Notes on formal deformations of abelian categories

Abelian length categories of strongly unbounded type

We discuss the notion of strongly unbounded type for abelian length categories; this is closely related to the Second Brauer-Thrall Conjecture for artin algebras. A new ingredient is the space of characters in the sense of Crawley-Boevey.

Characteristic subobjects in semi-abelian categories

We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a group hold in the general semi-abelian context, or in stronger ones.

Derived decompositions of abelian categories I

Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and suf…

CS-Rickart and dual CS-Rickart objects in abelian categories

We introduce (dual) relative CS-Rickart objects in abelian categories, as common generalizations of (dual) relative Rickart objects and extending (lifting) objects. We study direct summands and (co)products of (dual) relative CS-Rickart objects as well as classes all of whose objects are (dual) self-CS-Rickart. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.

Abelian model structures on categories of quiver representations

Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding-Chen rings is given.

Spectra of small abelian categories

We investigate the Ziegler and Zariski topologies on the lattice of Serre subcategories of a small abelian category.

CS-Baer and dual CS-Baer objects in abelian categories

We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.

局部可表示阿贝尔DG范畴的余导出范畴与反导出范畴

Coderived and contraderived categories of locally presentable abelian DG-categories

本文在局部可表示阿贝尔DG范畴框架下构造了反导出范畴与余导出范畴及其模型结构，证明了这些导出范畴的良生成性，并在局部凝聚情形下证明了余导出范畴由有限可表示对象的绝对导出范畴紧生成，推广了DG环上分次内射DG模及拟凝聚矩阵分解的相关结果。

An Embedding theorem for abelian monoidal categories

We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.

Split objects with respect to a fully invariant short exact sequence in abelian categories

We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.

Compactness in abelian categories

We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact.

阿贝尔范畴中的稳定函子与上同调理论

Stable functors and cohomology theory in abelian categories

该研究通过预包络/预覆盖子范畴上的稳定函子构建阿贝尔范畴的相对完备上同调，并据此刻画多种有限同调维数。

Stability Conditions and Maximal Green Sequences in Abelian Categories

In this paper we study the stability functions on abelian categories introduced by Rudakov in \cite{Ru} and their relation with torsion classes and maximal green sequences. Moreover we introduce a new kind of stability function which is compatible with the wall and chamber of the category.

具有正规投影的范畴中的阿贝尔对象

Abelian objects in categories with normal projections

本文证明在具有正规投影的任意范畴中，内部阿贝尔群仍保持简单行为，即与减法代数等价、结构唯一且态射保持结构，将幺半与减法范畴的结论推广。

拟阿贝尔范畴中的 Koszul 幺半群

Koszul Monoids in Quasi-abelian Categories

本文在具有足够平坦投射对象的双完备闭对称幺半拟阿贝尔范畴中，以无元素方式定义 Koszul 与二次幺半群，并借助左心嵌入证明其分次模导出范畴子范畴的对偶等价。

Strongly CS-Rickart and dual strongly CS-Rickart objects in abelian categories

We introduce (dual) strongly relative CS-Rickart objects in abelian categories, as common generalizations of (dual) strongly relative Rickart objects and strongly extending (lifting) objects. We give general properties, and we study direct summands, (co)products of (dual) strongly relative CS-Rickart objects and classes all of whose objects are (dual) strongly self-CS-Rickart.

阿贝尔范畴与三角范畴的2-范畴的精确性

Exactness of the 2-categories of abelian and triangulated categories

本文引入$2$-同调范畴概念，证明双正合、三角、带点、加性及阿贝尔范畴的$2$-范畴均为$2$-同调，并给出统一商构造准则。

Abelian quotients of the categories of short exact sequences

We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let $(\mathcal{C},\mathcal{E})$ be an exact category. We denote by $\mathcal{E}(\mathcal{C})$ the category of bounded complexes whose objects are given by short exact sequences in $\mathcal{E}…

Abelian right perpendicular subcategories in module categories

We show that an abelian category can be exactly, fully faithfully embedded into a module category as the right perpendicular subcategory to a set of modules or module morphisms if and only if it is a locally presentable abelian category with a projective generator, or in other words, the category of models of an additive algebraic theory of possibly infinite bounded arity. This includes the categories of contramodules over topological rings and other examples. Various versions of the definition of the right perpendicular subcategory are considered, all leading to the same class of abelian cate…

Relative Commutator Theory in Semi-Abelian Categories

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.

The derived and extension dimensions of abelian categories

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $Λ$, we give the upper bounds of the extension dimension of $Λ$ in terms of the radical layer length of $Λ$ and certain relative projective (or injective) dimension of some simple $Λ$-modules, from which some new upper bounds of the derived dimension of $Λ$ are induced.

On maximal green sequences in abelian length categories

In this article, we study the relationship among maximal green sequences, complete forward hom-orthogonal sequences and stability functions in abelian length categories. Mainly, we firstly give a one-to-one correspondence between maximal green sequences and complete forward hom-orthogonal sequences via mutual constructions, and then prove that a maximal green sequence can be induced by a central charge if and only if it satisfies crossing inequalities. As applications, we show that crossing inequalities can be computed by $c$-vectors for finite dimensional algebras; finally, we give the Rotati…

阿贝尔范畴中拟投射维数的性质

Properties of quasi-projective dimension over abelian categories

本文研究阿贝尔范畴中拟投射维数的基本性质，引入左Noether环的拟整体维数，并通过Nakayama代数实例展示其与整体维数的差异。

Rigid objects, triangulated subfactors and abelian localizations

We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categ…

阿贝尔范畴图表上的表示 I：全局结构与同调对象

Representations over diagrams of abelian categories I: Global structure and homological objects

本文系统研究阿贝尔范畴图表上的表示，探讨表示范畴的Grothendieck结构、重要函子与伴随关系，并刻画特殊同调对象，为后续工作奠定基础。

阿贝尔范畴中的相对倾斜理论 II：$n$-$\mathcal{X}$-倾斜理论

Relative tilting theory in abelian categories II: $n$-$\mathcal{X}$-tilting theory

本文在阿贝尔范畴中引入了$n$-$\mathcal{X}$-相对倾斜理论，统一了多种已有倾斜概念，并展示了其在正合范畴、extriangulated范畴及箭图表示范畴中构建新倾斜理论和余挠对的应用。

Prime spectra of abelian 2-categories and categorifications of Richardson varieties

We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…

n-阿贝尔范畴中宽子范畴的谱与支撑理论

Spectra and Support Theory for Wide Subcategories in $n$-Abelian Categories

本文针对本质小的Krull–Schmidt n-阿贝尔范畴的宽子范畴建立支撑理论，构造相关谱，刻画素宽子范畴，证明宽子范畴可由素宽子范畴恢复，还给出不存在分类支撑数据的2-阿贝尔例子。

Extension groups between atoms in abelian categories

We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual duals of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space t…

On the Rosenberg-Zelinsky sequence in abelian monoidal categories

We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the …