Gorenstein Homological Dimensions Research Articles

Gorenstein projective, injective, flat modules and dimensions over polynomial rings

We investigate Gorenstein projective, injective, flat modules and dimensions over a polynomial ring using the equivalent category of the module category over the polynomial ring. Several estimates of Gorenstein homological dimensions of modules over the polynomial ring are obtained.

M-Periodic Gorenstein objects

We present and study the concept of m-periodic Gorenstein objects relative to a pair (A,B) of classes of objects in an abelian category, as a generalization of m-strongly Gorenstein projective modules over associative rings. We prove several properties when (A,B) satisfies certain homological conditions, like for instance when (A,B) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established.

GC -projectivity and injectivity under Frobenius extensions

In this article, we prove that Gorenstein projectivity and injectivity with respect to a semidualizing module are preserved under Frobenius extensions. Furthermore, we obtain that the GC -projective and injective dimensions of modules are invariant under Frobenius extensions. As corollaries, we get some Gorenstein homological dimensions are preserved under Frobenius extensions, which show that Gorenstein homological dimensions have better invariant properties than the classical homological dimensions under Frobenius extensions.

Gorenstein Homological Dimensions for Extriangulated Categories

Let \(({\mathcal {C}},{\mathbb {E}},{\mathfrak {s}})\) be an extriangulated category with a proper class \(\xi \) of \({\mathbb {E}}\)-triangles. In a previous work, we introduced and studied the \(\xi \)-\({\mathcal {G}}\)projective and the \(\xi \)-\({\mathcal {G}}\)injective dimension for any object in \({\mathcal {C}}\). In this paper, we first give some characterizations of \(\xi \)-\({\mathcal {G}}\)projective dimension by using derived functors on \({\mathcal {C}}\). Consequently, we show that the following equality holds under some assumptions: $$\begin{aligned} \sup \{\xi \text {-}{\mathcal {G}}\mathrm{pd}M \ | \ \text {for} \ \text {any} \ M\in {{\mathcal {C}}}\}=\sup \{\xi \text {-}{\mathcal {G}}\mathrm{id}M \ | \ \text {for} \ \text {any} \ M\in {{\mathcal {C}}}\}, \end{aligned}$$where \(\xi \text {-}{\mathcal {G}}\mathrm{pd}M\) (resp., \(\xi \text {-}{\mathcal {G}}\mathrm{id}M\)) denotes the \(\xi \)-\({\mathcal {G}}\)projective dimension (resp., \(\xi \)-\({\mathcal {G}}\)injective dimension) of M. As an application, our main results generalize the work by Bennis–Mahdou and Ren–Liu, which are new for an exact category case. Moreover, our proof is far from the usual module or triangulated case.

Finitistic dimension and orthogonal classes of Gorenstein projective modules with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. It is proved that , which is a generalization of Holm, Gorenstein homological dimensions. J. Pure Appl Algebra 189(1–3):167–193, Theorem 2.28. Also, it is given a condition to verify certain relation between GC-projectives and GC-flats. Indeed, it is shown that every GC-projective module is GC-flat if and only if for every C-injective module I, the character module is in .

Gorenstein homological dimensions and some duality results

Let [Formula: see text] be a local ring and [Formula: see text] denote the Matlis duality functor. Assume that [Formula: see text] possesses a normalized dualizing complex [Formula: see text] and [Formula: see text] and [Formula: see text] are two homologically bounded complexes of [Formula: see text]-modules with finitely generated homology modules. We will show that if G-dimension of [Formula: see text] and injective dimension of [Formula: see text] are finite, then [Formula: see text] Also, we prove that if Gorenstein injective dimension of [Formula: see text] and projective dimension of [Formula: see text] are finite, then [Formula: see text] These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.

Relative Igusa-Todorov Functions and Relative Homological Dimensions

We develope the theory of ${\mathcal {E}}$ -relative Igusa-Todorov functions in an exact I T-context $({\mathcal {C}},{\mathcal {E}})$ (see Definition 2.1). In the case when ${\mathcal {C}}={\text {mod}}\, ({\Lambda })$ is the category of finitely generated left Λ-modules, for an artin algebra Λ, and ${\mathcal {E}}$ is the class of all exact sequences in ${\mathcal {C}},$ we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov’s results. Furthermore, we introduce the ${\mathcal {E}}$ -relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.

Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

For a commutative ring $R$ and a faithfully flat $R$-algebra $S$ we prove, under mild extra assumptions, that an $R$-module $M$ is Gorenstein flat if and only if the left $S$-module $\tp{S}{M}$ is Gorenstein flat, and that an $R$-module $N$ is Gorenstein injective if and only if it is cotorsion and the left $S$-module $\Hom{S}{N}$ is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.

GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES UNDER BASE CHANGE

Abstract. Transfer of homological properties under base change is aclassical field of study. Let R → S be a ring homomorphism. The re-lations of Gorenstein projective (or Gorenstein injective) dimensions ofunbounded complexes between U⊗ LR X (or RHom R (X,U)) and X areconsidered, where X is an R-complex and U is an S-complex. In ad-dition, some sufficient conditions are given under which the equalitiesG-dim S (U⊗ LR X) = G-dim R X + pd S U and Gid S (RHom R (X,U)) =G-dim R X+ id S U hold. IntroductionIt is well known that Gorenstein homological dimensions are refinements ofthe classical homological dimensions. In a different direction, homological di-mensions have been extended to complexes. Avramov and Foxby [2] definedprojective, injective and flat dimensions for unbounded complexes of left mod-ules over associative rings. The concepts of Gorenstein projective, injectiveand flat dimension for homologically bounded complexes were introduced byChristensen [3]. Veliche [10] extended the concept of Gorenstein projectivedimension of homologically bounded complexes to the setting of unboundedcomplexes over associative rings. Dually, Asadollahi and Salarian [1] intro-duced the concept of Gorenstein injective dimension of unbounded complexesover associative rings.Transfer of homological properties along ring homomorphisms is a classi-cal field of study (see, for instance, [9] and its references). Let R and S becommutative Noetherian rings. It was shown in [9] that if ϕ : R → S is aring homomorphism such that every S-module of finite flat dimension is offinite projective dimension over R via ϕ, X is an R-complex and U is an S-complex with finite projective dimension, then one has Gid

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].

Gorenstein homological dimensions of modules over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.

Gorenstein categories $\mathcal G(\mathscr X,\mathscr Y,\mathscr Z)$ and dimensions

Let $\mathscr {A}$ be an abelian category and $\mathscr {X},\mathscr {Y},\mathscr {Z}$ additive full subcategories of $\mathscr {A}$. We introduce and study the Gorenstein category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$ as a common generalization of some known modules such as Gorenstein projective (injective) modules \cite {EJ95}, strongly Gorenstein flat modules \cite {DLM} and Gorenstein FP-injective modules \cite {DM}, and prove the stability of $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$. We also establish Gorenstein homological dimensions in terms of the category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$.

GORENSTEIN DIMENSIONS MODULO A REGULAR ELEMENT

Let $R$ be a commutative ring. In this paper we study the behavior of Gorenstein homological dimensions of a homologically bounded $R$-complex under special base changes to the rings $R_{x}$ and $R/xR$, where $x$ is a regular element in $R$. Our main results refine some known formulae for the classical homological dimensions. In particular, we provide the Gorenstein counterpart of a criterion for projectivity of finitely generated modules, due to Vasconcelos.

Gorenstein theory for n-th differential modules

In this paper the definition of \(n\)-th differential modules is introduced. It is shown that an \(n\)-th differential module \((M,\delta _{M},n)\) is Gorenstein projective (resp. injective) if and only if \(M\) is Gorenstein projective (resp. injective). It is established that the relations between Gorenstein homological dimensions of an \(n\)-th differential module and the ones of its underlying module.

Gorenstein homological dimensions for triangulated categories

Let C be a triangulated category with a proper class E of triangles. Asadollahi and Salarian introduced and studied E-Gprojective and E-Ginjective objects, and developed a relative homological algebra in C.In this paper, we further study Gorenstein homological dimensions for triangulated categories. First, we discuss the finiteness of E-Gprojective and E-Ginjective dimensions. Then we define E-Gorenstein derived functors GExtEi(−,−), and characterize E-Gprojective and E-Ginjective dimensions of objects in C by vanishing of such functors. Consequently we can prove the equality sup{E-GpdM|for any M∈C}=sup{E-GidM|for any M∈C}, which is used to define the global E-Gorenstein dimension of C. Finally, E-Gphantom tower and E-Gcellular tower for objects in C with finite E-Gprojective dimension are constructed.

Gorenstein Homological Dimensions of Complexes with Respect to a Semidualizing Module

In this article, we study finite Gorenstein homological dimensions of complexes with respect to a semidualizing module. A special attention is paid to the dimensions of homologically bounded complexes which have nice functorial descriptions. Moreover, we use the resolutions of G C -homological modules to prove some stability results about G C -homological dimensions, where C is a semidualizing module.

On the Second Change of Rings Theorems for the Gorenstein Homological Dimensions

This paper is concerned with the second change of rings theorems for the Gorenstein homological dimensions. More precisely, we give the ultimate versions of these theorems relatively to these new invariants.

Generalized Local Cohomology Modules and Homological Gorenstein Dimensions

Let 𝔞 be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd𝔞(M, N) denote the supremum of the i's such that . First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd𝔞(M, N). Next, over a Cohen–Macaulay local ring (R, 𝔪), we show that provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne–Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.

On (Strongly) Gorenstein (Semi)Hereditary Rings

In this paper, we introduce and study the rings of Gorenstein homological dimensions less than or equal to 1. We call these Gorenstein (semi)hereditary rings and call a particular subclass of these strongly Gorenstein (semi)hereditary rings.

Excellent extensions and Gorenstein homological dimensions

Let R be a ring and S an excellent extension of R. In this paper we study the Gorenstein homological dimensions over R and S. We show that if M is an S-module, then MR is Gorenstein projective if and only if MS is Gorenstein projective. Secondly, we prove that the right Gorenstein dimension of R is equal to the right Gorenstein dimension of S. Some results in [4] and [6] are extended to the settings of Gorenstein projective modules and Gorenstein projective dimensions, respectively.