Alexey Elagin, Valery A. Lunts
In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $R$ be a commutative ring. We introduce the notion of support of objects in an $R$-linear triangulated category. As an application, we study the non-existence of Bridgeland stability conditions on $R$-linear triangulated categories.
Uniqueness of enhancement for triangulated categories
本文包含一般结果的唯一性DG增强三角范畴。作为结果，我们得到这样的唯一性的无界范畴的拟凝聚层，为三角范畴的完美复形，并为有界的导出范畴的拟射影概形上的凝聚层。如果一个计划是投射的，那么我们也证明了一个强唯一性的三角范畴的完美复形和有界导出范畴的相干层。这些结果直接意味着来自凝聚层的有界导出范畴和射影概形上的完美复形的三角化范畴的完全忠实函子可以用乘积上的对象来表示。
Triangulated categories of singularities and equivalences between Landau-Ginzburg models
在本文中，我们证明了存在的三角形范畴之间的一些类型的等价性的奇异品种的不同尺寸。这类等价推广了所谓的克诺尔周期性。因此，我们得到了不同维数的Landau-Ginzburg模型上的B型D-膜范畴之间的等价性。
Triangulated categories of singularities and D-branes in Landau-Ginzburg models
尽管标题中有物理学术语，但这篇论文是纯数学的。其目的是引入与代数簇的奇点相关的三角范畴，并建立这些范畴与Landau-Ginzburg模型中的D-膜的联系。
Rouquier's cocovering theorem and well-generated triangulated categories
本文研究了三角范畴在Rouquier意义下的上覆盖，证明了对任意正则基数$α$，在Neeman意义下，$α$ -紧性的条件是局部的.这是鲁基耶为普通紧性建立的。我们的结果产生了一个新的技术证明一个给定的三角范畴是良生的。作为应用，我们描述了拟紧半分离概型的无界导出范畴中的$α$ -紧对象.
Remarks on generators and dimensions of triangulated categories
本文证明了光滑拟射影曲线上凝聚层的有界导出范畴的维数等于1。我们还讨论了这些范畴的维数谱。
Affine space over triangulated categories: a further invitation to Grothendieck derivators
本文在稳定导子框架下，给出三角范畴上仿射空间（即多项式环）的构造，拓展 Grothendieck 导子理论。
Scalar extensions of triangulated categories
给定域$K$和域扩张$L/K$上的一个三角范畴，研究如何构造$L$上的一个三角范畴.我们的方法产生的基础变化计划$X_L$的导出范畴，如果一个开始的范畴是一个光滑的投射簇$X$在$K$和域扩张有限和伽罗瓦的有界导出范畴。我们还研究了三角范畴的维数在标量扩张下的行为。
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。
Triangulated structures induced by simplicial descent categories
本文研究了单下降范畴（D，s，E）（arXiv：0808.3684v2）的同伦范畴.证明了D模同伦的商范畴上的等价类E有左分式演算。我们研究了由（共）单纯下降结构诱导的纤维/上纤维序列。这样的纤维/cofiber序列的例子推导出（交换）微分分次代数，单纯集或拓扑空间。证明了稳定单纯下降范畴的同伦范畴是三角化的。此外，这些三角化结构还可以推广到D的图范畴的同伦范畴。作为推论，我们得到了以下范畴的三角结构：阿贝尔范畴的（滤）导出范畴，DG-范畴上DG-模的导出范畴，保序谱的稳定导出范畴和混合Hodge复形的定域范畴.
Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
本文建立了具有齐次超势W的Landau-Ginzburg模型中B型分次D-膜范畴与零上W纤维奇点的三角化范畴之间的等价性。主要结果是一个定理，表明分次三角范畴的奇异锥上的射影品种是通过一个完全忠实函子连接到有界导出范畴的相干层的基础上的锥。这意味着具有齐次超势W的Landau-Ginzburg模型中B型分次D-膜范畴通过一个完全忠实函子与方程W=0定义的投射簇上的相干层的导出范畴相连。
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.
Presheaves of triangulated categories and reconstruction of schemes
对于任何具有张量积$(K,\otimes)$的三角范畴，我们借助于$K$的厚子范畴，象Hopkins-Neeman-Neeman一样，把拓扑空间$Spc(K,\otimes)$联系起来.此外，对于$Spc(K,\otimes)$的每个开子集$U$，我们关联一个三角范畴$K(U)$，产生可以被认为是三角范畴的预层。将这一点应用于诺特模式$X$上的完全复形的导出范畴$(K,\otimes):=(D^{perf}(X),\otimes^L)$，拓扑空间$Spc(K,\otimes)$证明是$X$的基础拓扑空间;而且，对于每个开$U\subset X$，范畴$K(U)$自然等价于$D^{perf}(U)$。作为应用，我们给出了一种从完备复形$D^{perf}(X)$的导出范畴重构任意约化Noether概型$X$的方法，把后者看作是一个具有$\otimes^L$的张量三角范畴.
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.
Triangulated Categories of Big Motives via Enriched Functors
本文利用丰富函子的Grothendieck范畴同调代数，为Voevodsky的合理对应大动机范畴构造了两个模型。
Linearisations of triangulated categories with respect to finite group actions
给定一个有限群在一个三角范畴上的作用，我们研究在什么条件下可以用DG-增强构造一个线性化的三角范畴。特别地，如果群是光滑投射簇的自同构的有限群，范畴是凝聚层的有界导出范畴，则我们的构造分别产生光滑商簇上凝聚层的有界导出范畴.堆栈我们还考虑了张量积与挠典范丛的作用以及有限群在球面对象生成的范畴上的作用。
Completions of triangulated categories
本文为奥胡斯大学硕士课程讲义，系统介绍三角范畴的完备化理论，涵盖基本概念与核心方法。
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。
Homological support of big objects in tensor-triangulated categories
本文借助同调剩余域为张量三角范畴中的大对象引入支撑概念，并建立相应的张量积公式。
A tour of support theory for triangulated categories through tensor triangular geometry
这些说明试图给一个简短的调查的方法支持理论和研究格的三角形子类通过机械张量三角几何。一个主要目的是介绍陈述和证明局部到整体原理所需的材料。特别是，我们讨论了巴尔默的建设的频谱，广义Rickard幂等元和支持companies产生的三角范畴，张量三角范畴和行动。一路上也举了几个例子。这些笔记是基于2015年春季在巴塞罗那CRM的“表示论、代数拓扑和交换代数之间的相互作用”（IRTATCA）课程中的一系列讲座。
A Note on Support in Triangulated Categories
在本文中，我定义了一个概念，一个三角范畴中的一个紧支撑的对象。我证明了一些命题与此有关的传统概念的支持，并给出了应用程序的理论推导森田等价。我还讨论了一个连接到超对称规范理论所产生的D膜在一个奇点。
We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier-Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.
Triangulated categories of Gorenstein cyclic quotient singularities
我们证明了三角范畴之间的等价性Orlov的三角范畴的奇异性Gorenstein循环商奇点和导出范畴的表示的一个关系，这是从McKay的符号，通过删除一个顶点和一半的箭头。
We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the…
Triangulated categories of motives over fs log schemes
我们引入了日志motivic三角范畴的概念，这是理解FS日志计划的上同调理论的motivic方面的理论框架。然后研究了对数motivic三角范畴的Grothendieck六运算形式。
Pre-triangulated categories are triangulated
我们证明了一个更强的版本的八面体公理在一个预三角范畴。证明使用了一个新的引理，这是基于一个弱逆蛇引理的点加性范畴中的正合序列。
We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.
An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively. We also recover…
Functors on triangulated tensor categories
定义并研究了每个三角张量范畴的函子谱。证明了一个与Balmer定理类似（并基于Balmer定理）的拓扑Noether格式的重构结果。本文给出了Bondal-Orlov关于具有充足（反）典范丛的光滑射影簇的重构定理的另一种证明。
Support theory via actions of tensor triangulated categories
给出了张量三角化范畴T在三角化范畴K上的作用的定义。在这种情况下，T是刚性，compensaries产生和K是compensaries产生的，我们表明这引起了一个概念的支持，其中归类工作的本森，艾扬格，克劳斯和扩展工作的巴尔默和法维。我们证明了一个合适的版本的本地到全球的原则持有非常普遍。提出了一个相对的望远镜猜想，并给出了它成立的充分条件。
Triangular spectrum of some triangulated categories
本文计算了两类代数几何中常见的张量三角化范畴的三角谱。其中之一是有限群$G$作用在$X$上的光滑概形$X$上的$G$ -等变层的导出范畴.另一类是分裂超方案的导出范畴。
On some spectral spaces associated to tensor triangulated categories
本文考虑了张量三角范畴$(\mathcal K,\otimes,1)$上的模的三角范畴$\mathcal M$的厚$\mathcal K$ -子模空间$SMod(\mathcal M)$上的有限型闭包算子$c$.我们的目的是证明算子$c$的不动点空间$SMod^c(\mathcal M)$是一个谱空间，它也具有拓扑幺半群的结构。
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。
Length of triangulated categories
本文提出三角范畴的合成列概念以推广完全例外序列，并证明其对部分射影簇与有限维代数导出范畴的不变量性质，同时揭示某些情形下长度唯一而另一些情形下长度可变。
Universal support for triangulated categories
本文重新审视 Gratz–Stevenson 的理论，探讨无张量积时三角范畴对象支撑所依托的泛空间。
Relative Stability Conditions on Triangulated Categories
本文提出三角范畴相对于左容许子范畴的相对稳定条件，并借助粘合稳定条件的形变证明其形变性质，以回应 Bridgeland 稳定性与 dHYM 度量存在性关系的问题。
We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Lurie's theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts--Orlov and Canonaco--Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable infinity-categories. Finally, we propose a theory of stable $n$-categories which would interpolate between triangulated categories and stable inf…
Lectures on derived and triangulated categories
这些注释旨在快速介绍三角分类。我们从加性范畴的定义开始，以倾斜理论的一系列讨论结束。包括一些练习。
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.
In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric. We specialize these general constructions to triangulated categories, and restrict our attention to "good metrics". And the remarkable new theorem is that, when we start with a triangulated category $\mathcal S$ with a good metric, its completion $\mathfrak{L}(\mathcal{S})$ contains an interesting subcategory $\mathfrak{S}(\mathcal{S})$ which is always trian…
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject -- more precisely to articles by Illusie in SGA6, way back in the early 1970s. What's new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analysing mor…
Dimensions of triangulated categories
我们定义了一个三角范畴的维数。本文证明了有限维三角范畴上某类函子的一个表示定理。本文研究了代数或概型的有界导出范畴的维数，特别证明了各种凝聚层的有界导出范畴具有有限维数.对于自内射代数，Auslander表示维数的下界由稳定范畴的维数给出。我们用它来计算外代数的表示维数。这提供了第一个已知的表示维数>3的例子。本文证明了一个有限群的F_2上的群代数的Loewy长度在群的2-秩下有严格界（Benson的一个猜想）。
Construction of triangulated categories of motives using the localization property
利用局部化性质，我们构造了一个拟投射T-格式上的三角形动机范畴，其中T是Noether分离格式，并证明了Grothendieck六操作形式.我们还构建了动机的完整的童话实现。
Bounded $t$-structures, finitistic dimensions, and singularity categories of triangulated categories
本文引入三角范畴有限维数的新概念并沿完备化提升 $t$-结构，证明了三角范畴上有界 $t$-结构的存在性及其不变性的一般性结果，推广了 Neeman 的相关定理并给出了有界 $t$-结构存在性的范畴障碍。
Homological Aspects of Separable Extensions of Triangulated Categories
研究三角范畴在可分离扩张下的同调行为，证明有限全局维数、格氏性质和正则性等同调不变量在扩张下保持不变，并建立奇异范畴之间的关系，证明可分离扩张的奇异范畴等价于其奇异范畴的可分离扩张。
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。
On equivariant triangulated categories
考虑作用在三角范畴$\mathcal T$上的有限群$G$。本文研究了$\mathcal T$中$G$ -等变对象范畴$\mathcal T^G$上的三角结构.我们证明（在一定的技术条件下），这种结构存在。假设对$\mathcal T$的作用是由对$\mathcal T$的某些DG增强的DG作用引起的，我们构造了$\mathcal T^G$的DG增强。另外，我们还证明了“关于有限阿贝尔群作用成为等变范畴”的关系在幂等完备可加范畴上是对称的。
In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects. In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is approximable whenever $X$ is a quasicompact separated scheme. As corollaries of this (seemingly technical) statement one can prove striking improvements on old theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much smaller) categories $D^{\text{perf}}(X)$ and $D^b_…
On exact categories and applications to triangulated adjoints and model structures
我们表明，奎伦的小对象参数工程的确切类别在非常温和的条件下。这直接应用于余挠对及其与某些三角化伴随函子和模型结构的存在性的关系。特别是，准相干层的复合体范畴上不同精确结构的相互作用导致了Estrada，Gillespie，Guil Asensio，Hovey，Jørgensen，Neeman，Murfet，Prest，Trlifaj和其他人最近获得的结果的简化和推广版本。
Homologically finite-dimensional objects in triangulated categories
本文研究三角范畴的同调有限维对象，定义相关范畴性质并讨论无亏格收缩，通过几何代数例子说明概念，还证明奇异簇导出范畴半正交分解与其光滑化导出范畴的双射。
Approximable Triangulated Categories and Reflexive DG-categories
本文利用可逼近三角范畴刻画 proper DG-范畴的自反性，给出其完备化的新描述，并应用于概形、连通 DG-代数与 Azumaya 代数。
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 goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $\mathrm{Spec}R$, called the big cosupport.
Relative singularity categories, Gorenstein objects and silting theory
我们研究奇异范畴通过Gorenstein对象在三角范畴和淤积理论。设$ω$是三角范畴$\mathcal{T}$的半自对角（或预淤）子范畴。引入了$ω$ -Gorenstein对象的概念，它是三角范畴中Gorenstein投射模和Gorenstein内射模的进一步推广.证明了稳定范畴$\underline{\mathcal{G}_ω}$是一个三角范畴，其中$\mathcal{G}_ω$是所有$ω$-Gorenstein对象的子范畴，并且在一定条件下，它与$\mathcal{T}$关于$ω$的相对奇异范畴是三角等价的.
Braid cobordisms, triangulated categories, and flag varieties
我们认为，各种辫群行动的三角范畴应扩展到投射行动的范畴辫协边，并说明这是如何工作的例子。我们还构造了仿射辫群和辫配边范畴对全旗簇余切丛上相干层的导出范畴的作用。
Zeta functions in triangulated categories
证明了Voevodsky范畴DM中运动zeta函数在特殊三角形中的合理性的3取2性质。作为应用，我们证明了DM中拟射影曲线的基模所生成的厚三角化monoidal子范畴中的所有簇的基模zeta函数的合理性。结合P.O'Sullivan的结果，给出了一类动机非有限维而动机zeta函数有理的簇的例子。
Triangulated surfaces in triangulated categories
对于具有2-周期dg-增强的三角范畴A和三角定向标记曲面S，我们引入了一个dg-范畴F（S，A），它将A中的正合三角形系参数化为S的三角形.我们的主要结果是，F（S，A）是独立的选择S的三角形基本上是唯一的森田等价。特别地，它容许映射类群的一个典范作用。证明是基于循环2-Segal空间的一般性质。在最简单的情况下，当A是向量空间的2-周期复形范畴时，F（S，A）是曲面S的Ruppaya范畴的纯拓扑模型。因此，我们的建设可以被看作是实现一个2维的实例Kontsevich的程序本地化的类沿着奇异拉格朗日棘。
Triangulated categories of matrix factorizations for regular systems of weights with $ε=-1$
我们构造了一个完整的强例外集合在三角范畴的分次矩阵分解的多项式相关联的一个非退化的正则系统的重量，其最小指数等于-1。在相应的Grothendieck群中，强例外集合定义了符号为（1，0，2）的广义根系的根基和有穷级的Coxeter元，其本征向量是关于外尔群的IV型扩展对称域中的正则元.
We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal{A}$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal{A}$-modules and of $C$-comodules. When $\mathcal{A}$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。