Claudia Chaio, Victoria Guazzelli, Pamela Suarez
Let $A$ be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of $Q_A$ are suficient to be consider in order to compute the nilpotency index of the radical of the module category of $A$. In many cases, we give a formula to compute such index taking into account the ordinary quiver of the given algebra.
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A generalization of the nilpotency index of the radical of the module category of an algebra
本研究推广CGS的结果，确定了代数闭域上有限维表示有限的单项式代数与toupie代数中，计算其模范畴根的幂零指数时$Q_A$需考虑的顶点，放宽了Auslander-Reiten箭图需为具长度分支的限制。
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.
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 …
Koszul duality and covering theory are combined to realise the bounded derived category D of an algebra with radical square zero as a certain orbit category of the bounded derived category of finitely presented representations of an associated infinite quiver. As a consequence, the possible shapes of the connected components of the Auslander--Reiten quiver of D are described.
In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly s…
Let $\La$ be an elementary locally bounded linear category over a field with radical squared zero. We shall show that the bounded derived category $D^b(\ModbLa)$ of finitely supported left $\La$-modules admits a Galois covering which is the bounded derived category of almost finitely co-presented representations of a gradable quiver. Restricting to the bounded derived category $D^b({\rm mod}^b\hspace{-2pt}\La)$ of finite dimensional left $\La$-modules, we shall be able to describe its indecomposable objects, obtain a complete description of the shapes of its Auslander-Reiten components, and cl…
On representations of algebras with radical square zero
本文研究了具有零平方根理想代数的表示与物种表示之间的对应关系，并展示了如何通过物种的Auslander-Reiten quiver来重建代数的Auslander-Reiten quiver。
Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical ${\mathfrak r}$ and nilpotent radical ${\mathfrak n}=[{\mathfrak g},{\mathfrak r}]$. Given a finite dimensional ${\mathfrak g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\subset U({\mathfrak n}^m)=U$ is defined so that $U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U$, $U({\mathfrak n}^2)/U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U/U({\mathfrak n}^1)$, and so on. We say that $U$ is linked if each factor of it…
We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-r…
In abelian categories like the category of R-modules and even in the category S-Act 0 of S-acts with a unique zero, idempotent radicals and torsion theories are equivalent, and the τ-torsion and τ- torsion free classes of a torsion theory τ are closed under coproducts. These are not necessarily true in the category S-Act of S-acts. In this paper, we prove that torsion theories are equivalent with the Kurosh- Amitsur radicals. We, also, show that the class of Kurosh-Amitsur radicals is a reflective subcategory of Hoehnke radicals, as a poset.
Let $Λ$ be an artin algebra. We obtain that $Λ$ is syzygy-finite when the radical layer length of $Λ$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the category $\mod Λ$ of finitely generated right $Λ$-modules in terms of the projective of certain class of simple right $Λ$-modules and also get the left big finitistic dimension conjecture holds.
To an artin algebra with radical square zero, a regular algebra in the sense of von Neumann and a family of invertible bimodules over the regular algebra are associated. These data describe completely, as a triangulated category, the singularity category of the artin algebra. A criterion on the Hom-finiteness of the singularity category is given in terms of the valued quiver of the artin algebra.
On the stable radical of the module category for special biserial algebras
本文证明了关于特殊双列代数模范畴稳定根的猜想，给出了计算稳定根的算法，并构造了具有任意给定稳定根的 tame 代数。
We describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description of left, right, and two-sided cells.
The radical of functorially finite subcategories
本文研究Artin代数函子有限子范畴的根，证明其有加法生成元当且仅当无限根消失，并给出有限表示型的多种判据及根的幂零指数。
The Tannakian radical and the mantle of a braided fusion category
本文定义了辫子融合范畴的Tannakian根及其对应的局部化mantle，证明了mantle具有平凡Tannakian根并允许一个作为完备不变量的典范中心扩张，同时研究了此类reductive范畴的性质、结构与分类结果。
Various generalizations of the concept of injectivy, in particular injectivy with respect to a specific class of morphisms, have been intensively studied throughout the years in different categories. One of the important kinds of injectivy studied in the category R-Mod of R-modules is $τ$-injectivy, for a torsion theory $τ$, or in the other words r-injectivy, where r is the induced idempotent radical by $τ$. In this paper, we introduce the notion of r-injectivy, for a Hoehnke radical r in the category S-Act of S-acts and we study the main properties of this kind of injectivy. Indeed, we show t…
Let $Q$ be a finite quiver and $Λ$ be the radical square zero algebra of $Q$ over a field. We give a full and dense functor from the category of reduced differential projective modules over $Λ$ to the category of representations of the opposite of $Q$. If moreover $Q$ has oriented cycles and $Q$ is not a basic cycle, we prove that the algebra of dual numbers over $Λ$ is not virtually Gorenstein.
We show that for an artin algebra $Λ$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod-$Λ$ being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical.
Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical
本文研究了在几何参数空间源于 Levi 子群作用于抛物子代数阿贝尔幂零根基的情形下，对 Lusztig 幂零表示范畴中具有整数无穷小特征的酉表示进行分类，并将其组织成微局部 Arthur 包，作为验证相关酉表示猜想描述的测试用例。
Thick subcategories over weakly symmetric algebras with radical cube zero
本文分类了根立方零弱对称代数的厚子范畴，发现谱半径为2时可由Serre子范畴分类，否则仅平凡厚子范畴，并首次构造了非完全交的交换诺特Gorenstein优势局部环。
Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization
本文研究了根方零Nakayama代数上双模2-范畴的简单传递2-表示分类问题，通过定义2-表示的局部化并利用构造的泛性质，证明了具有有限顶点的简单传递2-表示均等价于胞腔2-表示的局部化，从而实现了完整分类。
A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category $\mathcal {O'}$ over complex generalised reductive Lie algebras. Then we make a preliminary research on the highest weight modules and the projective modules in this new category $\mathcal {O'}$, and generalize some conclusions in the classical case. As a critical difference from the complex semisimple Lie algebra case, we prove that there is no projective module…
We study the homotopy category $ K(\Inj A)$ of all injective modules over a finite dimensional algebra $A$ with discrete derived category. We give a classification of the indecomposable objects of $ K(\Inj A)$ for any radical square zero self-injective algebra $A$. In particular, every indecomposable object is endofinite.
The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the module category in terms of the induced natural transformations between representable functors on this graded category. When the ground ring is a perfect field, and the given morphism behaves nicely with respect to covering theory (as do irreducible morphisms with indecomposable domain or indecomposable codomain), it is shown that the degree of the morphism is f…
Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, that is the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$ that are bi-invariant for $K^1$ and whose supports are a finite union of $K^1$-double cosets. We produce a presentation by generators and relations of this algebra. Finally we prove that the level-$0$ subcategory o…
For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic c…
Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.
Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of $\mathscr{R}_R(G)$ is equivalent to a level-$0$ block (a block in which every object has non-zero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of $\mathscr{R}_R(G')$, where $G'$ is a direct product of groups of …
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…
We describe the representation theory of finitely generated indecomposable modules over artin algebras which do not lie on cycles of indecomposable modules involving homomorphisms from the infinite Jacobson radical of the module category.
On Riedtmann's well-behaved functors and applications to composites of irreducible morphisms
本文综述了 mesh 范畴与 Riedtmann 良态函子的相关结果，并说明如何利用这些函子研究不可约态射复合非零且落入根的第 (n+1) 次幂的判定问题。
We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
Let S(n) be the category of invariant subspaces of nilpotent operators with nilpotency index at most n. Such submodule categories have been studied already in 1936 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra of type A_n; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Lecl…
We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.
Separable commutative rings in the stable module category of cyclic groups
本文刻画有限循环 p-群稳定模范畴中的可分交换环对象，并给出任意有限群两类范畴中 Kelly 根的张量闭包。
We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\su…
We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.
This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$ can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to …
A tensor-triangular property for categories of representations of restricted Lie algebras
本文基于上同调支撑与张量三角几何为限制李代数定义了一种新性质，探讨了群概结构变化下极小根式厚张量理想的 Green 环结构，并猜想该性质等价于具有有限或驯顺表示型。
Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras
本文研究拟约化李超代数的Whittaker范畴，证明Backelin函子的不可约性，构造主有限W-超代数并建立Skryabin型等价，给出Soergel结构定理的超类比。
Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In this paper, we first study some basic properties of pseudo-kernel of morphisms in $C$. When $C$ has pseudo-kernels, mod($C$) is abelian and then, in this case, we study radical functors, half exact functors, left exact functors and injective objects in mod($C$). At last, we extend the results for Mod($C$).
The dg Leavitt algebra, singular Yoneda category and singularity category
本文证明任意由箭图与关系给出的有限维代数的 dg 奇异性范畴拟等价于 dg Leavitt 路径代数的完美 dg 导出范畴，通过引入奇异 Yoneda dg 范畴并利用显式 dg 局部化与商范畴操作实现该结果。
We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $Λ$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $Λ$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(Λ)$ contains a $d$-cluster tilting subcategory that is clos…
Representation theory of hereditary artin algebras of finite representation type
本文研究了有限表示型遗传artin代数的表示理论，通过确定Auslander-Reiten箭图中的所有hammock，提出了一种基于ext-quiver的有效构造方法，并计算了相关范畴中不可分解对象的数目及根的幂零指数。
We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.
We give an alternative proof to the fact that if the square of the infinite radical of the module category of an Artin algebra is equal to zero then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of $Δ$-good modules of a quasi-hereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type.
Simple algebras and exact module categories
本文验证了Etingof-Ostrik猜想：有限张量范畴中代数对象精确当且仅当为简单代数的有限直积，并引入Jacobson根类比，应用于不可压缩有限对称张量范畴。
When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimple for almost all $n$. In this work, using cellular methods, we give explicit generating functions for the dimensions of all the simple $TL_n(q)$-modules. Jones showed that if the order $|q^2|=\ell$ there is a canonical symmetric bilinear form on $TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in TL_{\ell-1}(q)\subseteq TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. Although th…
A remark on Ext groups for motives with maximal unipotent radicals
本文研究具有极大幂单根的动机的Tannakian子范畴中的Ext^1群，并应用于1-动机和混合Tate动机，推导出Grothendieck周期猜想的若干推论。
The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category O is equivalent to knowledge of the Kazhdan-Lusztig character of the irreducible object (we use this observation in joint work with Fishel-Manosalva). We then explore the extent to which Cherednik algeb…
Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or group scheme. We show that the negative Tate cohomology ring $\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of $kG$-modules for $G$ an extension of $H$. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero compone…
We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms $M_0 \stackrel{f_1}{\rightarrow} M_1 \to \cdots \to M_{r-1} \stackrel{f_r}{\rightarrow} M_r=M_0$ between indecomposable modules in mod A which do not belong to the infinite radical of mod A. In particular, we describe completely the structure of an arbitrary module category mod A whose all cycles are finite. The main structural results of the paper allow to derive several interesting combinato…
A ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups of order 2 and 3. We compare this to the situation in the derived category of a commutative ring. We also determine for which groups G the second power of the Jacobson radical of kG is stably isomorphic to a suspension of k.
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence…
We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $ψ$ be an arbitrary smooth character of $U$. Let $S \subset Irr(G)$ be a Zariski dense collection of irreducible representations of $G$. Then the span of the Bessel distributions $B_π$ attached to representations $π$ from $S$ is dense in the space $\mathcal S^*(G)^{U\times U,ψ\times ψ}$ of all $(U\times U,ψ\times ψ)$-equivariant distributions on $G.$ We base our proof on the following results: 1. The category of smooth…
In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $Λ${\rm -mod} that contain the Jacobson radical of $Λ$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}…
It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. More specifically, if k is a field whose characteristic divides the order of G, and $Φ$ is the ideal of phantom morphisms in the stable category k[G]-$\underline{\rm Mod}$ of modules over the group algebra k[G], then $Φ^{n-1} = 0,$ where n is the nilpotency index of the Jacobson radical J of k[G]. If $R$ is a semiprimary ring, with $J^n =0,$ and $Φ$ denotes the phantom ideal in the module category R-Mod, then $Φ^n$ is the ideal of morphisms that factor through a projective module.…
For a finite quiver $Q$ without sinks, we consider the corresponding finite dimensional algebra $A$ with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective $A$-modules. We call such a generator the injective Leavitt complex of $Q$. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of $Q$ is quasi-isomorphic to the Leavitt path algebra of $Q$. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra wi…
Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $Λ$-modules and the representation dimension as well as the right global dimension of $Λ$. In particular, we give an upper bound for …
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