Sondre Kvamme
Building on work of Jasso, we prove that any projectively generated $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category with enough projectives. This supports the claim that $d$-abelian categories are good axiomatizations of $d$-cluster tilting subcategories.
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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.
From triangulated categories to abelian categories--cluster tilting in a general framework
我们把聚类倾斜放在一个一般的框架中，证明了一个三角范畴模一个倾斜子范畴（即一个极大的一正交子范畴）的任何商都带有一个阿贝尔结构。这些交换代数是维数至多为1的Goreisten代数的模范畴。
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…
Tilting theory and functor categories I. Classical tilting
倾斜理论在有限和驯服表示类型的有限维代数的分类中，以及在许多其他数学分支中，一直是一个非常重要的工具。Happel [Ha]证明了广义倾斜诱导了模范畴之间的导出等价，Rickard [Ri]使用倾斜复形发展了导出范畴的一般森田理论。另一方面，M. Auslander和使用在他的证明第一布劳尔-萨尔猜想，后来，系统地使用在他的联合工作与I。重申稳定等价和许多其他应用。最近，函子范畴被用来研究有限维代数的Auslander-Reiten分支.本文的目的是将倾斜理论推广到任意函子范畴，并考虑到函子范畴Mod（modΛ）的应用，其中Λ是一个有限维代数。
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…
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 $…
Quotients of exact categories by cluster tilting subcategories as module categories
证明了正合范畴的某些子商范畴是交换的。这推广了Koenig-Zhu在（代数）三角范畴中的一个结果。作为一种特殊情况，如果一个具有足够投射和内射的正合范畴B有一个簇倾斜子范畴M，则B/M是阿贝尔范畴。更确切地说，它等价于M的稳定范畴上的可表示模范畴。
Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $τ$-rigid subcategory is contained in a support $τ$-tilting subcategory. We also show that support $τ$-tilting subcategories are in bijection with certain finitely generated torsion classes. Some applications of our main results are also given.
It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of $t$-structures. Firstly, we extend this picture into a hierarchy of $n$-quasi-abelian categories and $n$-tilting torsion classes. We prove that any $n$-quasi-abelian category admits a derived category endowed with a $n$-tilting pair of $t$-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these $t$-structures as quotient categories of coherent functors, genera…
As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quiver…
In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic $m$-cluster tilting objects. Under these bijections, flips of $(m+2)$-angulations correspond to mutations of $m$-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the $m$-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows…
Wakamatsu-tilting subcategories in extriangulated categories
本文在外三角范畴中定义Wakamatsu-tilting与cotilting子范畴，证明其与无穷版本重合，并研究特殊recollement下的粘合及其逆条件。
A bijection between support $τ$-tilting subcategories and $τ$-cotorsion pairs in extriangulated categories
该研究在外三角范畴框架下给出倾斜子范畴的新定义并验证其与已有定义等价，引入支撑$\tau$-倾斜子范畴与$\tau$-余挠对概念，建立二者的双射关系且诱导出倾斜子范畴与余挠对的双射。
Extended Module Categories in Higher Cluster Tilting Theory
本文通过研究三角范畴关于高丛倾斜子范畴的理想商，证明其自然DG增强是阿贝尔d-截断DG范畴，并利用Morita型定理将其等价于d-截断DG范畴上的d-扩展模范畴。
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…
Semi-abelian categories arising from pseudo cluster tilting subcategories
本文定义了外三角范畴中的伪丛倾斜子范畴，证明其商范畴构成半阿贝尔范畴，并在满足自正交条件时成为阿贝尔范畴，推广了正合范畴中的相关研究。
Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described in terms of Wakamatsu tilting (=semi-dualizing) subcategories. If moreover the exact category has higher kernels, then its image coincides with the category naturally associated with a cotilting subcategory up to summands. We apply these results to the representation theory of artin algebras. In particular, we show that the ideal quotient of a mo…
We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss…
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.
Tilting pairs in extriangulated categories
本文在外三角范畴中引入倾斜对概念，给出其Bazzoni刻画及Auslander-Reiten对应，推广了Wei和Xi对Artin代数模范畴的已知结果。
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.
Relative cluster tilting theory and $τ$-tilting theory
研究 Krull-Schmidt 三角范畴中两项弱相对丛倾斜子范畴的变异，证明几乎完全子范畴恰有两个完备化，并应用于函子范畴与 Abel 范畴的 $τ$-倾斜理论。
In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and tau-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with $τ$-rigid pairs, and also relative maximal rigid objects with support tau-tilting pairs under some assumptions. These results generalize their work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. Finally, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable…
Tilting theory for extended module categories
本文研究扩张模范畴的倾斜理论，建立 Happel-Reiten-Smalo 定理、s-挠对刻画及 Auslander-Reiten 理论，并证明 τ_[m]-倾斜对、silting 复形与函子有限 s-挠对间的双射。
Iyama-Solberg correspondence for exact dg categories
本文推广了簇倾斜对和 Auslander 正合 dg 范畴的概念，建立了预簇倾斜三元组与极小 Auslander-Gorenstein 正合 dg 范畴之间的双射，推广了模范畴的 Iyama-Solberg 对应。
Extriangulated length categories: torsion classes and $τ$-tilting theory
本文定义外三角长度范畴，证明其等价于容许 simple-minded system，并系统研究挠类格的结构、Hasse 箭图及支撑挠类与支撑 $τ$-倾斜子范畴间的双射。
For each positive integer $n$ we introduce the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which $n$-exangulated categories are $n$-exact in the sense of Jasso and which are $(n+2)$-angulated in the sense of Geiss-Keller-Oppermann. For extriangulated categories with enough projectives and injectives we introduce the notion of $n$-cluster tilting subcategories and show that under certain conditions such $n$-cluster tilting subcategories are $n$-exangulated.
Classification of abelian hereditary directed categories satisfying Serre duality
在一个正在进行的项目，以分类所有的遗传阿贝尔范畴，我们提供了一个分类的外有限有向遗传阿贝尔范畴满足Serre对偶导出等价。为了证明分类，我们将广泛地研究Auslander-Reiten分量的形状，并使用倾斜对象和坐标的适当概括，即部分倾斜集和准简单对象的探测。
Relative tilting, interlacing categories and rational Dyck posets
该研究构造了有理Dyck楼梯关联代数与A型高阶Auslander代数幂等拐角的显式倾斜等价，证明了互素正整数的Chapoton-Ladkani-Rognerud猜想，拓展了区间倾斜机制并实现了楼梯导出范畴的几何实现。
Higher extension closure and $d$-exact categories
本文证明任何弱幂等完备的$d$-正合范畴等价于一个弱幂等完备正合范畴的$d$-簇倾斜子范畴，并推广到代数$(d+2)$-角范畴，同时证明$d$-簇倾斜子范畴的环绕正合范畴在弱幂等完备假设下唯一，进而得到$d$-挠类和$d$-宽子范畴的$d$-簇倾斜性质。
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (noncompact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Ric…
Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs
本文通过引入相对（余）分解及 $\mathcal{X}$-完备对与 $\mathcal{X}$-遗传对，推广了 Auslander-Buchweitz 逼近理论，为发展 $n$-$\mathcal{X}$-倾斜理论提供了相对逼近存在性定理及同调维数等关键结果。
Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of abelian categories for artin algebras $Λ'$, $Λ$ and $Λ''$. Under certain conditions, we present an explicit construction of gluing of (support) $τ$-tilting modules in $\mbox{mod} Λ$ with respect to (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively; conversely, we study the construction of (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ obtained from (support) $τ$-tilting modules in $\mbox{mod} Λ$.
Silting mutation in triangulated categories
在代数的表示论中，“突变”的概念常常起着重要的作用，其中有两种情况是众所周知的，即“簇倾斜突变”和“例外突变”。本文主要研究“倾斜突变”，它的缺点是不能通过替换一个倾斜对象的某些被加数来得到一个新的倾斜对象。本文的目的是通过引入淤积对象的“淤积突变”作为“倾斜突变”的推广来消除这一缺点。我们将发展淤积突变的基本理论。特别地，我们通过推广Riedtmann-Schofield和Happel-Unger的理论，在淤积对象集合上引入了偏序，并建立了与“淤积突变”的关系。我们表明，迭代淤积突变作用于局部，遗传或典型代数淤积对象的集合上的传递。最后，我们给出了一个双射之间的淤积子类别和某些t-结构。
Rigid objects, triangulated subfactors and abelian localizations
本文研究了由刚性反变有限子范畴诱导的三角范畴的阿贝尔局部化，以及三角范畴的子因子范畴上的三角结构。在这方面，我们推广Buan-Marsh和Iyama-Yoshino最近的结果。我们还推广了Keller-Reiten关于Gorenstein和Calabi-Yau性质的基本结果，这些结果适用于由某些刚性（不一定是簇倾斜）子范畴产生的范畴，以及文献中关于三角范畴的2-簇倾斜子范畴与相干函子的相关阿贝尔范畴的倾斜子范畴之间的联系的一些结果。最后，我们沿着这些线刻画2-集群倾斜子范畴。
Let C be triangulated category and X a cluster tilting subcategory of C. Koenig and Zhu showed that the quotient category C/X is Gorenstein of Gorenstein dimension at most one. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let C be extriangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C. In this article, we show that under certain conditions the quotient category C/X is Gorenstein of Gorenstein dimension at most one. As an a…
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides with the class of relatively flat objects of $\mathcal{C}$ for some relative projectively generated exact structure, while $\mathcal{B}$ coincides with the class of relatively divisible objects of $\mathcal{C}$ for some relative injectively generated exact structure. We exhibit Galois connections between relative cotorsion pairs, relative projectively generat…
Large silting mutation in extriangulated categories
本文在具有集合指标(余)积的扩张三角范畴中发展了无限维倾斜突变理论，并应用于任意环上的n-余倾斜复形和有限整体维数环上的无限维n-(余)倾斜模的突变理论。
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.
In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this categories and show that every $n\mathbb{Z}$-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an $n\mathbb{Z}$-abelian (resp., $n\mathbb{Z}$-exact) category. Also we show that every small $n\mathbb{Z}$-abelian category arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.
We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable categ…
Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules
本文在幂零锥及抛物 Springer 解消的凝聚层导出范畴上构造 co-$t$-结构，用以刻画外尔奇偶对象推进的不可分解性，并证明 $A$ 型、$p>h$ 时相对 Humphreys 猜想的概形论形式。
Intervals of torsion pairs and generalized Happel-Reiten-Smalø tilting
本文研究了广义HRS折叠，通过torsion对的区间与心的双射关联，推广了HRS折叠方法，并展示了$t$-结构的扩展应用。
Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories
本文引入阿贝尔n-截断范畴作为高维构造的统一框架，将余挠对的心构造推广到任意三角范畴上的n-余挠对，证明所得扩张心兼具兼容的预三角化与外三角化结构，且可实现为外三角化局部化。
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…
We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories $\mathcal{T}$ of $\mathcal{A}$ closed under extensions and direct summands are precisely those for which $(_{}^{\perp_1}\mathcal{T},\mathcal{T})$ is a right complete cotorsion pair, where $_{}^{\perp_1}\mathca…
Recollements from partial tilting complexes
研究了由自正交紧对象诱导的dg-代数的导出范畴的元，得到了Rickard定理的推广。特别是环上部分倾斜模的情形，我们推广了[B2]和[BMT]中关于三角等价的结果。最后，我们着重讨论了环的导出范畴、双反射子范畴和广义泛局部化之间的关系。
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…
The completion of $d$-abelian categories
本文研究$d$-阿贝尔范畴的完备化Ind$(\u27e8M\u27e9)$的性质，证明其等价于左$d$-正合函子构成的满子范畴，探讨其成为$d$-丛倾斜子范畴的条件，并在$d=2$时建立与有限型的等价关系，关联Iyama的有限性问题。
The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category $\C$ has Auslander-Reiten $\E-$triangles and the relations of the Grothendieck group $K_{0}(\C)$ are generated by the Auslander-Rieten $\E-$triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category $\C$ with a cluster tilting subcategory, the relations of the Grothendieck group $K_0(\C)$ are gener…
Higher exact dg-categories
本文引入n-精确微分格范畴概念，证明其在特定条件下具有n-exangulated结构，揭示了n-精确微分格范畴对n-exangulated类的dg增强作用。
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…
Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this result generalizes work by Beligiannis.
Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S$ the endomorphism ring. We prove that $T$ induces an equivalence between stable categories of repetitive algebras of $R$ and $S$.
Let $\mathcal C$ be a concrete category. We prove that if $\mathcal{C}$ admits a universally free object $\mathsf F$, then there is a projectively universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such that for any $B\in \mathcal{C}$ and $τ\in {\rm Mor}(B)$ there exists an epimorphism $π\in {\rm Mor}(\mathsf F, B)$ such that $πτ= u π$. This builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math. Soc. 145 (2017)) who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical sys…
General heart construction for twin torsion pairs on triangulated categories
在上一篇文章中，我们从三角范畴上的任意挠对构造了一个阿贝尔范畴。这推广了一个$t$ -结构的心脏和一个集群倾斜子范畴的理想商。最近，Buan和Marsh通过一个簇倾斜子范畴推广了商，证明了在一定条件下，从一个三角范畴中的刚性对象可以构造一个积分Preabel范畴作为商.本文通过考虑一对挠对，将这两种结构同时推广。
Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator
本文研究带有紧倾斜生成元的张量三角化范畴，定义了平坦维数与$M$-幽灵维数，证明其与心范畴维数的关系，推广了Hovey-Lockridge定理。
We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory as well as in the equivariant, motivic, and parametrized variant thereof. As an application of these equivalences we obtain abstract tilting results for trees valid in all these situations, hence generalizing a result of Happel. The main tools introduced for the construction of these reflection f…
General heart construction on a triangulated category (I): unifying t-structures and cluster tilting subcategories
Keller和Reiten的文章证明了一个三角范畴（在一定条件下）与一个簇倾斜子范畴的商成为一个阿贝尔范畴。之后，Koenig和Zhu详细地展示了如何在更抽象的背景下给出这个商范畴上的交换结构。另一方面，正如自20世纪80年代以来所熟知的，任何$t$ -结构的中心是阿贝尔的。我们统一这两个建设使用的概念，一个cotorsion对。对于三角范畴中的任一余挠对，当余挠对分别来自于一个簇倾斜子范畴或一个$t$ -结构时，我们可以自然地将一个阿贝尔范畴与之联系起来，它分别给出上述两个阿贝尔范畴中的每一个。
Cluster tilting and complexity
我们研究的概念，积极和消极的复杂性对对象的集群类别。第一个主要结果表明，出现的最大复杂性是一个，两个或无限的，这取决于底层遗传代数的表示类型。在第二个结果中，我们研究了簇倾斜代数的有界导出范畴，并证明了当代数是有限型或驯服型时，出现的最大复杂性要么为零，要么为一。
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