Universal derived equivalences of posets

扭类偏序集的导出等价

Derived equivalence of posets of torsion classes

研究具有单投射模的有限维代数的扭类偏序结构，证明1-APR倾斜下扭类偏序集通过翻转-翻转变换相关，进而其关联代数导出等价。

有限区间多重集与偏序集表示的 universal 分类

Multisets of finite intervals and a universal category of poset representations

本文提出一个通用构造，将任意偏序集分配为一个有限余完备的加法范畴，并探讨其在有限偏序集下的阿贝尔性质。

三角等价于Gorenstein铺砌阶环与有限偏序集的 incidence 算代数

Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets

本文研究了Gorenstein铺砌阶环与有限偏序集的incidence算代数之间的三角等价关系，并探讨了相关偏序集在分级Morita等价和覆盖下的行为，最后对满足特定条件的Gorenstein铺砌阶环进行了分类。

偏序集与分数 Calabi-Yau 范畴

Posets and Fractional Calabi-Yau Categories

本文研究偏序集模的导出范畴与拟齐次孤立奇点三角范畴的关系，以几何范畴为中介提出导出等价的启发式方法，并强调 Weight 概念的核心作用。

Algebraically equipped posets

We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped posets and $p$-equipped posets, for a prime number $p$. We study their categories of representations and establish equivalences with some module categories, categories of morphisms and a subcategory of representations of a differential tensor algebra. Through this, we obtain matrix representations and its corresponding matrix classification problem.

连续偏序集上的短暂模与Scott层

Ephemeral Modules and Scott Sheaves on a Continuous Poset

本文利用域理论将短暂模推广到连续偏序集，证明持久模关于短暂模Serre子范畴的商范畴等价于Scott拓扑上的层范畴，并借此研究持久模的度量性质。

类型 $\mathbb{D}$ 偏序集的socle-投射范畴的几何实现

A geometric realization of socle-projective categories for posets of type $\mathbb{D}$

本文引入类型 $\mathbb{D}$ 偏序集，证明其socle-投射范畴有限表示型，并通过带孔多边形弧范畴与模范畴的等价给出几何实现，空alien箭头时cluster子代数等于完整cluster代数。

相对区间倾斜、高阶Auslander楼梯拐角与有理Dyck偏序集

Relative tilting, interlacing categories and rational Dyck posets

该研究构造了有理Dyck楼梯关联代数与A型高阶Auslander代数幂等拐角的显式倾斜等价，证明了互素正整数的Chapoton-Ladkani-Rognerud猜想，拓展了区间倾斜机制并实现了楼梯导出范畴的几何实现。

Kostant section, universal centralizer, and a modular derived Satake equivalence

We prove analogues of fundamental results of Kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic (with mild assumptions), and for integral coefficients. As an application, we use these results to obtain a "mixed modular" analogue of the derived Satake equivalence of Bezrukavnikov-Finkelberg.

万有形变环与导出等价

Universal deformation rings and derived equivalences

本文证明导出等价的稳定函子保持有限维k-代数上有限生成Gorenstein-投射模的versal形变环的同构类，并推广到具有层次的Morita型奇异等价。

On derived equivalences of lines, rectangles and triangles

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are "componentwise" tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizing the ADE-chain related to singularity theory, incidence algebras of posets and certain Auslander algebras or more generally endomorphism algebras of initial preprojective modules over path algebras of …

On Singular Equivalences of Morita Type and Universal Deformation Rings for Gorenstein Algebras

Let $Λ$ be a finite-dimensional algebra over a fixed algebraically closed field $\mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $Λ$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. The third author also proved that if $Λ$ is a Gorenstein $\mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $Λ$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$, then …

Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable finitely generated non-projective Gorenstein-projective left $Λ$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$. In this article, we prove that the universal deformation rings $R(Λ,V)$ and $R(Λ,Ω_ΛV)$ are isomorphic, where $Ω_ΛV$ denotes the first syzygy of $V$ as a left $Λ$-module. We also prove the following result. Assume that $Λ$ is Gorenstein and that $Γ$ is another Gorenstein $\mathbf{k}$-algebra such that there exists $\ell …