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达描述物理世界的方程，并提供工具来求解描述物理世界的方程。通常，这些方程本质上是代数方程，例如描述直线、圆和抛物线的方程。通常，方程有无限多个解——想想定义圆的方程——通常不可能写下完整的解列表。相反，目标是找到研究解集集合结构的方法，这称为多样性。一种富有成效的方法是考虑簇的（代数）函数。该研究项目的一部分解决了这一努力中出现的问题。数学在描述和研究与对称性相关的现象方面也取得了巨大的成功。这导致了一种称为群的数学结构。有趣的是，在某些情况下，有一种方法可以将品种附加到一个群体中，并且在过去几年中，包括研究者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相关的问题。当前项目的一部分涉及这些方面。这项研究植根于交换代数，也应用于正特性的有限群格式的表示论。该项目的第一部分涉及交换环上模的同调方面。提出的问题范围广泛，从学科内部发展产生的问题到更广泛的数学背景中最新进展所启发的问题，特别是代数表示论和同伦理论。该项目第二部分的重点是交换代数的 Hochschild 上同调。经典上，霍克希尔德上同调一直很受关注，因为它是捕获对角态射性质的两个主要上同调理论之一。霍克希尔德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涉及这两方面。近年来，交换环理论的概念和技术已被证明在解决正特征域上的有限群表示论问题方面非常有效，并为它提供了新的启示。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