Let $H$ be a Hopf algebra over a field $k$ with a bijective antipode. It is proved that the Gorenstein global dimension of $H$ coincides with the Gorenstein projective dimension of the trivial left (or right) $H$-module $k$. Then, $H$ is finite dimensional if and only if the Gorenstein projective dimension of $k$ is trivial. Although monoidal Morita-Takeuchi equivalence of Hopf algebras does not preserve the global dimension, we demonstrate that it does preserve the Gorenstein global dimension and the Artin-Schelter Gorenstein property; this supports Brown-Goodearl's question of whether every noetherian (affine) Hopf algebra is AS Gorenstein. Finally, for $H$ and an $H$-Galois object $B$, we show the categories of modules $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$ are monoidal model categories regarding Gorenstein projective model structure, provided that the Gorenstein global dimension of $H$ is finite. The corresponding stable categories are tensor triangulated categories.
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 $A$ 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.
We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.
Gorenstein homological algebra is a kind of relative homological algebra which has been developed to a high level since more than four decades. In this report we review the basic theory of Gorenstein homological algebra of artin algebras. It is hoped that such a theory will help to understand the famous Gorenstein symmetric conjecture of artin algebras. With only few exceptions all the results in this report are contained in the existing literature. We have tried to keep the exposition as self-contained as possible. This report can be viewed as a preparation for learning the newly developed th…
Let $R$ be a two-sided noetherian ring and $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_R(M)$. Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_R(M)$. We characterize Gorenstein projective $T_R(M)$-modules in terms of $R$-modules.
In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $\mathcal{A}$, we define a group homomorphism $\mathrm{Hdet}: \mathrm{Aut}_{dg}(\mathcal{A})\to k^{\times},$ called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $\mathcal{A}^G$ to be Gorensten, where $\mathcal{A}$ i…
We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra $A$, there exists a connected self-injective Nakayama algebra $A'$ such that there is a sequence of left retractions linking $A$ to $A'$; in particular, the singularity category of $A$ is triangle equivalent to the st…
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 study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying …
We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Forbenius extensions.
We generalise notions of Gorenstein homological algebra for rings to the context of arbitrary abelian categories. The results are strongest for module categories of rngs with enough idempotents. We also reformulate the notion of Frobenius extensions of noetherian rings into a setting which allows for direct generalisation to arbitrary abelian categories. The abstract theory is then applied to the BGG category O for Lie superalgebras, which can now be seen as a "Frobenius extension" of the corresponding category for the underlying Lie algebra and is therefore "Gorenstein". In particular we obta…
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance …
Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open problems of Gorenstein homological algebra by showing that so-called Gorenstein projective resolutions exist over quite general rings, thereby enabling the definition of a Gorenstein version of derived fun…
In this paper we develop the homological properties of the $(\mathcal{L}, \mathcal{A})$-Gorenstein flat $R$-modules $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A} \subseteq \mathrm{Mod} (R^{op})$ sometimes corresponds to a duality pair $(\mathcal{L}, \mathcal{A})$. We study the weak global and finitistic dimensions that comes with $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ and show that over a $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R -$ is left balanced over $\mathrm{Mod} (R^{op}) \times \mathrm{Mod} (R)$ by …
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$_…
We present and study the concept of $m$-periodic Gorenstein objects relative to a pair $(\mathcal{A,B})$ of classes of objects in an abelian category, as a generalization of $m$-strongly Gorenstein projective modules over associative rings. We prove several properties in some cases where $(\mathcal{A,B})$ satisfies certain homological conditions, like for instance when $(\mathcal{A,B})$ is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established.
Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$ω$-cotorsionfree modules and a subclass of the class of $ω$-adstatic modules. Also we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of ${\rm Hom}(ω,M)$. By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a …
n-recollements of triangulated categories and n-derived-simple algebras are introduced. The relations between the n-recollements of derived categories of algebras and the Cartan determinants, homological smoothness and Gorensteinness of algebras respectively are clarified. As applications, the Cartan determinant conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry conjecture is reduced to 2-derived-simple algebras.
Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants
School of Mathematics Sciences, Sichuan Normal University（四川师范大学数学科学学院）；School of Mathematics and Statistics, Tianshui Normal University（天水师范学院数学与统计学院）
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 any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodu…
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.
This paper brings together two theories in algebra that have had been extensively developed in recent years. First is the study of various homological dimensions and what information such invariants can give about a ring and its modules. A collection of intermediate C-Gorenstein dimensions is defined and this allows generalizations of results concerning C-Gorenstein dimension and certain Serre-like conditions. Second is the theory of linkage first introduced by Peskine and Szpiro and generalized to modules by Martinskovsky and Strooker. Using the further generalization of module linkage of Nag…
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…
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.
We review the theory of Co-Gorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization". We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.
Let $Λ$ and $Γ$ be left and right noetherian rings and $_ΛU$ a Wakamatsu tilting module with $Γ={\rm End}(_ΛT)$. We introduce a new definition of $U$-dominant dimensions and show that the $U$-dominant dimensions of $_ΛU$ and $U_Γ$ are identical. We characterize $k$-Gorenstein modules in terms of homological dimensions and the property of double homological functors preserving monomorphisms. We also study a generalization of $k$-Gorenstein modules, and characterize it in terms of some similar properties of $k$-Gorenstein modules.
The homological spectrum for monoidal triangulated categories
University of Georgia（佐治亚大学）；University at Albany, SUNY（纽约州立大学奥尔巴尼分校）；Northeastern University（东北大学）；International Center for Mathematical Sciences, Institute of Mathematics and Informatics Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所国际数学科学中心）
In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.
We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ Λ) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(Λ)$ for a finite dimensional algebra $Λ$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } Λ< \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (Λ)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(Λ)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(Λ)$ is replaced with a finitely graded IG algebra $A$…
We introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring--the Gorenstein flat-cotorsion dimension--and prove that it, unlike the Gorenstein flat dimension, behaves as one expects of a homological dimension without extra assumptions on the ring. Crucially, we show that it coincides with the Gorenstein flat dimension for complexes where the latter is finite, and for complexes over right coherent rings--the setting where the Gorenstein flat dimension is known to behave as expected.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, …
Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over $RH$, we show that the skew group rings $RG$ and $RH$ share some properties such as being $n$-Gorenstein, $n$-perfect, $n$-coherent, $(n,d)$, Ding-Chen or IF-rings.
For a recollement $(\mathcal{D}B,\mathcal{D}A,\mathcal{D}C)$ of derived categories of algebras, we investigate when the functor $j^*:\mathcal{D}A\rightarrow\mathcal{D}C$ is an eventually homological isomorphism. In this context, we compare the algebras $A$ and $C$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.
Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.
We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category Mon(B, A-Gproj) to describe some Gorenstein projective bimodules over the tensor product of $A$ and $B$. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for Mon(B, A-Gproj) being the category of all Gorenstein projective bimodules. In addition, If both $A$ and $B$ are Gorenstein, we can describe the category of all Gorens…
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with $\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for $\mathcal{A}$ to be Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively…
