Homological Aspects of Commutative Algebra and Applications to Modular Representation Theory

交换代数的同调方面及其在模表示理论中的应用

• 批准号: 1700985
• 负责人: Srikanth Iyengar
• 金额: $ 30万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700985&HistoricalAwards=false
• 关键词: Homological; Aspects; Commutative; Algebra; Applications

One of the myriad functions of mathematics is to provide language to formulate, and tools to solve, equations that describe the physical world. Often these equations are algebraic in nature, such as those describing lines, circles, and parabolas. Typically, the equations have infinitely many solutions -- think about the equation defining a circle -- and it is usually not possible to write down a complete list of solutions. Rather, the objective is to find ways to study the structure of the collection of the solution set, which is called a variety. A fruitful approach has been to consider the (algebraic) functions on the variety. One part of this research project addresses questions that have emerged in this endeavor. Mathematics has also been remarkably successful in describing and studying phenomenon related to symmetry. This leads to a mathematical structure called a group. Intriguingly, in certain contexts, there is a way to attach a variety to a group, and in the past few years various researchers, including the investigator, have been able to solve problems related to groups using tools that had been developed to study varieties. A part of the current project deals with these aspects.This research is rooted in commutative algebra, with applications also to the representation theory of finite group schemes in positive characteristic. The first part of the project concerns the homological aspects of modules over commutative rings. The problems posed range from those that have arisen from the internal developments in the subject to ones inspired by recent advances in the broader mathematical context, notably the representation theory of algebras and homotopy theory. The focus in the second part of the project is on the Hochschild cohomology of commutative algebras. Classically, Hochschild cohomology has been of interest for it is one of two main cohomology theories that capture properties of the diagonal morphism. A different aspect of Hochschild cohomology has begun to play an increasingly important role: Its action on derived categories, and various triangulated categories. The problems proposed in the second part of the project address both aspects. In recent years, notions and techniques from commutative ring theory have proved to be remarkably effective in solving problems in, and shedding new light on, the representation theory of finite groups over a field of positive characteristic. This project will develop these connections in new directions.

数学的众多功能之一是提供语言来表述描述物理世界的方程，并提供工具来求解这些方程。通常，这些方程本质上是代数的，例如描述直线、圆和抛物线的方程。通常，方程有无限多个解--想想定义一个圆的方程--通常不可能写下完整的解列表。相反，我们的目标是找到研究解集集合的结构的方法，这被称为多样性。一种卓有成效的方法是考虑簇上的(代数)函数。这项研究项目的一部分解决了在这一努力中出现的问题。数学在描述和研究与对称性有关的现象方面也取得了显著的成功。这导致了一种称为群的数学结构。耐人寻味的是，在某些情况下，有一种方法可以将品种与一个群体联系起来，在过去的几年里，包括调查人员在内的各种研究人员已经能够使用为研究品种而开发的工具来解决与群体有关的问题。这项研究植根于交换代数，并应用于正特征有限群方案的表示理论。该项目的第一部分涉及交换环上的模的同调方面。提出的问题从学科内部发展引起的问题到受到更广泛的数学背景下的最新进展的启发，特别是代数的表示理论和同伦理论。本项目第二部分的重点是交换代数的Hochschild上同调。经典的Hochschild上同调一直是人们感兴趣的，因为它是两个主要的上同调理论之一，它捕捉到了对角态射的性质。Hochschild上同调的另一个方面已经开始发挥越来越重要的作用：它对派生范畴和各种三角范畴的作用。项目第二部分提出的问题解决了这两个方面。近年来，交换环理论中的概念和技巧被证明在解决正特征域上有限群的表示理论的问题和揭示有限群的表示理论方面是非常有效的。该项目将在新的方向上发展这些联系。

项目成果

Openness of the Regular Locus and Generators for Module Categories

模块类别正则轨迹和生成器的开放性

• DOI: 10.1007/s40306-018-0294-8
• 发表时间: 2019
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: Iyengar, Srikanth B.;Takahashi, Ryo
• 通讯作者: Takahashi, Ryo

Base change along the Frobenius endomorphism and the Gorenstein property

• DOI: 10.1016/j.jalgebra.2021.12.001
• 发表时间: 2019-10
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Pinches Dirnfeld

The Nakayama functor and its completion for Gorenstein algebras

• DOI: 10.24033/bsmf.2849
• 发表时间: 2020-10
• 期刊: Bulletin de la Société mathématique de France
• 作者: S. Iyengar;H. Krause

Rigid Ideals in Gorenstein Rings of Dimension One

• DOI: 10.1007/s40306-018-00315-0
• 发表时间: 2018-04
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: C. Huneke;S. Iyengar;R. Wiegand

Locally Complete Intersection Maps and the Proxy Small Property

局部完整的交叉路口地图和代理小属性

• DOI: 10.1093/imrn/rnab041
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Briggs, Benjamin;Iyengar, Srikanth B;Letz, Janina C;Pollitz, Josh
• 通讯作者: Pollitz, Josh