For a self-orthogonal module $T$, the relation between the quotient triangulated category $D^b(A)/K^b({\rm add} T)$ and the stable category of the Frobenius category of $T$-Cohen-Macaulay modules is investigated. In particular, for a Gorenstein algebra, we get a relative version of the description of the singularity category due to Happel. Also, the derived category of a Gorenstein algebra is explicitly given, inside the stable category of the graded module category of the corresponding trivial extension algebra, via Happel's functor $F: D^b(A) \longrightarrow T(A)^{\mathbb{Z}}{-}\underline{\rm mod}$.
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 define mutation pair in a pseudo-triangulated category. We prove that under certain conditions, for a mutation pair in a pseudo-triangulated category, the corresponding quotient category carries a natural triangulated structure. This result unifies many previous constructions of quotient triangulated categories.
We give a method to realize Verdier quotients as triangulated subfactors of an arbitrary triangulated category. We show that Iyama-Yoshino triangulated subfactors are Verdier quotients under suitable conditions.
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
We give a sufficient condition for a Verdier quotient $\ct/\cs$ of a triangulated category $\ct$ by a thick subcategory $\cs$ to be realized inside of $\ct$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot--Guo--Keller.
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…
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 introduce the notion of a prile of one-sided triangulated categories. Roughly speaking, a prile consists of two one-sided triangulated categories having a common full subcategory which inherits a pretriangulated structure from these ambient categories. The main example arises from exact model categories. This allows us to recover the pretriangulated structure of the homotoy category of an exact model category.
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…
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.
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).
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
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.
In this paper, as an analogue of the spectrum of a tensor triangulated category introduced by Balmer, we define a spectrum of a triangulated category which does not necessarily admit a tensor structure. We apply it for some triangulated categories associated to a commutative noetherian ring.
The basic properties of locally finite triangulated categories are discussed. The focus is on Auslander--Reiten theory and the lattice of thick subcategories.
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.
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.
We give a characterisation of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object $X$ in the extriangulated category, the morphism $0 \rightarrow X$ is a deflation and the morphism $X \rightarrow 0$ is an inflation.
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.
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.
The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial $t$-structures, and in particular, full abelian subcategories need not arise as hearts of a $t$-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.
In this article, we investigate the category $\mathcal{A}^G$ of equivariant objects of an additive category $\mathcal{A}$ with respect to an action of a finite group $G$. We show that if $G$ is solvable then we can reconstruct $\mathcal{A}$ from $\mathcal{A}^G$ via a finite sequence of equivariantization. We also consider the possibility of a triangulated structure on $\mathcal{A}^G$ canonical in certain sense when $\mathcal{A}$ is triangulated and give several instances in which $\mathcal{A}^G$ is indeed canonically triangulated.
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 additivity of traces in certain tensor triangulated categories for endomorphisms of finite order of distinguished triangles is investigated. For the identity endomorphism this has been fully established by J. P. May ("The additivity of traces in triangulated categories", Adv. Math., 2001, 163, 34-73). By imposing extra conditions on the coefficients we show how May's result implies a stronger additivity.
We define the Hall algebra associated to any triangulated category under some finiteness conditions with the $t$-periodic translation functor $T$ for odd $t>1.$ This generalizes the results in \cite{Toen2005} and \cite{XX2006}.
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…
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.
The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent to an ideal quotient of a subcategory of C. On the other hand, Aslak Buan and Robert Marsh proved that this module category is also equivalent to some localisation of C. In this paper, we give a conceptual interpretation, inspired from homotopical algebra, of this double description. Our main aim, yet to be acheived, is to generalise Buan-Marsh's result to the case o…
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…
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.
This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.
We give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism algebras as the Auslander algebras. For the second one, we introduce the notion of $[1]$-additive generators and consider their graded endormorphism algebras as the Auslander algebras. We give a homological characterization of the Auslander algebras for each class. Along the way, we also show that the algebraic triangle structures on the homotopy categories ar…
The concept of a morphism determined by an object provides a method to construct or classify morphisms in a fixed category. We show that this works particularly well for triangulated categories having Serre duality. Another application of this concept arises from a reformulation of Freyd's generating hypothesis.
We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.
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.
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 show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These inclu…
We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.
Let C be a finite EI category and k be a field. We consider the category algebra kC. Suppose K(C)=D^b(kC-mod) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category and we compute its spectrum in the sense of Balmer. When C=G*P is a finite transporter category, the category algebra becomes Gorenstein so we can define the stable module category \CM k(G*P), of maximal Cohen-Macaulay modules, as a quotient category of K(G*P). Since \CM k(G*P) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor …
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 show that the homotopy category of injective $A$-modules is generically trivial if and only if the derived category of all modules is generically trivial for an algebra $A$. Moreover we show some connections between the generic objects, locally finiteness and Krull-Gabriel dimension.
We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions.
We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2-Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.
In this article, we investigate the condition for the hearts of twin cotorsion pairs to be equivalent, compatibly with the associated functors. This is related to the vanishing of components of pairs through the associated functors.
We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.
We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated $k$-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of self-injective Nakayama algebras, determining this way the self-injective Nakayama algebras admitting indecomposable CY modules. In particular, this result recovers the algebras whose stable categories are Calabi-Yau, which have been obtained in [BS].
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.
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…
