We give a characterization of radical square zero bound quiver algebras $\mathbf{k} Q/\mathcal{J}^2$ that admit $n$-cluster tilting subcategories and $n\mathbb{Z}$-cluster tilting subcategories in terms of $Q$. We also show that if $Q$ is not of cyclically oriented extended Dynkin type $\tilde{A}$, then the poset of $n$-cluster tilting subcategories of $\mathbf{k} Q/\mathcal{J}^2$ with relation given by inclusion forms a lattice isomorphic to the opposite of the lattice of divisors of an integer which depends on $Q$.
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 …
We give a characterization of $n$-cluster tilting subcategories of representation-directed algebras based on the $n$-Auslander-Reiten translations. As an application we classify acyclic Nakayama algebras with homogeneous relations which admit an $n$-cluster tilting subcategory. Finally, we classify Nakayama algebras of global dimension $d<\infty$ which admit a $d$-cluster tilting subcategory.
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 give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type $A_{ \infty }$.
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…
Let $Λ$ be an artin algebra. In this paper, the notion of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be introduced. It is shown that every $n\mathbb{Z}$-cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is $n\mathbb{Z}$-Gorenstein if and only if $Λ$ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an $n\mathbb{Z}$-Gorenstein cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is an $n\mathbb{Z}$-cluster tilting subcategory of the exact category ${\rm{Gprj}}{\mbox{-}}Λ$, the subcategory of all Gorenstein projective objects of ${\rm{mod}}{\mbox{-}}Λ$. So…
$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-clust…
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.
For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting $A$-modules and the tilting $B$-modules, and we prove that there is a one-one correspondence between the basic $T$-tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$. Moreover, we show that there is a one-one correspondence between the $T$-contravariantly finite $T$-resolving subcategories of $T^{\perp}$ and the basic $T$-t…
Given $n\leq d<\infty$, we investigate the existence of algebras of global dimension $d$ which admit an $n$-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras $A$ and $B$, a projective $A$-module $P$ and an injective $B$-module $I$ satisfying certain conditions, we show how we can construct a new representation-directed algebra $Λ$ in such a way that the representation theory of $Λ$ is completely described by the representation theories of $A$ and $B$. Next we introduce $n$-fractured subcategori…
Let $Λ$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$Λ$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $\mathcal{M}$ if and only if every effaceable functor $\mathcal{M}\rightarrow Ab$ has finite length. As a consequence we show that if mod$Λ$ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of $Λ$.
We present a new way to construct $n$-cluster tilting subcategories of abelian categories. Our method takes as input a direct system of abelian categories $\mathcal{A}_i$ with certain subcategories and, under reasonable conditions, outputs an $n$-cluster tilting subcategory of an admissible target $\mathcal{A}$ of the direct system. We apply this general method to a direct system of module categories $\text{mod}Λ_i$ of representation-directed algebras $Λ_i$ and obtain an $n$-cluster tilting subcategory $\mathcal{M}$ of a module category $\text{mod}\mathcal{C}$ of a locally bounded Krull-Schmid…
In this article, we study the Gorenstein property of abelian quotient categories induced by fully rigid subcategories on an exact category B. We also study when d-cluster tilting subcategories become fully rigid. We show that the quotient abelian category induced by such d-cluster tilting subcategories are hereditary.
A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If $Φ$ is a finite dimensional algebra, then each functorially finite wide subcategory of $\operatorname{mod}( Φ)$ is of the form $ϕ_{ * }\big( \operatorname{mod}( Γ) \big)$ in an essentially unique way, where $Γ$ is a finite dimensional algebra and $Φ\stackrel{ ϕ}{ \longrightarrow } Γ$ is an algebra epimorphism satisfying $\operatorname{Tor}^{ Φ}_1( Γ,Γ) = 0$. Let ${\mathcal F}…
Let $\mathcal B$ be an extriangulated category with enough projectives $\mathcal P$ and enough injectives $\mathcal I$, and let $\mathcal R$ be a contravariantly finite rigid subcategory of $\mathcal B$ which contains $\mathcal P$. We have an abelian quotient category $\mathcal H/\mathcal R\subseteq \mathcal B/\mathcal R$ which is equivalent ${\rm mod}(\mathcal R/\mathcal P)$. In this article, we find a one-to-one correspondence between support $τ$-tilting (resp. $τ$-rigid) subcategories of $\mathcal H/\mathcal R$ and maximal relative rigid (resp. relative rigid) subcategories of $\mathcal H$,…
Let $\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\mathcal{R}$ a rigid subcategory of $\mathcal{C}$. We introduce the notions of two-term $\mathcal{R}[1]$-rigid subcategories, two-term (weak) $\mathcal{R}[1]$-cluster tilting subcategories and two-term maximal $\mathcal{R}[1]$-rigid subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\mathcal{R}[1]$-rigid subcategories of $\mathcal{C}$ and the set of $τ$-rigid subcategories of $\mod\mathcal{R}$, which induces a one-to-one correspondence b…
For an exact category $\mathcal{E}$ with enough projectives and with a $d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $\mathcal{E}$ and $\underline{\mathcal{E}}$. We also deduce that the Gorenstein projectives of $\mathcal{E}$ admit a $d\mathbb{Z}$-cluster tilting subcategory under some assumptions. Finally, we compute the $…
Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is hereditary and has finite length. In this article, we study the relation between support tilting subcategories of B/K and maximal relative rigid subcategories of B. More precisely, we show that the image of any cluster tilting subcategory of B is support tilting in B/K and any support tilting subcategory in B/K can be lifted to a unique relative maximal rigid subcat…
If (A,B) and (A',B') are co-t-structures of a triangulated category, then (A',B') is called intermediate if A \subseteq A' \subseteq ΣA. Our main results show that intermediate co-t-structures are in bijection with two-term silting subcategories, and also with support tau-tilting subcategories under some assumptions. We also show that support tau-tilting subcategories are in bijection with certain finitely generated torsion classes. These generalise results by Adachi, Iyama, and Reiten.
The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent mod…
A notion of $n$-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this article. We show that there exists a one-to-one correspondence between $n$-cotorsion pairs and $(n+1)$-cluster tilting subcategories. As an application, this result generalizes the work by Huerta, Mendoza and Pérez in an abelian case. Finally, we give some examples illustrating our main result.
In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over an algebraically closed field. Firstly, we give a complete list of symmetric algebras with radical cube zero having only finitely many isomorphism classes of two-term tilting complexes in terms of their associated graphs. Secondly, we enumerate the number of two-term tilting complexes for each case in the list.
Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to give new results about the radical series of some tilting modules for $SL_4(K)$, where $K$ is a field of positive characteristic.
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…
We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.
In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcatego…
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.
In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster-tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category B, a functorially finite, extension closed subcategory T of B is cluster til…
We define tilting subcategories in arbitrary exact categories to archieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. Discuss standard results for tilting subcategories: Auslander correspondence, Bazzoni description of the perpendicular category. Secondly: We treat the question of induced derived equivalences separately - given a tilting subcategory T, we ask if a functor on the perpendicular category induces a derived equivalence to a (certain) functor category over T. If this is the case, we call the tilting subcategory ideq tilt…
Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}Λ$, where $Λ$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(Λ)$, the submodule category of $Λ$, 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 quotie…
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…
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and injectives has a cluster tilting subcategory M, then B/M is abelian. More precisely, it is equivalent to the category of finitely presented modules over the stable category of M.
We introduce the concept of a pseudo-cluster tilting subcategory from the viewpoint of the fact that the quotient of an exact category by a cluster tilting subcategory is an abelian category. We prove that the quotients in the case of pseudo-cluster tilting are always semi-abelian. In addition, it is abelian if and only if some self-orthogonal conditions are satisfied. We revisit the abelian quotient category of conflations by splitting ones, and get that there exists a unique exact substructure such that it is a cluster quotient.
From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $Λ$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$Λ$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$Λ$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$Λ$ if and only if $\mathfrak{m}$ has an additive generator…
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…
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 $\langle \mathc…
An algebra is said to be \emph{$τ$-tilting finite} provided it has only a finite number of $τ$-rigid objects up to isomorphism. We associate a category to each such algebra. The objects are the wide subcategories of its category of finite dimensional modules, and the morphisms are indexed by support $τ$-tilting pairs.
Let $B$ be a finite dimensional algebra and $A=B[P_0]$ be the one-point extension algebra of $B$ with respect to the finitely generated projective $B$-module $P_0$. The categories of $B$-modules and $A$-modules are related by two adjoint functors $\mathcal{R}$ and $\mathcal{E}$, called the restriction and the extension functors, respectively. Based on the nice homological properties of these two functors, restriction and extension of some notions such as tilting and $τ$-tilting modules have been studied in the category of finitely presented modules, i.e. small mod. In this paper, we investigat…
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 $\mathcal{T}$…
We generalize the notions of $n$-cluster tilting subcategories and $τ$-selfinjective algebras into $n$-precluster tilting subcategories and $τ_n$-selfinjective algebras, where we show that a subcategory naturally associated to $n$-precluster tilting subcategories has a higher Auslander--Reiten theory. Furthermore, we give a bijection between $n$-precluster tilting subcategories and $n$-minimal Auslander--Gorenstein algebras, which is a higher dimensional analog of Auslander--Solberg correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of $n$-Auslander correspondence (Iyama,…
The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $τ_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\MM_n$ of preinjective-like modules obtained by applying $τ_n$ to injective modules repeatedly. We call a finite dimensional algebra $Λ$ \emph{$n$-complete} if $\MM_n=\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\End_Λ(M)$ is $(n+1)$-comp…
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 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…
