Thick subcategories of finite algebraic triangulated categories
Claudia Köhler
本文给出了代数三角化标准范畴中含有100多个不可分解对象的厚子范畴的分类。
We classify the thick subcategories of an algebraic triangulated standard category with finitely many indecomposable objects.
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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 a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
Codistinguished abelian subcategories
引入三角范畴的共区分阿贝尔子范畴，推广了Jorgensen的真阿贝尔子范畴并与Linckelmann的区分阿贝尔子范畴对偶，证明其具有由环境三角范畴诱导的非平凡正合结构且所有阿贝尔范畴的正合结构均由此产生，并证明完全忠实伴随三元组可生成新的共区分阿贝尔子范畴。
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。
Localisation and colocalisation of triangulated categories at thick subcategories
给定一个厚的子范畴的三角范畴，我们定义了一个共定位和一个自然的长正合序列，涉及到原来的类别和它的本地化和共定位在子范畴。类似地，我们构造了一个自然长正合列，它包含关于厚子范畴的同调函子和它的全导出函子之间的典范映射。
For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces…
The passage among the subcategories of weakly approximable triangulated categories
本文证明弱可逼近三角范畴的经典全子范畴之间的包含关系是内在的，推广了Rickard定理，使其适用于更一般的范畴和概形。
Report on locally finite triangulated categories
讨论了局部有限三角范畴的基本性质。重点是Auslander-Reiten理论和厚子范畴格。
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…
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。
We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category.
We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general result is then applied to algebraic geometry.
Central support for triangulated categories
本文引入了本质小三角范畴厚子范畴格的中心概念，由此定义了中心支撑，其相对版本统一了张量三角范畴支撑理论与上同调支撑，并为交换子范畴对建立了Mayer-Vietoris序列。
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcateg…
We define the notion of right $n$-angulated category, which generalizes the notion of right triangulated category. Let $\mathcal{C}$ be an additive category or $n$-angulated category and $\mathcal{X}$ a covariantly finite subcategory, we show that under certain conditions the quotient $\mathcal{C}/\mathcal{X}$ is a right $n$-angulated category. This result generalizes some previous work.
Ore's theorem for thick subcategories
本文研究了有限群在有界导出范畴中厚子类格的分布性质，证明p- nilpotent群是唯一满足该性质的群，并给出了有限维代数的有界导出范畴和完美复形中厚子类格分布的充要条件。
Auslander-Buchweitz approximation theory for triangulated categories
我们引入并发展了一个类似的Auslander-Buchweitz逼近论（见\cite{AB}）在三角范畴的背景下，通过使用一个版本的相对同源性在这种设置。我们还证明了三角化范畴$\T,$中的相对同调代数的几个结果，这些结果是基于某些子范畴在有限分解和Hom-space消失下的行为。举例来说：我们建立了$\T.$的某些三角化子范畴中的preenvelope（和precover）的存在性。结果类似于Auslander和Buchweitz的各种构造和结果，并且集中于探索配备有一对$(\X,ω),$的三角化范畴$\T$的结构，其中$\X$在扩张下是封闭的，而$ω$通常在附加条件下是$\X,$中的弱共生成元。这减少，除其他事项外，存在杰出的三角形享有特殊性质，和行为（适当定义）（共同）决议，射影或内射维度的对象的$\T$和形成正交子类别。最后，讨论了三角范畴中Rouquier维数的一些关系.
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated st…
Approximating triangulated categories by spaces
本文研究本质小三角范畴的厚子范畴格，证明其分配性蕴含空间frame性质，并构造两个具有不同函子性的非交换谱以实现空间化泛逼近。
We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory is generated by a partial silting object. In particular, every such smashing subcategory admits a silting t-structure.
An assortment of properties of silting subcategories of extriangulated categories
本文研究外三角范畴的silting子范畴，证明其诱导Grothendieck群的基，引入silting突变并探讨有限射影维数子范畴的性质，最终恢复Auslander--Reiten关于tilting模与分解子范畴双射的结果。
We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the cardinality and the height of the poset. We explicitly describe spherical subcategories and their poset structure for derived categories of certain finite-dimensional algebras.
Model theory in compactly generated (tensor-)triangulated categories
本文阐述了紧致生成三角范畴与张量-三角范畴中的模型理论，描述了 pp 公式与 pp 型等概念，证明了量词消去与虚元消去，并探讨了可定义子范畴、挠理论及内部对偶性。
We classify t-structures and thick subcategories in discrete cluster categories $\mathcal{C}(\mathcal{Z})$ of Dynkin type $A$, and show that the set of all t-structures on $\mathcal{C}(\mathcal{Z})$ is a lattice under inclusion of aisles, with meet given by their intersection. We show that both the lattice of t-structures on $\mathcal{C}(\mathcal{Z})$ obtained in this way and the lattice of thick subcategories of $\mathcal{C}(\mathcal{Z})$ are intimately related to the lattice of non-crossing partitions of type $A$. In particular, the lattice of equivalence classes of non-degenerate t-structur…
Stratifying triangulated categories
对赋分次交换诺特环R作用的任何紧生成的三角范畴T，引入了分层的概念.这个概念的效用是通过建立不同的后果，遵循T是分层的R。其中包括T的局部化子范畴根据R中素理想集合的子集的分类; T中紧对象子范畴的厚子范畴的分类;以及关于同态Hom_T^*（C，D）的R-模的支撑的结果，导致了群的模表示的支撑簇的张量积定理的类似。
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…
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 $…
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…
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。
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.
Are all localizing subcategories of stable homotopy categories coreflective?
我们证明了，在一个三角范畴与组合模型，每个局部化子范畴是共反射和每个共局部化子范畴是反射的，如果某个大基数公理（Vopenka的原则）是假设为真。由此可见，在相同的假设下，正交性在局部化子范畴和上局部化子范畴之间建立了一个双射对应。这样一个双射的存在性被Hovey、Palmieri和Strickland在他们对稳定同伦范畴的公理化研究中以及Neeman在良生成的三角范畴的背景下作为一个开放问题留下。
Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category
对于具有一个$d-$簇倾斜子范畴$\mathcal{T}$的Calabi-Yau维的Calabi-Yau三角化范畴$\mathcal{C}$，证明了$\mathcal{C}$的分解由$\mathcal{T}$的特殊分解即$\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$决定，其中$\mathcal{C}_i, i\in I$是三角化子范畴，当且仅当$\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$其中$\mathcal{T}_i, i\in I$是具有$Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$和$i\not= j.$的子范畴。这导致了在$2-$ Calabi-Yau三角化范畴中的任何两个群集倾斜对象同时连接或不连接。作为应用，我们证明了不可分解的$2-$ Calabi-Yau三角范畴不具有非平凡的t-结构和非平凡的余t-结构。这使我们能够在这个三角范畴中给出余挠对的分类。此外，Nakaoka意义下的余挠对的心等价于余挠对的核的自同态代数上的模范畴。
We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.
Localization theorems for weakly approximable triangulated categories
本文研究弱近似可三角化范畴的定位现象，证明在弱假设下，此类范畴的Recollement诱导出若干自然子范畴的短正合列，应用于环的导出范畴、DG代数和概形。
Weakly approximable triangulated categories and enhancements: a survey
本文综述弱可逼近三角范畴自然子范畴的内在性及三角范畴增强唯一性结果，并展示其相互作用如何大幅推广 Rickard 关于概形与环的导出不变性定理。
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.
Localization of triangulated categories with respect to extension-closed subcategories
本文构建三角范畴相对于扩张闭子范畴的外三角局部化框架，统一了Verdier商等局部化现象，刻画了局部化范畴为三角或正合范畴的等价条件，并应用于上同调函子相关结构的理解。
We prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated $C$ that is compactly generated by a single object $G$ is weakly approximable if $C(G,G[i])=0$ for $i>1$ (we say that $G$ is weakly negative if this assumption is fulfilled; the case where the equality $C(G,G[1])=0$ is fulfilled as well was mentioned by Neeman himself). Moreover, if $G\cong \bigoplus_{0\le i\le n}G_i$ and $C(G_i,G_j[1])=0$ whenever $i\le j$ then $C$ is also approximable. The latter result can be useful since (under a few more additional assumptions) i…
We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we construct an order-preserving map from silting objects in $\C$ to bounded $t$-structures on $\D$ and show that the map is bijective if and only if $\C$ is silting-discrete if and only if $\D$ is $t$-discrete. Based on a work of Qiu and Woolf, the above result is applied to show that if $\C$ is silting-discrete then the stability space of $\D$ is contractible. Thi…
We classify thick subcategories $\mathcal T \subset D^b(\mathrm{coh}\,C)$ for smooth projective curves $C$ over an algebraically closed field.
We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non-zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect complexes over a commutative ring without non-trivial idempotents has no proper non-zero thick subcategories that are strongly generated.
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during t…
Exactness of the 2-categories of abelian and triangulated categories
本文引入$2$-同调范畴概念，证明双正合、三角、带点、加性及阿贝尔范畴的$2$-范畴均为$2$-同调，并给出统一商构造准则。
We define the Grothendieck group of an $n$-exangulated category. For $n$ odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete subcategories of an $n$-exangulated category with an $n$-(co)generator in terms of subgroups of the Grothendieck group. This unifies and extends results of Thomason, Bergh--Thaule, Matsui and Zhu--Zhuang for triangulated, $(n+2)$-angulated, exact and extriangulated categories, respectively. We also introduce the notion of an $n$-exangulated subcategory and prove …
The homological spectrum via definable subcategories
本文提出用可定义子范畴研究张量三角范畴的同调谱，证明其同胚于 Ziegler 谱的商，并刻画同调剩余域内射对象，从而以纯性视角联系同调谱与 Balmer 谱。
The left (right) triangulated structures of the stable categories
Bkillannis和Marmarlannis [ \emph{Comm. in Algebra,} 22（12）（1994），5021-5036]在由同调有限子范畴导出的加法范畴的稳定范畴上构造了左、右三角结构。我们将他们的结果扩展到稍微更一般的设置。作为我们结果的应用，我们给出了一些新的例子，稳定范畴的左或右三角结构的阿贝尔模型范畴。一个有趣的结果是，我们可以通过稳定范畴的同伦范畴来描述阿贝尔模型范畴的同伦范畴的预三角结构。
A note on thick subcategories of stable derived categories
对于具有足够多投射对象的正合范畴，我们建立了包含投射对象的厚子范畴与稳定导出范畴的厚子范畴之间的双射.使用这种双射，我们分类厚的子类别的有限生成模在当地的完整的交叉和生产的类别的相干层的一个单独的诺特计划与一个丰富的家庭的发电机。
Metrics on triangulated categories and their enhancements
研究弱可逼近三角化范畴自然子范畴增强的唯一性，通过在∞-范畴层面增强优秀度量理论，推广了增强唯一性结果，回答开放问题，还证明子范畴等价关系可提升，完善前文并扩展相关猜想结果。
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。
Localizing and colocalizing subcategories on schemes
本文研究概形上导出范畴的⊗-理想局部化子范畴和H-余理想共局部化子范畴，证明在点生成凝聚概形上它们由子集分类，且共局部化子范畴是局部化子范畴的正交补。
Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently. In this paper, we prove that the Gabriel-Zisman localization $\mathcal B/({\rm thick}\mathcal W)$ of an extriangulated category $\mathcal B$ with respect to a presilting subcategory $\mathcal W$ satisfying certain condition can be realized as a subfactor category of $\mathcal B$. This generalizes the result by Iyama-Yang for silting reduction on triangulated categories. Then we discuss the relation between silting subcategories and tilting subcategories in extriangulated categori…
Formal completions and idempotent completions of triangulated categories of singularities
本文的主要目的是证明两个方案的三角化奇点范畴的幂等完备化是等价的，如果这些方案沿奇点的形式完备化是同构的。我们还讨论了稠密子范畴上的Kesason定理以及它与负K-理论的关系。
We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact $R$-linear functor between $R$-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a f…
From $(n+1)$-term subcategories to $(n+1)$-term complexes
研究从代数三角范畴的n - 刚性子范畴生成的(n + 1)项子范畴到(n + 1)项复形全子范畴的外三角函子，给出其为满的条件及相关双射，还应用于(n + 1)-卡拉比 - 丘范畴的n - 簇倾斜情况。
On derived categories of module categories over multiring categories
本文证明了在特定条件下，多环范畴上模范畴的导出范畴等价性等价于原模范畴的等价性，并探讨了相关局部化理论。
Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative …
Lifting of recollements and gluing of partial silting sets
本文研究三角范畴中再余裂的提升与倾斜集的粘合，提出提升判据并给出Artin代数情形下粘合经典倾斜对象的显式构造。
We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As…
We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit in…
Frobenius condition on a pretriangulated category, and triangulation on the associated stable category
如Happel所示，从任何Frobenius正合范畴，我们可以构造一个三角范畴作为稳定范畴。另一方面，Iyama和Yoshino证明了，如果三角范畴中的一对子范畴$\mathcal{D}\subseteq\mathcal{Z}$满足某些条件（即，$(\mathcal{Z},\mathcal{Z})$是$\mathcal{D}$ -突变对），则$\mathcal{Z}/\mathcal{D}$成为三角分类。在这篇文章中，我们考虑这两个结构的同时推广。
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