Hongxing Chen, Steffen Koenig
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.
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Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.
In the paper, we mainly connect the Gorenstein derived equivalence and stable functors of Gorenstein projective modules. Specially, we prove that a Gorenstein derived equivalence between CM-finite algebras A and B can induce a stable functor between the factor categories A-mod/A-Gproj and B-mod\B-Gproj. Furthermore, the above stable functor is an equivalence when A and B are Gorenstein.
Homotopy Equivalences induced by Balanced Pairs
在阿贝尔范畴中引入了可加子范畴平衡对的概念。给出了平衡子范畴对产生复形的等价同伦范畴的充分条件。作为应用，我们证明了左Gorenstein环的Gorenstein投射模的同伦范畴与Gorenstein内射模的同伦范畴之间存在三角等价，这限制于投射模的同伦范畴与内射模的同伦范畴之间的三角等价.在交换Gorenstein环的情况下，我们证明了我们的等价扩展到一个自然的同构Iyengar-Krause的等价。
Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras
设$Λ$是一个整体维数为2的Auslander的1-Gorenstein Artin代数.若$Λ$允许$\modΛ$的平凡极大1-正交子范畴，则对任何不可分解模$M \in \mod Λ$，我们有：$M$的投射维数等于1当且仅当它的内射维数等于1;若$M$的投射维数等于2，则$M$是内射的.在这种情况下，我们进一步得到$Λ$是一个倾斜代数。
Thick subcategories and virtually Gorenstein algebras
一个Artin代数定义为虚Gorenstein，如果与所有Gorenstein投射模右正交（关于Ext^*（-，-））的模类与与所有Gorenstein内射模左正交的模类重合。我们用有限生成模的概念给出了一个新的刻画。此外，代数的一个例子，这是不是几乎Gorenstein。
Invariants with respect to recollements of the stable category of Gorenstein projective A-modules over an algebra A and stable equivalences are investigated. Specifically, the Gorenstein rigidity dimension is introduced. It is shown that the Gorenstein rigidity dimension is invariant with respect to both Morita equivalences and the stable equivalences of Gorenstein projective modules. As a consequence, the Gorenstein rigidity dimension is shown the invariant of derived equivalences. The Gorenstein rigidity dimension is compared along the recollements of the stable category of Gorenstein projec…
We consider the equivalence from the stable module category to a subcategory $\mathcal{L}_A$ of the homotopy category constructed by Kato. This equivalence induces a correspondence between distinguished triangles in the homotopy category and perfect exact sequences in the module category. We show that an exact equivalence between categories $\mathcal{L}_A$ and $\mathcal{L}_B$ induces a stable equivalence of Morita type between two finite dimensional algebra A and B under a separability assumption. Moreover, we provide further sufficient conditions for a stable equivalence induced by an exact f…
From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, …
Motivated by understanding the Nakayama conjecture which states that algebras of infinite dominant dimension should be self-injective, we study self-orthogonal modules with virtually Gorenstein endomorphism algebras and prove the following result: Given a finitely generated, self-orthogonal module over an Artin algebra with an orthogonal condition on its Nakayama translation, if its endomorphism algebra is virtually Gorenstein, then the module is projective. As a consequence, we re-obtain a recent result: the Nakayama conjecture holds true for the class of strongly Morita, virtually Gorenstein…
We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.
Singular equivalences induced by ring extensions
本文研究有限维代数扩张下奇点范畴与Gorenstein缺陷范畴的三角等价，给出充分条件并应用于平凡扩张、Morita环和三角矩阵代数，提供约化方法。
Relative Auslander--Gorenstein Pairs
本文引入相对Auslander--Gorenstein对与相对Auslander对的概念，用倾斜-余倾斜模的相对支配/余支配维数刻画其性质，推广统一已有结果并给出范畴对应应用与实例。
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 …
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。
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 …
Gorenstein categories and separable equivalences
本文在可分等价下建立了左$Λ$-模加法子范畴的正交类关系，证明了Gorenstein范畴与Wakamatsu倾斜模保持不变，并讨论了$G_{C}$-投射（内射）模及Auslander（Bass）类的不变性。
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。
For $Λ$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $ΛQ$ is a Gorenstein algebra and the category ${\rm Gproj}ΛQ$ of Gorenstein-projective $ΛQ$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}ΛQ\to \underline{\rm Gproj}ΛQ(v)$ between the stable categories modulo projectives, where $Q(v)$ is obtained from $Q$ by changing the direction of each arrow ending in $v$. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that $F(v)$ is an equ…
Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements
给定一个具有幂等元$a$的Artin代数$Λ$，我们比较了代数$Λ$和$aΛa$的Gorensteinness，奇异范畴和Hochschild上同调的有限生成条件Fg.特别是，我们确定假设的幂等元$a$，确保$Λ$是Gorenstein的当且仅当$aΛa$是Gorenstein的，即奇异类别的$Λ$和$aΛa$是等价的，Fg为$Λ$保持当且仅当Fg为$aΛa$保持。我们的做法的问题，通过使用的阿贝尔范畴，我们证明的结果有关Gorensteinness和奇异类别在这个一般设置。这些结果被应用于Cohen-Macaulay模的稳定范畴、三角形矩阵代数和路代数的等价类。
Singular equivalences induced by homological epimorphisms
我们证明了两个代数之间的某个同调满射导致它们的奇异范畴之间的三角等价。将所得结果应用于矩阵代数的一个构造，刻画了一些非Gorenstein代数的奇异范畴。
Relative Singularity Categories and Gorenstein-Projective Modules
引入了关于阿贝尔范畴的自正交子范畴$ω$的相对奇异范畴的概念。引入了$ω$ -Cohen-Macaulay对象的Frobenius范畴，并在一定条件下证明了$ω$ -Cohen-Macaulay对象的稳定范畴与相对奇异范畴是三角等价的.作为应用，我们将Gorenstein-投射模的稳定范畴与环的奇异范畴联系起来.证明了Gorenstein环的Gorenstein-投射模的稳定范畴是紧生成的，并且它的紧对象与有限生成的Gorenstein-投射模在直和项下一致.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein alge…
Applications of Gorenstein projective $τ$-rigid modules
本文引入 $CM$-$\ au$-倾斜自由代数并研究其同调性质，通过等价关系建立 Gorenstein 投射 $\ au$-刚性模与某些模的双射，并部分解决了 Tachikawa 第一猜想。
Homotopy category of projective complexes and complexes of Gorenstein projective modules
设$R$是有单位元的环，$\C(R)$表示$R$ -模的复形范畴.本文研究了由射影（分别是）产生的同伦范畴。单射）复合体以及Gorenstein投射（分别）。Gorenstein内射）模。我们证明了$R$上的投射复形的同伦范畴（记为$\KPC$）总是良生成的，并且当$\KPR$是良生成的时，它是良生成的.在此基础上，证明了当$R$是有限Krull维数的交换Noether环时，Gorenstein投射复形类是预覆盖的.进一步证明了在这样的环上包含函子$ι: \K(\RGPrj)\hookrightarrow \KR$有一个右伴随$ι_ρ$，其中$\K(\RGPrj)$是Gorenstein投射$R$模的同伦范畴.类似的，或者说是双重的，结果为内射（分别）。Gorenstein单射）复合物。如果$R$有一个对偶复形，则将给出投射复形和内射复形的同伦范畴之间的三角等价。作为应用，我们得到了三角范畴$\K(\RGPrj)$和$\K(\RGInj)$之间的一个等价性，当$R$是交换的、Noether的且允许一个对偶复形时，这个等价性限制在$\KPR$和$\KIR$之间.
We study Gorenstein flat objects in the category ${\sf Rep}(Q,R)$ of representations of a left rooted quiver $Q$ with values in ${\sf Mod}(R)$, the category of all left $R$-modules, where $R$ is an arbitrary associative ring. We show that a representation $X$ in ${\sf Rep}(Q,R)$ is Gorenstein flat if and only if for each vertex $i$ the canonical homomorphism $φ_i^X: \oplus_{a:j\to i}X(j)\to X(i)$ is injective, and the left $R$-modules $X(i)$ and ${\rm Coker}φ_i^X$ are Gorenstein flat. As an application of this result, we show that there is a hereditary abelian model structure on ${\sf Rep}(Q,R…
Homological theory of orthogonal modules
本研究从稳定模范畴视角研究自入射Artin代数上的有限生成正交生成元，建立M-相对稳定范畴的粘合、刻画紧对象，给出Tachikawa第二猜想的等价刻画，还引入Gorenstein-Morita代数并证明Nakayama猜想对其成立。
On universal deformation rings and stable equivalences of Gorenstein-projective modules
研究在奇异Morita型等价下，Gorenstein-投射模的泛形变环的同构性，推广了Gorenstein代数情形的结果。
Quillen equivalence for chain homotopy categories induced by balanced pairs
研究平衡对诱导的链同伦范畴的三角等价条件，通过模型范畴的同调范畴实现，并应用于余内射态射、Gorenstein射影与内射模及纯射影与纯内射对象。
Gorensteinness, homological invariants and Gorenstein derived categories
建立了Gorenstein导出范畴、Gorenstein亏损范畴和Gorenstein稳定范畴之间的关系。利用这些结果，研究了代数$A$的Gorenstein性和关于$A$的有界Gorenstein导出范畴$D^{b}_{gp}(A\mbox{-}{\rm mod})$的元素的不变量.具体地说，$A$的Gorenstein性以三种方式来表征：$D^{b}_{gp}(A\mbox{-}{\rm mod})$中Auslander-Reiten三角形的存在; $D^{b}_{gp}(A\mbox{-}{\rm mod})$的等价;以及Gorenstein导出的等价。证明了Cohen-Macaulay型的有限性和有限维数的有限性对于$D^{b}_{gp}(A\mbox{-}{\rm mod})$的元是不变的.
We show that singular equivalences of Morita type with level between finite-dimensional Gorenstein algebras over a field preserve the (Fg) condition.
The Existence of Maximal $n$-Orthogonal Subcategories
对于具有整体维度$n$的$(n-1)$ -Auslander代数$Λ$，我们根据具有投射维度$n-1$或$n$的简单$Λ$ -模的性质，给出了$Λ$允许极大$(n-1)$ -正交子范畴的一些必要条件。对整体维数为2的几乎遗传代数$Λ$，证明了$Λ$存在极大1-正交子范畴当且仅当对任意非投射不可分解$Λ$ -模$M$，$M$是内射的等价于$M$的约化阶等于2.给出了Gorenstein对称猜想与共倾模$T$的$^{\bot}T$的极大$n$ -正交子范畴的存在性之间的联系.对Gorenstein代数，证明了极大$n$ -正交模的所有非投射直和都是$Ω^nτ$ -周期的.此外，我们还研究了模的复杂性与两个有限维代数的张量积的极大$n$ -正交子范畴的存在性之间的关系。
Maximal self-orthogonal modules and a new generalization of tilting modules
本文提出了射影Wakamatsu倾斜模作为倾斜模的新推广，并在有限性条件下证明了多种自正交模的等价性，同时引入Bongartz补全并探讨了相关同调猜想。
Homotopy categories, Leavitt path algebras and Gorenstein projective modules
对于一个无源无汇的有限维代数，证明了相应的根为零的有限维代数上内射模的无圈复形的同伦范畴三角等价于Leavitt路代数的导出范畴，该路代数被看作是一个具有平凡微分的微分分次代数，它进一步三角等价于von Neumann正则代数的平凡扩张代数上的Gorenstein投射模的稳定范畴。给出了投射模的无圈复形的同伦范畴的一个相关但不同的结果。限制这些等价紧对象，我们得到各种描述的奇异范畴的有限维代数根式平方零，其中包含以前的结果。
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ Λ) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(Λ)$ for a finite dimensional algebra $Λ$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } Λ< \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (Λ)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(Λ)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(Λ)$ is replaced with a finitely graded IG algebra $A$…
Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules
在Cohen-Macaulay环上的诺特代数上，证明有限生成Gorenstein投射模构成遗传cotorsion对的左半部分，并回答Takahashi的问题，刻画左弱Gorenstein性质。
It is shown that a 4-recollement of derived categories of CM-finite algebras induces a 2-recollement of the corresponding Cohen-Macaulay Auslander algebras, which generalises the main theorem of Pan [S. Y. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012), 355{363]. Moreover, both Auslander- Reiten conjecture and Gorenstein projective conjecture are shown invariant under 3 (or 4)-recollement of unbounded derived categories of algebras.
Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-sy…
Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms
本文修改最终同调同构定义，引入n-伴随型奇异等价，发展相关理论并应用于同调猜想简化与性质保持，同时建立其与三角函子的对应关系，纳入3-伴随型奇异等价框架。
Ascent and descent of Gorenstein homological properties
本文研究有限代数环同态下 Gorenstein 同调性质的上升与下降，给出判据并用以检测环的 Gorenstein 性质及刻画稳定范畴的三角等价。
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance …
For a finite dimensional algebra $Λ$ and a non-negative integer $n$, we characterize when the set $\tilt_nΛ$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $Λ$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}Λ$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $Λ$ and its factor algebra $Γ=Λ/(e)$, we show that there is a bij…
Singularity categories, preprojective algebras and orthogonal decompositions
本文利用Minamoto和Amiot，Iyama，Reiten的结果，构造了一类分次Gorenstein代数的分次奇异范畴在其射影概形上嵌入到凝聚层的导出范畴.这些分次代数是使用Herschend，Iyama和Oppermann定义的$d$ -表示无限代数的预投射代数构造的。我们将这种嵌入的构造的半正交分解的导出范畴的相干层的射影概形的Gorenstein代数参数1所描述的Orlov。
Recollements of Cohen-Macaulay Auslander algebras and Gorenstein derived categories
设$A$、$B$和$C$是有单位元的结合环.利用Koenig的一个结果，我们证明了如果我们有一个$\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$级元，把$A$写成$B$和$C$，那么我们得到一个由$A$，$B$和$C$的模范畴归纳出的某些函子范畴的$\mathbb{D}^-({\rm{Mod\mbox{-}}} )$级元.作为应用，我们推广了Pan [Sh. Pan，Derived equivalences for Cohen-Macaulay Auslander algebras，J. Pure Appl. Algebra，216（2012），355-363]在Gorenstein artin algebras的等价性方面。证明了Gorenstein型和有限Cohen-Macaulay型是虚Gorenstein代数的$\mathbb{D}^{\rm{b}}_{{{\mathcal{G}p}}}({\rm{mod\mbox{-}}})$水平元的不变量，其中$\mathbb{D}^{\rm{b}}_{{{\mathcal{G}p}}}$表示Gorenstein导出范畴.
Singular equivalences induced by bimodules and quadratic monomial algebras
本文研究双模张量函子诱导奇异等价的条件，并构造显式双模以建立二次单项式代数与根平方零相关代数之间的带层级奇异等价。
Derived equivalences for Cohen-Macaulay Auslander algebras
设$A$和$B$是Cohen-Macaulay有限型的GorensteinArtin代数.证明了如果$A$和$B$是导出等价的，则它们的Cohen-Macaulay Auslander代数也是导出等价的.
Auslander and Bridger introduced the notions of n-spherical modules and n-torsionfree modules. In this paper, we construct an equivalence between the stable category of n-spherical modules and the category of modules of grade at least n, and provide its Gorenstein analogue. As an application, we prove that if R is a Gorenstein local ring of Krull dimension d>0, then there exists a stable equivalence between the category of (d-1)-torsionfree R-modules and the category of d-spherical modules relative to the local cohomology functor.
Constructions of symmetric separable equivalences and their applications
该研究针对对称可分等价的Artin代数，证明模的自同态代数间存在对称可分等价，给出从已知构造新等价的方法，讨论刚性维数的不变性，并证明其保持Frobenius有限型等多个代数性质。
We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X, Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the form $I x + \sum_{j=1}^i B_j y_j, $ where $I$ is the identity matrix and $B_j$ is an square matrix with entries in the residue field, $k$. From there, we prove that there exists a bijection between the set of isomorphism classes of totally reflexive modules (without projective summands) over $S_i$ which are minimal generated by $n$ elements and the set of $i$-…
A higher dimensional Auslander-Iyama-Solberg correspondence
本文将 Auslander–Iyama–Solberg 对应推广到高维情形，用任意具有有限投射与内射维数的自正交模替代极小忠实投射内射模，建立相对 Auslander–Gorenstein 对与推广的预簇倾斜模之间的新双射，并应用于一般线性群的模表示论。
Purity, ascent, and periodicity for Gorenstein flat cotorsion modules
本文研究Gorenstein平坦余挠模稳定范畴的纯结构及标量扩张函子，并将Knörrer周期性推广到完全超曲面奇点与其二重双分支覆盖的大奇点范畴之间。
Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be a finitely generated $Λ$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined versal deformation ring $R(Λ,V)$. If the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$, they also proved under the additional assumption that $Λ$ is self-injective that $R(Λ,V)$ is universal. In this paper, we prove instead that if $Λ$ is arbitrary but $V$ is Gorenstein-projective then $R(Λ,V)$ is also universal when the stable endomorphism ri…
Singularity Categories of Bäckström Orders
本文研究Bäckström顺序的奇点范畴，通过von Neumann正则代数和双模给出显式描述，并建立了与有限维根数平方零代数的奇异等价，同时分类了多种相关代数结构。
We prove the equivalence of two presentations of the Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ and we also show the equivalence with a third presentation when $\mathfrak{g}$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.
On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras
本文研究与Gorenstein代数相关的不可分解$\n$分次顶点代数的结构，证明局部性、不可分解性与半共形顶点代数结构条件的等价性，分析$V_0$、$V_1$的结构特征及双线性型的影响，给出秩一Heisenberg顶点算子代数的嵌入条件。
Characterizing higher Auslander(-Gorenstein) Algebras
本文利用挠理论方法将经典 Auslander-Tachikawa 定理推广到高阶情形，用投射维数小于 n 的模范畴的 Abel 性质刻画 n-Auslander 代数，并给出高阶 Auslander-Gorenstein 代数的范畴刻画及应用。
Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences
本文研究局部Φ-Beilinson-Green代数，利用n-exangulated范畴中的对称逼近序列和倾斜复形构造导出等价，推广了Chen-Xi结果并得到新的导出等价代数类。
Realizing stable categories as derived categories
本文讨论了分次自内射代数的表示论与有限整体维数代数的表示论之间的关系。对于正分次自内射代数$A$，使得$A_0$具有有限的整体维数，我们构造了两类三角等价.首先证明了有限整体维数的$\mathbb{Z}$ -分次$A$ -模的稳定范畴与某个代数$Γ$的导出范畴之间存在三角等价。其次，证明了若$A$具有Gorenstein参数$\ell$，则$\mathbb{Z}/\ell\mathbb{Z}$ -分次$A$ -模的稳定范畴与$Γ$的导轨范畴之间存在三角等价，导轨范畴是导出范畴的轨范畴的三角壳.
We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $f \colon X \to \mathrm{Spec}\ R$, where $X$ has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $A_{\mathrm{con}}$ known as the contraction algebra. As a corollary of our main result, th…
Let $G$ denote a possibly discrete topological group admitting an open subgroup $I$ which is pro-$p$. If $H$ denotes the corresponding Hecke algebra over a field $k$ of characteristic $p$ then we study the adjunction between $H$-modules and $k$-linear smooth $G$-representations in terms of various model structures. If $H$ is a Gorenstein ring we single out a full subcategory of smooth $G$-representations which is equivalent to the category of all Gorenstein projective $H$-modules via the functor of $I$-invariants. This applies to groups of rational points of split connected reductive groups ov…
Categorical actions and derived equivalences for finite odd-dimensional orthogonal groups
本文证明了 Broué 的阿贝尔亏群猜想对有限奇数维正交群成立，通过约化定理、构造 Kac-Moody 代数的范畴作用及研究孤立 RoCK 块，最终建立了块及其 Brauer 对应间的导出等价。
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