Commutative algebra: homological and homotopical aspects

这是交换代数中从上同调和同伦理论的角度提出的一个建议，与有限维代数的表示理论有很强的联系。许多被提议的研究主题都受到同伦理论的启发，同伦理论也可以作为处理同伦理论的直觉和技术的来源。从广义上讲，项目中反复出现的主题是三个主题的相互作用：交换代数；微分梯度代数与模；还有三角分类。虽然微分梯度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将交换代数中的经典构造和方法（例如局部上同调和派生补全）应用于微分梯度代数和三角范畴，在有限维代数的表示理论中取得了惊人的成果，特别是群代数。以及交换代数本身。本提案旨在进一步发展这些，并在新的方向。许多相当多样的代数结构——这意味着，宽泛地说，不直接涉及微积分中的任何概念，如连续性或变化率——已经在数学中发展起来，以模拟（物理）世界中的现象。与这一建议特别相关的两个例子是群及其表示，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（如流形或多项式方程的解集）上作为函数环出现的环。然而，直到几年前，人们才意识到，如果我们用微积分中最原始的结构，即导数来增强一个环，那么就有可能统一地捕获这些不同代数结构中编码的大部分信息。有趣的是，这些仍然相当神秘的混合结构，被称为微分梯度代数，早在20世纪50年代初就已经成为代数拓扑学的重要工具。微分梯度代数可以看作是连接数学和数学物理中各种代数和几何背景的桥梁。其结果是，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，并且发现了它们之间的新联系。一般来说，微分梯度代数是相当复杂的，但有一些有趣的类似乎适用于交换环理论中开发的方法，交换环理论是一个具有大量发达的工具和技术的经典主题。这个建议的广泛目的是从这个角度来研究微分梯度代数。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，表示理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.