相对于g(X)的导出范畴

设A是有足够投射对象的Abel范畴,(x,y)是A中的完备遗传余挠对.本文引入Gorenstein x-导出范畴,记作Dg(x)(A),并且从不同的角度研究D_g(x)(A).当(x,y)取作特殊的余挠对时,得到一些已知的导出范畴,如Gorenstein导出范畴和Gorenstein平坦导出范畴等.通过与g(X)有关的同伦范畴K~(-,gxb)(g(x))描述有界Gorenstein x-导出范畴D~b_(g(x))(A)和有界导出范畴D~b(A),并给出一些三角等价.     

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.     

m-循环复形范畴的Bridgeland-Hall代数的余代数结构

设A是一个遗传Abel范畴且■是A的投射对象构成的满子范畴.本文主要研究胁循环复形范畴C_m(■)的Bridgeland-Hall代数的余代数结构(其中m≥2).受Yanagida工作的启发,我们在C_m(■)上定义一个新的正合结构,由此得到了其Bridgeland-Hall代数的余代数结构.同时,证明了存在A的扩展Ringel-Hall代数到m-循环复形范畴C_m(■)的Bridgeland-Hall代数的余代数嵌入.     

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.     

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.     

Koszul微分分次模

引入了Koszul微分分次模的概念. 给定Koszul微分分次代数上的一个下有界的微分分次模, 如果这个模到平凡模的Ext-\!群是有界的分次空间, 则它必定包含一个微分分次子模, 其在适当的截断和移位下是Koszul微分分次模; 这样的模还可以通过一系列Koszul微分分次模来逼近(参见本文推论3.6). 设$A$是一个Koszul微分分次代数, $D^c(A)$是微分分次右$A$-\!模范畴的导出范畴中由对象$A_A$生成的满三角子范畴. 如果平凡微分分次模$k_A$落在范畴$D^c(A)$中, 则三角范畴$D^c(A)$的标准$t$-\!结构的中心, 作为Abel范畴, 与某个有限维代数上的有限生成模范畴对偶. 进一步, 可推得三角范畴$D^c(A)$等价于它的标准$t$-\!结构的中心的有界导出范畴.     

正合范畴的整体Gorenstein维数

设A是有足够多投射对象和足够多内射对象的正合范畴.本文研究了A的整体Gorenstein维数和A中的Gorenstein导出函子.利用同调的方法,证明了:如果A有可数直和与可数直积,那么sup{GpdM|M∈A}=sup{GidM|M∈A};对A中的对象M, N,若Gp M ∞, GidN ∞,则对任意的i≥0, Ext_(GP)~i(M, N)≌Ext_(GI)~i(M, N).     

关于余挠三元组的余星子范畴和余倾斜子范畴

设$\mathcal{A}$ 是一个Abel范畴,且 $(\mathcal{X}, \mathcal{Z},\mathcal{Y})$ 是一个完全遗传余挠三元组.介绍 $\mathcal{A}$ 的 $n$-$\mathcal{Y}$-余倾斜子范畴的定义,并给出 $n$-$\mathcal{Y}$-余倾斜子范畴的一个刻画,类似于 $n$-余倾斜模的 Bazzoni 刻画.作为应用,证明了在一个几乎 Gorenstein 环 $R$ 上, 如果 $\mathcal{GP}$ 是 $n$-$\mathcal{GI}$-余倾斜的, 那么 $R$ 是一个 $n$-Gorenstein 环, 其中 $\mathcal{GP}$ 表示 Gorenstein 投射 $R$-模组成的子范畴且 $\mathcal{GI}$ 表示 Gorenstein 内射 $R$-模组成的子范畴. 进而, 研究 任意环$R$上的$n$-余星子范畴, 以及关于余挠三元组 $(\mathcal{P}, R$-Mod, $\mathcal{I})$ 的 $n$-$\mathcal{I}$-子范畴与 $n$-余星子范畴之间的关系, 其中 $\mathcal{P}$ 表示投射左 $R$-模组成的子范畴且 $\mathcal{I}$ 表示内射左 $R$-模组成的子范畴.     

三角范畴中Gorenstein对象的稳定性

设C是带有三角真类ξ的三角范畴.Asadollahi和Salarian引入并研究了ξ-Gorenstein投射和ξ-Gorenstein内射对象,并将Gorenstein同调代数发展到了三角范畴C中.本文继续研究三角范畴的Gorenstein同调性质.将对ξ-Gorenstein投射对象给出一些等价刻画,作为应用,得到了所有的ξ-Gorenstein投射对象构成的子范畴GP(ξ)有很好的稳定性.     

半粘合诱导的t-结构（英文）

本文研究了对于给定的一个三角范畴的上(下)粘合(C′,C,C″),如何由C的一个t-结构诱导C′和C″的t-结构的问题.利用左(右)t-正合函子的概念,给出了由C的一个t-结构可诱导出C′和C″的t-结构的充分条件.将粘合的一些相关结果推广到了上(下)粘合的情形.     

A right triangulated version of Gentle-Todorov’s theorem

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We prove a right triangulated version of Gentle-Todorov’s theorem by introducing the notion of right homotopy cartesian square.     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.     

Recollements,idempotent completions and t-structures of triangulated categories

We first prove that the idempotent completion of a right or left recollement of triangulated categories is still a right or left recollement, then show that the t-structure on a triangulated category is compatible with taking idempotent completion. Finally, an application of the main theorem is given, which is focused on the boundedness and nondegeneration of the t-structure induced by a recollement and its idempotent completion.     

The Grothendieck group of a cluster category

For the cluster category of a hereditary or a canonical algebra, or equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an admissible triangulated structure.     

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Algebras and Representation Theory - Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this...     

On Moduli Spaces for Abelian Categories

We show that if 𝒜 is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.     

Subcategories of fixed points of mutations by exceptional objects in triangulated categories

We first prove that the subcategory of fixed points of mutation determined by an exceptional object E in a triangulated category coincide with the perpendicular category of E. Based on this characterisation, we prove that the subcategory of fixed points of mutation in the derived category of the coherent sheaves on weighted projective line with genus one is equivalent to the derived category of a hereditary algebra. Meanwhile, we induce two new recollements by left and right mutations from a given recollement.     

Local coherence of hearts associated with Thomason filtrations

Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [19], [20]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [39], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a useful recursive characterisation in case of a finite length filtration. Low length cases involve hereditary torsion classes of finite type of the ring, and even their Happel–Reiten–Smalø hearts; in these cases, the relevant characterisations are given by few module-theoretic conditions.     

General Heart Construction on a Triangulated Category (I): Unifying <Emphasis Type="Italic">t</Emphasis>-Structures and Cluster Tilting Subcategories

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.