Homotopical methods and cohomological supports in local algebra

本研究项目通过各种同调结构的透镜来研究交换代数中的奇点。交换代数作为代数几何的局部模型；后者是现代数学的一个中心分支，其重点是多项式方程组(在整个数学中普遍存在的对象)的解集的几何性质。在交换代数中，人们研究被称为(局部)环的代数结构，它提供了对代数几何中探索的解集上的光滑点和奇点的见解。自20世纪50年代S创立以来，同调代数一直被用来描述局部交换代数中的奇点，提供了有价值的环论见解。这个项目的目的是利用同调代数的工具来更深入地了解交换环，从而揭示局部交换代数和代数几何中的奇点。此外，这一策略通过借鉴代数拓扑学和表示论中的丰富思想来促进交换代数的发展，这些领域严重依赖于开发在各自领域中应用的同调方法，并(进一步)揭示了交换代数与这些领域之间的联系。该奖项还将用于资助对拟议研究项目感兴趣的研究生。更具体地说，PI将应用一系列同调工具来收集对易代数的见解；两个核心工具是同伦方法和上同调支持。这两个理论在交换代数中的应用都是深远的，拟议的研究计划将进一步磨练这一机制，着眼于局部代数的突破。特别是，这个项目的一个主要焦点将是研究在交换代数中出现的某些三角范畴的结构性质。在这个方向上的计划包括在Quillen的一个长期存在的猜想上获得牵引力，通过理解有界导出范畴中的生成元来统一素数特征交换代数中的结果，以及将某些上同调支撑的维度与交换代数中的经典不变量联系起来，希望在Avramov和Jacobson的问题上取得进展。决议的结构属性是该项目的另一个主要重点。在这个方向上，PI将局部代数中的Koszul对偶现象扩展到包括非二次代数和非分次代数。这将使用决议上的A-无穷结构来引入和研究一类环(或更一般的环映射)，从而推广经典Koszul代数类。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.