商范畴的Recollement

证明了三角范畴的recollement可以自然诱导其商范畴的recollement.特别地,得到类似于群同态第二基本定理的结果,即若U是三角范畴D的局部化(或余局部化)子范畴,V是U的三角满子范畴,则U/V是D/V的局部化(或余局部化)子范畴,并且有三角等价(D/V)/(U/V)≌D/U.同理,对Abel范畴的recollement也有相应的结果.     

从三角范畴的recollement到Abel范畴的recollement

研究了三角范畴的recollement与Abel范畴的recollement的关系.证明了：若三角范畴D允许关于三角范畴D和D的recollement,则Abel范畴D/T允许关于Abel范畴D/i^＊（T）和D/j^＊（T）的recollement,其中T为D的cluster-倾斜子范畴,且满足i＊i^＊（T）＊T,j^＊j^＊（T）^＊T.     

广义Comma范畴的Recollement

本文研究广义Comma范畴上Recollement问题.利用Abel范畴上Recollement及其伴随函子,诱导出广义Comma范畴,并利用比较函子构造出广义Comma范畴上的Recollement.这些结果推广了一般Abel范畴上的Recollement,丰富了Comma范畴研究.     

Godement积和函子范畴的recollement（英文）

对给定的两个伴随对,本文利用Godement积给出其上函子范畴的伴随对.进一步地,本文证明:如果一个预加范畴的recollement满足加法函数j~*是满的,则该recollement可以自然诱导其上模范畴的一个阿贝尔范畴的recollement.该结果从范畴的角度给出了三角矩阵环可自然具有recollement的一个新解释.     

三角范畴的coherent函子范畴的recollement

文章研究了三角范畴D及其coherent函子范畴A(D)的recollement之间的关系.利用D的recollement可以诱导A(D)的prerecollement,文章证明了该prerecollement是recollement的充分必要条件是D的recollement是可裂的;并且D的recollement可以诱导A(D)的prerecollement.     

三角范畴的coherent函子范畴的recollement（英文）

文章研究了三角范畴D及其coherent函子范畴A(D)的recollement之间的关系.利用D的recollement可以诱导A(D)的prerecollement,文章证明了该prerecollement是recollement的充分必要条件是D的recollement是可裂的;并且D的recollement可以诱导A(D)的prerecollement.     

函子范畴上加性函子的一个表示

研究函子范畴ModC上加性函子的表示,把一个Abel群作成范畴ModC上的一个左C-模,构造出一个Hom函子和一个函子态射,证明了从函子范畴ModC到范畴Ab的任意变和为积的反变左正合可加函子都与某个Hom函子自然等价.所得结论在函子范畴上,推广了Watts定理.     

三角范畴和Abel范畴的Torsion理论

本文主要研究了三角范畴在Abel化过程中torsion理论的保持问题.利用三角范畴的coherent函子范畴是Abel范畴,证明了T的coherent函子范畴A(T)是A(D)的thick子范畴;若(X,Y)是D的torsion理论,且D=X*Y的扩张是可裂的,那么(A(X),A(Y))是A(D)的torsion理论.     

Abel范畴的平凡扩张、Recollement和应用

讨论在一定条件下Abel范畴的recollement经过范畴的平凡扩张可诱导出一个新recollement的问题.将结果应用到环上的模范畴,得到平凡(单点)扩张环具有Morita等价不变性;结合加法范畴的幂等完备化,构造出一个幂等完备化范畴关于范畴平凡扩张的recollement.     

拟Abelian范畴上函子范畴的局部化

通过拟Abelian范畴的局部类构造出函子范畴的局部类,进一步研究函子范畴的局部化范畴与局部化范畴的函子范畴之间的关系.     

A recollement construction of Gorenstein derived categories

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.     

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 example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Generalized lax epimorphisms in the additive case

In this paper we call generalized lax epimorphism a functor defined on a ring with several objects, with values in an abelian AB5 category, for which the associated restriction functor is fully faithful. We characterize such a functor with the help of a conditioned right cancellation of another functor, constructed in a canonical way from the initial one. As consequences we deduce a characterization of functors inducing an abelian localization and also a necessary and sufficient condition for a morphism of rings with several objects to induce an equivalence at the level of two localizations of the respective module categories.     

Abel范畴的右双-Giraud粘合

我们在本文中引入了Abel范畴的右双-Giraud粘合定义.我们证明了右双-Giraud粘合与余遗传和遗传的挠对存在着双射对应.此外,我们通过模范畴中特定的幂等理想刻画了这类挠对.     

粘合、倾斜同调维数与高维Auslander-Reiten理论

本文主要研究阿贝尔范畴粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中$\mathscr{A}$, $\mathscr{B}$与$\mathscr{C}$之间的倾斜同调维数关系. 特别地,对遗传的阿贝尔范畴$\mathscr{B}$,给出了粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中的范畴之间的$n$-几乎可裂序列间的联系.     

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.     

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.     

Liftings of normal functors in the category of compacta to categories of topological algebra and analysis

We prove that the liftings of a normal functor F in the category of compact Hausdorff spaces to the categories of (abelian) compact semigroups (monoids) are determined by natural transformations F(?)×F(?) → F(?×?) satisfying requirements that correspond to associativity, commutativity, and the existence of a unity. In particular, the tensor products for normal monads satisfy (not necessarily all) these requirements. It is proved that the power functor in the category of compacta is the only normal functor that admits a natural lifting to the category of convex compacta and their continuous affine mappings.