Derived invariants of commutative rings

交换环的导出不变量

• 批准号: 0602498
• 负责人: Srikanth Iyengar
• 金额: $ 13.9万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602498&HistoricalAwards=false
• 关键词: Derived; invariants; commutative; rings

The focus of this project is derived categories arising in commutative algebra. The investigator and his collaborators bring to bear a point of view on triangulated categories fashioned in algebraic topology to establish new results on, and unearth surprising connections among, classical invariants of modules over commutative rings. Another aspect of the proposed research concerns derived categories as a whole, and is inspired by recent spectacular developments in the representation theory of finite dimensional algebra, due to Rouquier: extending earlier work of Konstevich, Bondal and Van den Bergh, and others, on derived categories of coherent sheaves, Rouquier settled a long standing question concerning Auslander's representation dimension, and also an important conjecture of Benson concerning modular representations of finite groups. To this end, Rouquier introduced a notion of a dimension for derived categories, and demonstrated its relevance to representation theory and algebraic geometry. A crucial step Rouquier's work is a lower bound on the derived category of the exterior algebra on a vector space over a field of characteristic two, which is an example of a zero-dimensional commutative complete intersection ring. The project will investigate the dimension of derived categories of general commutative rings, with emphasis on locally complete intersections. This brings in, and contributes to the study of, Hochschild cohomology of commutative algebra. The results obtained will serve to clarify that relationship between dimension and classical invariants of rings. Techniques developed here would have an impact on developments in representation theory and algebraic geometry.It has long been recognized that derived categories, which were introduced by Verdier in the mid-sixties and in the context of algebraic geometry and commutative algebra, provide a convenient milieu for doing homological algebra. However, it has been realized over the last fifteen years that derived categories, and more generally, triangulated categories, are also exceptionally well-suited for expressing (and proving!) results in diverse subjects in mathematics and mathematical physics. This has had the effect that methods developed in one field have often profoundly influenced a host of others, and generated new interactions among them. One of the main components of the research outlined in this proposal seeks to exploit a elementary notion arising in algebraic topology, namely, the idea of `building' objects out of a given collections of objects in a triangulated category, and bring them to be bear on problems in commutative algebra. The project promises to have an impact on the study of derived categories in general, and, in particular, on geometry, representation theory, and algebraic topology.

该项目的重点是在交换代数中引起的类别。 调查员及其合作者对代数拓扑中制定的三角类别的观点提出了一种观点，以建立新的结果，并出现令人惊讶的连接，这是通勤戒指的古典模块不变的人之间的连接。拟议的研究的另一个方面涉及整个类别，并受到有限维代数的代表理论的最新壮观发展的启发，这是由于rouquier而引起的：扩展了Konstevich，Bondal和van den Bergh的早期工作，以及其他人，以及其他较长的构造范围，以及一个重要的构造界面，并构成了一个重要的构造，并构成了一个重要的构想。 Benson关于有限基团的模块化表示。为此，Rouquier引入了衍生类别的维度的概念，并证明了其与表示理论和代数几何形状的相关性。至关重要的步骤rouquier的作品是在特征二的字段上的矢量空间上外部代数的派生类别的下限，这是零维交换完整交叉环的一个示例。 该项目将研究一般通勤环的派生类别的维度，重点是局部完整的交叉点。这带来了霍基柴尔德共同学代数的研究并有助于研究。 获得的结果将有助于阐明尺寸与环的经典不变性之间的关系。这里开发的技术将对代表理论的发展和代数几何形状产生影响。长期以来，人们已经认识到，衍生的类别是由Verdier在六十年代中期和代数的几何形状和交换代数的背景下引入的，为在同级Algebra方面提供了便利的MILIEU。但是，在过去的十五年中，它已经实现了，更普遍地，三角类别的类别也非常适合表达（并证明！）导致数学和数学物理学中的多样化学科。 这产生了一种在一个领域开发的方法经常影响了其他许多人，并在其中产生了新的互动。该提案中概述的研究的主要组成部分之一旨在利用代数拓扑中产生的基本概念，即，从三角形类别中给定的对象集合中“构建”对象的概念，并将它们带入通勤代数中的问题。该项目有望对一般的派生类别的研究产生影响，尤其是对几何，表示理论和代数拓扑的影响。

项目成果

Stratifying modular representations of finite groups

• DOI: 10.4007/annals.2011.174.3.6
• 发表时间: 2008-10
• 期刊: arXiv: Representation Theory
• 作者: D. Benson;S. Iyengar;H. Krause

Local cohomology and support for triangulated categories

• DOI: 10.24033/asens.2076
• 发表时间: 2007-02
• 期刊: Annales Scientifiques De L Ecole Normale Superieure
• 影响因子: 1.9
• 作者: D. Benson;S. Iyengar;H. Krause