Stephen Zito
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. In particular, we study which $τ$-rigid C-modules are also $τ$-rigid B-modules.
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We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.
We give an example of a cluster-tilted algebra A with quiver Q, such that the associated cluster algebra has a denominator vector which is not the dimension vector of any indecomposable A-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra A, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid A-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomp…
We provide an affirmative answer for the question raised almost twenty years ago concerning the characterization of tilted artin algebras by the existence of a sincere finitely generated module which is not the middle of a short chain.
This is an appendix to the Handbook of Tilting Theory, edited by Angeleri-Huegel, Happel and Krause, to be published soon. Part 1 of the appendix provides an outline of the core of tilting theory. Part 2 is devoted to topics where tilting modules and tilted algebras have shown to be relevant. Both Parts 1 and 2 contain historical annodations and reminiscences. The final Part 3 is a short report on some striking recent developments which are motivated by the cluster theory of Fomin and Zelevinsky. In particular, we will guide the reader to the basic properties of cluster tilted algebras, to the…
The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.
We determine the minimal lower bound $n$, with $n \geq 1$, where the $n$-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of $m$, $m \ge 2$, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the $(m+1)$-th power of …
We study the projective dimension of finitely generated modules over cluster-tilted algebras End(T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End(T)-modules are of the form Hom(T,M) for some object M in C, and since End(T) is Gorenstein of dimension 1, the projective dimension of Hom(T,M) is either zero, one or infinity. We define in this article the ideal I_M of End(T[1]) given by all endomorphisms that factor through M, and show that the End(T)-module Hom(T,M) has infinite projective dimension precisely when I_M is non-zero. Moreover, we apply the…
Let $Λ$ be a hereditary algebra, $B_0=End_Λ(T_0)$ be a tilted algebra. We will construct tilting $B_0$-modules from tilting $Λ$-modules and use this result to show how tilting quivers of BB-tilted algebras can be obtained from those of $Λ$
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.
On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type
本文证明了驯顺型cluster-tilted代数上不同的不可分解$τ$-刚性模具有不同的维数向量，并以此为基础建立了驯顺型cluster algebras的弱版本分母猜想，同时利用Iyama-Yoshino构造描述了相关的三角范畴subfactors。
In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras.
We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq \module(A,\dd)$, the moduli space $\M(C)^{ss}_θ$ is either a point or just $\mathbb P^1$ whenever $θ$ is an integral weight for which $C^s_θ\neq \emptyse…
We characterize the indecomposable transjective modules over an arbitrary cluster-tilted algebra that do not lie on a local slice, and we provide a sharp upper bound for the number of (isoclasses of) these modules.
Every cluster-tilted algebra $B$ is the relation extension $C\ltimes \text{Ext}^2_C(DC,C)$ of a tilted algebra $C$. A $B$-module is called induced if it is of the form $M\otimes_C B$ for some $C$-module $M$. We study the relation between the injective presentations of a $C$-module and the injective presentations of the induced $B$-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced $B$-module. In the case where the $C$-module, and hence the $B$-module, is projective, our construction yields an injective resolution. In par…
We propose a new approach to study the relation between the module categories of a tilted algebra $C$ and the corresponding cluster-tilted algebra $B=C\ltimes E$. This new approach consists of using the induction functor $-\otimes_C B$ as well as the coinduction functor $D(B\otimes_C D-)$. We show that $DE$ is a partial tilting and a $τ$-rigid $C$-module and that the induced module $DE\otimes_C B$ is a partial tilting and a $τ$-rigid $B$-module. Furthermore, if $C=\text{End}_A T$ for a tilting module $T$ over a hereditary algebra $A$, we compare the induction and coinduction functors to the Bu…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If $C$ is a representation-finite cluster-tilted algebra, then the indecomposable $C$-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra $C$, we are going to describe the dimension vectors of the inde…
We study the projective dimensions of the restriction of functors Hom(-,X) to a contravariantly finite rigid subcategory T of a triangulated category C. We show that the projective dimension of Hom(-,X)|T is at most one if and only if there are no non-zero morphisms between objects in T[1] factoring through X, when the object X belongs to a suitable subcategory of C. As a consequence, we obtain a characterisation of the objects of infinite projective dimension in the category of finitely presented contravariant functors on a cluster-tilting subcategory of C.
For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.
Quot spaces in tilted hearts and Hall algebra modules
本文通过对应叠构造了曲线上固定斜率的半稳定凝聚层的Hall代数在倾斜心脏中某些商空间的导出范畴上的双边范畴作用。
We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch.
We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander-Reiten component then its representation dimension is equal to three.
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the bounded derived category $\mathcal{D}^{b}(H)$, and of the cluster-tilting objects, in terms of the category $\modΓ$\ of finitely generated modules over a suitable tilted algebra $% Γ.$ Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $τ$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $τ$-tilting theory. Afterwards we define $τ$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $τ$-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with $τ$-slices.
Derived equivalences of self-injective 2-Calabi--Yau tilted algebras
本文证明2-Calabi--Yau稳定范畴中具有自入射自同态代数的极大刚性对象的自同态代数导出等价，并给出诱导该等价的双边倾斜复形描述。
Let $Λ$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $Λ$ form a maximal forward hom-orthogonal sequence of $Λ$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $Λ$. Thus we give a proof of Igusa-Todorov's conjecture.
The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension -- rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types $A$ and $E$ are given.
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.
We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of Geiss-Leclerc-Schroer. In addition, we prove a general result about relative 3-Calabi-Yau duality ove…
In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections. We then give an algorithmic procedure for constructing the tubes of a cluster-tilted algebra of euclidean type. Our main result characterizes quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type.
We define an operation which associates to a pair (B,M) where B is a cluster-tilted algebra and M is a B-module which lies in a local slice of B, a new cluster-tilted algebra B'. In terms of the quivers, this operation corresponds to adding one vertex (and arrows).
Weakly holonomic equivariant $\mathcal{D}$-modules on rigid analytic spaces
针对p进李群作用下的刚性解析空间$\u0001d49f$-模，研究者建立余可允许等变模的维数理论，提出弱全纯等变$\u0001d49f$-模范畴并研究其对偶与运算保持性质。
Bijections between $τ$-rigid modules
本文研究两个基本τ-倾斜模的不可分解直和项，构造并描述它们之间的一种特殊双射。
Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type. In type A, we establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel's Theorem. Also,…
We show that the quotient of the continuous cluster category $\mathcal C_π$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-Chapoton-Schiffler and Buan-Marsh-Reiten for cluster categories to the continuous cluster category of type $A$. These results will be generalized in a series of forthcoming joint papers of the two authors with Job Rock.
Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to give new results about the radical series of some tilting modules for $SL_4(K)$, where $K$ is a field of positive characteristic.
It is well-known that any maximal Cohen-Macaulay module over a hypersurface has a periodic free resolution of period 2. Auslander, Reiten and Buchweitz have used this periodicity to explain the existence of periodic projective resolutions over certain finite-dimensional algebras which arise as stable endomorphism rings of Cohen-Macaulay modules. These algebras are in fact periodic, meaning that they have periodic projective resolutions as bimodules and thus periodic Hochschild cohomology as well. The goal of this article is to generalize this construction of periodic algebras to the context of…
We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin-Zelevinsky mutation of the extended exchange matrix, (2) a forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author's work with P.J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers …
We prove that all gentle 2-Calabi-Yau tilted algebras (over an algebraically closed field) are Jacobian, moreover their bound quiver can be obtained via block decomposition. Related families of gentle $(m+1)$-Calabi-Yau tilted algebras are the $m$-cluster-tilted algebras of type $\mathbb{A}$ and $\widetilde{\mathbb{A}}$. For these algebras we prove that a module $M$ is stable Cohen-Macaulay if and only if $Ω^{m+1} τM \simeq M$.
$τ$-rigid modules are essential in the $τ$-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $τ$-rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is $τ$-rigid. Finally, we give a $τ$-tilting theorem on homological dimension which is an analog to that of classical tilting modules.
Mutating ordered $τ$-rigid modules with applications to Nakayama algebras
本文通过 TF-有序 $τ$-刚性模解释 $τ$-例外序列的突变，并利用显式组合描述证明 Nakayama 代数上突变的传递性。
Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.
Sequences of ICE-closed subcategories via preordered $τ^{-1}$-rigid modules
本文通过引入cogen-预序τ^{-1}-刚性模，建立了其与无挠类区间序列的双射，并结合Sakai的结果，分类了反变有限ICE-序列及同调决定通道的中间t-结构。
Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid objects in the corresponding orbit category C(Q), in terms of bipartite noncrossing graphs (with loops) in a circle. We also describe the endomorphism algebras of the maximal rigid objects, and we prove that a certain class of these algebras are iterated tilted algebras of type A.
Given a finite dimensional algebra $A$ over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra $A$ and the induced morphism in the module category of the endomorphism algebra of a tilting $A$-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.
Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in Kb(proj Λ), nu-stable support tau-tilting Λ-modules and nu-stable functorially finite torsion classes in mod Λ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in C if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category C. As an application, we give a necessary condition such that selfinjective algebras are 2-CY ti…
For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_λ in commutative ring Z[Λ], we show that f_λ is just the formal character of the Weyl module V(λ) if f_λ satisfy several natural conditions. Hence we give a necessary and sufficient condition for constructing a family of g_l-modules from a family of g_{l-1}-modules.
We define coadmissible equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces and relate them to admissible locally analytic representations of semisimple $p$-adic Lie groups.
Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $Ω$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to …
We give an overview of the theory of $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces and its applications to admissible locally analytic representations of $p$-adic Lie groups.
We prove that the category of coadmissible D-cap-modules on a smooth rigid analytic space supported on a closed smooth subvariety is naturally equivalent to the category of coadmissible D-cap-modules on the subvariety, and use this result to construct a large family of pairwise non-isomorphic simple coadmissible D-cap-modules.
Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.
We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-Økland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.
On the global dimension three endomorphism algebras of the minimal generator-cogenerator
本文研究极小生成-余生成子自同态环整体维数为三的代数（表示遗传代数），证明其无挠有限性及特定条件下的同调性质，并给出其属于特定倾斜代数类的充分条件。
We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.
In this paper, we show that for an algebra $Λ$ with radical square zero and an indecomposable $Λ$-module $M$ such that $Λ$ is Gorenstein of finite type or $τM$ is $τ$-rigid, $M$ is $τ$-rigid if and only if the first two projective terms of a minimal projective resolution of $M$ have no on-zero direct summands in common. We also determined all $τ$-tilting modules for Nakayama algebras with radical square zero. Moreover, by giving a construction theorem we show that a basic connected radical square zero algebra admitting a unique $τ$-tilting module is local.
An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$
针对Dynkin型𝔸ₙ箭图路代数的极大几乎刚性模，引入非标准正合结构将其转化为新结构下的极大刚性模（等价于倾斜模），证明该结构下模范畴为0-Auslander范畴，并讨论向𝔻型箭图与gentle代数的推广。
Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B…
We characterize the syzygies and co-syzygies over 2-Calabi-Yau tilted algebras in terms of the Auslander-Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen-Macaulay modules, the representation dimension of algebras and the Igusa-Todorov functions. In particular, we prove that the Igusa-Todorov dimensions of d-Gorenstein algebras are equal to d. For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen-Macaulay category in terms of tagged arcs in the punctured disc. We also describe th…
A characterization of 2-fold torsion classes induced by $τ$-rigid modules
本文定义阿贝尔范畴的$n$重挠（无挠）类，并刻画有限维代数上由$τ$-刚性模诱导的2重挠类。
Let $Q$ be a finite quiver of Dynkin type and $Λ=Λ_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\mathcal {T}_Λ$ be the mutation graph of maximal rigid $Λ$ modules. Geiss, Leclerc and Schr$\ddot{\rm o}$er conjectured that $\mathcal {T}_Λ$ is connected, see [C.Geiss, B.Leclerc, J.Schröer, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $Λ$ is of representation finite type or tame type. Moreover, we also prove that $\mathcal {T}_Λ$ is isomorphic to the tilting graph of ${…
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