高遗传代数的Serre函子的熵

Projective-injective modules, Serre functors and symmetric algebras

We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.

Derived equivalences for hereditary Artin algebras

We study the role of the Serre functor in the theory of derived equivalences. Let $\mathcal{A}$ be an abelian category and let $(\mathcal{U}, \mathcal{V})$ be a $t$-structure on the bounded derived category $D^b \mathcal{A}$ with heart $\mathcal{H}$. We investigate when the natural embedding $\mathcal{H} \to D^b \mathcal{A}$ can be extended to a triangle equivalence $D^b \mathcal{H} \to D^b \mathcal{A}$. Our focus of study is the case where $\mathcal{A}$ is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exi…

Auslander-Gorenstein algebras from Serre-formal algebras via replication

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate…

Serre 函子与挠对

Serre functor and torsion pairs

本文研究阿贝尔范畴中挠对的可消去性，证明在 Serre 对偶条件下可消去蕴含 Serre 函子封闭性，并在遗传代数情形给出等价刻画。

Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions

A triangulated category is said to be Calabi-Yau of dimension d if the dth power of its suspension is a Serre functor. We determine which stable categories of self-injective algebras A of finite representation type are Calabi-Yau and compute their Calabi-Yau dimensions. We achieve this by studying the minimal projective resolution of the stable Auslander algebra of A over its enveloping algebra, and use covering theory to reduce to (generalized) preprojective algebras of Dynkin graphs. We also describe how this problem can be approached by realizing the stable categories in question as orbit c…

τ - 霍赫希尔德（上）同调、塞尔双模的平方以及塔马尔金 - 齐甘演算的考克斯特自同构

$τ$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus

该研究关联有限维代数\(\Lambda\)的霍赫希尔德理论的两种丰富内容，即\(\tau\) - 霍赫希尔德（上）同调与考克斯特自同构\(\sigma_{\Lambda}\)。通过中山函子等方法揭示了相关复形及性质，证明两种细化横向，提出组合不变量等，还探讨了导出不变性问题。

高维 Auslander-Gorenstein 代数与 Gabriel 拓扑

Higher Auslander-Gorenstein Algebras And Gabriel Topologies

本文通过 Gabriel 拓扑刻画高维 Auslander(-Gorenstein) 代数，给出 Auslander-Buchsbaum-Serre 定理的高维版本，并揭示模的投射维数与零化子理想投射维数之间的联系。