Higher Dimensional Homological Algebra

高维同调代数

• 批准号: EP/P016014/1
• 负责人: Peter Jorgensen
• 金额: $ 30.73万
• 依托单位: Newcastle University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP016014%2F1
• 关键词: Higher; Dimensional; Homological; Algebra

Classic homological algebra had its origin some 60 years ago in algebraic topology. Since then, it has grown to a theory with applications to many areas of mathematics, including combinatorics, geometry, and representation theory. Classic homological algebra is often phrased as the theory of abelian and triangulated categories.Higher dimensional homological algebra is a new development of the last decade. It is the theory of n-abelian categories and (n+2)-angulated categories, where n is a positive integer. In these categories, the role previously played by 1-extensions is taken over by n-extensions. Note that the case n=1 gives ordinary abelian and triangulated categories, hence classic homological algebra. We refer to (n+2)-angulated categories because the case n=1 gives triangulated categories. Higher dimensional homological algebra is currently very active. It has applications to algebraic geometry, combinatorics, and the representation theory of finite dimensional algebras. There are substantial contributions from a number of strong mathematicians (Iyama, Keller, Reiten). Some of the combinatorial structures which appear, like higher dimensional cyclic polytopes, are novel to homological algebra and representation theory.This project will provide three key items currently missing in higher dimensional homological algebra: Tilting objects, higher dimensional derived categories, and higher dimensional model categories. We will define tilting objects in higher dimensional homological algebra and show a version of the Ingalls-Thomas bijections between support tilting objects, torsion classes, intermediate t-structures, and non-crossing partitions. This will enhance and illuminate the links to combinatorics. We will define higher dimensional derived categories and higher dimensional model categories. This will provide the right context for higher dimensional tilting theory, and give a comprehensive framework for higher dimensional homological algebra.

经典同调代数起源于大约60年前的代数拓扑学。从那时起，它已经发展成为一个理论，应用于许多数学领域，包括组合数学，几何和表示论。经典同调代数常被称为交换范畴和三角范畴理论，高维同调代数是近十年来的新发展。它是n-阿贝尔范畴和（n+2）-角范畴的理论，其中n是正整数。在这些范畴中，先前由1-扩张所扮演的角色被n-扩张所取代。注意，n=1的情况给出了普通的阿贝尔范畴和三角范畴，因此是经典的同调代数。我们指的是（n+2）-角化范畴，因为n=1的情况给出了三角化范畴。高维同调代数是当前非常活跃的研究领域。它应用于代数几何、组合学和有限维代数的表示论。有大量的贡献，一些强大的数学家（井山，凯勒，赖滕）。一些组合结构的出现，如高维循环多面体，是新的同调代数和表示理论。这个项目将提供目前在高维同调代数中缺少的三个关键项目：倾斜对象，高维导出类别，和高维模型类别。我们将在高维同调代数中定义倾斜对象，并展示支撑倾斜对象、扭转类、中间t结构和非交叉分区之间的英格尔斯-托马斯双射的一个版本。这将增强和阐明组合学的联系。我们将定义更高维度的派生类别和更高维度的模型类别。这将为高维倾斜理论提供正确的背景，并为高维同调代数提供一个全面的框架。

项目成果

Friezes, weak friezes, and T-paths

饰带、弱饰带和 T 形路径

• DOI: 10.48550/arxiv.2005.06230
• 发表时间: 2020
• 作者: Canakci I

Infinite friezes and triangulations of annuli

无限的饰带和环带的三角剖分

• DOI: 10.1142/s0219498824502074
• 发表时间: 2023
• 期刊: Journal of Algebra and Its Applications
• 影响因子: 0.8
• 作者: Baur K

Lattice bijections for string modules, snake graphs and the weak Bruhat order

字符串模块、蛇图和弱 Bruhat 阶的格双射

• DOI: 10.48550/arxiv.1811.06064
• 发表时间: 2018
• 作者: Canakci I

Addendum and Erratum: Mapping cones for morphisms involving a band complex in the bounded derived category of a gentle algebra

附录和勘误：涉及温和代数有界派生范畴中带复形的态射的映射锥

• DOI: 10.48550/arxiv.2001.06435
• 发表时间: 2020
• 作者: Canakci I

Transport of structure in higher homological algebra

• DOI: 10.1016/j.jalgebra.2021.01.019
• 发表时间: 2020-03
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Raphael Bennett-Tennenhaus;Amit Shah