}\,(R_{\mathfrak {p}})\geq 2 $ . As the ring R itself is a semidualizing module, this result gives a generalization of a natural setting for extension of results due to Serre and Samuel (see Czech. Math. J. 62(3) (9) 663–672 and Beitrage Algebra Geom. 50(2) (3) 353–362). In addition, it is shown that over ring R with $\dim R \leq n$ , where n≥2 is an integer, $G_{D}-\dim _{R} (\text {Hom}\,_{R} (M,D)) \leq n-2$ for every finitely generated R-module M and a dualizing R-module D.

BALANCE FOR RELATIVE HOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

We derive in the paper the tensor product functor -<TEX>${\otimes}_R$</TEX>- by using proper <TEX>$\mathcal{GP}_C$</TEX>-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of <TEX>$\mathcal{G}_C$</TEX>-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.

Relative Gorenstein Dimensions

In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of GC-projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of GC-injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of GC-projective modules and the Bass class associated to C on one side, and the class of G\({_{C^{\vee}}}\) -injective modules (C∨ = HomR (C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other.