Homological systems and bocses

Derived equivalences and stable equivalences of Morita type, I

For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this paper, we first show that each derived equivalence $F$ between the derived categories of Artin algebras $A$ and $B$ arises naturally a functor $\bar{F}$ between their stable module categories, which can be used to compare certain homological dimensions of $A$ with that of $B$; and then we give a sufficient condition for the functor $\bar{F}$ to be an equivalence.…

导出等价、新矩阵等价与同调猜想

Derived equivalences, new matrix equivalences, and homological conjectures

该研究基于域上有限维代数与双矩阵中心化子的同构性，引入三种新矩阵等价关系，刻画了中心化子矩阵代数间的Morita、导出等等价关系，并证明有限维数猜想等对其成立。

稳定等价与同调维数

Stable equivalences and homological dimensions

本文研究了中心化矩阵代数之间的稳定等价，并证明了相关同调维数的保持以及Alperin--Auslander/Auslander--Reiten猜想的成立。

正交模的同调理论

Homological theory of orthogonal modules

本研究从稳定模范畴视角研究自入射Artin代数上的有限生成正交生成元，建立M-相对稳定范畴的粘合、刻画紧对象，给出Tachikawa第二猜想的等价刻画，还引入Gorenstein-Morita代数并证明Nakayama猜想对其成立。

Reduction techniques of singular equivalences

It is shown that a singular equivalence induced by tensoring with a suitable complex of bimodules defines a singular equivalence of Morita type with level, in the sense of Wang. This result is applied to homological ideals and idempotents to produce new reduction techniques for testing the properties of syzygy-finite and injectives generation of finite dimensional algebras over a field.

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence o…

Igusa-Todorov距离

Igusa-Todorov distances

本文引入Igusa-Todorov距离衡量Artin代数偏离Igusa-Todorov代数的程度，证明其有限性并给出上界，进而用于界定奇点范畴维度，并证明该距离在多种代数等价关系下保持不变。

代数有界扩张的范畴性质与同调猜想

Categorical properties and homological conjectures for bounded extensions of algebras

本文定义了代数的有界扩张，证明其满足带有级别的 Morita 型奇异等价及相关稳定范畴等价，并探讨了多种同调猜想，给出了实用函子准则及在平凡扩张和三角矩阵代数上的应用。

高阶稳定dg范畴与簇Morita理论的Auslander对应

Auslander correspondence for higher stable dg categories and cluster Morita theory

该研究建立高阶稳定dg范畴的Auslander对应，引入d-簇dg范畴并证明其包含原d-稳定dg范畴作为d-簇倾斜子范畴，发展簇Morita理论并证明Amiot猜想的Morita变体与Calabi-Yau对应。

Derived equivalences of gentle algebras via Fukaya categories

Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(Σ_A, Λ_A; η_A)$, where $Σ_A$ is an oriented smooth surface with non-empty boundary, $Λ_A$ is a set of stops on $\partial Σ_A$ and $η_A$ is a line field on $Σ_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(Σ_A, Λ_A ;η_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $Σ_A$ on the h…

积分系数环与代数的同调维数

Integral coefficient rings and homological dimensions of algebras

该研究定义了模的积分轮廓与有限维复代数的积分系数环，证明积分轮廓是模的完全不变量，还得出积分系数环的Morita不变性及代数整体维数、有限维数有限的等价条件等结果。

Contravariant finiteness and iterated strong tilting

Let $\mathcal{P}^{<\infty} (Λ$-mod$)$ be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra $Λ$. We develop an applicable criterion that reduces the test for contravariant finiteness of $\mathcal{P}^{<\infty} (Λ$ -mod$)$ in $Λ$-mod to corner algebras $e Λe$ for suitable idempotents $e \in Λ$. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of $\mathcal{P}^{<\infty} (Λ$-mod$)$. The consequences pursued hinge on the fact that this finiteness condition is known to be equ…