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Let R be a right noetherian ring. We introduce the concept of relative singularity category $$\Delta _{\mathcal {X} }(R)$$ of R with respect to a contravariantly finite subcategory $$\mathcal {X} $$ of $${\text {{mod{-}}}}R.$$ Along with some finiteness conditions on $$\mathcal {X} $$ , we prove that $$\Delta _{\mathcal {X} }(R)$$ is triangle equivalent to a subcategory of the homotopy category $$\mathbb {K} _\mathrm{{ac}}(\mathcal {X} )$$ of exact complexes over $$\mathcal {X} $$ . As an application, a new description of the classical singularity category $$\mathbb {D} _\mathrm{{sg}}(R)$$ is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.

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:...

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$., Comment: Accepted for publication in Forum Mathematicum

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.

Using a relative version of Auslander’s formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander (1971), to categorical resolutions.

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

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 C be a G-category for some group G. We show that a G-covering functor p : C ⟶ C / G induces G-precovering of their bounded derived categories, singularity categories and Gorenstein defect categories. Then we provide a Gorenstein version of Gabriel's theorem. Using this result we investigate the number of summands in a decomposition of the middle term of almost split sequences over monomial algebras.

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory $\mathcal{S}_{\mathcal{X}}(\Lambda)$ of the morphism category $\rm{H}(\Lambda)$ consisting of all monomorphisms $(A\stackrel{f}\rightarrow B)$ with $A, B$ and $\rm{Cok} \ f$ in $\mathcal{\mathcal{X}}$. Since $\mathcal{S}_{\mathcal{X}}(\Lambda)$ is closed under extensions then it inherits naturally an exact structure from $\rm{H}(\Lambda)$. We will define two other different exact structures else than the canonical one on $\mathcal{S}_{\mathcal{X}}(\Lambda)$, and the indecomposable projective (resp. injective) objects in the corresponding exact categories completely classified. Enhancing $\mathcal{S}_{\mathcal{X}}(\Lambda)$ with the new exact structure provides a framework to construct a triangle functor. Let $\rm{mod}\mbox{-}\underline{\mathcal{X}}$ denote the category of finitely presented functors over the stable category $\underline{\mathcal{X}}$. We then use the triangle functor to show a triangle equivalence between the bounded derived category $\mathbb{D}^{\rm{b}}(\rm{mod}\mbox{-}\underline{\mathcal{X}})$ and a Verdier quotient of the bounded derived category of the associated exact category on $\mathcal{S}_{\mathcal{X}}(\Lambda)$. Similar consideration is also given for the singularity category of $\rm{mod}\mbox{-}\underline{\mathcal{X}}$., Comment: 37 pages, any comments are welcome. arXiv admin note: text overlap with arXiv:1802.03683 by other authors

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, we show that the homotopy category of N-complexes of projective R-modules is triangle equivalent to the homotopy category of projective T_{N-1}(R)- modules where T_{N-1}(R) is the ring of triangular matrices of order N-1 with entries in R. We also define the notions of N-singularity category and N-totally acyclic complexes. We show that the category of N-totally acyclic complexes of finitely generated projective R-modules embeds in the N-singularity category, which is a result analogous to the case of ordinary chain complexes., 20

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.

We study total acyclicity for complexes of projective and injective representations of quivers. A classification is given for such complexes in terms of associated vertex-complexes. When the base ring or the quiver is nice enough, this classification is used to prove the existence of Gorenstein projective precovers in the category of representations of quivers. Furthermore, we exploit this local description to obtain some criteria for the category of representations of a quiver to be Gorenstein or virtually Gorenstein. If Λ is an artin algebra it is proved that, for an arbitrary quiver 𝒬, the representation category Rep(𝒬, Λ) is virtually Gorenstein whenever Λ is virtually Gorenstein. A description of Gorenstein projective and Gorenstein injective representations of quivers over general rings is also provided.

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.

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.

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.

We study totally acyclic complexes of projective modules over triangular matrix rings and then use it to classify Gorenstein projective modules over such rings. We also use this classification to obtain some information concerning Cohen-Macaulay finite and virtually Gorenstein triangular matrix artin algebras., This paper is now accepted for publication in the Algebra Colloquium

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

This note is devoted to the study of cotorsion theory in the category of quiver representations. Let [Formula: see text] be a quiver, ξ be a class of representations, and let Vξ denote the class of all modules each of which appears in a vertex of an element of ξ. If the pair (⊥Vξ, Vξ) (respectively, (Vξ, Vξ⊥)) is a cotorsion theory, we investigate conditions under which the pair (⊥ξ, ξ) (respectively, (ξ, ξ⊥)) is a cotorsion theory and vice versa. We show that in certain cases, completeness can be transferred between these cotorsion theories.