A triangulated category $\mathcal{T}$ whose suspension functor $Σ$ satisfies $Σ^m \simeq \mathrm{Id}_{\mathcal{T}}$ as additive functors is called an $m$-periodic triangulated category. Such a category does not have a tilting object by the periodicity. In this paper, we introduce the notion of an $m$-periodic tilting object in an $m$-periodic triangulated category, which is a periodic analogue of a tilting object in a triangulated category, and prove that an $m$-periodic triangulated category having an $m$-periodic tilting object is triangulated equivalent to the $m$-periodic derived category of an algebra under some homological assumptions. As an application, we construct a triangulated equivalence between the stable category of a self-injective algebra and the $m$-periodic derived category of a hereditary algebra.
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are a generalization of cluster-tilting objects. When $\D$ is $2$-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let $\la={\rm End}^{op}_{\D}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\D$ and support $τ$-tilting $\la$-modules, which gener…
We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of [KR], we prove that any maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via mutations, then their endom…
In the theory of triangulated categories, we propose to replace hearts of $t$-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories. In support of our proposal, we show that proper abelian subcategories with a few vanishing negative self extensions permit a tilting theory which is a direct generalisation of Happel-Reiten-Smalø tilting of hearts.
We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.
Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $τ$-tilting theory: Inspired by the work of Iyama, Jørgensen and Yang \cite{ijy}, we introduce the notion of $τ$-tilting subcategories and tilting subcategories of $\mod\T$. We show that there exists …
In this paper, we study the conjecture II.1.9 of Cluster structures for 2-Calabi-Yau categories and unipotent groups, which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster tilting object in a connected Hom-finite triangulated 2-CY category C. We obtain some conditions equivalent to the conjecture, and using them we proved the conjecture.
We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categ…
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a…
In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we int…
Let T be a locally finite triangulated category with an autoequivalence F such that the orbit category T/F is triangulated. We show that if X is an m-cluster tilting subcategory, then the image of X in T/F is an m-cluster tilting subcategory if and only if X is F-perodic. We show that for path-algebras of Dynking quivers δone may study the periodic properties of n-cluster tilting objects in the n-cluster category Cn(kδ) to obtain information on periodicity of the preimage as n-cluster tilting subcategories of Db(kδ). Finally we classify the periodic properties of all 2-cluster tilting objects …
Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $Γ$ the endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq \mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \ie $R[1]$-rigid objects of $\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\operatorname{\mathsf{pr}}(R)$ are in bijection with $τ$-rigid $Γ$-modules, and the maximal $R[1]$-rigid objects with respect to $\operatorname{\mathsf{pr}}(R)$ are in bijecti…
Motivated by its links to $τ$-tilting theory, we introduce a generalization of cotorsion pairs in module categories. Such pairs are also linked to co-t-structures in corresponding triangulated categories, and to cotorsion pairs in certain extension closed (and hence extriangulated) subcategories, which we call two-term categories.
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
By using the approach in \cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in \cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories.
In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this $d$-cluster category.
For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of $\mathcal{T}$, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that the Gabriel …
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.
Let {\cal T} be a triangulated category, {\cal A} a full subcategory of {\cal T} and {\cal X} a functorially finite subcategory of {\cal A}. If {\cal A} has the properties that any {\cal X}-monomorphism of {\cal A} has a cone and any {\cal X}-epimorphism has a cocone. Then the subfactor category {\cal A/[X]} admits a pretriangulated structure in the sense of [BR]. Moreover the above pretriangulated category {\cal A/[X]} with ({\cal X},{\cal X}[1]) = 0 becomes a triangulated category if and only if ({\cal A},{\cal A}) forms an {\cal X}-mutation pair and {\cal A} is closed under extensions.
We derive an algorithm for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras coming from standard algebraic 2-CY triangulated categories with a finite number of indecomposables. These form a class of algebras that satisfy the setup for our mutation algorithm.
We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this bijection restricts to one between equivalence classes of self-small partially silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products. We descri…
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell…
For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/C is abelia…
Let $k$ be a field and $\mathcal{C}$ a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of $\mathcal{C}$, denoted $K_0(\mathcal{C})$, can be expressed as a quotient of the split Grothendieck group of a higher-cluster tilting subcategory of $\mathcal{C}$. Assume that $n\geq 2$ is an even integer, $\mathcal{C}$ is $n$-Calabi Yau and has an $n$-cluster tilting subcategory $\mathcal{T}$. Then, for every indecomposable $M$ in $\mathcal{T}$, there is an Auslander-Reiten $(n+2)$-angle in $\mathcal{T}$…
This short note surveys the constructions of 3-Calabi--Yau triangulated categories with simple-minded collections due to Ginzburg and Kontsevich--Soibelman and the constructions of 2-Calabi--Yau triangulated categories with cluster-tilting objects due to Buan--Marsh--Reineke--Reiten--Todorov and Amiot, and includes a discussion on the normal form of 2-Calabi--Yau triangulated categories with cluster-tilting objects.
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.
For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.
Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in Kb(proj Λ), nu-stable support tau-tilting Λ-modules and nu-stable functorially finite torsion classes in mod Λ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in C if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category C. As an application, we give a necessary condition such that selfinjective algebras are 2-CY ti…
A finite directed category is a $k$-linear category with finitely many objects and an underlying poset structure, where $k$ is an algebraically closed field. This concept unifies structures such as $k$-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.
Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (or cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (cotilting) subcategories and coresolving covariantly (resolving contravariantly, resp.) finite subcatgories which are closed under direct summands and satisfies some cogenerating (generating, resp.) conditons. App…
The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly 2 cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exa…
We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. Under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. We show that the quotient functor preserves the AR-structure. As an application we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions J is a cluster-tilting subca…
We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\mathcal{A}_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\mathcal{A}_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to c…
Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. In this article, we introduce and study the notion of $ξ$-tilting object in an extriangulated category, where $ξ$ is a proper class of $\mathbb{E}$-triangles. This extends results in \cite{YGH_2020}.
In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We study this systematically. In particular we show that the degeneration of the zero object actually induces all other degenerations by homotopy pullback, that degeneration of $0$ is closely linked, but not equivalent, to having zero image in the Grothendieck group.
If $k$ is a field, $A$ a finite dimensional $k$-algebra, then the simple $A$-modules form a simple minded collection in the derived category $\operatorname{D}^b( \operatorname{mod} A )$. Their extension closure is $\operatorname{mod} A$; in particular, it is abelian. This situation is emulated by a general simple minded collection $\mathcal{S}$ in a suitable triangulated category $\mathcal{C}$. In particular, the extension closure $\langle \mathcal{S} \rangle$ is abelian, and there is a tilting theory for such abelian subcategories of $\mathcal{C}$. These statements follow from $\langle \mathc…
Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U, and that, in turn, support tilting subcategories of D/U can be lifted uniquely to maximal 1-orthogonal subcategories of D.
We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara's characterisation of degeneration in module varieties.
Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions.
We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.
A notion of mutation of subcategories in a right triangulated category is defined in this paper. When (Z,Z) is a D-mutation pair in a right triangulated category C, the quotient category Z/D carries naturally a right triangulated structure. More-over, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category Z/D becomes a triangulated category. When C is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen respectively.
We develop the basic properties of $w$-simple-minded systems in $(-w)$-Calabi-Yau triangulated categories for $w \geq 1$. The main result is a reduction technique for negative Calabi-Yau triangulated categories. We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. Our construction provides an inductive technique for constructing simple-minded systems.
This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster structure, and we classify these subcategories in terms of what one may call `triangulations of the infinity-gon'. This is reminiscent of the cluster category C of type A_n which is a 2-Calabi-Yau triangulated category whose Auslander-Reiten quiver is a quotient of ZA_n. The cluster tilting subcategories of C form a cluster structure and they are classified in terms of triangulations of the (n+3)-gon. T…
Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories have a natural extriangulated structure and are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category.
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 $\mathbb{X}$.
Let $\mathscr T$ be a $2$-Calabi--Yau triangulated category, $T$ a cluster tilting object with endomorphism algebra $Γ$. Consider the functor $\mathscr T( T,- ) : \mathscr T \rightarrow \mod Γ$. It induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support $τ$-tilting pairs. This is due to Adachi, Iyama, and Reiten. The notion of $( d+2 )$-angulated categories is a higher analogue of triangulated categories. We show a higher analogue of the above result, based on the notion of maximal $τ_d$-rigid pairs.
We give a criterion for cohomological symmetry in a triangulated category. As an application, we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra.
In this article, we define relative resolutions and coresolutions in extriangulated categories. By studying this relative resolutions and coresolutions, we get a generalization of the Auslander-Buchweitz approximation theory. Finally, we develop some theories of relative tilting objects in extriangulated categories.
We present a simplified and more intuitive proof of a theorem of Peng and Xiao, which constructs a Lie algebra from any 2-periodic triangulated k-category (satisfying some finiteness assumptions).
We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.
Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories
Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）；Department of Mathematical Sciences, University of the Ryukyus（冲绳大学数学科学系）；Faculty of Engineering Science, Kansai University（关西大学工程科学部）
This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.
We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate $t$-structure whose heart has enough projectives.
Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this article. Let $\cal A$ be an extension closed subcategory of an extriangulated category $\cal C$. Then the quotient category $\cal M:=\cal{A}/\cal{X}$ carries naturally a triangulated structure whenever $(\cal A,\cal A)$ forms an $\cal X$-mutation pair. This result unifies many previous constructions of triangulated quotient categories, and using it gives a classif…
