Frobenius外三角范畴的稳定范畴中silting子范畴的一个新刻画

Extending ($τ$-)tilting subcategories and (co)silting modules

Let $B$ be a finite dimensional algebra and $A=B[P_0]$ be the one-point extension algebra of $B$ with respect to the finitely generated projective $B$-module $P_0$. The categories of $B$-modules and $A$-modules are related by two adjoint functors $\mathcal{R}$ and $\mathcal{E}$, called the restriction and the extension functors, respectively. Based on the nice homological properties of these two functors, restriction and extension of some notions such as tilting and $τ$-tilting modules have been studied in the category of finitely presented modules, i.e. small mod. In this paper, we investigat…

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.

d-扩张模范畴中的半砖与宽子范畴

Semibricks and wide subcategories in extended module categories

该研究在三角范畴的d-扩张心中定义半砖与宽子范畴并证明其与有限长度宽子范畴的双射，在d-扩张模范畴中推广对应关系至(d+1)-项单态集合，还刻画了(d+1)-项silting复形的突变性质。

逗号范畴中的倾斜对象与挠对

Silting objects and torsion pairs in comma categories

本文刻画了逗号范畴中的倾斜对象，并证明了在特定条件下，两个子范畴对为挠对等价于其在特殊逗号范畴中诱导的对为挠对。

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。

Semibricks in extriangulated categories

Let $\mathcal{X}$ be a semibrick in an extriangulated category $\mathscr{C}$. Let $\mathcal{T}$ be the filtration subcategory generated by $\mathcal{X}$. We give a one-to-one correspondence between simple semibricks and length wide subcategories in $\mathscr{C}$. This generalizes a bijection given by Ringel in module categories, which has been generalized by Enomoto to exact categories. Moreover, we also give a one-to-one correspondence between cotorsion pairs in $\mathcal{T}$ and certain subsets of $\mathcal{X}$. Applying to the simple minded systems of an triangulated category, we recover a …

Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories

Let $(\mathfrak{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear extriangulated category with $k$ a commutative artinian ring. We define an additive subcategory $\mathfrak{C}_r$ (respectively, $\mathfrak{C}_l$) of $\mathfrak{C}$ in terms of the representable functors from the stable category of $\mathfrak{C}$ modulo $\mathfrak{s}$-injectives (respectively, $\mathfrak{s}$-projectives) to $k$-modules, which consists of all $\mathfrak{s}$-projective (respectively, $\mathfrak{s}$-injective) objects and objects isomorphic to direct summands of finite direct sums of all th…

Silting mutation in triangulated categories

In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we int…

斜倾约简、相对AGK构造与Higgs构造

Silting reduction, relative AGK's construction and Higgs construction

本文基于斜倾约简理论，研究了相对AGK构造和Higgs构造在冰淬图与势能设定下的应用，证明了这两种构造将斜倾约简转化为斜倾约简。

Reductions of triangulated categories and simple-minded collections

Silting and Calabi-Yau reductions are important process in representation theory to construct new triangulated categories from given one, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi-Yau) reduction. For a triangulated category $\cal T$ with a pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new triangulated category $\cal U$ such that the SMCs in $\cal U$ bijectively correspond to those in $\cal T$ containing $\cal R$. Secondly, we give an analogue of Buchweit…

三角范畴上的度量与(t-结构)的限制

Metrics on triangulated categories and restrictions of (co)-$t$-structures

本文研究了带有度量的三角范畴中硅亭诱导的t-结构和co-t-结构的限制行为，通过建立硅亭子范畴在预完成中的反变有限性等价于标准t-结构限制到预完成的条件，得出右余合环的纯类别刻画。

torsion pairs 的区间与广义 Happel-Reiten-Smalø 折叠

Intervals of torsion pairs and generalized Happel-Reiten-Smalø tilting

本文研究了广义HRS折叠，通过torsion对的区间与心的双射关联，推广了HRS折叠方法，并展示了$t$-结构的扩展应用。

通过表示论的聚类结构：聚类集合、热带对偶性、聚类特征和量化

Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation

本文通过表示论发展聚类结构理论，结合热带对偶性和量子化，揭示聚类集合与特征的范畴化方法及应用。