Gorenstein Injective Research Articles

Flat model structures and Gorenstein objects in functor categories

We construct a flat model structure on the category ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ of additive functors from a small preadditive category $\mathcal {Q}$ satisfying certain conditions to the module category ${_{R}\mathsf {Mod}}$ over an associative ring $R$ , whose homotopy category is the $\mathcal {Q}$ -shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$ , an object in ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal {Q}$ , and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.

N-Cotorsion pairs

Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of $n$-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as $m$-cluster tilting subcategories.

Totally acyclic complexes

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. The cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. 6. Every acyclic complex of flat modules is F-totally acyclic. 7. The cycles of every acyclic complex of Gorenstein flat modules are Gorenstein flat. Thus we improve slightly on a result of Iyengar and Krause; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if R is Gorenstein. We replace the dualizing complex hypothesis by the finiteness of the Krull dimension, and add more equivalent conditions.In the second part of the paper we focus on the noncommutative case. We prove that for a two sided noetherian ring R of finite finitistic flat dimension that satisfies the Auslander condition the following are equivalent: 1. Every complex of injective (left and respectively right) R-modules is totally acyclic. 2. R is Iwanaga–Gorenstein.

On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht M𝔞. We prove that injdim M < ∞ if and only if [Formula: see text] and that [Formula: see text]. We also prove that if R has a dualizing complex and Gid RM < ∞, then [Formula: see text]. Moreover if R and M are Cohen-Macaulay, then Gid RM < ∞ whenever [Formula: see text]. Next, for a finitely generated R-module M of dimension d, it is proved that if [Formula: see text] is Cohen-Macaulay and [Formula: see text], then [Formula: see text]. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.

Gorenstein Injective and Injective Complete Cohomological Dimensions of Groups

Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group Γ, called the injective complete cohomological dimension of Γ, denoted by iccd Γ. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group Γ, which is denoted by Gid Γ. We show that Gid Γ is related to iccd Γ, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd Γ is a refinement of Gid Γ. In addition, we show that silp Γ = spli Γ < ∞ if and only if the Shapiro lemma holds for injective complete cohomology.

Vanishing of Tate Cohomology and Gorenstein Injective Dimension

In this note, some sufficient conditions are given for the equality Gid R RHom R (M, N) = G-dim R M + Gid R N to hold via vanishing of Tate cohomology. For instance, let R be a commutative Gorenstein local ring, and let M and N be finite R-modules. If one has for all i ∈ ℤ, then Gid R RHom R (M, N) = G-dim R M + Gid R N. In addition, some Auslander–Buchsbaum type width (depth) formulas are given.

Gorenstein conditions over triangular matrix rings

A ring is left Gorenstein regular if the classes of left modules with finite projective dimension and finite injective dimension coincide and the injective and projective finitistic left dimensions are finite. Let A and B be rings and U a (B,A)-bimodule such that UB has finite projective dimension and UA has finite flat dimension. In this paper we characterize when the ring T=(A0UB) is left Gorenstein regular and, over such rings, when a left T-module is Gorenstein projective or Gorenstein injective. As applications of these results, we characterize when T is left CM-free and give a necessary condition for existence of an infinite cardinal λ such that each Gorenstein projective module is a direct sum of λ<-generated modules.

Gorenstein Injective and Projective Complexes with Respect to a Semidualizing Module

Let R be a commutative (possibly non-Noetherian) ring (in order to make things less technical) and C a semidualizing R-module. In this article, we introduce and investigate the notion of G C -injective (G C -projective) complexes. This extends Enochs and García Rozas's notion of Gorenstein injective (Gorenstein projective) complexes. We then show that a complex X is G C -injective (G C -projective) if and only if X m is a G C -injective (G C -projective) module for each m ∈ ℤ.

(Weakly) Gorenstein injective and (weakly) Gorenstein coflat comodules

In this paper, we introduce Gorenstein coat comodules and weakly Gorenstein injective (resp. coat) comodules over any coalgebra C. We prove that, for a left semiperfect coalgebra C, a left C-comodule M is (weakly) Gorenstein coat if and only if M is (weakly) Gorenstein injective. Some properties of weakly Gorenstein injective comodules are studied.

GORENSTEIN-INJECTORS, GORENSTEIN-FLATORS

Over a ring R, let <TEX>$P_R$</TEX> be a finitely generated projective right R-module. Then we define the G-injector (G-projector) if <TEX>$P_R$</TEX> preservers Gorenstein injective modules (Gorenstein projective modules), the Gflator if <TEX>$P_R$</TEX> preservers Gorenstein flat modules. G-injector (G-flator) and G-injector are characterized focus primarily on the cases where R is a Gorenstein ring, and under this condition we also study the relations between the injector (projector, flator) and the G-injector (G-projector, G-flator). Over any ring we also give the characteristics of G-injector (G-flator) by the Gorenstein injective (Gorenstein flat) dimensions of modules.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In addition, we prove that if R → S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules, then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical. We also give some applications of these results.

A Bass Formula for Gorenstein Injective Dimension

In this article a generalized version of the Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension over a commutative Noetherian ring.

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings#

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.

The Gorenstein Injective Envelope of the Residue Field of a Local Ring#

Let R be a commutative Noetherian local Gorenstein ring with residue field k. We show that G(k), the Gorenstein injective envelope of k, is an artinian R-module, and we compute G(k) in the case where R = k[[S]] is a semigroup ring and S is symmetric. We also show that a certain subring of the endomorphism ring of G(k) is a complete local (but possibly non-commutative) ring.

Gorenstein homological dimensions

In basic homological algebra, the projective, injective and flat dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein flat dimensions are studied. There is a variety of nice results about Gorenstein dimensions over special commutative noetherian rings; very often local Cohen–Macaulay rings with a dualizing module. These results are done by Avramov, Christensen, Enochs, Foxby, Jenda, Martsinkovsky and Xu among others. The aim of this paper is to generalize these results, and to give homological descriptions of the Gorenstein dimensions over arbitrary associative rings.

Covers and envelopes over Gorenstein rings

A module over a Gorenstein ring is said to be Gorenstein injective if it splits under all modules of finite projective dimension. We show that over a Gorenstein ring every module has a Gorenstein injective envelope. We apply this result to the group algebra $\hat{Z}_{\rho}G$ (with $G$ a finite group and $\hat{Z}_{p}$ the ring of $p$-adic integers for some prime $p$) and show that ever finitely generated $\hat{Z}_{p}G$-module has a cover by a lattice. This gives a way of lifting finite dimensional representations of $G$ over $Z/(p)$ to modular representations of $G$ over $\hat{Z}_{p}$.

Model structures on modules over Ding-Chen rings

An n-FC ring is a left and right coherent ring whose left and right self-FP-injective dimension is n. The work of Ding and Chen shows that these rings possess properties which generalize those of n-Gorenstein rings. In this paper we call a (left and right) coherent ring with finite (left and right) self-FP-injective dimension a Ding-Chen ring. In the case of Noetherian rings, these are exactly the Gorenstein rings. We look at classes of modules we call Ding projective, Ding injective and Ding flat which are meant as analogs to Enochs' Gorenstein projective, Gorenstein injective and Gorenstein flat modules. We develop basic properties of these modules. We then show that each of the standard model structures on Mod-R, when R is a Gorenstein ring, generalizes to the Ding-Chen case. We show that when R is a commutative Ding-Chen ring and G is a finite group, the group ring R[G] is a Ding-Chen ring.