Free Resolutions, K-Theory and dg-Categories

本研究项目涉及交换代数和同调代数相关的主题，并着眼于形式方程组。虽然处理这些方程的规则与高中代数中的规则相同，但研究它们的一般设置产生了强大的工具，这些工具将交换代数置于许多纯数学的核心，并应用于许多其他研究领域，例如数论（研究整数的性质）和代数几何（研究多项式方程系统解的几何性质）。同调代数是代数的一个分支，与代数拓扑（研究拓扑空间，即“形状”）领域有关。同调代数研究的一个中心对象是模的复合体，它可以被认为是拓扑空间（“形状”）概念的抽象。这个项目的一个具体目标是解决一些长期以来关于这种复合体和类似拓扑空间的猜想。该奖项还将支持从事相关课题研究的研究生。更详细地说，本研究项目旨在解决关于交换环上有限秩自由模的有界复形的各种猜想，特别是关于在这种具有有限长度同调的复形中出现的模的可能秩。这些题目涉及到Halperin和Carlsson关于拓扑空间上环面和初等阿贝尔p群的自由作用的猜想。这些课题研究的成功将促进我们对交换环上同调代数、代数拓扑以及两者相互作用的理解。本研究还将探讨关于光滑和适当的dg-范畴的几个猜想，重点是与具有孤立奇点的超曲面相关的矩阵分解的dg-范畴的例子。光滑范畴和性质范畴可以被认为是光滑和固有代数变体的非交换类似物。这一领域目标的成功将促进我们对经典代数几何中著名猜想的非交换类似物的理解。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 concerns topics related to commutative and homological algebra and looks at formal systems of equations. While the rules for manipulating such equations are the same as the ones learned in high-school algebra, the general setting in which they are studied yields powerful tools that put commutative algebra at the heart of much of pure mathematics with applications to many other areas of study, such as number theory (the study of properties of the integers) and algebraic geometry (the study of geometric properties of solutions to systems of polynomial equations). Homological algebra is the branch of algebra related to the field of algebraic topology (the study of topological spaces, i.e."shapes"). A central object of study in homological algebra is that of a complex of modules, which can be thought of an abstraction of the notion of a topological space ("shape"). One specific goal of this project is to settle some long-standing conjectures concerning such complexes and the analogous topological spaces. The award will also support graduate students working on affiliated topics. In more detail, this research project aims to settle various conjectures concerning bounded complexes of finite rank free modules over commutative rings, in particular concerning the possible ranks of the modules appearing in such complexes that have finite length homology. These topics relate to conjectures of Halperin and Carlsson concerning free actions of tori and elementary abelian p-groups on topological spaces. The success of the proposed research on these topics will thus advance our understanding of homological algebra over commutative rings, of algebraic topology, and of the interaction of the two. This research also will pursue several conjectures about smooth and proper dg-categories, focusing on the example of the dg-category of matrix factorizations associated to a hyper-surface with an isolated singularity. Smooth and property dg-categories can be thought of as non-commutative analogues of smooth and proper algebraic varieties. The success of the goals in this area will advance our understanding of the non-commutative analogues of well-known conjectures in classical algebraic geometry.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.