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Let Λ be an Artin algebra. In this paper, the notion of n Z -Gorenstein cluster tilting subcategories will be introduced. It is shown that every n Z -cluster tilting subcategory of mod-Λ is n Z -Gorenstein if and only if Λ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an n Z -Gorenstein cluster tilting subcategory of mod-Λ is an n Z -cluster tilting subcategory of the exact category Gprj - Λ , the subcategory of all Gorenstein projective objects of mod-Λ. Some basic properties of n Z -Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are n-resolving, a higher version of resolving subcategories.

Let Λ be an artin algebra and X be a quasi-resolving subcategory of mod-Λ which is of finite type. Let SX(Λ) be the full subcategory of the morphism category H(Λ) consisting of all monomorphisms f:...

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.

In this paper, (higher) Gorenstein flat phantom morphisms over rings will be introduced and studied. To study their relationship, a characterization of Gorenstein flat objects in morphism category ...

A relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Dual version will be treated. Several examples and applications will be provided. In particular, we show that under certain circumstances, if relative Auslander algebras of artin algebras $\Lambda$ and $\Lambda'$ are Morita equivalent, then $\Lambda$ and $\Lambda'$ are also Morita equivalent., Comment: Section 6 of the previous version is deleted completely. Title is changed and references are modified, accordingly

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes., Comment: Published in \emph{Nagoya Journal of Mathematics}

Background: Crimean-Congo hemorrhagic fever (CCHF) is an acute, feverous disease that is either caused by tick bites or the direct contact of humans with the blood and tissues of infected livestock. The majority of patients suffering from this disease are slaughterhouse workers (including butchers), farmers, veterinarians and hospital staff. Thus, this study aimed to investigate the health behaviors of butchers regarding CCHF and study factors affecting such behaviors based on the health belief model.Methods: This is a descriptive cross-sectional study conducted on 500 butchers in Ardabil Province in 2020 by a multistage sampling method. The participants of the study completed the researcher-made questionnaire of health belief model and health behaviors model relevant to CCHF.The collected data were then analyzed by descriptive statistical tests and linear regression analysis. Results: The mean (SD) age of the participants was 44.4 (10.5), and 96% were males. Only xxx% of the participants displayed acceptable disease-preventive behaviors. The validity and reliability of the developed questionnaire were confirmed by the Cronbach-alpha index. The results of the exploratory factor analysis showed that the constructs of the model explained 84% of the total variance. The results of the study revealed that among the variables of the health belief model, perceived susceptibility (p-value = 0.006, β = 0.152) and perceived barriers (p-value = 0.05, β = 0.14) were the strongest factors predicting disease-preventive behaviors regarding CCHF.Conclusion: The results of the study showed that the health belief model can predict preventive behaviors for CCHF.Therefore, designing and executing interventions based on the results of this study may encourage such preventive behaviors in butchers.

In this note, the notion of cotorsion classes is introduced into the higher homological algebra. Our results motivate the definition, showing that this notion of $n$-cotorsion classes satisfies usual properties one could expect. In particular, a higher version of Wakamatsu's Lemma is proved. Connections with wide subcategories are also studied., Comments are welcome

Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}\Lambda$, where $\Lambda$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(\Lambda)$, the submodule category of $\Lambda$, consisting of all monomorphisms in $\mathcal{M}$. We construct two functors from $\mathcal{S}(\mathcal{M})$ to ${\rm mod}\mbox{-}\underline{\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotient categories of the submodule category of $\mathcal{M}$ modulo their respective kernels. Moreover, they are related by a syzygy functor on the stable category of ${\rm mod}\mbox{-}\underline{\mathcal{M}}$. These functors can be considered as a higher version of the two functors studied by Ringel and Zhang [RZ] in the case $\Lambda=k[x]/{\langle x^n \rangle}$ and generalized later by Eir\'{i}ksson [E] to self-injective artin algebras. Several applications will be provided.

The main theme of this paper is to study different “Gorenstein defect categories” and their connections. This is done by studying rings for which 𝕂 a c ( Prj- R ) = 𝕂 t a c ( Prj- R ) , that is, rings enjoying the property that every acyclic complex of projectives is totally acyclic. Such studies have been started by Iyengar and Krause over commutative Noetherian rings with a dualizing complex. We show that a virtually Gorenstein Artin algebra is Gorenstein if and only if it satisfies the above mentioned property. Then, we introduce recollements connecting several categories which help in providing categorical characterizations of Gorenstein rings. Finally, we study relative singularity categories that lead us to some more “Gorenstein defect categories”.

Let ${\mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact category $({\mathscr{A}}, {\mathscr{E}})$, where $n \geq 1$ is an integer. It is proved by Jasso that if $n> 1$, then ${\mathscr{C}}$ although is no longer exact, but has a nice structure known as $n$-exact structure. In this new structure conflations are called admissible $n$-exact sequences and are ${\mathscr{E}}$-acyclic complexes with $n+2$ terms in ${\mathscr{C}}$. Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called `ideal approximation theory' into `higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of `higher ideal approximation theory'., Comments are Welcome

Let Mod- S denote the category of S -modules, where S is a small pre-additive category. Using the notion of relative derived categories of functor categories, we generalize Rickard's theorem on derived equivalences of module categories over rings to Mod- S . Several interesting applications will be provided. In particular, it will be shown that derived equivalence of two coherent rings not only implies the equivalence of their homotopy categories of projective modules, but also implies that they are Gorenstein derived equivalent. As another application, it is shown that a good tilting module produces an equivalence between the unbounded derived category of the module category of the ring and the relative derived category of the module category of the endomorphism ring of it.

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_{n}}$ and $R$ is a $k$-Gorenstein ring, where $n$ is the number of vertices of $\mathcal{Q}$., Accepted for publication in Comm. Algebra

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided., Comment: To appear in Communications in Algebra

In this paper, ideal balanced pairs in an abelian category will be introduced and studied. It is proved that every ideal balanced pair gives rise to a triangle equivalence of relative derived categories. We define complete ideal cotorsion triplets and investigate their relation with ideal balanced pairs.

This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category $\mathcal {S}$ and a maximal object s of $\mathcal {S}$ and construct a recollement of $\text {Mod-}\mathcal {S}$ in terms of $\text {Mod-End}_{\mathcal {S}}(s)$ and $\text {Mod-}(\mathcal {S}\setminus \{s\})$ in four different levels. In case $\mathcal {S}$ is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N-complexes.

We study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A^{{\rm op}})$ and ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A)$. If $A=\Lambda$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda^{\rm op})$ and ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{{\rm{b}}}({\rm mod}{\mbox{-}}\Lambda)$ admits a Serre functor and hence has Auslander-Reiten triangles., Comment: arXiv admin note: text overlap with arXiv:1605.04745

The main objective of this paper is to study the relative derived categories from various points of view. Let A be an abelian category and C be a contravariantly finite subcategory of A . One can define C -relative derived category of A , denoted by D C ⁎ ( A ) . The interesting case for us is when A has enough projective objects and C = GP - A is the class of Gorenstein projective objects, where D C ⁎ ( A ) is known as the Gorenstein derived category of A . We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickardʼs theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories.

According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to ${\mathcal{C}}$. Recently Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander formula including the versions for all modules and for unbounded derived categories. We apply them to include some results concerning recollements of triangulated categories.

Let R be a ring and Q be a quiver. We study the homotopy categories K ( Prj Q ) and K ( Inj Q ) consisting, respectively, of projective and injective representations of Q by R-modules. We show that, for certain quivers, these triangulated categories are compactly generated and provide explicit descriptions of compact generating sets. Moreover, in case R is commutative and noetherian with a dualizing complex D, the dualizing functor D ⊗ R − : K ( Prj R ) → K ( Inj R ) is extended to a triangulated functor K ( Prj Q ) → K ( Inj op Q ) which is an equivalence of triangulated categories. This functor, establishes an equivalence on K ( Prj Q ) and K ( Inj Q ) , whenever Q is finite.

Recently the notions of sfliΓ, the supremum of the flat lengths of injective Γ-modules, and silfΓ, the supremum of the injective lengths of flat Γ-modules have been studied by some authors. These homological invariants are based on spli and silp invariants of Gedrich and Gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups. In this paper we will study these invariants. It turns out that, for any group Γ, the finiteness of silfΓ implies the finiteness of sfliΓ, but the converse is not known. We investigate the situation in which sfliΓ

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.

Let [Formula: see text] be a ring and [Formula: see text] be a finite quiver. We give an explicit formula for the injective envelopes and projective precovers in the category [Formula: see text] of bound representations of [Formula: see text] by left [Formula: see text]-modules, where [Formula: see text] is a combination of monomial and commutativity relations. Some applications will be provided. In particular, it is shown that if [Formula: see text] is acyclic and [Formula: see text] is an Iwanaga-Gorenstein ring, then so are these bound quiver algebras.

Using Auslander’s G-dimension, we assign a numerical invariant to any group Γ . It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ . We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.

We develop and study Tate and complete cohomology theory in the category of sheaves of O X -modules. Different approaches are included. We study the properties of these theories and show their power in reflecting the Gorensteinness of the underlying scheme. The connection of these two theories will be discussed.

We introduce and study a complete cohomology theory for complexes, which provides an extended version of Tate–Vogel cohomology in the setting of (arbitrary) complexes over associative rings. Moreover, for complexes of finite Gorenstein projective dimension a notion of relative Ext is introduced. On the basis of these cohomology groups, some homological invariants of modules over commutative noetherian local rings, such as Martsinkovsky’s ξ -invariants and relative and Tate versions of Betti numbers, are extended to the framework of complexes with finite homology. The relation of these invariants with their prototypes is explored.

The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.

Let (R, 𝔪) be a commutative Noetherian local ring. We exhibit certain modules T over R which test G-dimension of a finitely generated R-module M with finite G-dimension in the following sense: if for all j ≥ i, where i is a positive integer, then G-dim R M

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

We define two new homological invariants for a finitely generated module M over a commutative Noetherian local ring R, its Buchsbaum dimension B-dim R M, and its Monomial conjecture dimension MC-dim R M. It will be shown that these new invariants have certain nice properties we have come to expect from homological dimensions. Over a Buchsbaum ring R, every finite module M has B-dim R M

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.

A new homological dimension, called GCM-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. GCM-dimension (short for Generalized Cohen–Macaulay dimension) characterizes Generalized Cohen–Macaulay rings in the sense that: a ring R is Generalized Cohen–Macaulay if and only if every finitely generated R-module has finite GCM-dimension. This dimension is finer than CM-dimension and we have equality if CM-dimension is finite. Our results will show that this dimension has expected basic properties parallel to those of the homological dimensions. In particular, it satisfies an analog of the Auslander–Buchsbaum formula. Similar methods will be used for introducing quasi-Buchsbaum and Almost Cohen–Macaulay dimensions, which reflect corresponding properties of their underlying rings.

Let R be a commutative Noetherian local ring with maximal ideal and let M be a finitely generated R-module. The local cohomology modules are known to be Artinian, so is finitely generated for all i...

In this paper, we study the category of sheaves over an infinite partially ordered set with its natural topological structure. Totally acyclic complexes in this category will be characterized in terms of their stalks. This leads us to describe Gorenstein projective, injective and flat sheaves. As an application, we get an analogue of a formula due to Mitchell, giving an upper bound on the Gorenstein global dimension of such categories. Based on these results, we present situations in which the class of Gorenstein projective sheaves is precovering as well as situations in which the class of Gorenstein injective sheaves is preenveloping.

Let A be a Noetherian ring, let M be a finitely generated A-module and letbe a system of ideals of A. We prove that, for any ideal a in �, if, for every prime ideal p of A, there exists an integer k(p), depending on p, such that a k(p) kills the general local cohomology module H j �p (Mp) for every integer j less than a fixed integer n, where �p :={ap : a2 �}, then there exists an integer k such that a k H j (M) = 0 for every j < n.

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 K ( Prj C ( R ) ) , is compactly generated provided K ( Prj R ) 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. Furthermore, it turns out that over such rings the inclusion functor ι : K ( GPrj R ) ↪ K ( R ) has a right adjoint ι ρ , where K ( GPrj R ) is the homotopy category of Gorenstein projective R modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If R has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories K ( GPrj R ) and K ( GInj R ) , that restricts to an equivalence between K ( Prj R ) and K ( Inj R ) , whenever R is commutative, noetherian and admits a dualising complex.

We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that if $\mathcal {X}$ is a sufficiently nice subcategory of an abelian category, then the derived dimension of $\mathcal {A}$ is at most $\mathcal {X}$ -dim $\mathcal {A}$ , provided that $\mathcal {X}$ -dim $\mathcal {A}$ is greater than one. We provide some applications.

1. IntroductionThe notion of representation dimension of an artin algebra Λ, denoted rep.dimΛ, hasbeen introduced by Auslander in his Queen Mary Notes in 1971. His hope was to provide“a reasonable way of measuring how far the category of finitely generated modules over anartin algebra is from being of finite representation type”. Recall that an artin algebra Λis called of finite representation type if, up to isomorphisms, there are only finitely manyfinitely generated indecomposable Λ-modules. Such algebras are called representation finite.The importance of this class of artin algebras for the whole representation theory of artinalgebras is well understood, see e.g. [Au], [Ga] and [Ta].Auslander proved that Λ is of finite representation type if and only if its representationdimension is at most two. But it turned out to be hard to compute the actual value ofthe representation dimension of a given algebra. Moreover, all the algebras that one couldcompute explicitly their representation dimensions, had representation dimension at mostthree. So a natural question arises: are there artin algebras of representation dimensiongreater than three. Of course such algebras are of infinite representation type. On the otherhand, another question was whether this invariant is always finite. These questions has beenanswered positively, at almost the same time. Iyama [I] in 2003 proved that it is alwaysfinite and Rouquier [Rq1] in 2005 showed that any natural number n, except 1, can occuras the representation dimension of some algebra. In fact, he showed that the representationdimension of the exterior algebra Λk