Commutative algebra: homological and homotopical aspects

这是一个建议，在交换代数的观点，上同调和同伦理论，具有很强的连接表示理论的有限维代数。许多被提出的研究课题都受到同伦理论的启发，同伦理论也是处理这些问题的直觉和技术的来源。广义上讲，该项目中经常出现的主题是三个主题的相互作用：交换代数;微分分次代数和模块;和三角范畴。虽然微分分次代数和三角范畴方法在交换代数中已经成功地应用了很长时间，但PI和他的合作者一直在采用交换代数的经典结构和方法（例如，局部上同调，并导出完成）的背景下，微分分次代数和三角范畴，与惊人的回报表示理论的二维代数，尤其是群代数和交换代数本身。这一建议旨在进一步发展这些，并在新的方向。一些相当不同的代数结构--这意味着，松散地说，不直接涉及任何概念，从微积分一样的连续性或变化率--已经在数学中开发的模型现象（物理）世界。两个与这个提议特别相关的例子是群及其表示，它们非常适合捕捉涉及对称性的现象，以及环，特别是那些在各种几何对象上出现的函数环，如流形或多项式方程的解集。然而，直到几年前，人们才意识到，如果我们通过微积分中最原始的结构来增强环，即导数，那么就有可能统一地捕获编码在这些不同代数结构中的大部分信息。有趣的是，这些仍然相当神秘的混合结构，称为微分分次代数，在20世纪50年代初已经成为代数拓扑学的重要工具。微分分次代数可以被看作是联系数学和数学物理中各种代数和几何背景的桥梁。这就产生了这样的效果，即在一个领域中发展起来的方法深刻地影响了许多其他领域，并发现了它们之间的新联系。微分分次代数，一般来说，相当复杂，但有一些有趣的类，似乎是服从交换环理论，一个经典的主题与大量的和发达的工具和技术开发的方法。这个建议的主要目的是从这个角度研究微分分次代数。除了加深我们对它们的认识外，所提出的研究预计将对交换代数，表示论和相关领域产生影响。

This is a proposal in commutative algebra from the point of view of cohomology and homotopy theory, with strong connections to representation theory of finite dimensional algebras. Many of the proposed topics of research are inspired by homotopy theory, which also serves as a source of intuition, and of techniques, for dealing with them. Broadly speaking, the recurrent theme in the project is the interplay of three topics: commutative algebra; differential graded algebras and modules; and triangulated categories. While differential graded algebras and triangulated category methods have long been used successfully in commutative algebra, the PI and his collaborators, among others, have been adapting classical constructions and methods from commutative algebra (for example, local cohomology, and derived completions) to the context of differential graded algebras and triangulated categories, with striking returns in the representation theory of finitely dimensional algebra, notably, group algebras, and in commutative algebra itself. This proposal seeks to further develop these, and in new directions.A number of rather diverse algebraic structures---this means, loosely speaking, not directly involving any notions from calculus like continuity or rate of change---have been developed in mathematics to model phenomenon in the (physical) world. Two examples particularly relevant to this proposal are groups and their representations, that are remarkably well-adapted to capture phenomenon involving symmetry, and rings, especially those that arise as rings of functions on various geometric objects like manifolds or solution sets of polynomial equations. However, it was only a few years ago that it was realized that if we enhance a ring by the most primitive structure from calculus, namely, a derivative, then it becomes possible to uniformly capture much of the information encoded in these various algebraic structures. Interestingly, these still rather mysterious hybrid structures, called differential graded algebras, emerged as important tools in algebraic topology already in the early 1950s. Differential graded algebras can be seen as bridges that relate various algebraic and geometric contexts in mathematics and mathematical physics. This has had the effect that methods developed in one field have profoundly influenced a host of others, and new connections among them have been discovered. Differential graded algebras are, in general, rather complicated, but there are interesting classes that appear to be amenable to methods developed in commutative ring theory, a classical topic with a large and well-developed body of tools and techniques. The broad aim of this proposal is to investigate differential graded algebras from this perspective. Besides deepening our knowledge of them, the proposed research is expected to have impact on commutative algebra, representation theory, and related fields.