Homotopical methods and cohomological supports in local algebra

本研究计划透过各种同调构造的透镜来探讨交换代数中的奇点。交换代数是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系统（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年代成立以来，同调代数在描述局部交换代数中的奇点方面发挥了重要作用，提供了有价值的环理论见解。本项目旨在利用同调代数的工具来更深入地理解交换环，从而揭示局部交换代数和代数几何中的奇点。此外，该策略通过借鉴代数拓扑和表示理论中的丰富思想来推进交换代数，这些领域在各自领域的应用中大量依赖于开发同调方法，并（进一步）揭示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交换代数的见解；两个中心工具是同调方法和上同支持。这两种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，并着眼于局部代数的突破。特别地，这个项目的主要焦点将是研究交换代数中出现的某些三角化范畴的结构性质。在这个方向上的项目包括对Quillen的一个长期猜想的推动，通过理解有界派生范畴中的生成器来统一素特征交换代数的结果，以及将某些上同调支持的维数与交换代数中的经典不变量联系起来，以期在Avramov和jacobson的问题上取得进展。分辨率的结构属性是该项目的另一个主要关注点。在这个方向上，PI将扩展局部代数中的Koszul对偶现象，使其包括非二次代数和非分级代数。这将使用分辨率上的a -∞结构来引入和研究一类环（或更一般的环映射），推广经典Koszul代数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.