We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.
Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel's characterisation of idempotent ideals occuring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
In this paper, we study (complete) cotorsion pairs in extriangulated categories. First, we study a relationship between an interval of the poset of cotorsion pairs and the poset of cotorsion pairs in the heart associated to the interval. Secondly, we establish a bijection between bounded hereditary cotorsion pairs and silting subcategories in extriangulated categories.
We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobiński & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composi…
We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.
We give necessary and sufficient conditions for torsion pairs in a hereditary category to be in bijection with $t$-structures in the bounded derived category of that hereditary category. We prove that the existence of a split $t$-structure with nontrivial heart in a semiconnected Krull-Schmidt category implies that this category is equivalent to the derived category of a hereditary category. We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding $t$-structures. Finally, we classify…
Quasi-hereditary were introduced by L. Scott \cite{Scott, CPS1,CPS2} in order to deal highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups, and they have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander [A], [AQM] and used in his proof of the first Brauer-Thrall conjecture [A2] and later on used systematically in his joint work with I. Reiten on stable equivalence [AR], [AR2] and many other applications. R…
In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
A Jordan Hölder theorem is established for derived module categories of piecewise hereditary algebras. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.
We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the triangulated category is algebraical, we may replace the equivalence by a triangle equivalence. We give two intrinsic characterizations of hereditary triangulated categories using a certain full subcategory and the non-existence of certain paths. We apply them to piecewise hereditary algebras.
In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification, we will study the shapes of the Auslander-Reiten components extensively and use appropriate generalisations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.
An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.
We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver.
In this paper we give an explicit algorithm to construct the ordinary quiver of a finite EI category for which the endomorphism groups of all objects have orders invertible in the field k. We classify all finite EI categories with hereditary category algebras, characterizing them as free EI categories (in a sense which we define) for which all endomorphism groups of objects have invertible orders. Some applications on the representation types of finite EI categories are derived.
We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.
Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for non-hereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In this paper we study the image if the natural functor from the bounded derived category to the cluster category, that is we investigate how far the orbit category is from being the cluster category. We show that for wide classes of non-piecewise hereditary algebras the orbit category is never equal to the clus…
In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include homotopy categories of 2-term complexes with projective components, which are related to silting mutation, and cluster categories (with relevant relative extriangulated structures) where cluster tilting mutation take place. We prove that there is a theory of irreducible mutation for maximal rigid objects and subcategories in hereditary extriangulated categor…
Model structure arising from one hereditary complete cotorsion pair on extriangulated categories
School of Mathematics, Hangzhou Normal University（杭州师范大学数学学院）；School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学科学学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
Koenig, Külshammer and Ovsienko showed that Morita equivalence classes of quasi-hereditary algebras are in one-to-one correspondence with equivalence classes of the module categories over directed bocses. In this article, we extend this result to $Δ$-filtered algebras and $\overlineΔ$-filtered algebras.
Following the work of Beilinson, Bernstein and Deligne, we study restriction and induction of t-structures in triangulated categories with respect to recollements. For derived categories of piecewise hereditary algebras we give a necessary and sufficient condition for a bounded t-structure to be induced from a recollement by derived categories of algebras. As a corollary we prove that for hereditary algebras of finite representation type all bounded t-structures can be obtained in this way.
In this paper, we prove that the lower triangular matrix category $Λ=\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary $\mathrm{Hom}$-finite Krull-Schmidt $K$-categories and $M$ is a $\mathcal U\otimes_K \mathcal T^{op}$-module that satisfies suitable conditions, is quasi-hereditary in the sense of \cite{LGOS1} and \cite{Martin}. Moreover, we solve the problem of finding quotients of path categories isomorphic to the lower triangular matrix category $Λ$, where $\mathcal T=K\mathcal{R/J}$ and $\mathc…
Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category $\mathcal O$. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category $\mathcal{F}(Δ)$ of modules with standard (Verma, Weyl, \dots) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg alg…
Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.
The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a $K(π,1)$ for the associated picture group, thereby allowing for the computation of the homology of picture groups of finite type as carried out in [7] for the case of $A_n$. In this paper we show that the cluster morphism category is a $CAT(0)$-category for hereditary algebras of finite or tame type with only small tubes. As a consequence, we get that the classifying space of the cluster morphism category is a locally $CAT(0)$ s…
Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposa…
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a hereditary abelian category $\mathcal{H}$, we verify that the tilting functor Hom$_\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\mathcal{C(H)}$ to the cluster category $\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\mathc…
We study IE-closed subcategories of a module category, subcategories which are closed under taking Images and Extensions. We investigate the relation between IE-closed subcategories and torsion pairs, and characterize $τ$-tilting finite algebras using IE-closed subcategories. For the hereditary case, we show that IE-closed subcategories can be classified by twin rigid modules, pairs of rigid modules satisfying some homological conditions. Moreover, we introduce mutation of twin rigid modules analogously to tilting modules, which gives a way to calculate all twin rigid modules for the represent…
We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum $T\oplus\cdots\oplus T[-(d-2)]$ has hereditary endomorphism algebra $H$ is triangle equivalent to the orbit category $\mathscr{D}^b(\mathrm{\mathop{mod}}\, H)/τ^{-1/(d-1)}[1]$ of the derived category of $H$ for a naturally defined $(d-1)$-st root $τ^{1/(d-1)}$ of the AR translation, provided $H$ is of non-Dynkin type. We also show that hereditaryness of $H$ follo…
We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects, and classify its projective modules in this case. As applications, we completely classify transporter categories, orbit categories, and Quillen categories with left hereditary category algebras over a field.
We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $Δ$. If a quasi-hereditary algebra $Ł$ is Koszul with respect to $Δ$, then $Ł$ and the Yoneda extension algebra of $Δ$ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of $Δ$ is Koszul in the classical sense.
In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal U}\oplus R_{\mathcal U}/R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ is a recollement of the derived module category $\D{R}$ of $R$ and the derived module category $\D{{\mathbb A}_{\mathcal{U}}}$ of the adèle ring ${\mathbb A}_{\mathcal{U…
In this article, we generalize the concept of torsion pairs and study its structure. As a trial of obtaining all torsion pairs, we decompose torsion pairs by projective modules and injective modules. Then we calculate torsion pairs on the algebra KAn and tub categories. At last we try to find all torsion pairs on the module categories of finite dimensional hereditary algebras.
In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively.
Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting modules, we introduce (m,n)-almost hereditary algebras. These are the algebras with global dimension m+n and such that any indecomposable module has projective dimension at most m, or else injective dimension at most n. We relate these two classes of algebras, among which (m,1)-almost hereditary ones play a special role. For these, we prove that any indecomposabl…
Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）；Department of Pure Mathematics Faculty of Mathematics and Statistics University of Isfahan（伊斯法罕大学数学与统计学院纯数学系）；Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）
Let R be a ring and let T be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by T into the derived category of R has a right adjoint, which is a colocalization. Benson has recently shown how to compute this right sdjoint when R is the group ring of a finite group over a prime field and T is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where T is any hereditary torsion class on R. This yields an explicit formula for the colocalization with respect to T, using an injective cogenerator for…
We consider the relationship between the relative stable category of Benson, Iyengar, and Krause and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism…
In this paper, we consider the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits silting objects. Secondly, for $A=KQ/I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mod^{\mathbb Z} A)$ has tilting objects. As a consequence, $D_{sg}(\mod^{\mathbb Z}A)$ is triangulated equivalent to the derived category of a hereditary algebra $H$ which is of finite repre…
Let $R$ be a right notherian ring. We introduce the concept of relative singularity category $Δ_{\mathcal{X}}(R)$ of $R$ with respect to a contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along with some finiteness conditions on $\mathcal{X}$, we prove that $Δ_{\mathcal{X}}(R)$ is triangle equivalent to a subcategory of the homotopy category $\mathbb{K}_{\rm{ac}}(\mathcal{X})$ of exact complexes over $\mathcal{X}$. As an application, a new description of the classical singularity category $\mathbb{D}_{\rm{sg}}(R)$ is given. The relative singularity categories are applie…
We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of $\mathop{\rm Ext}\nolimits$-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded $t$-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension a…
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the bounded derived category $\mathcal{D}^{b}(H)$, and of the cluster-tilting objects, in terms of the category $\modΓ$\ of finitely generated modules over a suitable tilted algebra $% Γ.$ Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.
For a given stability condition $σ$ on a triangulated category we define a $σ$-exceptional collection as an Ext-exceptional collection, whose elements are $σ$-semistable with phases contained in an open interval of length one. If there exists a full $σ$-exceptional collection, then $σ$ is generated by this collection in a procedure described by E. Macrì. Constructing $σ$-exceptional collections of length at least three in $D^b(\mathcal A)$ from a non-semistable exceptional object, where $\mathcal A$ is a hereditary hom-finite abelian category, we introduce certain conditions on the Ext-nontriv…
In this paper we focus on the relations between the derived categories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We consider derived discrete Koszul algebras, and we give necessary and sufficient conditions for these Koszul algebras to be derived equivalent to their Yoneda algebras. Finally, we look at the Koszul algebras such that they are derived equivalent to a hereditary algebra. In the case tha…
This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.
Given a torsion pair $\mathbf{t} = (\mathcal{T} ;\mathcal{F})$ in a module category $R$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smalø$\text{ }$ t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\mathbf{t}}$ which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino-Kato-Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs $\mathbf{t}$ for which $\mathcal{H}_{\mathbf{t}}$ is a module category in the following cases: i) …
We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations of quivers with potentials (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algeb…
We consider hereditary Artin algebras over arbitrary fields and prove that there is a natural bijection between the Weyl groups and the sets of full additive cofinite submodule closed subcategories of the module categories. While Oppermann, Reiten and Thomas have shown this for algebraically closed fields and finite fields, we give a different method of proof that holds independently of the field. In particular, we show a relatively simple way to construct all modules that contain a given preinjective module as a submodule.
