We prove that if a positively-graded ring $R$ is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme ${\rm Tails}(R)$ is a Gorenstein category in the sense of \cite{EEG}. Moreover, under this condition, a (right) recollement relating Gorenstein-injective sheaves in ${\rm Tails}(R)$ and (graded) Gorenstein-injective $R$-modules is given.
In this paper, we introduce the notions of Gorenstein weak injective and weak flat modules respectively in terms of weak injective and weak flat modules, which is larger than classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we also discuss a relation between weak cosyzygy and Gorenstein weak cosyzygy of a module, and the stability of Gorenstein weak injective modules.
In this paper, we study the relation between $m$-strongly Gorenstein projective (resp. injective) modules and $n$-strongly Gorenstein projective (resp. injective) modules whenever $m \neq n$, and the homological behavior of $n$-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of $n$-strongly Gorenstein flat modules. Then we study the homological behavior of $n$-strongly Gorenstein flat modules, and the relation between these modules and $n$-strongly Gorenstein projective (resp. injective) modules.
For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $\mathcal{A}$. We use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $\mathcal{DI}$ for Ding injective modules, and $\mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are …
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.
We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible with the notion of $n$-coherent rings introduced by Bravo-Perez: So a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of …
We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left $R$-module is at most the flat dimension of the injective envelope of $_RR$. Then we get that the injective envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the injective envelope of every flat left $R$-module is (Gorenstein) flat, if and only if…
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 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.
Absolutely clean and level $R$-modules were introduced in [BGH13] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring $R$. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective $R$-modules. Here we study these concepts in the category of chain complexes of $R$-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category $\text{Ch}(R)$ of chain complexes has a cofibrantly generated model structure where every object …
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…
In this note, we mainly extend some Gorenstein homological properties from special rings (Noetherian or coherent rings ) to arbitrary rings by introducing the notions of Gorenstein weak injective and weak projective modules respectively.
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…
Let A be a noncommutative noetherian with dualizing complex R. We study the multiplicities of indecomposable injectives in a minimal injective resolution of R. In particular when A is a Gorenstein ring we get information on minimal injective resolutions of A.
Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.
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.
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.
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.
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…
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.
Let $Λ$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $Λ$ admits a trivial maximal 1-orthogonal subcategory of $\modΛ$, then for any indecomposable module $M \in \mod Λ$, we have that the projective dimension of $M$ is equal to one if and only if so is its injective dimension and that $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $Λ$ is a tilted algebra.
Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated sc…
This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…
Let $Λ$ and $Γ$ be artin algebras and $_ΛU_Γ$ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of $k$-Gorenstein modules with respect to $_ΛU_Γ$ and then characterize it in terms of the $U$-resolution dimension of some special injective modules and the property of the functors ${\rm Ext}^{i}({\rm Ext}^{i}(-, U), U)$ preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of $_ΛU_Γ$ with finite left or righ…
In this paper, we describe the structure of the localization of Ext^{i}_{R}(R/P,M), where P is a prime ideal and M is a module, at certain Ore sets X. We first study the situation for FBN rings, and then consider matters for a large class of Auslander-Gorenstein rings. We need to impose suitable homological regularity conditions to get results in the more general situation. The results obtained are then used to study the shape of minimal injective resolutions of modules over noetherian rings.
In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_…
Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_R^i(M, R)=0$ for any $1…
Let $R$ be any ring with identity and Ch($R$) the category of chain complexes of (left) $R$-modules. We show that the Gorenstein AC-projective chain complexes are the cofibrant objects of an abelian model structure on Ch($R$). The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when $R$ is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly g…
In this article we investigate the relations between the Gorenstein projective dimensions of $Λ$-modules and their socles for minimal n-Auslander-Gorenstein algebras $Λ$ in the sense of Iyama and Solberg \cite{IS}. First we give a description of projective-injective $Λ$-modules in terms of their socles. Then we prove that a $Λ$-module $N$ has Gorenstein projective dimension at most n iff its socle has Gorenstein projective dimension at most n iff $N$ is cogenerated by a projective $Λ$-module. Furthermore, we show that minimal n-Auslander-Gorenstein algebras can be characterised by the relation…
Let $Λ$ be the path algebra of a finite quiver $Q$ over a finite-dimensional algebra $A$. Then $Λ$-modules are identified with representations of $Q$ over $A$. This yields the notion of monic representations of $Q$ over $A$. If $Q$ is acyclic, then the Gorenstein-projective $\m$-modules can be explicitly determined via the monic representations. As an application, $A$ is self-injective if and only if the Gorenstein-projective $\m$-modules are exactly the monic representations of $Q$ over $A$.
The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$ under the natural action of a finite subgroup of $SL_n(k)$ is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras $A$. We develop …
In this paper, we focus on $n$-syzygy modules and the injective cogenerator determined by the minimal injective resolution of a noether ring. We study the properties of $n$-syzygy modules and a category $R_n(\mod R)$ which includes the category consisting of all $n$-syzygy modules and their applications on Auslander-type rings. Then, we investigate the injective cogenerators determined by the minimal injective resolution of $R$. We show that $R$ is Gorenstein with finite self-injective dimension at most $n$ if and only if $\id R\leq n$ and $\fd \bigoplus_{i=0}^n I_i(R)< \infty$. Some known res…
Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective (respectively, $\mathscr{T}$-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if $\mathscr{T}$ contains all projective (respectively, injective) objects of $\mathscr{A}$, then the above assertion holds true if…
For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we …
Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point out that such objects generalize the usual Gorenstein projective objects and others generalizations appearing in the literature as Ding-projective, Ding-injective, $\mathcal{X}$-Gorenstein projective, Gorenstein AC-projective and $G_C$-projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein proje…
We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension t…
In this paper we study the finiteness of global Gorenstein AC-homological dimensions for rings, and answer the questions posed by Becerril, Mendoza, Pérez and Santiago. As an application, we show that any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category, which improves the known result by Beligiannis.
In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolution for ordinary modules. These are different from DG-projective and DG-injective resolutions which is…
For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic c…
For a ring $A$, we consider the question whether every bounded above cochain complex of injective $A$-modules which is acyclic is null-homotopic. We show that if $A$ is left and right noetherian and has a dualizing complex, then this implies that the finitistic dimension of $A$ is finite. In the appendix, Nakamura and Thompson show that the opposite holds over any ring. Our results give several new necessary and sufficient conditions for a ring to have finite finitistic dimension in a very general setting. Applications include a generalization of a recent result of Rickard about relations betw…
