William Wong
We will be defining a type of perverse equivalence that always corresponds to a derived equivalence with two-term tilting complexes. We are going to show that the tilting considered by Okuyama and Yoshii for the proof of Broué's conjecture for $SL(2,q)$ in defining characteristic is a composition of such perverse equivalences.
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In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
Perverse equivalences, BB-tilting, mutations and applications
我们涉及的BB-倾斜和反常导出等价在一个顶点的概念。基于这些概念，我们定义了代数的突变，导致导出等价的。我们目前的应用程序的自同态代数簇倾斜对象在2-Calabi-Yau范畴和代数的整体维数最多为2。
Induction in perverse schobers and cluster tilting theory
本文在带边曲面的反常schobers框架内，通过研究限制函子及其伴随归纳函子，展示了精确∞-范畴中cluster tilting子范畴的粘合性质，并提供了来自高阶拓扑Fukaya范畴和带穿孔标记曲面的新实例。
Inspired by the works of Rickard on splendid equivalences and of Chuang and Rouquier on perverse equivalences, we are here interested in the combination of both, a splendid perverse equivalence. This is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups, as a splendid equivalence induces local derived equivalences via the Brauer functor. We prove that under certain conditions, we have an equivalence between a perverse equivalence between the homotopy category of p-permutation modules and local derived pervers…
Tilting exercises
这是倾斜物体理论在反常层几何设置中的应用。我们表明，这一理论是一个自然的框架贝林森的胶合反常层建设。在旗簇的Schubert分层的特殊情况下，我们得到了Soergel的“Strukturnel”的一个简短证明，并描述了（根据Kapranov的一个猜想）范畴O的Serre函子。我们的一些结果是由Rouquier独立获得的。
In this article, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been able to obtain a new type of autoequivalence. This autoequivalence has some uncommon features. It is more conveniently conceived and proved using the representation theory of its Brauer correspondence but at the same time it can be very neatly described, using a type of derived equivalence called perverse equivalence, in global settings.
Morse theory and tilting sheaves
利用分层莫尔斯理论、环面作用和附近的圈，我们给出了倾斜反常层的尺规作图法。
Stacks similar to the stack of perverse sheaves
在拓扑空间X上引入一类阿贝尔范畴的栈，我们称之为“P型栈”。“这类栈包括反常层的栈（任何反常的，相对于固定的分层都是可构造的），并且被相当无害的公理挑选出来。我们发现，一些基本的结构理论的反常层持有的一般堆栈类型P：这样的堆栈是局部等价的MacPherson-Vilonen建设，并在一定的连通性条件下，其范畴的全球对象是等价于范畴的模在有限维代数。为了证明这些结果，我们发展了一个基本的倾斜形式主义的堆栈类型P-另一种意义上说，这些堆栈是“类似于堆栈的反常层。"
Rigidity of tilting complexes and derived equivalence for self-injective algebras
基于倾斜复形的刚性，证明了代数闭域上的有限维自内射代数类在导出等价下是闭的。
We show that perverse equivalences between module categories of finite-dimensional algebras preserve rationality. As an application, we give a connection between some famous conjectures from the modular representation theory of finite groups, namely Broué's Abelian Defect Group conjecture and Donovan's Finiteness conjectures.
Perverse Equivalences and Broué's Conjecture
在Broué猜想的基础上，基于逆等价，给出了构造有限群块之间导出等价的一种新方法.我们特别提供了本地和全球反常数据描述的主块和派生的等价有限单群与Sylow子群C3 × C3。我们还研究了一般情况下的自同构群的扩展。
Universal derived equivalences of posets of tilting modules
我们表明，对于两个箭图没有定向循环相关的BGP反射，其倾斜模块的偏序集相关的一个简单的组合结构，我们称之为触发器。证明了无定向圈箭图的导等价路代数的倾斜模的偏序集是泛导等价的。
Equivalence classes of Wakamatsu tilting modules and preenveloping and precovering subcategories
研究Wakamatsu倾斜模的等价类，扩展Mantese-Reiten定理至任意结合环。
From Jantzen to Andersen Filtration via Tilting Equivalence
范畴O中从投射对象到韦尔马模的同态空间继承了韦尔马模上的Jantzen滤子的导出滤子。另一方面，在[Soe 07]中描述了从韦尔马模到倾斜模的同态空间上的Andersen过滤。[Soe 98]的倾斜等价导出了这类Hom-空间的同构。我们将证明，这种等价性甚至可以识别两个过滤。
Perverse Equivalences and Broué's Conjecture II: The Cyclic Case
利用最近的反常等价理论和Deligne-Lusztig簇研究了Lie型群的Broué交换亏群猜想.我们的方法是直接分析某些Deligne-Lusztig簇（Broué猜想的几何形式）引起的反常等价;这使用了这些簇的上同调，以及来自分圆Hecke代数的信息。我们从一个猜想上同调的这些Deligne-Lusztig品种，证明各种理想的性质，然后用它来证明存在的反常等价预测的几何形式的Broué猜想每当缺陷组是循环的（除了可能为两个块的布劳尔树是未知的）。这是一个必要的第一步，以证明布鲁埃猜想一般，作为反常等价建立归纳从各种列维小组。这篇文章是最新的一系列拉斐尔Rouquier和作者的最终目的是证明Broué猜想的幂幺块群的李型。
Following Craven and Rouquier's computational method to tackle Broué's abelian defect group conjecture, we present two algorithms implementing that procedure in the case of principal blocks of defect $D \cong C_{\ell} \times C_{\ell}$ for a prime $\ell$; the first describes a stable equivalence between $B_0(G)$ and $B_0(N_G(D))$, and the second tries to lift a such stable equivalence to a perverse derived equivalence. As an application, we show that Broué's conjecture holds for the principal $5$-block of the simple group $Ω^{+}_8(2)$.
On a common generalization of Koszul duality and tilting equivalence
本文给出了分次代数的Koszulity的一个新定义，其中零度部分具有有限的整体维数，但不一定是半单的。标准的Koszul对偶定理在这种情况下成立。我们给出了BGG范畴$\mathcal O$的多重自由块所产生的代数的一个应用。
On tilting complexes providing derived equivalences that send simple-minded objects to simple objects
给定一组“头脑简单”的对象在一个导出范畴中，里卡德构造了一个复数，它在一个对称代数上提供了一个派生等价，将“头脑简单”的对象发送到简单的对象。当这个复合物是一个倾斜复合物时，我们用t-结构表示，表明存在一个相关的自然$t$ -结构，并且我们用A-无穷结构表示这个复合物的另一种结构。我们的方法和凯勒·尼古拉斯的方法相似
We define a generalization of the Mullineux involution on multipartitions using the theory of crystals for higher level Fock spaces. Our generalized Mullineux involution turns up in representation theory via two important derived functors on cyclotomic Cherednik category $\mathcal{O}$: Losev's "$κ=0$" wall-crossing, and the Ringel duality.
Chuang and Rouquier describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension $-1$. We develop an analogue of their theory for Calabi-Yau categories of dimension $w<0$ and show it is equivalent to the mutation theory of $w$-simple-minded systems. Given a non-positively graded, finite-dimensional symmetric algebra $A$, we show that the differential graded stable category of $A$ has negative Calabi-Yau dimension. When $A$ is a Brauer tree algebra, we construct a combinatorial model of the dg-stable category and show that perverse equivalen…
We construct derived equivalences between group graded self-injective algebras, starting from equivalences between their 1-components, obtained via a construction of J. Rickard and S. Al-Nofayee.
For a balanced wall crossing in geometric invariant theory, there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterise when this is an intersection cohomology complex of a local system on…
Tilting mutations as generalized Kauer moves for (skew) Brauer graph algebras with multiplicity
本文将广义Kauer移动与倾斜突变诱导导出等价的结果推广到任意重数Brauer图代数及斜Brauer图代数，并证明重数为1的斜Brauer图代数是斜gentle代数的平凡扩张。
Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
我们证明了一个广义的扭曲几何佐竹等价的Finkelberg-Lysenko的背景下，可分解的格拉斯曼的约化群G相对于一条平滑曲线X，类似于Gaitsgory的推广“在德容猜想”的佐竹等价本身。为了表达这个结果对可分解性的依赖，我们考虑这个格拉斯曼算子上的一个2-范畴格贝，我们称之为“sf格贝”;这个定理则涉及被这样的格贝扭曲的反常层。这种扭曲的概念不同于，但在特殊情况下是等同于，认为芬克尔伯格-李森科。我们根据G的余权格上的二次型和X上的格给出了这些格的分类。此外，我们还将泛局部无环性（Universal Local Acyclicity，简称CCN）的技术手段引入到佐竹等价的证明中，并将其与Beilinson的反常层的邻近圈胶合相结合，将Mirkovic-Vilonen等价证明中的所有主要构造简化为CCN情形，使它们具有简单的描述，其性质具有简单的证明.我们还给出了一个几何证明，证明了普通仿射格拉斯曼算子上的球面反常层范畴是半单的。
Tilting mutation of weakly symmetric algebras and stable equivalence
我们考虑了弱对称代数在单模子集上的倾斜突变。相原这些突变被定义为长度为1的某些倾斜复形的自同态环。本文从弱对称代数A出发，用一个带关系的代数表示，详细地描述了在A的代数的一个无环顶点上突变所得到的代数的代数结构和关系。在这种形式中，突变过程似乎类似于，虽然比德克森，韦曼和泽列文斯基的突变过程复杂得多。根据定义，由倾斜突变序列连接的弱对称代数是等价的，因此是稳定等价的。本文的第二个目的是通过Okuyama描述简单模象的一个结果来研究这些稳定等价。作为应用，我们回答了Asashiba关于一类有限表示型自内射代数的导出Picard群的问题。最后，我们通过引入三角范畴中的正交砖的极大系统的突变过程，这是出于倾斜突变对稳定范畴中的简单模块的集合的影响。
We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.
$ω$-left approximation dimensions under Stable equivalence
本文研究了无节点及半单直和项的稳定等价Artin代数模的$ω$-左逼近维数传递性质，并据此证明了基础Wakamatsu倾斜模的一一对应关系及该倾斜猜想的保持性。
Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group
在math.RT/0201073中，我们构造了一个简单代数群的旗簇的余切丛上的等变凝聚层的导出范畴与Langlands对偶群的仿射旗簇上的可构造层的范畴（的商）之间的等价。下面我们证明这个等价的某些性质;为幂零锥上的等变相干层范畴提供类似的“朗兰兹对偶”描述;并推导出Lusztig和Ostrik的一些推论。
Universal derived equivalences of posets of cluster tilting objects
我们表明，对于两个颤抖没有定向周期相关的BGP反射，偏序集的集群倾斜对象相关的一个简单的组合建设，我们称之为触发器。证明了无定向圈箭图的导等价路代数的簇倾斜对象的偏序集是泛导等价的。特别是，所有寒武纪晶格对应于同一Dynkin图的各种取向是普遍导出等价的。
The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that…
Derived equivalence for the simple flop of type $D_4$ via tilting bundles
本文针对 $D_4$ 型简单翻折的局部模型，通过构造两侧的倾斜丛证明其导出等价，并由此推导出次数为 $12$ 的一般 $K3$ 曲面间的导出等价。
The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character $χ$ of $\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a perfect derived category of dg-modules $D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(χ)) - \text{dgMod})$ and the triangulated category generated by the corresponding perverse sheaves. The main step in this con…
Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good chara…
Cluster equivalence and graded derived equivalence
本文介绍了一种新的方法来组织整体维数不超过2的代数。我们引入集群等价的概念，这些代数的基础上，他们的广义集群范畴是否是等价的。我们特别感兴趣的问题有多少信息是保留在其广义簇范畴的代数，或者，换句话说，如何密切相关的两个代数，如果他们有等价的广义簇范畴。我们的方法利用了广义聚类范畴中的聚类倾斜对象：我们首先观察到广义聚类范畴中的聚类倾斜对象与导出范畴的聚类倾斜子范畴是自然双射，然后证明了后者的一个识别定理.利用这个识别定理，我们给出了两个簇等价代数导出等价的精确判据。对于一个给定的代数，我们进一步描述了所有导出的等价代数，具有相同的标准集群倾斜对象在其广义集群范畴。最后，我们证明了在一般情况下，如果两个代数是簇等价的，那么（在一定条件下）这两个代数可以分次，使得分次模的范畴是导出等价的.为此，我们引入了具有势的分次箭图的突变，并证明了这一概念反映了导出范畴的突变。
We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type $A_1^{(1)}$ to the category of equivariant perverse coherent sheaves on the nilpotent cone of type $A$. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann-Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type $A_1^{(1)}$ given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver-…
On derived equivalence classification of gentle two-cycle algebras
我们分类，直到导出（等价地，倾斜cotilting）等价所有的非退化温柔的双循环代数。在退化情形下，我们给出了一个部分分类并提出了一个猜想。
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 $S$.
In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.
We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of (-2)-curves, we obtain an equivalence with modules over a well known algebra.
Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be the Happel-Reiten-Smalø tilt of $\mathcal{A}$ with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between $\mathcal{B}$ and $\mathcal{A}$, which is compatible with the inclusion of $\mathcal{B}$ into the derived category of $\mathcal{A}$. In particular, any splitting torsion pair induces a derived equivalence. We prove that for the realization functor of any bounded $t$-structure, its denseness implies its fully-faithfulness.
To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algeb…
Tilting modules and tilting torsion pairs - filtrations induced by tilting modules
本文研究倾斜模如何通过诱导滤子揭示模块范畴的整体结构，探讨其在代数与表示论中的应用与贡献。
In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we …
Derived equivalences between symmetric special biserial algebras
突变的概念在代数的表示论中起着至关重要的作用。两种突变是众所周知的：倾斜/淤积突变和抖动突变。本文主要研究对称代数的倾斜突变问题。引入SB箭图的突变，给出了对称特殊双曲代数倾斜突变的一个组合刻画。作为应用，我们推广了Rickard星定理.我们还介绍了Brauer图的翻转，并将我们的结果应用到Brauer图代数。
On the Cohomology of Deligne-Lusztig Varieties
本文提出了李型群的特殊Deligne-Lusztig簇的上同调的幂单特征标的度的一个猜想，并得到了它的一些推论，这些度是Broué交换亏群猜想的几何形式中的一个必要数据，可以用来在新的情况下验证这个几何猜想. Broué猜想的几何版本应该产生一个更组合定义的导出等价，称为反常等价。我们证明了我们的代数度是一个整数（这是不明显的），并有一个完美的等距正确的奇偶性，并验证了它诱导一个反常等价的所有幂幺块的群的李型循环亏损群，只要形状的布劳尔树是已知的（即，不是E7和E8）。它也被用来寻找一些非循环情况下的反常等价。本文是对李型群Broué猜想的一个导出等价证明的精确形式的数学描述的一个贡献。
Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforwa…
Equivalences between cluster categories
[BMRRT]和[BMR]发展了遗传代数簇范畴的倾斜理论。这些结果被推广到遗传阿贝尔范畴的簇范畴。进一步，对于遗传阿贝尔范畴$\mathcal{H}$中的任意倾斜对象$T$，证明了倾斜函子Hom $_\mathcal{H}(T,-)$诱导了从簇范畴$\mathcal{C(H)}$到簇范畴$\mathcal{C}(A)$的三角等价，其中，$A$是准倾斜代数End $_{\mathcal{H}}T.$在遗传交换范畴$\mathcal{H},$ $\mathcal{H}'$的一个导范畴是三角形等价于一个遗传代数的基础上，利用簇倾斜代数（定义为簇范畴中倾斜对象的自同态代数）与相应的拟倾斜代数之间的精确关系，证明了簇范畴$\mathcal{C(H)}$与$\mathcal{C(H')}$是三角等价的当且仅当$\mathcal{H}$与$\mathcal{H}'$是导出等价的.作为应用，我们通过取$\mathcal{H}$为有值Dynkin算子的表示范畴，取$T$为BGP-倾斜（或APR-倾斜），给出了Fomin-Zelevinsky定义的“截断单反射”在相应类型的几乎正根集[FZ 2，FZ 5]上的一个实现.
Tilting theoretical approach to moduli spaces over preprojective algebras
利用预投射代数$Lambda$上的倾斜理论研究了$Lambda$ -模的模空间.本文利用$Lambda$上的倾斜模定义了半稳定模的范畴，并给出了它们之间的一个等价，即所谓的反射函子。此外，我们证明了等价性诱导了模空间之间代数簇的同构。特别地，我们研究了当模空间与Kleinian奇点相关时的情形。本文推广了Crawley-Boevey的一个结果，它是Ito-Nakamura型McKay对应的又一证明。
Derived equivalences of triangular matrix rings arising from extensions of tilting modules
一个三角形矩阵环A由三元组（R，S，M）定义，其中R和S是环，M是S-R-双模.在本文的主要定理中，我们证明了：如果T是倾斜S-模，则在M作为S-模的某些同调条件下，可以将T推广到A上的倾斜复形，从而导出A与由（S '，R，M'）指定的另一个三角形矩阵环之间的导出等价，其中环S'和R-S'-双模M'仅依赖于T和M，并且S'导出等价于S。注意，在环R上不需要任何条件。当S是一个有限整体维数的Artin代数且M是一个S-模的逆生成时，这些条件都得到了满足。在这种情况下，有$(S',R,M')=(S,R,DM)$，其中D是可生成S-模范畴上的对偶。当S是任意的，M作为S-模有有限投射分解，且对所有$n>0$有$\Ext^n_S(M,S)=0$时，它们也是成立的。本例中为$(S',R,M')=(S,R,\Hom_S(M,S))$。
Holomorphic Lagrangian branes correspond to perverse sheaves
设X是紧致复流形，$D_c^b(X)$是$X$上可构造层的有界导出范畴，$Fuk(T^*X)$是$T^*X$的有界导范畴. $Fuk(T^*X)$中的拉格朗日膜是全纯的，如果底层拉格朗日子流形在$T^*X_{\mathbb{C}}$中是复解析的，$X$的全纯余切丛。我们证明了在[NaZa09]和[Nad09]中建立的$D^b_c(X)$和$DFuk(T^*X)$之间的拟等价性下，具有适当阶化的全纯拉格朗日膜对应于反常层.
Derived equivalence classification of m-cluster tilted algebras of type A
我们利用A型$m$ -簇复形的极大面把A型m-簇倾斜代数描述为具有关系的箭图。然后，我们分类的连通组件的m-集群倾斜代数A型导出等价使用倾斜复合物直接相关的组合的m-集群复杂的A型。这推广了Buan和Vatne的一个结果。
Elementary equivalences for blocks with normal elementary abelian defect group of rank 2
本文研究了秩为2的正规初等阿贝尔亏群的块在 Okuyama 初等等价下的行为，给出了初等倾斜复形不可约分量间齐次映射的完整目录，并证明当子集为区间时该目录足以描述所有齐次映射。
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
We study necessary and sufficient conditions for a dg bimodule to yield triangle equivalences between (quotients of) the corresponding derived categories. This is related to recent work by Bazzoni-Mantese-Tonolo, Yang, Angeleri Hügel-Koenig-Liu, Chen-Xi, Bazzoni-Pavarin,... on large tilting modules, homological epimorphisms and recollements.
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However,…
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。
Quasi-equivalence of relabeled positroid cluster structures
本文证明了Fraser--Sherman-Bennett猜想，通过构造倾斜模和Frobenius范畴化，建立了重标记正丛簇结构的拟等价性，并利用导出等价和幺模变换证明了坐标环上的恒等映射是拟簇同构。
For a certain Wakamatsu-tilting bimodule over two artin algebras $A$ and $B$, Wakamatsu constructed an explicit equivalence between the stable module categories over the trivial extension algebra of $A$ and that of $B$. We prove that Wakamatsu's functor is a triangle functor, thus a triangle equivalence.
To a big n-tilting object in a complete, cocomplete abelian category A with an injective cogenerator we assign a big n-cotilting object in a complete, cocomplete abelian category B with a projective generator, and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of A and B. Under various assumptions on A, which cover a wide range of examples (for instance, if A is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that B is the abelian category of contramodules over a topological rin…
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…
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