Homological Aspects of Commutative Algebra and Applications to Modular Representation Theory

数学的众多功能之一就是提供表述描述物理世界的方程的语言和求解方程的工具。通常这些方程本质上是代数的，比如那些描述直线、圆和抛物线的方程。通常，方程有无限多个解——想想定义一个圆的方程——通常不可能写出一个完整的解列表。相反，目标是找到方法来研究解集集合的结构，这被称为多样性。一个富有成效的方法是考虑（代数）函数上的变化。该研究项目的一部分解决了在这一努力中出现的问题。数学在描述和研究与对称有关的现象方面也取得了显著的成功。这就产生了一个叫做群的数学结构。有趣的是，在某些情况下，有一种方法可以将一个品种与一个群体联系起来，在过去的几年里，各种研究人员，包括调查员，已经能够使用研究品种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。本研究植根于交换代数，并应用于正特征有限群方案的表示理论。项目的第一部分关注交换环上的模的同调方面。所提出的问题范围从那些产生于学科内部发展的问题到最近在更广泛的数学背景下取得进展的问题，特别是代数的表示理论和同伦理论。项目第二部分的重点是交换代数的Hochschild上同调。在经典上，Hochschild上同是捕捉对角态射性质的两种主要上同理论之一，一直受到人们的关注。Hochschild上同的另一个方面开始发挥越来越重要的作用：它对派生范畴和各种三角化范畴的作用。项目第二部分提出的问题解决了这两个方面。近年来，交换环理论中的概念和技术被证明在求解正特征域上有限群的表示理论方面是非常有效的，并对有限群的表示理论有了新的认识。这个项目将在新的方向上发展这些联系。

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.