Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the non-commutative realm, $k$--algebras with a dualizing complex.
We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.
The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the princ…
We prove that, for any $n \geq 2$, the classes of $\rm{FP}_{n}$-injective modules and of $\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\rm{FP}_{\infty}$-injective and $\rm{FP}_{\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gor…
Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.
In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of Gorenstein projective, injective, and flat modules and the notion of the usual projective, injective, and flat modules.
We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.
We give a sufficient condition for the class of Gorenstein injective modules be precovering: if $R$ is right noetherian and if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under filtrations, then $\mathcal{GI}$ is precovering in $R-Mod$. The converse is also true when we assume that $\mathcal{GI}$ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in $Ch(R)$. We also give a sufficient condition for the exist…
Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is regular if and only if the injective dimension of $k$ is finite. In this paper it is shown that $R$ is Gorenstein if and only if the Gorenstein injective dimension of $k$ is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. It also improves Christensen's generalized Bass formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000).
This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call, respectively, strongly n-Gorenstein projective, injective and flat modules. These three classes of modules give us a new characterization of the first modules, and they are a generalization of the notions of strongly Gorenstein projective, injective, and flat modules respectively.
There are nice relations between graded homological dimensions and ordinary homological dimensions. We study the Gorenstein injective dimension of a complex of graded modules denoted by $^*\Gid$, and derive its properties. In particular we prove the Chouinard's like formula for $^*\Gid$, and compare it with the usual Gorenstein injective dimension.
In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.
Let $(R,\frak m)$ be a commutative Noetherian local ring and let $M$ and $N$ be finitely generated $R$-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper we prove a generalization of Ischebeck Formula, that is $\depth_RM+\sup\{i| {0.1cm}\Ext_R^i(M,N)\neq 0\}=\depth R.$
Let R be a graded ring and n > 1 an integer. In this paper, We introduce the notions of Ding n-gr-injective and Ding n-gr-flat modules by using of special finitely presented graded modules . Then, some properties of Ding n-gr-injective and Ding n-gr-flat modules are obtained. On n-gr-coherent rings, we investigate the relationships among Ding n-gr-injective and Ding n-gr-flat modules and also, we prove that any graded module in R-gr (resp. gr-R) admits an Ding n-gr-injective (resp. Ding n-gr-flat) cover and preenvelope.
Let $R \to S$ be a local ring homomorphism and $N$ a finitely generated $S$-module. We prove that if the Gorenstein injective dimension of $N$ over $R$ is finite, then it equals the depth of $R$.
Let R be a ring, X a class of R-modules and n>1 an integer. In this paper, via special finitely presented modules, we introduce the concepts of Gorenstein n-X-injective and n-X-flat modules. And aside, we obtain some equivalent properties of these modules on n-X-coherent rings. Then, we investigate the relations among Gorenstein n-X-injective, n-X-flat, injective and flat modules on X-FC-rings (i.e., self n-X-injective and n-X-coherent rings). Several known results are generalized to this new context
Let R be a ring. In this paper, Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules and some of their basic properties are studied. Moreover, some characterizations of rings over Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules are given. Also, some known results can be obtained as corollaries
Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective $R$-modules need not be closed under tensor products, even when $R$ is local a…
A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module.
Duality properties are studied for a Gorenstein algebra that is finite and projective over its center. Using the homotopy category of injective modules, it is proved that there is a local duality theorem for the subcategory of acyclic complexes of such an algebra, akin to the local duality theorems of Grothendieck and Serre in the context of commutative algebra and algebraic geometry. A key ingredient is the Nakayama functor on the bounded derived category of a Gorenstein algebra, and its extension to the full homotopy category of injective modules.
We show that a differential module is Gorenstein projective if and only if its underlying module is Gorenstein projective. Dually, a differential module is Gorenstein injective if and only if its underlying module is Gorenstein injective.
Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jorgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair $\mathfrak{D} = (\mathcal{L},\mathcal{A})$. This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo-Gillespie-Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfe…
A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.
Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, 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…
Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely ge…
Let $R$ be a local ring of positive characteristic and $X$ a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of $X$ via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then $R$ is Gorenstein.
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…
We characterize Ding modules and complexes over Ding-Chen rings. We show that over a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Ding flat) R-modules coincide with the Gorenstein projective (resp. Gorenstein injective, resp. Gorenstein flat) modules, which in turn are nothing more than modules appearing as a cycle of an exact complex of projective (resp. injective, resp. flat) modules. We prove a similar characterization for chain complexes of R-modules: A complex is Ding projective (resp. Ding injective, resp. Ding flat) if and only if each component is Ding projective …
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 ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modu…
Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently prese…
Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted $\KPC$, is always well generated and is compactly generated provided $\KPR$ is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension…
For a given class of modules $\A$, we denote by $\widetilde{\A}$ the class of exact complexes $X$ having all cycles in $\A$, and by $dw(\A)$ the class of complexes $Y$ with all components $Y_j$ in $\A$. We consider a two sided noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (flat, projective respectively) $R$-modules. We prove (Theorem 1) that the following are equivalent: 1. Every exact complex of injective modules is totally acyclic. 2. Every exact complex of Gorenstein injective modules is in $\widetilde{\mathc…
Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…
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 S\otimes 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.
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $\mathcal{F}_C$-projective dimension, and finite $\mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
In this paper, we prove that the global Gorenstein projective dimension of a ring $R$ is equal to the global Gorenstein injective dimension of $R$, and that the global Gorenstein flat dimension of $R$ is smaller than the common value of the terms of this equality.
We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.
The so-called 'change-of-ring' results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.
In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring $R$. For an $R$-module $M$, we denote by ${\rm Gpd}_RM$ and ${\rm Gfd}_R M$ the Gorenstein projective and flat dimensions of $M$, respectively. We show that ${\rm Gpd}_RM<\infty$ if and only if ${\rm Gfd}_RM<\infty$ provided the Krull dimension of $R$ is finite. Moreover, in the case that $R$ is local, we correspond to a dualizing complex ${\bf D}$ of $\hat{R}$, the classes $A'(R)$ and $B'(R)$ of $R$-modules. For a module $M$ over a local ring $R$, we show that $M\in A'(R)$ if and only if ${\rm Gp…
Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that $Γ_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective $R$-module $G$. Let $(R,\frak m)$ be a local ring and $M$ be a finitely generated $R$-module. We show that ${\rm Gid}{\bf R}Γ_{\frak m}(M)<\infty$ if and only if ${\rm Gid}_{\hat{R}}(M\otimes_R\hat{R})<\infty$. We also show that if ${\rm Gfd}_R{\bf R}Γ_{\frak m}(M)<\infty$, then ${\rm Gfd}_RM<\infty$. Let $(R,\frak m)$ be a Cohen-Macaulay local ring and $M$ be a Cohen-Macaulay…
Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we investigate the relations between the Gorenstein projective dimension and the Gorenstein flat dimension.
A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C) \simeq A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C-Gorenstein projective, C-Gorenstein injective, and C-Gorenstein flat dimension, and investigate the properties of these dimensions.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with bounded complexes of projective and injective modules, respectively. As an application, we investigate the Gorenstein version of the New Intersection Theorem. The second chapter studies the notion of cofiniteness for modules and complexes set forth by Hartshorne. Recruiting techniques of derived category, we study this notion thoroughly, obtain novel results, an…
