Jianmin Chen, Yanan Lin, Shiquan Ruan
We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2,2,2,2;λ), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra associated with X of type (2,2,2,2).
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Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
We introduce the notions of Gorenstein projective $τ$-rigid modules, Gorenstein projective support $τ$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $τ$-tilting modules. More precisely, for an algebra $Λ$, We prove that there is a bijection between the set of Gorenstein projective support $τ$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$τ$-tilting finite algebras and show that $Λ$ is CM-$τ$-tilting finite if and only if…
Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $τ$-tilting $B$-module is a support $τ$-tilting $A$-module; and, conversely, the restriction of a given support $τ$-tilting $A$-module is a support $τ$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $τ$-tilting modules.
We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.
We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category O.
In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $Mod(C)$, from a skeletally small preadditive category $C$ to the category of abelian groups. We introduced the notion of a a generalized tilting category $T$, and extended Happel's theorem to $Mod(C)$. We proved that there is an equivalence of triangulated categories $D^b (Mod(C))\cong Db (Mod(T))$. In the case of dualizing varieties, we proved a version of Happel's theorem for the categories of finitely presented functors. We also proved in this paper, that there exists a relation b…
In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. These results are well-known by several specialists, but seem difficult to find in the existing literature.
Tilting Objects via Recollements and $p$-Cycles on Weighted Projective Lines
该研究提出借助recollement在三角范畴中构造倾斜对象的新方法，利用$p$-循环给出加权射影线向量丛稳定范畴中线丛与扩张丛的显式表达式，证明相关倾斜对象并构造出新的倾斜对象。
We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.
We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each reduced expression $\bf{w}$ of $w$ and give a sufficient condition on $\bf{w}$ such that $M(\bf{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\bf{w})$. Moreover, we compare it with a triangle equivalence given by Amiot-Reiten-Todorov for a cluster category.
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.
$n\mathbb{Z}$-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…
An open question for $d$-tilting bundles on Geigle-Lenzing projective spaces
本研究利用迭代2-APR突变构造了型为$(2,2,p,q)$的Geigle-Lenzing射影空间上的一类2-倾斜丛，并以此为相关公开问题提供了反例，推进了对应空间的倾斜丛研究。
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…
D.Happel and L.Unger defined a partial order on the set of basic tilting modules. We study the poset of basic pre-projective tilting modules over path algebra of infinite type. First we will give a criterion for Ext-vanishing for pre-projective modules. And by using this we will give a combinatorial characterization of the poset of basic pre-projective tilting modules. Finally we will see a structure of a pre-projective part of tilting quivers.
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.
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 $\mathscr{E}$.
We study the t-structure induced by an n-tilting module T in the derived category D(R) of a ring R. Our main objective is to determine when the heart of the t-structure is a Grothendieck category. We obtain characterizations in terms of properties of the module category over the endomorphism ring of T and as a main result we prove that the heart is a Grothendieck category if and only if T is a pure projective $R$-module.
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $τ^2 T\cong T$, where $τ$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we…
We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some "Koszul-like" assumptions (which we prove are satisfied for example for the blocks of the category $\mathcal{O}$), we obtain a non-degenerate pairing between certain graded homomorphism and graded extension …
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.
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and …
Projectively Wakamatsu Tilting Modules over One-Point Extensions
本文研究单点延伸代数上的射影Wakamatsu投射模，通过添加新投射模提升模，并在特定同调条件下证明提升过程保持突变关系。对表示有限代数的源点延伸，给出PWT模的完整分类，建立PWT(Γ)与PWT(Λ)和RPWT(Λ, S_i)的双射。
We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras, whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generali…
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…
Let T be a locally finite triangulated category with an autoequivalence F such that the orbit category T/F is triangulated. We show that if X is an m-cluster tilting subcategory, then the image of X in T/F is an m-cluster tilting subcategory if and only if X is F-perodic. We show that for path-algebras of Dynking quivers δone may study the periodic properties of n-cluster tilting objects in the n-cluster category Cn(kδ) to obtain information on periodicity of the preimage as n-cluster tilting subcategories of Db(kδ). Finally we classify the periodic properties of all 2-cluster tilting objects …
We show that there is a reflection type bijection between the indecomposable summands of two multiplicity free tilting modules $X$ and $Y$. This bijection fixes the common indecomposable summands of $X$ and $Y$ and sends indecomposable projective (resp., injective) summands of exactly one module to non-projective (resp., non-injective) summands of the other. Moreover, this bijection interchanges the two possible non-isomorphic complements of an almost complete tilting module.
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.
Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.
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…
Let $T$ be a $1$-tilting module whose tilting torsion pair $({\mathcal T}, {\mathcal F})$ has the property that the heart ${\mathcal H}_t$ of the induced $t$-structure (in the derived category ${\mathcal D}({\rm Mod} \mbox{-} R)$ is Grothendieck. It is proved that such tilting torsion pairs are characterized in several ways: (1) the $1$-tilting module $T$ is pure projective; (2) ${\mathcal T}$ is a definable subcategory of ${\rm Mod} \mbox{-} R$ with enough pure projectives, and (3) both classes ${\mathcal T}$ and ${\mathcal F}$ are finitely axiomatizable. This study addresses the question of …
Motivated by $τ$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $Λ$ with action by a finite group $G$, we introduce the notion of $G$-stable support $τ$-tilting modules. Then we establish bijections among $G$-stable support $τ$-tilting modules over $Λ$, $G$-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective $Λ$-modules, and $G$-stable functorially finite torsion classes in the category of finitely generated left $Λ$-modules. In the case when $Λ$ is the endomorphism of a $G$-stable cluster-tilt…
The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposabl…
In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and …
In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided …
We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tu…
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。
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…
Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.
Let $Λ$ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $τ$-tilting $Λ$-modules. In this paper, we will show that it can be restricted to $τ$-tilting modules.
In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke categor…
Let $k$ be a field. In this short note we give an example of a $2$-abelian $k$-category, realized as a $2$-cluster-tilting subcategory of the category $\operatorname{mod}\,A$ of finite dimensional (right) $A$-modules over a finite dimensional $k$-algebra $A$, for which the naive idea for constructing its "bounded derived category" as $2$-cluster-tilting subcategory of the bounded derived category of $\operatorname{mod}\,A$ cannot work.
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 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…
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。
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
Tilting representations of finite groups of Lie type
本文研究了有限型李群的倾斜表示，通过Deligne-Lusztig范畴O构造了生成集，并计算了特征标以验证Dudas-Malle的猜想。
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…
On the Wakamatsu tilting conjecture
本文证明广义 Auslander–Reiten 猜想蕴含 Wakamatsu 倾斜猜想，并利用有限投射维数张量忠实倾斜模的投射性，确立其在左 Artin 局部环及有限群群环上成立。
Tilting objects in periodic triangulated categories
本文在 $m$-周期三角范畴中引入 $m$-周期倾斜对象概念，并证明具有该对象的范畴在同调假设下三角等价于代数的 $m$-周期导出范畴，同时构建了自内射代数稳定范畴与遗传代数导出范畴间的三角等价。
We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate $t$-structure whose heart has enough projectives.
We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster algebra, as done by Fu and Keller, we exploit intrinsic properties of $\mathcal{C}$. For this purpose, we define an ice quiver associated to each cluster tilting object in $\mathcal{C}$. In Dynkin case $\mathbb{A}_n$, we also prove that the mutation class of the ice quiver associated to the cluster tilting object given by the direct sum of all projective objects …
In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded d…
Geometric-combinatorial approaches to tilting theory for weighted projective lines
本研究针对(2,2,n)型加权射影线的凝聚层范畴，构建带标记点柱面的几何-组合模型，建立不可分解层与斜曲线、伪三角剖分与倾斜层的双射，证明倾斜图连通性，为倾斜理论提供新组合视角。
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…
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…
ACM tilting bundles on a Geigle-Lenzing projective plane of type $(2,2,2,p)$
本文研究型 $(2,2,2,p)$ 的 Geigle–Lenzing 射影平面上的 ACM 倾斜丛，刻画线丛倾斜丛、给出构造程序，并获得其分类结果。
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