Ryo Kanda
For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object.
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For an exact category $\mathcal{E}$, we study the Butler's condition "AR=Ex": the relation of the Grothendieck group of $\mathcal{E}$ is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that $\mathcal{E}$ has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker…
We introduce systematic methods to construct Grothendieck categories from colored quivers and develop a theory of the specialization orders on the atom spectra of Grothendieck categories. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster's result in commutative ring theory. We also show that there exists a Grothendieck category which has empty atom spectrum but has nonempty injective spectrum.
Module-theoretic approach to dualizable Grothendieck categories
本文通过模论方法证明可对偶Grothendieck范畴满足Ab6和Ab4*条件，结合Stefanich结果，完全回答了Brandenburg等人提出的修正猜想。
We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.
In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty}$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.
We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander-Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.
We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As…
Locally finitely presented Grothendieck categories with a flat generator
本文研究了局部有限呈现的格罗滕迪克范畴中平坦生成元的存在性问题，并通过反例和等价改述提供了否定解答。
Presentations of Grothendieck constructions
本文给出交换环$\Bbbk$上从小范畴到$\Bbbk$ -范畴的2-范畴函子的Grothendieck构造.
Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.
Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories
我们计算的Grothendieck群的某些2-Calabi-丘三角范畴自然出现在研究的联系之间的表示和Fomin-Zelevinsky的集群代数。在这个设置中，我们还证明了一个推广的Fomin-Zelevinsky的突变规则。
Classifying Grothendieck rings of pivotal fusion categories of rank four
本文对复数域上秩四枢轴融合范畴的Grothendieck环进行了完整分类，确定恰好十五个且均可酉范畴化，并利用新障碍与穷举方法给出维数上界3600。
Deconstructibility and the Hill lemma in Grothendieck categories
Grothendieck范畴的一个满子范畴称为可解构的，如果它由某个对象集的所有超限扩张组成。这个概念为结构理论和Grothendieck范畴的子范畴的近似构造提供了一个方便的框架。它还允许在Grothendieck范畴上的复范畴上构造模型结构和t-结构。在本文中，我们的目标是建立解构类的基本结果，并概述如何将这些应用在上述领域。这与Gillespie、Enochs、Estrada、Guil Asensio、Murfet、Neeman、Prest、Trlifaj等人最近的工作有关。
Finiteness of homological dimensions in triangulated categories
本文研究三角范畴中的有限维数、大有限维数和全局维数，建立重合范畴中这些维数与外范畴中维数的关系不等式，为扩展已知环的同调维数结果到弱近似三角范畴提供统一框架。
The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category $\C$ has Auslander-Reiten $\E-$triangles and the relations of the Grothendieck group $K_{0}(\C)$ are generated by the Auslander-Rieten $\E-$triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category $\C$ with a cluster tilting subcategory, the relations of the Grothendieck group $K_0(\C)$ are gener…
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}…
Wide subcategories and brick-finiteness for length categories
该研究将有限维代数砖块有限性特征推广到有限秩阿贝尔长度范畴，证明范畴砖块有限的条件，基于交换关系给出证明，还作为推论扩展了第一个砖块布劳尔 - 索尔猜想的有效性。
Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense (co)resolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the classification of dense triangulated subcategories of triangulated categories due to Thomason.
A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories
该研究从挠论角度刻画模层，分析有向范畴上Grothendieck拓扑的结构，分类特殊诺特EI范畴的拓扑并推广表示的局部自内射性质，给出群表示论应用方向。
A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an exam…
We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study tor…
Let $k$ be an algebraically closed field of characteristic zero, and let $\mathcal{C} = \mathcal{R}-mod$ be the category of finite-dimensional modules over a fixed Hopf algebra over $k$. One may form the wreath product categories $\mathcal{W}_{n}(\mathcal{C}) = (\mathcal{R} \wr S_n)-mod$ whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in $\mathcal{W}_{n}(\mathcal{C})$ allows one to demonstrate stability of structure constants in the Grothendieck rings (approp…
Co- Versus Contravariant Finiteness of Categories of Representations
这篇文章补充了作者最近的工作。(1)给出了有限维代数$Λ$的满子范畴$Λ\text{-mod}$的协变有限性失效的一个判据.将该准则应用于有限投射维数的有限生成$Λ$ -模范畴${\cal P}^{\infty}(Λ\rm{-mod})$，否定了$Λ\text{-mod}$中${\cal P}^{\infty}(Λ\rm{-mod})$是否总是协变有限的问题.第二部分讨论了${\cal P}^{\infty}(Λ\rm{-mod})$的逆变有限性。给出了一个例子，这个条件失败，失败，但是，可通过一个序列的一点扩展。特别是，这个例子表明，固化失败的逆变有限性的${\cal P}^{\infty}(Λ\rm{-mod})$通常涉及权衡其他所需的质量的代数。
Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category $\mathcal{W}_n(\mathcal{C})$. It was shown in \cite{Ryba} that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty$. The resulting "limit" ring, $\mathcal{G}_\infty^{\mathbb{Z}}(\mathcal{C})$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by \cite{Mori}. This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ which …
$t$-Structures with Grothendieck hearts via functor categories
研究带余积三角范畴中 $t$-结构的心成为 AB5 或 Grothendieck 范畴的条件，通过函子范畴将心表示为 Serre 商并分类保余积同调函子，证明标准良生成三角范畴中纯内射对象总余生成 $t$-结构。
Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture
本文研究局部有限呈现的纯半单格罗滕迪克范畴与左纯半单环之间的等价关系，并扩展了Auslander和Ringel-Tachikawa对应到左纯半单环类。
Grothendieck groups of repetitive cluster categories
本文计算 A_n 与 D_n 型重复丛范畴的 Grothendieck 群，推广经典丛范畴结果，并揭示参数 p 带来的结构及其与 AR 理论、Coxeter 变换的联系。
The first author constructed a $q$-parameterized spherical category $\sC$ over $\mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.
A finiteness property for braided fusion categories
我们介绍了编织融合范畴的有限性属性，描述了一个猜想，将具有这一特点的类别，并验证了一些重要的情况下的猜想。特别地，我们说一个范畴有F，如果相关的辫群表示因子在一个有限群上，并建议积分Frobenius-Perron维数的范畴正是具有性质F的范畴。
Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.
Constructions of Waldhausen categories via Grothendieck opfibrations
本文提出利用Grothendieck opfibration结合纤维与基范畴的Waldhausen结构构造总范畴的Waldhausen结构，并证明左根箭图表示范畴在特定条件下为Waldhausen范畴。
The Grothendieck groups of n-cluster categories of type A_{\infty}
本文研究 A_{\infty} 型 n-丛范畴的 Grothendieck 群，证明其同构于整数群，推广了 Murphy 的结论。
We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.
A category of FI type is one which is sufficiently similar to finite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck fibration to finite sets and injections. We begin by carefully reviewing the theory of fibrations of categories with motivating examples relevant to algebra and representation theory. We classify which functors between FI type categories are fibrations, and thus obtain sufficient conditions for an FI type category to be the result of a Grothendieck construction.
A note on Grothendieck groups of periodic derived categories
本文确定了周期导出范畴的 Grothendieck 群，证明了其结构取决于周期 $m$ 的奇偶性，并作为应用证明了严格周期倾斜对象的非同构直和项数量与对象选择无关。
We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander-Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander-Reiten triangles. An application is given to the homotopy category of perfect complex…
Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $(\mathcal D,H)$-Hopf modules over a Hopf comodule category $\mathcal D$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $Hom$ objects and higher derived func…
We introduce the notion of asymptotic Grothendieck groups for abelian and triangulated categories that are both AB4 and AB4*. We study when the asymptotic Grothendieck group of the heart of a triangulated category with a t-structure is isomorphic to the asymptotic Grothendieck group of the triangulated category itself. We also explain a connexion with the notion of topological Grothendieck group from Achar-Stroppel. Finally, we compute the asymptotic Grothendieck group of some positive (dg-)algebras having a cone bounded, locally finite dimension.
Completions of Grothendieck groups
对于某一类阿贝尔范畴，我们证明了如何理解无限投射分解（或者更一般地说，某些只在上面有界的链复形）的“欧拉示性数“，通过传递到Grothendieck群的一个适当的完备化。我们还证明了右正合函子（或其左导出函子）诱导这些完备Grothendieck群的连续同态，并讨论了来自范畴化的例子和应用。
Grothendieck-Verdier functors
引入Grothendieck-Verdier范畴间的函子，证明其2-范畴等价于带否定的线性分配范畴，并建立提升定理，应用于Hopf幺半群和代数模结构。
We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in sta…
Purity in compactly generated derivators and t-structures with Grothendieck hearts
本文在强稳定导子的紧生成三角范畴框架下，通过定向同伦余极限刻画纯三角与可定义子范畴，给出左非退化t-结构具有Grothendieck心的多组等价条件，并建立心的有限性与相关纯内射部分余倾斜对象纯度的关联。
Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fac…
On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness
研究Gorenstein维数为1的Iwanaga-Gorenstein代数的合冲范畴，通过幂等元商约化给出其稳定范畴的显式关系，并刻画范畴保持不变的条件，应用于两类CM有限代数。
Auslander-Reiten (n+2)-angles and local finiteness
本文针对$(n+2)$-角范畴，为偶数$n$的Auslander-Reiten (n+2)-角与局部有限性的蕴含关系建立充分条件，给出该开放问题的部分肯定答案。
The classification problem for simple $\mathfrak{sl}(2)$-modules leads in a natural way to the study of the category of finite rank torsion free $\mathfrak{sl}(2)$-modules and its subcategory of rational $\mathfrak{sl}(2)$-modules. We prove that the rationalization functor induces an identification between the isomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along theway several new re…
Let $k$ be a field and $\mathcal{C}$ a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of $\mathcal{C}$, denoted $K_0(\mathcal{C})$, can be expressed as a quotient of the split Grothendieck group of a higher-cluster tilting subcategory of $\mathcal{C}$. Assume that $n\geq 2$ is an even integer, $\mathcal{C}$ is $n$-Calabi Yau and has an $n$-cluster tilting subcategory $\mathcal{T}$. Then, for every indecomposable $M$ in $\mathcal{T}$, there is an Auslander-Reiten $(n+2)$-angle in $\mathcal{T}$…
Grothendieck groups and tilting objects
设C是代数闭域k上具有Serre函子的连通Noether遗传Abel Ext-有限范畴，具有有限维同态和扩张空间.利用math.RT/9911242中对这类范畴的分类，我们证明了：如果C有某个无限长的对象，则C的Grothendieck群是非线性生成的当且仅当C有一个倾斜的对象.
We define the Grothendieck group of an $n$-exangulated category. For $n$ odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete subcategories of an $n$-exangulated category with an $n$-(co)generator in terms of subgroups of the Grothendieck group. This unifies and extends results of Thomason, Bergh--Thaule, Matsui and Zhu--Zhuang for triangulated, $(n+2)$-angulated, exact and extriangulated categories, respectively. We also introduce the notion of an $n$-exangulated subcategory and prove …
We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1…
Weak Waldhausen categories and a localization theorem
本文引入弱Waldhausen范畴，通过修改Waldhausen公理以更好地与extriangulated范畴理论相匹配，并定义其Grothendieck群。该结构允许单侧extriangulated局部化理论，从而提取出当前理论无法获得的Grothendieck群右精确序列，同时给出了三个应用实例，包括对Enomoto-Saito局部化定理的新证明以及对三角范畴中n-簇状子范畴索引的新的同构证明。
The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. In particular, we determine the Grothendieck mon…
We study liftings of abelian model structures to categories of chain complexes and construct a realization functor from the derived category of a Grothendieck abelian category equipped with a cofibrantly generated, hereditary abelian model structure to the homotopy category of that model structure.
This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…
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