Comparison of Abelian categories recollements

Recollements of extriangulated categories

We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement $(\mathcal{A}$, $\mathcal{B}$, $\mathcal{C})$ of extriangulated categories, we show that cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$ induce cotorsion pairs in $\mathcal{B}$ under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomena when it is applied to abelian categories.

Torsion Pairs in Recollements of Abelian Categories

For a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of abelian categories, we show that torsion pairs in $\mathcal{A}$ and $\mathcal{C}$ can induce torsion pairs in $\mathcal{B}$; and the converse holds true under certain conditions.

Torsion pairs and recollements of extriangulated categories

In this article, we prove that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then torsion pairs in $\mathcal A$ and $\mathcal C$ can induce torsion pairs in $\mathcal B$, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.

同构再分配与奇异等价

Equivariant recollements and singular equivalences

本文研究了可微再分配与奇异等价，探讨了可除和三角范畴的再分配特性，并应用结果构建了导出范畴的可微再分配及斜群代数间的奇异等价。

Properties of abelian categories via recollements

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…

Lifting recollements of abelian categories and model structures

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwa…

Recollements of Module Categories

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.

Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements

Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified $\infty$-category in terms of stratumwise dualizability and a projection formula for the links. Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism $\i…

外三角范畴的重构中的维数与余挠对

Dimensions and cotorsion pairs in recollements of extriangulated categories

本文研究外三角范畴重构中的余解维数界，并证明遗传余挠对在特定条件下可诱导其他范畴中的遗传余挠对，进而构造阿贝尔范畴的重构。

$n$-余挠对与外三角范畴的再收集

$n$-cotorsion pairs and recollements of extriangulated categories

本文研究外三角范畴的再收集，证明了 $n$-余挠对在相邻范畴间的诱导与反诱导，并给出伪簇倾斜子范畴条件使加法商形成半 Abel 范畴的再收集。

再余裂的提升与部分倾斜集的粘合

Lifting of recollements and gluing of partial silting sets

本文研究三角范畴中再余裂的提升与倾斜集的粘合，提出提升判据并给出Artin代数情形下粘合经典倾斜对象的显式构造。

Recollements arising from cotorsion pairs on extriangulated categories

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by Chen-Liu-Yang in a triangulated case. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications to illustrate our main results. In particular, we obtain the Krause's recollement whose the proofs are both elementary and very general.

Types of Serre subcategories of Grothendieck categories

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$ and for each $(m, -n)$ in this list, there exists a Serre subcategory such that its type is $(m, -n)$. This uses right (left) recollements of abelian categories, Tachikawa-Ohtake [TO] on strongly hereditary torsion pairs, and Geigle-Lenzing [GL] on localizing subcategories. If all the functors in a recollement of abelian categories are exact, then the recollem…

微分分次范畴的一个recollement

A recollement of differential graded categories

本文利用富范畴工具证明了带余积封闭满微分子范畴的微分分次范畴存在典范recollement，并将函子dg-范畴的recollement推广到微分分次三角矩阵范畴，推广了Chen和Zheng的已有结果。

abelian categories 的 recollements 的 Igusa-Todorov 属性

Igusa-Todorov properties of recollements of abelian categories

本文研究了 abelian categories 的 recollements 下 Igusa-Todorov 属性的行为，并应用于 Artin algebras 和 Morita context rings。

沿重摄胶合弱可近似性

Gluing weak approximability along recollements

该研究探讨由单个对象紧生成的三角范畴的重摄是否胶合弱可近似性，并将其应用于拟紧拟分离代数空间的复形导出范畴。

外三角分解系统、$s$-挠对与回收

Extriangulated factorization systems, $s$-torsion pairs and recollements

该研究在外三角范畴中引入外三角分解系统，证明其与$s$-挠对的双射关系，并探讨回收下二者的粘合问题，为范畴论相关结构研究提供新视角。

Recollements与$n$-余挠对

Recollements and $n$-cotorsion pairs

研究recollement中三个阿贝尔范畴的n-余挠对关系，给出特定条件下中间范畴n-余挠对的显式粘合构造，同时研究由中间范畴n-余挠对得到两侧范畴n-余挠对的构造。

Triangular Matrix Categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbfΛ=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ initiated in [21]. First, given an additive category $\mathcal{C}$ and an ideal $\mathcal{I}_{\mathcal{B}}$ in $\mathcal{C}$, we prove a well known result that there is a canonical recollement $\xymatrix{\mathrm{Mod}(\mathcal{C}/\mathcal{I}_{\mathcal{B}})\ar[r]_{} & \mathrm{Mod}(\mathcal{C})\ar[r]_{}\ar@<-1ex>[l]_{}\ar@<1ex>[l]_{} & \mathrm{Mod}(\mathcal{B})\ar@<-1ex>[l]_{}\ar@<1ex>[l]_{}}$. We show that given a recollement betwee…

Recollement for Differential Graded Algebras

A recollement of triangulated categories describes one such category as being "glued together" from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a Differential Graded Algebra in terms of two other such categories.

函子范畴、Recollements 与三角矩阵范畴

Functor categories, Recollements, Triangular Matrix category

本文通过阿贝尔范畴的 recollements 理论研究三角矩阵范畴，构造了两个典范 recollements，并证明在相关函子正合时中间项范畴仍为三角矩阵范畴，推广了 Liping Li 的结果；同时给出模范畴具有非平凡挠对时原范畴等价于三角矩阵范畴的判定。

Balanced pairs and recollements in extriangulated categories with negative first extensions

A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $ξ$ with enough $ξ$-projectives and enough $ξ$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $\mathcal B$ can induce balanced pairs in $\mathcal A$ and $\mathcal C$ under natural assumptions. As a a…

外三角范畴中的Recollements、余积和积

Recollements, coproducts and products in extriangulated categories

本文在外三角范畴中引入类似阿贝尔范畴AB4/AB4*的AET4/AET4*条件，给出其等价陈述并推广阿贝尔范畴结果，应用于Recollements和t-结构，还附录研究伴随对性质。

外三角化范畴的recollement中的n-余挠对

$n$-cotorsion pairs in a recollement of extriangulated categories

本文研究外三角化范畴的recollement中n-余挠对的构造与诱导关系，证明了从两端范畴的n-余挠对可构造中间范畴的n-余挠对，并在适当条件下实现反向诱导。

Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements

Given an artin algebra $Λ$ with an idempotent element $a$ we compare the algebras $Λ$ and $aΛa$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element $a$ which ensure that $Λ$ is Gorenstein if and only if $aΛa$ is Gorenstein, that the singularity categories of $Λ$ and $aΛa$ are equivalent and that Fg holds for $Λ$ if and only if Fg holds for $aΛa$. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorenstei…

通过三角范畴的Recollement构造理想cotorsion对

Construction of ideal cotorsion pairs via recollements of triangulated categories

本文通过三角范畴的Recollement结构，研究了如何从两个完整理想cotorsion对诱导出三角范畴中的一组完整理想cotorsion对。

Recollements from partial tilting complexes

We consider recollements of derived categories of dg-algebras induced by self orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results on triangle equivalences proved in [B2] and [BMT]. In the end we focus on the connection between recollements of derived categories of rings, bireflective subcategories and generalized universal localizations".

由左Frobenius对诱导的重粘合

Recollements induced by left Frobenius pairs

本文研究阿贝尔范畴中左Frobenius对在逗号范畴中的诱导条件，并据此从其稳定范畴构造右三角范畴与三角范畴的重粘合，应用于完备遗传余挠对和Gorenstein投射对象。

Gluing n-tilting and n-cotilting subcategories

Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions.

三角范畴的再分解与三角矩阵范畴的奇点范畴

Recollements of Triangulated Categories and the Singularity Category of a Triangular Matrix Category

利用再分解框架，在特定同调条件下证明三角矩阵范畴的奇点范畴等价于其右下子范畴的奇点范畴，推广了Pin Liu和Ming Lu的结果。

Homological theory of $k$-idempotent ideals in dualizing varieties

In this work, we develop the theory of $k$-idempotent ideals in the setting of dualizing varieties. Several results given previously in \cite{APG} by M. Auslander, M. I. Platzeck, and G. Todorov are extended to this context. Given an ideal $\mathcal{I}$ (which is the trace of a projective module), we construct a canonical recollement which is the analog to a well-known recollement in categories of modules over artin algebras. Moreover, we study the homological properties of the categories involved in such a recollement. Consequently, we find conditions on the ideal $\mathcal{I}$ to obtain quas…

Idempotent completion of extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.

Realisation functors in tilting theory

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (noncompact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Ric…

Support $τ$-tilting modules and recollements

Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of abelian categories for artin algebras $Λ'$, $Λ$ and $Λ''$. Under certain conditions, we present an explicit construction of gluing of (support) $τ$-tilting modules in $\mbox{mod} Λ$ with respect to (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively; conversely, we study the construction of (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ obtained from (support) $τ$-tilting modules in $\mbox{mod} Λ$.

Q形导出范畴的Krause型重分

A Krause-type recollement for Q-shaped derived categories

本文推广Krause与Liu的工作，对满足特定条件的k代数A，构造Q形导出范畴的Krause型重分，利用完全遗传余挠对完成构造，明确紧生成相关性质。

Models for singularity categories

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krause's recollement for the stable derived category. In the special case of curved mixed Z-graded complexes, we show that one of our singu…

阿贝尔范畴的局部相干性与粘合

Local coherence and recollement of abelian categories

本文研究阿贝尔范畴局部相干性在粘合下的传递条件，引入拟射影伴随函子对，推广相干准则，将Stenström的刻画扩展到紧组装Grothendieck范畴，汇集环相干性的等价条件。

通过Nakaoka-Palu局部化从三角范畴构造阿贝尔范畴

Abelian categories from triangulated categories via Nakaoka-Palu's localization

本文改进Nakaoka-Palu的HTCP局部化，扩展Abe-Nakaoka的心构造，统一涵盖三角范畴中刚性对象自同态环模范畴的两种实现，并涉及稳定范畴、三角范畴与阿贝尔范畴的粘合等情形。

阿贝尔逗号范畴上的稳定性条件

Stability Conditions on Abelian Comma Categories

该研究针对逗号范畴，给出其为阿贝尔范畴的充分条件，计算其Grothendieck群并给出诺特、阿廷性的充要判据，还定义了阿贝尔逗号范畴上的稳定性条件并实现反向诱导，丰富了范畴论相关理论。

Scalar Extension of Abelian and Tannakian Categories

We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories invo…

离散簇范畴完备化中的torsion对、t-结构和余t-结构

Torsion pairs, t-structures, and co-t-structures for completions of discrete cluster categories

本文对离散簇范畴的Paquette-Yildirim完备化中的torsion对、t-结构和余t-结构进行分类，证明其aisle与非交叉划分（附加数据）双射，并分类recollement。

Definability and approximations in triangulated categories

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.

阿贝尔范畴的分层

Stratifications of abelian categories

本文研究可通过迭代重摄分解的阿贝尔范畴即分层，给出其等价于有限维代数有限维模范畴及为ε-分层范畴的充要条件，推广了相关已有结论。

Abelian categories versus abelian 2-categories

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.

Universal rigid abelian tensor categories and Schur finiteness

We study the construction of arXiv:2111.11217 [math:AG] in more detail, especially in the case of Schur-finite rigid $\otimes$-categories. This leads to some groundwork on the ideal structure of rigid additive and abelian $\otimes$-categories.

Rickart and dual Rickart objects in abelian categories: transfer via functors

We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.

Abelian 1-Calabi-Yau Categories

In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence.

A classification of nullity classes in abelian categories

We give a classification of nullity classes (or torsion classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories.

Comparison of the definitions of Abelian 2-categories

In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined. One is the relatively exact 2-category, and the other(s) is the (2-)abelian Gpd-category. We compare these notions. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories.

Torsion pairs and quasi-abelian categories

We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.

Negative K-groups of abelian categories

We prove that negative K-groups of small abelian categories are trivial.

Representation of n-abelian categories in abelian categories

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.

From triangulated categories to abelian categories--cluster tilting in a general framework

We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.

通过DG范畴对阿贝尔范畴的高维推广

Higher-dimensional generalization of abelian categories via DG-categories

本文提出阿贝尔n-截断DG范畴作为阿贝尔范畴的高维推广，统一了普通阿贝尔范畴与稳定DG范畴，并证明其同伦范畴具有外三角和预三角结构，同时建立了满-单分解等经典理论的高维类比。