arXiv math/06014182006-01math.CTmath.RT

Classification of abelian hereditary directed categories satisfying Serre duality

Adam-Christiaan van Roosmalen

备注：: 30 pages with 12 figures; incorporated referee's comments

英文摘要：

In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification, we will study the shapes of the Auslander-Reiten components extensively and use appropriate generalisations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Noetherian hereditary categories satisfying Serre duality

In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

2001-12-04math.CTmath.RT

Hereditary Categories with Serre Duality which are generated by Preprojectives

We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver.

2010-04-09math.CTmath.RT

Hereditary uniserial categories with Serre duality

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.

2010-11-28math.CTmath.RT

The Quiver of Projectives in Hereditary Categories with Serre Duality

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.

2009-09-23math.CTmath.RT

Numerically finite hereditary categories with Serre duality

Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.

2015-01-13math.AGmath.CT

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…

2016-11-13math.CTmath.RT

更多相关论文收起更多论文

Directed hereditary species and decomposition spaces

We introduce the notion of directed hereditary species and show that they have associated monoidal decomposition spaces, comodule bialgebras, and operadic categories. The notion subsumes Schmitt's hereditary species, Gálvez--Kock--Tonks directed restrictions species, and a directed version of Carlier's construction of monoidal decomposition spaces and comodule bialgebras. In addition to all the examples of Schmitt, Gálvez--Kock--Tonks and Carlier, the new construction covers also the Fauvet--Foissy--Manchon comodule bialgebra of finite topological spaces, the Calaque--Ebrahimi-Fard--Manchon co…

2022-11-27math.ATmath.COmath.CT

Types of Serre subcategories of Grothendieck categories

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$ and for each $(m, -n)$ in this list, there exists a Serre subcategory such that its type is $(m, -n)$. This uses right (left) recollements of abelian categories, Tachikawa-Ohtake [TO] on strongly hereditary torsion pairs, and Geigle-Lenzing [GL] on localizing subcategories. If all the functors in a recollement of abelian categories are exact, then the recollem…

2017-05-09math.CT

Abelian hereditary fractionally Calabi-Yau categories

As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will be called fractionally Calabi-Yau is its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are: the category of finite dimensional representations of a Dynkin quiver, the category of finite dimensional nilpotent representations of a cycle, and the cate…

2010-10-23math.CT

2-Calabi-Yau categories with a directed cluster-tilting subcategory

As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quiver…

2016-11-11math.CTmath.RT

Balance in stable categories

We study when the stable category of an abelian category modulo a full additive subcategory is balanced and, in case the subcategory is functorially finite, we study a weak version of balance. Precise necessary and sufficient conditions are given in case the subcategory is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.

2006-10-03math.CT

Q形导出范畴的Krause型重分

A Krause-type recollement for Q-shaped derived categories

Lanzhou Jiaotong University（兰州交通大学）；Northeast Normal University（东北师范大学）

本文推广Krause与Liu的工作，对满足特定条件的k代数A，构造Q形导出范畴的Krause型重分，利用完全遗传余挠对完成构造，明确紧生成相关性质。

2026-10-16math.CTmath.RAmath.RT

模层的挠论解释与有向范畴上的Grothendieck拓扑

A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories

该研究从挠论角度刻画模层，分析有向范畴上Grothendieck拓扑的结构，分类特殊诺特EI范畴的拓扑并推广表示的局部自内射性质，给出群表示论应用方向。

2025-07-29math.CTmath.RT