Gorenstein Modules Research Articles

When are the classes of Gorenstein modules (co)tilting?

When are the classes of Gorenstein modules (co)tilting?

Gorenstein modules and dimension over large families of infinite groups

Gorenstein modules and dimension over large families of infinite groups

Strongly $\mathcal{VW}$-Gorenstein $N$-complexes

Let $\mathcal{V},\mathcal{W}$ be two classes of $R$-modules. The notion of strongly $\mathcal{VW}$-Gorenstein $N$-complexes is introduced, and under certain mild hypotheses on $\mathcal{V}$ and $\mathcal{W}$, it is shown that an $N$-complex $X$ is strongly $\mathcal{VW}$-Gorenstein if and only if each term of $X$ is a $\mathcal{VW}$-Gorenstein module and $N$-complexes ${\rm Hom}_{R}(V,X)$ and $ Hom_{R}(X,W)$ are $N$-exact for any $V\in\mathcal{V}$ and $W\in\mathcal{W}$. Furthermore, under the same conditions on $\mathcal{V}$ and $\mathcal{W}$, it is proved that an $N$-exact $N$-complex $X$ is $\mathcal{VW}$-Gorenstein if and only if $\mathrm{Z}_{n}^{t}(X)$ is a $\mathcal{VW}$-Gorenstein module for each $n\in\mathbb{Z}$ and each $t=1,2,\ldots,N-1$. Consequently, we show that an $N$-complex $X$ is strongly Gorenstein projective (resp., injective) if and only if $X$ is $N$-exact and ${\rm Z}^t_{n}(X)$ is a Gorenstein projective (resp., injective) module for each $n\in\mathbb{Z}$ and $t=1,2,\ldots,N-1$.

Generalized Gorenstein modules with respect to duality pairs over triangular matrix rings

Let A and B be rings and [Formula: see text] with M an A-B-bimodule. We first construct a semi-complete duality pair [Formula: see text] over T using duality pairs over A and B, respectively. Then we characterize when a left T-module is Gorenstein projective, injective or flat with respect to the duality pair [Formula: see text]. As applications, we investigate when a left T-module is projectively coresolved Gorenstein flat, Ding injective or Gorenstein flat. In addition, we give an estimate of Gorenstein flat dimension of a left T-module with respect to [Formula: see text]. Finally, we construct model structures and recollements relative to these generalized Gorenstein modules.

A UNIFIED APPROACH TO VARIOUS GORENSTEIN MODULES

Let 𝒱,𝒲,𝒴,𝒳 be four classes of modules. We introduce a notion of (𝒱,𝒲,𝒴,𝒳)-Gorenstein modules which unifies a number of well-known Gorenstein modules. Under certain mild assumptions on the classes 𝒱,𝒲,𝒴,𝒳, we first study properties of (𝒱,𝒲,𝒴,𝒳)-Gorenstein modules. Then, we apply the obtained results to establish the Foxby equivalences associate to (𝒱,𝒲,𝒴,𝒳)-Gorenstein modules. Many results on the Gorenstein modules involved in this general setting can be obtained as corollaries.

Homological properties of trivial ring extensions

Let [Formula: see text] be a trivial extension of a ring [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text]. We first study the transfer of some rings such as left [Formula: see text]-rings, left Kasch rings, left PS rings between [Formula: see text] and [Formula: see text]. Then we investigate how to construct [Formula: see text]-tilting modules and Gorenstein modules in [Formula: see text]-Mod from [Formula: see text]-Mod. Finally we exhibit the connections between preenvelopes (respectively, precovers) in [Formula: see text]-Mod and preenvelopes (respectively, precovers) in [Formula: see text]-Mod.

Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

Strongly ($$\mathscr {X},\mathscr {Y},\mathscr {Z}$$)-Gorenstein Modules and Applications

In this paper, we introduce strongly ($$\mathscr {X},\mathscr {Y},\mathscr {Z}$$)-Gorenstein modules, where $$\mathscr {X},\mathscr {Y},\mathscr {Z}$$ are additive full subcategories of R-$$\mathrm Mod$$. These modules provide a new characterization of $$\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$$-modules, that every $$\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$$-module is a direct summand of a certain strongly ($$\mathscr {X},\mathscr {Y},\mathscr {Z}$$)-Gorenstein module. Another application in global dimensions is given: the global left $${\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})}$$-dimension is equal to the global right $${\mathcal {G}(\mathscr {\mathscr {X^{'}},\mathscr {Y^{'}},\mathscr {Z^{'}}})}$$-dimension over any ring R, where $$\mathscr {X^{'}},\mathscr {Y^{'}},\mathscr {Z^{'}}$$ are additive full subcategories of R-$$\mathrm{Mod}$$ related to $$\mathscr {X},\mathscr {Y},\mathscr {Z}$$.

The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.

Foxby equivalences associated to strongly Gorenstein modules

In order to establish the Foxby equivalences associated to strongly Gorenstein modules, we introduce the notions of strongly $\mathcal{W}_P$-Gorenstein, $\mathcal{W}_I$-Gorenstein and $\mathcal{W}_F$-Gorenstein modules and discuss some basic properties of these modules. We show that the subcategory of strongly Gorenstein projective left $R$-modules in the left Auslander class and the subcategory of strongly $\mathcal{W}_P$-Gorenstein left $S$-modules are equivalent under Foxby equivalence. The injective and flat case are also studied.

Strongly Gorenstein graded modules

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.

(𝒲,𝒴,𝒳)-Gorenstein complexes

ABSTRACTLet 𝒲,𝒴,𝒳 be three classes of left R-modules. In this paper, we introduce and study (𝒲,𝒴,𝒳)-Gorenstein complexes as a common generalization of completely 𝒲-resolved complexes [26], Gorenstein projective (resp., injective) complexes [8], Ding projective (resp., injective) complexes [32] and Gorenstein AC-projective (resp., AC-injective) complexes [4]. It is shown that under certain hypotheses, a complex C is (𝒲,𝒴,𝒳)-Gorenstein if and only if each Cn is a (𝒲,𝒴,𝒳)-Gorenstein module and are exact for any . This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of (𝒲,𝒴,𝒳)-Gorenstein complexes and modules are explored.

Gorenstein complexes and recollements from cotorsion pairs

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modules have finite projective dimension we obtain the duals. These results extend to a general ring R by replacing the Gorenstein modules with the Gorenstein AC-modules introduced in [9]. We also see that in any abelian category with enough injectives, the Gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair.

On co-Gorenstein modules, minimal flat resolutions and dual Bass numbers

The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if $R$ is a local $U$-ring and $M$ is an Artinian $R$-module, then $M$ is a co-Gorenstein $R$-module if and only if the complex ${\rm H

Gorenstein-duality for one-dimensional almost complete intersections – with an application to non-isolated real singularities

AbstractWe give a generalisation of the duality of a zero-dimensional complete intersection for the case of one-dimensional almost complete intersections, which results in a Gorenstein module M = I/J. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity f, with a one-dimensional critical locus, we relate the signature on the Jacobian module I/Jf to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ℙ2(ℝ) of even degree is given.

STRONGLY WP -GORENSTEIN MODULES

In this paper, for a xed semidualizing bimodule C =S CR, we introduce the notion of strongly WP -Gorenstein modules and give the foxby equivalence between the categories of DP(R)\AC(R) and DCP(R)\BC(S). Then we investigate the properties of strongly WP -Gorenstein modules and dimensions. Key words and phrases: semidualizing module; stronglyWP -Gorenstein module; C-projective module.

G-Gorenstein Modules

Let R be a commutative Noetherian ring. In this paper, we study those finitely generated R-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of G-Gorenstein modules are given and a class of such modules is determined. It is shown that the class of G-Gorenstein modules strictly contains the class of Gorenstein modules. Also, we provide a Gorenstein injective resolution for a balanced big Cohen-Macaulay R-module. Finally, using the notion of a G-Gorenstein module, we obtain characterizations of Gorenstein and regular local rings.

Gorenstein modules of finite length

In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Symmetrically Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers. We show that their graded minimal free resolution is selfdual in a strong sense. Applications include a proof of the dependence of the monoid of Betti tables of Cohen-Macaulay modules on the characteristic of the base field. Moreover, we give a new proof of the failure of the generalization of Green's Conjecture to characteristic 2 in the case of general curves of genus 2n −1. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Formula omitted]-Gorenstein modules

Let W be a self-orthogonal class of left R-modules. We introduce and study W -Gorenstein modules as a common generalization of some known modules such as Gorenstein projective (injective) modules (Enochs and Jenda, 1995 [7]) and V-Gorenstein projective (injective) modules (Enochs et al., 2005 [12]). Special attention is paid to W P -Gorenstein and W I -Gorenstein modules, where W P = { C ⊗ R P | P is a projective left R -module } and W I = { Hom S ( C , E ) | E is an injective left S -module } with C R S a faithfully semidualizing bimodule.

First, Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.