Homotopical methods and cohomological supports in local algebra

局部代数中的同伦方法和上同调支持

• 批准号: 2302567
• 负责人: Joshua Pollitz
• 金额: $ 15.35万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302567&HistoricalAwards=false
• 关键词: Homotopical; methods; cohomological; supports; local

This research project investigates singularities in commutative algebra through the lens of various homological constructions. Commutative algebra serves as a local model for algebraic geometry; the latter is a central branch of modern mathematics, where the focus is on the geometric properties of solutions sets to systems of polynomial equations (objects ubiquitous throughout mathematics). In commutative algebra, one examines algebraic structures known as (local) rings, which provide insights into both smooth and singular points on the solution sets explored in algebraic geometry. Since its inception in the 1950's, homological algebra has been instrumental in describing singularities in local commutative algebra, offering valuable ring-theoretic insights. This project aims to leverage tools from homological algebra to gain a deeper understanding of commutative rings, thereby shedding light on singularities in local commutative algebra and algebraic geometry. Moreover, this strategy advances commutative algebra by drawing from the wealth of ideas in algebraic topology and representation theory, areas that have leaned heavily on developing homological methods for application in their respective fields, and (further) revealing connections between commutative algebra and these areas. The award will also be used to fund graduate students interested in the proposed research program. More specifically, the PI will apply an array of homological tools to glean insights in commutative algebra; the two central tools being homotopical methods and cohomological support. Applications of both theories have been far-reaching in commutative algebra, and the proposed research program will further hone this machinery with an eye toward breakthroughs in local algebra. In particular, a primary focus of this project will be on studying the structural properties of certain triangulated categories arising in commutative algebra. Projects in this direction include gaining traction on a long-standing conjecture of Quillen, unifying results in prime characteristic commutative algebra by understanding generators in the bounded derived category, and relating the dimension of certain cohomological supports to classical invariants in commutative algebra with the hopes of making progress on questions of Avramov and Jacobsson. The structural properties of resolutions is another primary focus of the project. In this direction, the PI will extend Koszul duality phenomena in local algebra to include non-quadratic and non-graded algebras. This will be achieved using A-infinity structures on resolutions to introduce and study a class of rings (or more generally ring maps) generalizing the class of classical Koszul algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目通过各种同源结构的角度研究了交换代数的奇异性。交换代数是代数几何形状的局部模型。后者是现代数学的中心分支，其中重点放在解决方案系统的几何特性上（整个数学中的对象）。在交换代数中，一个人研究了称为（局部）环的代数结构，这些结构可为代数几何形状探索的溶液集上的平滑和奇异点提供见解。自1950年代成立以来，同源代数一直在描述当地交换代数的奇异性，提供了宝贵的环理论见解。该项目旨在利用同源代数的工具来深入了解交换环，从而阐明当地交换代数和代数几何形状的奇异性。此外，该策略通过从代数拓扑和代表理论中的丰富思想中汲取丰富的思想来推动交换性代数，这些领域倾向于在其各自领域开发同源方法，并（进一步）揭示交换代数和这些领域之间的联系。该奖项还将用于资助对拟议研究计划感兴趣的研究生。更具体地说，PI将应用一系列同源工具来收集交换代数的见解。这两个中心工具是同位方法和共同体学支持。这两种理论的应用在交换代数方面都是深远的，拟议的研究计划将进一步磨练这种机械，并着眼于当地代数的突破。特别是，该项目的主要重点将是研究交换代数中某些三角类别的结构特性。朝这个方向的项目包括在长期以来对Quillen的长期猜想中获得牵引力，并通过了解有限的派生类别中的发电机，并将某些共同体支持的维度与交换代数的经典代数的维度联系起来，从而统一统一的结果，并将某些共同体支持的维度联系起来。决议的结构特性是该项目的另一个主要重点。在这个方向上，PI将将局部代数的Koszul二元现象扩展到包括非季度和非级代数。这将是使用关于决议的一个赋予结构来实现的，以介绍和研究一类戒指（或更常见的环地图）推广经典的Koszul代数类别。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。